polarization elements - p.otto sch¤rpf sj

Polarization elements.

O.ScharpfBerchmanskolleg

Kaulbachstrasse 31a80539 Munchen, Germany

Munchen, March 2, 2010

Contents

1 Experimenting with the neutron spin 51.1 Tools to prepare precessing polarization . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Polarization and spinor (|P | = 1 always) . . . . . . . . . . . . . . . . . . . . . 61.1.2 Polarization and density matrix (|P | < 1 expressible) . . . . . . . . . . . . . 71.1.3 precession coil to rotate spinor in direction Θ,Φ . . . . . . . . . . . . . . . . 11

1.1.3.1 Spin with (ϑ, ϕ) in constant field ~B . . . . . . . . . . . . . . . . . . 111.1.3.2 Rotation in spinor space . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.4 Time dependent Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . 151.1.4.1 The next steps:various flippers, CRYOPAD, SSPAD, . . . . . . . . 151.1.4.2 Spin in a homogeneous magnetic field . . . . . . . . . . . . . . . . . 16

2 Applications for measuring equipment 172.1 π-flipper and its use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Polarization measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 π/2-flipper and its use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Adjustment of the currents in a π/2-flipper . . . . . . . . . . . . . . . . . . . 202.2.2 Measurement of precessing spin polarization, CRYOPAD, SSPAD . . . . . . 21

2.3 Slowly varying magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Adiabatic spin rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Density matrix formalism 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Compilation of rules for the Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 some simple rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Compilation of some more sophisticated rules for Trσ . . . . . . . . . . . . . 323.2.3 Scattering matrix as link between in and out . . . . . . . . . . . . . . . . . . 33

3.3 Application: nuclear coherent and incoherent sc. . . . . . . . . . . . . . . . . . . . . 353.3.1 Spinless target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Correction for finite flipping ratio . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.3 Scattering of polarized neutrons by a target with only nuclear spin incoher-

ence: vanadium calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.4 Summary about the density matrix formalism . . . . . . . . . . . . . . . . . . 40

4 Theory of XYZ-difference method 434.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 XYZ-Difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.1 The ‖–⊥ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3

4 CONTENTS

4.3.1.1 Magnetic part of the neutron cross section . . . . . . . . . . . . . . 464.4 General case of multidetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Spin flip cross sections (magnetic part only) . . . . . . . . . . . . . . . . . . . 484.4.1.1 Polarization in x-direction . . . . . . . . . . . . . . . . . . . . . . . 484.4.1.2 Polarization in y direction . . . . . . . . . . . . . . . . . . . . . . . 504.4.1.3 Polarization in z direction . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.2 Non flip cross section for the magnetic part only . . . . . . . . . . . . . . . . 504.5 Application to a powder measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Derivation and limits of validity of P′ = −κ(κ ·P) . . . . . . . . . . . . . . . . . . . 534.7 Instrumental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Density Matrix formalism for magnetic scattering 615.1 General formulas for magnetic scattering only . . . . . . . . . . . . . . . . . . . . . 61

5.1.1 I↑↑ + I↑↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1.2 I↑↑ − I↑↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1.3 Express ~Q⊥ by (Mx,My,Mz) and α . . . . . . . . . . . . . . . . . . . . . . . 625.1.4 Derivation of the correlation functions Γν

ss′(~r, ω) . . . . . . . . . . . . . . . . 635.2 Nuclear-magnetic interference terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Visualization of the full formalism of the density matrix method for the 3d PA . . . 665.4 Application to paramagnetic powder scattering . . . . . . . . . . . . . . . . . . . . . 67

5.4.1 Derivation of ~P ′ = −~e(~e · ~P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 1

Experimenting with the neutron spin

The neutron has a magnetic moment of

µ = −1.913 · µN with µN = eh/2mp = 5.051 · 10−27JT−1 (1.1)

In the following we want to 1. define polarization of a neutron beam including also precessing spinpolarization 2. learn how to measure this polarization (also the precessing components), which is alinear process. 3. learn to understand,treat and measure the quadratic process of scattering, wherealso fluctuations are contributing.

We do this by experimentally preparing a precessing spin polarization and by preparing the toolsfor its measurement. This is done by visualizing the mathematical expressions like spinors, Paulimatrices, spin rotation operators, density matrix operations etc. by showing the correspondingexperimental physical device to realize it or to measure it.

1.1 Tools for the preparation of a precessing spin polarization

Quantum mechanics gives us a probability density to find a particle in a state with orbit angularmomentum, spin angular momentum, total angular momentum. But it gives neither informationabout the motion of the particle in its orbit, nor about the coordinate of the axis of rotation. Allwhat it says can be visualized as if the orbit ”precessed” in an unobservable way about the z-axisdescribable by a precession cone of the total ~J , ~L or ~S. In the case of the orbital behaviour one has atleast with ψ∗ψ a probability density of the particle in a state with the quantum number l,m. In thespin case only the fact is given that the particle such as the neutron or the electron has an internalangular momentum of h/2, where the three components sx, sy, sz obey the commutation relations[sx, sy] = ihsz or more generally [si, sj ] = ihεijksk, with i, j, k = x, y, z, but no probability densityof an orbit or of the spin directions or something similar is given. Such commutation relations arecommon to all angular momentum operators. In modern quantum mechanics angular momentaare often defined by such commutation relations, which are more general than the definition ~p× ~rwhich is not applicable to the spin. We want to consider some consequences of this commutator.What follows from this commutator with respect to the unobservable way the quantum mechanicalspin behaves (not the expectation value of it), which we will describe using the words ”as if it’precessed’ about the z-axis”? We want to understand the relationship between the indeterminacyrelationship, the commutator and the visualization of the spin of a particle.(omitted because ofspace limitations).

We want to become acquainted with the different theoretical tools like spinors, Pauli spinmatrices, density matrix formalism to describe the physical processes we are investigating. In

5

6 CHAPTER 1. EXPERIMENTING WITH THE NEUTRON SPIN

connection with this we want to understand the way how a spin flipper works in contrast to a spinrotator. These are the tools to be used in an instrument for polarization analysis measurements.An experimentalist must go to the limits of visualizability (= measureability) to be able to measuresomething. This is so as measurements always are in the range of classical physics, they representthe borderline between quantum mechanics and classical physics. For this reason the experimentalphysicist has to know this part of the theory better than the theoretician, who often does not needto care about measuring what he is treating formally. In the following we will always try to ask,what a certain formal process means physically, which is often just a sort of visualization.

1.1.1 Polarization and spinor

We use the Pauli two component spinor formalism . The wave function of the neutron with spin 12

is described in quantum mechanics by a spinor

ψ(r) =

(ψ1(r)ψ2(r)

)In textbooks about quantum mechanics examples can rarely be found where the two componentsψ1 and ψ2 are different functions of space coordinates . This is only the case if the Schrodingerequation includes terms which represent a strong force on magnetic moments, which is differentfor the two spin orientations. Examples for this are the Stern-Gerlach fields and also the magnetichelical structures of a Bloch wall for neutrons with gracing incidence [4, 5] or even the refractionof neutrons by a magnetic prism. This represents a sort of spin-orbit interaction. In this case thespin up and spin down components have different paths and the two components are separated inspace. In the case of the Bloch wall the solution includes plane waves with 8 different wave vectorsinside the wall. But those effects play a role only in very strong magnetic field inhomogeneities inferromagnets. These act as strong forces on the magnetic dipoles of the neutron spin and changethe path differently for the two components. In the following we use slowly varying magnetic fields,where the deviation of the neutron path is so small that the paths for both components are thesame. Then we can factorize space and spin parts. Slowly varying field in this case is related tothe wavelength of the neutron: If the size of the field inhomogeneities is on a length scale ordersof magnitude larger then the neutron wavelength then only the magnetic field at the respectivepoint or space region is sensed by the neutron. Space and time coordinates are then only coupledthrough the neutron velocity ~v with ~r = ~vt. Inhomogeneous magnetic fields along the neutron paththen appear in spin space only as simple time dependent quantities ~B(~vt). Then the spinors areonly functions of time.

First we want to visualize a constant spinor. We can write spin up |+〉 and spin down |−〉

|+〉 =

(10

)= χ+ (1.2)

|−〉 =

(01

)= χ− (1.3)

〈+| = (1, 0) = χ†+ (1.4)

〈−| = (0, 1) = χ†− (1.5)

χ = c+χ+ + c−χ− =

(c+c−

)(1.6)

Is there a relationship between the above used ~s = (sx, sy, sz) in three dimensions and thisspinor? This relationship between spinor space and cartesian space can be described using Pauli

1.1. TOOLS TO PREPARE PRECESSING POLARIZATION 7

spin matrices:

σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 −1

)(1.7)

(〈sx〉, 〈sy〉, 〈sz〉) =h

2(χ†σxχ, χ

†σyχ, χ†σzχ) (1.8)

which is the relationship between ~s and the spinor χ. Attention has to be paid to the fact that thisonly connects 〈sx〉, 〈sy〉, 〈sz〉 and not sx, sy, sz.

........

........

........

................................................................................

.....................

.........................

.................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................

..............................................................................................................................................................pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppp

pppppppppppppppppppppppppp


ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp〈s〉

x

y

z

Θ

Φ

Figure 1.1: Spherical coordinates (ϑ, ϕ)of the polarization direction of the spin forthe visualization of the spinor eq.(1.10)

If we wish to determine the χ which yields (see fig.1.1)

(〈sx〉, 〈sy〉, 〈sz〉) =h

2(cosϕ sinϑ, sinϕ sinϑ, cosϑ) (1.9)

we find

χ =

(cos ϑ

2 · e−i ϕ

2

sin ϑ2 · e

i ϕ2

)(1.10)

In order to see this we calculate(cos

ϑ

2ei

ϕ2 , sin

ϑ

2e−i ϕ

2

)(0 11 0

)(cos ϑ

2 e−i ϕ

2

sin ϑ2 e

i ϕ2

)= sinϑ cosϕ (1.11)

and similarly for y and z-components.These equations yield the relationships between the expectation values of the vector in the

direction (ϕ, ϑ) in spherical coordinates to the corresponding vector in Cartesian coordinates and tothe corresponding spinor in spinor space. In the following we will continuously use these relationsto visualize the spinor, and often without much calculation. The process of eq.(1.8) is a strictrelationship but it often results in unecessary complicated calculations which we can avoid byremembering the relationships (1.9) and (1.10).

1.1.2 Polarization and density matrix (|P | < 1 expressible)

How can one prepare a polarized beam? In textbooks this is nearly always done by a Stern-Gerlachmagnet. In practice we do it using a magnetized mirror. Such a mirror represents just a simplepotential well for one spin state. Inside of each material the neutrons of a neutron beam have


+µB

−µB

2πhmN

ρmbmnuclear potential

+ magnetic potential

-magnetic potential

-6

neutrons with spin up

?

neutrons with spin downare transmitted

are reflected

-

Figure 1.2: Nuclear and magnetic potential wells representing the way how a mirror polarizerworks

another velocity than in free space. This can be described by a potential energy. A layer ofmaterial acts on a neutron beam like a potential well. If the material is ferromagnetic there is anadditional contribution to this potential energy VM = ±µN ·B the sign depending on the orientationof the magnetic moment of the neutron in the magnetic induction B (see fig.1.2). In an unpolarizedbeam we have parallel and antiparallel spin components. As the spin and the magnetic momentof the neutron have opposite orientation those neutrons with parallel spin have a higher potentialenergy inside the magnetized layer. The others have a lower potential energy. If the size of themagnetic potential is just as high as the nuclear potential then only the spin state with a parallelspin sees a potential well. It is reflected, if its total energy is lower than the resulting potential wellof the nuclear plus the magnetic contribution. The other component is transmitted or absorbed, ifthe substrate is made of absorbing material.

If we have a completely polarized neutron beam, we have particles in the state χ+ or χ− inspinor space or particles with the spin direction (0,0,1) in cartesian space, the spin pointing inz-direction. This direction is determined by the quantization axes and is given by the direction ofthe magnetic field at the polarizer. The expectation values χ†χ for a spinor as given by eq.(1.10)is always normalized to one. The 3d-vector 1.9 has the direction of the polarization of the beam,but it is not clear how with these spinors to express an incomplete polarization. A common factorstill would describe a fully aligned spin, which would not be normalized. To describe an incompletepolarized beam one introduces a means of describing an ensemble of pure and mixed states by thedensity operator. We define with

χ =

(c1c2

)|c1|2 + |c2|2 = 1 (1.12)

a new operator which is the dyadic product of χ and χ†:

ρ =

(c1c2

)(c∗1, c

∗2) = χχ† (1.13)

ρ =

(|c1|2 c∗2c1c∗1c2 |c2|2

)(1.14)


=12

(1 + ~P · ~σ

)(1.15)

=12

(1 + Pz Px − iPy

Px + iPy 1− Pz

)(1.16)

=12

(1 +

(cosϑ sinϑe−iϕ

sinϑeiϕ − cosϑ

))(1.17)

with the properties

Trace ρ = Trρ =2∑

i=1

ρii = |c1|2 + |c2|2 = 1 (1.18)

ρ† = ρ (1.19)

i.e. ρ is a Hermitian operator. We can immediately see that

|c1|2 =12(1 + Pz) (1.20)

|c2|2 =12(1− Pz) (1.21)

Pz = |c1|2 − |c2|2 (1.22)Px = c∗1c2 + c∗2c1 = 2< (c∗1c2) = 2< (c∗2c1) (1.23)Py = c∗1c2 − c∗2c1 = 2= (c∗1c2) = 2= (c∗2c1) (1.24)

The density operator can be written as∑

m |m〉〈m| and if Trρ < 1 then the probability of eachstate can be included and it is ρ =

∑m |m〉pm〈m|. A completely polarized beam with polarization

in z-direction ~P = (0, 0, 1) yields the density matrix

ρ = |+〉〈+| =(

10

)(1, 0) =

(1 00 0

)(1.25)

A completely polarized beam in positive or negative x-direction with ~P = (±1, 0, 0) yields thedensity matrix

ρ =12

(1 + cosϑ sinϑe−iϕ

sinϑeiϕ 1− cosϑ

)=

(12 ±1

2±1

212

)(1.26)

An unpolarized beam with ~P = 0 has the density matrix

ρ =

(12 00 1

2

)(1.27)

A partially polarized beam consisting of 50% of a beam polarized in +x-direction and of 50%polarized in +y-direction has the density matrix

ρ = 0.5 ·(

12

12

12

12

)+ 0.5 ·

(12 − i

2i2

12

)=

(12

1−i4

1+i4

12

)(1.28)

Using eq.(1.17) we see that this corresponds to a degree of polarization of 1/√

2 = 0.70 in thedirection (1,1,0). (prepairing is hard, analysis is easier)

A powerful relationship of the density matrix is the following: The ensemble average of an op-erator A (sometimes also called expectation value, but it should not be mixed up with the quantum


mechanical expectation value alone, it contains both sorts of averages: quantum mechanical andstatistical, see below) is 〈A〉. In quantum mechanics it can be written as

〈A〉 = (c∗1, c∗2)

(A11 A12

A21 A22

)(c1c2

)(1.29)

= (c∗1, c∗2)

(A11c1 +A12c2A21c1 +A22c2

)(1.30)

= A11c∗1c1 +A12c

∗1c2 +A21c1c

∗2 +A22c2c

∗2 (1.31)

= Tr

(A11 A12

A21 A22

)(c1c

∗1 c1c

∗2

c∗1c2 c2c∗2

)(1.32)

= Tr

(A11c1c

∗1 +A12c

∗1c2 A11c1c

∗2 +A12c2c

∗2

A21c1c∗1 +A22c

∗1c2 A21c1c

∗2 +A22c2c

∗2

)(1.33)

= Tr(Aρ) = Tr(ρA) (1.34)

〈A〉 = Tr(ρA) = Tr(Aρ) (1.35)

The reason why it is so powerful is that the trace does not depend on the quantum mechanicalrepresentation. So the trace of ρA can be evaluated using any convenient basis.

The statistical average plays a role if one has a statistical ensemble of N systems, for examplewith each system having its own spinor χν and its own density matrix ρν . Then one gets an averageρ by

ρ =1N

N∑ν=1

ρν =12[

(1001

)+ ((

1N

∑ν

Pν) · σ))] (1.36)

=12(1 + (P · σ)) (1.37)

〈A〉 =1N

N∑ν=1

〈A〉ν =1N

N∑ν=1

Trσ(ρνA) (1.38)

= Tr(ρA) (1.39)

P = 〈σ〉 =1N

N∑ν=1

〈~σ〉ν =1N

∑ν

Tr(ρνσ) (1.40)

=12Trσ[(1 + P · σ)σ] (1.41)

As an example we determine the polarization for the mixture as given by eq.(1.28)

〈σx〉 = Tr(ρσx) = Tr

(12

1−i4

1+i4

12

)(0 11 0

)= Tr

(1−i4

12

12

1+i4

)=

12

(1.42)

〈σy〉 =12

(1.43)

〈σz〉 = 0 (1.44)

~P = (12,12, 0) (1.45)


Another example, which we need later, is a beam consisting of neutrons with polarizations equallydistributed over a range of φ = −π/10 to φ = +π/10 about the x-direction. ϑ shall be a fixed anglesay π/2. In this case the density matrix is

ρ =12

(1 〈e−iϕ〉

〈eiϕ〉 1

)=

12

1 e−〈ϕ2〉

2

e−〈ϕ2〉

2 1

=12

(1 0.98

0.98 1

)(1.46)

i.e. we have a reduced polarization of 98% even if the angular range of the polarization vector is36 degrees. For the averaging of 〈eiϕ〉 we use

〈eiϕ〉 = 〈cosϕ〉+ i〈sinϕ〉 = 〈cosϕ〉+ 0 (1.47)

= 1− 12〈ϕ2〉+

14!〈ϕ4〉 − · · · (1.48)

e−〈ϕ2〉

2 = 1− 12〈ϕ2〉+

18〈ϕ2〉2 − · · · (1.49)

〈φ2〉 =5π

∫ π/10

−π/10φ2dφ =

5π· 23

(π

10

)3

(1.50)

There are also many other useful relationships involving the density matrix

ρ|φ〉 = χχ†|φ〉 = χ(χ†φ) (1.51)ρ2 = ρ if ρ is a pure state (1.52)

~Pσχ = (2ρ− 1)χ = 2ρχ− χ = 2χχ†χ− χ = 2χ− χ = χ (1.53)

i.e. ~P points in the direction of the spin of the particle. The time evolution of the ensembles isdescribed by

ih∂ρ

∂t= −[ρ,H] (1.54)

The density matrix here is introduced because it is very powerful in the derivation of scatteringcross sections for polarized beams. To describe the behaviour of the spin in a polarized beamitself it seems to be more complicated than the above simple spinor description. We will show theusefulness of the density matrix in connection with the derivation of the formulas for scatteringcross sections. To understand the way a flipper or a spin rotator works it is easier to use the spinorformalism. But it can also be described using the density matrix formalism.

1.1.3 precession coil to rotate spinor in direction Θ, Φ

1.1.3.1 Spin with direction (ϑ,ϕ) in a constant magnetic field in z direction

We want now to apply the knowledge we have gained in the last section.We want to see the behaviour of the polarization if the vector of the polarization with orientation

Θ,Φ enters into a magnetic field with (0,0,Bz) (see fig.(1.5) First we have to investigate how one canrealize this. Normally the spin is kept parallel to the quantization axis by a guide field. Thereforeinitially the field and the spin are parallel behind the polarizer (see fig.1.5). As seen in figure 1.5we use a rectangular coil where two mutually orthogonal rectangular coils are wound just one ontop of another. In fig.1.5 this is indicated by showing pieces of these coils. The rest of it is cutaway to show the windings. These coils must be large enough so that within the cross section ofthe beam we can expect a homogeneous region of the respective field components. Unfortunately itis not possible to have a sudden transition also in the y direction, the field lines can not just begin


at a wall, they must be closed (div ~B = 0 cannot be avoided). (They can also not just begin at aMeissner shield!). By selecting the currents in each of these coils one can rotate the field inside thisrectangular coil in the xz plane into any desired direction. So we get an angle µ between field andpolarization at the entrance of the coil. But ϕ is either 0 or π.

The result of our following calculation will show that the spin precesses around this new fielddirection µ, 0 inside the coil (see fig.1.6). By selecting the field direction and the field strength asgiven in the figure caption 1.6 we can move the polarization direction into any wanted direction atthe exit of the sudden xz coil. There it suddenly will enter a field again in z direction. In this waywe have prepared a beam with the polarization in direction Θ,Φ in a constant magnetic field in zdirection at the exit of the coil. i.e. a spinor with the initial condition

χ =

(cos Θ

2 e−iΦ

2

sin Θ2 e

iΦ2

)(1.57)

or in Cartesian space(Px, Py, Pz) = (cos Φ sinΘ, sinΦ sinΘ, cos Θ) (1.58)

or with the density matrix

ρ =

(cos2 Θ

2 cos Θ2 sin Θ

2 e−iΦ

cos Θ2 sin Θ

2 eiΦ sin2 Θ

2

)=

12

(1 +

(cos Θ sin Θe−iΦ

sinΘeiΦ − cos Θ

))(1.59)

1.1.3.2 Rotation in spinor space

In order to be able to visualize the behaviour of the polarization in a magnetic field with an initialspin of eq.(1.57) we need a simple way to describe the rotation in spinor space. The following willnot be a rigorous mathematical demonstration but more an exercise to become acquainted withthe spinor space and its visualization. For rigorous mathematics you have to consult the book ofDirac [2] or a book about Quantum Mechanics such as that of Sakurai [12]. A rotation around thez-axis by an angle ϕ is affected by the following matrix

eiϕ2

σz ≡(ei

ϕ2 0

0 e−i ϕ2

)(1.60)

We demonstrate this by showing

1. that this identity is valid and

2. that it represents a rotation around the z axis by the angle ϕ.

........

........

........

................................................................................

.....................

.........................

.................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................

..............................................................................................................................................................pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppp



ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp〈s〉

x

y

z

Θ

ΦFigure 1.3: The initial conditions with po-larization direction ~〈s〉 in direction (ϑ, ϕ)and field ~B in direction z is what we want.


........

........

..............................................................

.......................

...............................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppp

pppppppppppppppppp

pppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppx

y

z

qqqqqqqqqqqqqqqqsqqqqqqqqqqqqqqqq

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppB

ϑ

Figure 1.4: Angles of spin ~s andfield ~B at the entrance of the sud-den xz coil

......................................... ............................................

..............................................

...................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................

................................................................................................................................................................................................ ................................

................................................

................................................

................................................

................ ................................

................................................

................ ................................

................ ................................

................................................

................ ................................

................

Polarizer

polarizedneutrons

π/2 flipped spin

part of z-coil

........................................................

................

.............................................. ................

................

sudden x-z coil

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

part of x coil........................... ................

....................................... ................

................

................

X

Y

Z.............................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................

Figure 1.5: Visualization of the possibility to realizethe initial condition that a polarized neutron begins ina defined way with a direction (Θ,Φ) inclined with re-spect to a magnetic field ~B. It enters suddenly throughthe walls of two orthogonal coils wound on top of eachother as indicated in the figure. (See text)

this identity is valid

eiϕ2

σz =

(1 00 1

)+ i

ϕ

2σz + i2

ϕ2

4· 12!

(1 00 1

)+

i3ϕ3

23 · 3!σz +

i4ϕ4

24 · 4!

(1 00 1

)(1.61)

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

..................................................................................................................................................................................................................................................................................................................................................

.....................

........................

............................

......................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

.............................................

.........................................................................................

............................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................

...............

....................................................................................................................................................................................

........

........

........

........

........

........

........

........

........

.

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

X

Z

qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqq

Θ

Φ

γµ

Y

ν

~B

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq

.......................................................................................

...........................

............................................................................................................................................................................................................................................................................................................................................

......................................................................

Φ

90−Θ/2µ

γ 90− ν

Nepers rule gives

cos Φ = cotµ · cot(90− Θ2

)

tanµ =tan Θ

2

cos Φ(1.55)

cos γ = sin Φ · cosΘ2

(1.56)

Figure 1.6: Elementary geometry gives µ and γ as functions of Θ, Φ (Neper’s rules). Adjustingthe coil first as a flipper coil gives the current Iπ for a precession of 180 deg and Ic to compensatethe guide field inside the coil. By selecting the current I = Iπ · 2γ

180 with Ix = I sinµ in the suddenx-coil and Iz = (I − Ic) cosµ in the z-coil we can manage any spin direction Θ,Φ at the exit of thecoils. Outside the coils ~B is in z direction.


=

(1 00 1

)cos

ϕ

2+ iσz sin

ϕ

2(1.62)

=

(cos ϕ

2 00 cos ϕ

2

)+ i

(sin ϕ

2 00 − sin ϕ

2

)(1.63)

=

(ei

ϕ2 0

0 e−i ϕ2

)(1.64)

it represents a rotation around the z axis by the angle ϕ

We can show this by using the fact that the relationship between the spinor space and the threedimensional space can be expressed by the Pauli spin matrices. If we want to transform the vector~P = (x, y, z) by the application of the above rotation operator to ~P ′ = (x′, y′, z′) then we have totransform it in the following way

~P ′σ =

(z′ x′ − iy′

x′ + iy′ −z′

)(1.65)

= eiϕ2

σz

(z x− iy

x+ iy −z

)e−i ϕ

2σz (1.66)

=

(ei

ϕ2 0

0 e−i ϕ2

)(z x− iy

x+ iy −z

)(e−i ϕ

2 00 ei

ϕ2

)(1.67)

=

(ei

ϕ2 0

0 e−i ϕ2

)(ze−i ϕ

2 (x− iy)eiϕ2

(x+ iy)e−i ϕ2 −zei

ϕ2

)(1.68)

=

(z (x− iy)eiϕ

(x+ iy)e−iϕ −z

)(1.69)

From this we see immediately

x′ = x cosϕ+ y sinϕ (1.70)y′ = −x sinϕ+ y cosϕ (1.71)z′ = z (1.72)

that it represents a rotation in three dimensional space around the z axis by an angle ϕ.We can also show it by the effect of the rotation operator on the spinor

eiγ2σz

(cos ϑ

2 e−i ϕ

2

sin ϑ2 e

i ϕ2

)=

(ei

γ2 0

0 e−i γ2

)(cos ϑ

2 e−i ϕ

2

sin ϑ2 e

i ϕ2

)=

cos ϑ2 e

−i(ϕ−γ)

2

sin ϑ2 e

i(ϕ−γ)

2

(1.73)

This shows that the application of the above rotation operator diminishes just the angle ϕ i.e. itrepresents a rotation around the z axis in the clockwise sense.

Similarly one gets for the rotation about x

eiχ2

σx = 1 cosχ

2+ iσx sin

χ

2=

(cos χ

2 i sin χ2

i sin χ2 cos χ

2

)(1.74)

and for the rotation about the y axis

eiϑ2σy = 1 cos

ϑ

2+ iσy sin

ϑ

2=

(cos ϑ

2 sin ϑ2

− sin ϑ2 cos ϑ

2

)(1.75)


Generally a rotation around an axis ~n by an angle ω we get the rotation operator

eiω2~n·~σ = 1 cos

ω

2+ i sin

ω

2~n · ~σ =

(1 00 1

)cos

ω

2+ i sin

ω

2

(cosϑ sinϑe−iϕ

sinϑeiϕ − cosϑ

)(1.76)

with~n = (cosϕ sinϑ, sinϕ sinϑ, cosϑ) (1.77)

Some exercises to become acquainted with these spinors

1. Determine the direction ϑ, ϕ for an arbitrary spinor(a1

b1

).

Solution:(a1

b1

)= c1

(cos ϑ

2 e−i ϕ

2

sin ϑ2 e

i ϕ2

)with tan

ϑ

2=|b1||a1|

and ϕ = arg b1 − arg a1 (1.78)

2. Given the spinor χ =(eiα cos δeiβ sin δ

). a) Determine the polarization vector. b) determine the matrix

which rotates it to(

10

)Solution:

ϑ = 2δ; and ϕ = β − α; with ~P = (sin 2δ cos(β − α), sin 2δ sin(β − α), cos 2δ) (1.79)

By inspection one sees that one has to rotate by β − α around the z-axis and then by 2δ around thex-axis. This yields the matrices

e−i β+α2

(cos δ sin δ− sin δ cos δ

)(ei β−α

2 00 e−i β−α

2

)= Q (1.80)

Applying this to χ results in the required spinor as one can see by inspection.

1.1.4 Time dependent Schrodinger equation

1.1.4.1 The next steps:various flippers, CRYOPAD, SSPAD,

Now we want to see the behaviour of the spinor inside the coil with the field in (ϑ, ϕ) directionwhich represents a constant homogeneous magnetic field inside the coil of fig.1.5. This will indicatethe way how various sorts of flippers work such as a π/2-flipper or a π-flipper. We will see howthis can be used to completely measure a precessing spin. This gives us the understanding of apolarimeter for neutrons, which is realized with the CRYOPAD at the ILL and can be also done onan instrument like D7 after the addition of a π/2 flipper coil behind an adjustable guide field sectionbetween sample and analyzer, which we could call simplified spherical polarization analysis device(SSPAD), because it avoids complicated cryotechniques. After this we will derive the behaviour ofa spin rotator which is a static field which changes the direction as a function of space coordinate(e.g. a helix). This is accomplished by the solution of the time dependent Schrodinger equation ina magnetic field with a helical geometry in space, which again can be realized by an appropriatelywound coil. This helps us to understand what happens if we change the direction of the guide fieldof the neutron spin. Then we will shortly apply our knowledge to the uncertainty relationship forthe spin and try to get out as much as possible from it (omitted because of space limitations).After this we have all the necessary elements to measure and to analyse the scattering of polarizedneutrons. We still require some introduction into the theoretical concepts of the different sorts ofscattering of polarized neutrons. This can be done by calculating the transition probabilities or byapplication of the density matrix formalism to polarized beam scattering.


1.1.4.2 Spin behaviour in a constant homogeneous magnetic field

The description of a spin in a constant B-field can be obtained by using the time dependentSchrodinger equation. For this we need the potential of the spin in a magnetic field. The magneticmoment of the neutron is µ · ~σ with µ given by eq.(1.1 on page 5) and ~σ the Pauli spin matrices.The potential energy in the field ~B is then V = −µ~σ · ~B. Omitting the dependence on ~r andconcentrating on the spin only we can write down the time dependent Schrodinger equation of thisproblem (reason that this is possible see 1.1.1 on page 6):

ih∂

∂t

(ψ1

ψ2

)= µ~σ · ~B

(ψ1

ψ2

)(1.81)

∂

∂t

(ψ1

ψ2

)=

µB

ih

(cosϑ sinϑe−iϕ

sinϑeiϕ cosϑ

)(ψ1

ψ2

)(1.82)

= −iωL

2

(cosϑ sinϑe−iϕ

sinϑeiϕ cosϑ

)(ψ1

ψ2

)(1.83)

= − i2

(ωz ωx − iωy

ωx + iωy −ωz

)(ψ1

ψ2

)(1.84)

with

ωL =2µBh

(1.85)

ωx = ωL sinϑ sinϕ (1.86)ωy = ωL sinϑ cosϕ (1.87)ωz = ωL cosϑ (1.88)

Separation of variables yields

d

(ψ1

ψ2

)(ψ1

ψ2

) = −iωL

2

(cosϑ sinϑe−iϕ

sinϑeiϕ cosϑ

)dt (1.89)

Integration gives

ln

(ψ1(t)ψ2(t)

)− ln

(ψ1(0)ψ2(0)

)= −iωL

2

(cosϑ sinϑe−iϕ

sinϑeiϕ cosϑ

)t (1.90)

= − i2

(ωz ωx − iωy

ωx + iωy −ωz

)t (1.91)

(ψ1(t)ψ2(t)

)= e

−iωLt

2

(cosϑ sinϑe−iϕ

sinϑeiϕ cosϑ

)(ψ1(0)ψ2(0)

)(1.92)

With our knowledge of rotations in spinor space we see immediately that this represents a rotationof ωLt around the axis ~n, which has the direction ϑ, ϕ in space, i.e. around the field ~B.

We include here only the spin but not the space behaviour. In [6, 7] also the space behaviour ofa wave packet and the beam as a quantum mechanical current is included in the description, buthere this would hide the essential behaviour under many practically empty formulas. If you haveunderstood the essential things you can add this formalism yourself.

Chapter 2

Applications for measuring equipment

2.1 Behaviour of the spin in a π flipper coil

We want to apply the solution eq.(1.92) for a field along the x-axis of our coil. For this we adjustthe currents in the xz-sudden-coil of fig.1.5 so that the guide field in z-direction is compensatedinside the coil and only a field in x-direction exists. This corresponds to ϑ = 900 and ϕ = 00. Withthis we obtain

~n~σ =

(0 11 0

)(2.1)

This yields the following behaviour of the spin |+ >=

(10

)inside the coil

(ψ1(t)ψ2(t)

)= e−i

ωLt

2σx

(10

)(2.2)

=

(cos ωLt

2 i sin ωLt2

i sin ωLt2 cos ωLt

2

)(10

)(2.3)

=

(cos ωLt

2i sin ωLt

2

)= ei

π4

(cos ωLt

2 e−i π4

sin ωLt2 ei

π4

)(2.4)

With our knowledge of the visualization of a spinor from eq.(1.9) and fig.1.1 we can immediatelyvisualize this as a precession of an incident spin |+ > with ϑ changing with time and ϕ = π/2fixed i.e. around the x-axis in the y-z plane with the time dependent angle ωLt (see fig.2.1). If weadjust the current in this coil so, that the time of flight through the coil is just the time neededfor half a Larmour precession around the x-axis, the neutron exits from the coil with spin directionreversed with respect to the direction of the guide field. This is exactly the behaviour of a flippercoil. For this we only need dc-currents and it has nothing to do with a random process, as theword insinuates. It is the result of a classical spin precession. The transition from the guide fieldto inside of this coil and the transition from the inside of the coil to the guide field on the outsideare sudden transitions, the neutron remains in its state that it had before.

The adjustment of the currents for such a flipper can be done in the easiest way by firstcalculating the necessary current for half a Larmor precession for the pathlength L in the coil andthe wavelength λ of the neutrons by

Hπ =67.825

λ[A]L[cm]Oerstedt =

67.825λ[nm]L[cm]

mT (2.5)

17

18 CHAPTER 2. APPLICATIONS FOR MEASURING EQUIPMENT

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqguide field guide field

π flipperfield

polarized neutronsflipped neutrons

spin direction precessinginside the flipper coilhalf a Larmor precession

bormal tofigure plane

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

Figure 2.1: Half a Larmor precession as a π flipper

Setting the x coil to this current value one then carries out a current scan for the field thatcompensates the guide field. This can then be fitted using a parabola and the minimum of theparabola is the best correction current. Then one can repeat this also for the x coil and then againfor the z-coil until it does not change. The advantage of this method is that it uses also count rateswhich are not too small to find the minimum.

2.1.1 Polarization measurement

Using such a flipper coil and a second polarizing mirror as analyzer we are now able to determinethe degree of polarization. For this we measure the intensity I↑↑ with spin not flipped and theintensity I↑↓ with spin flipped. Their ratio (after background subtraction) gives the flipping rationR = I↑↑/I↑↓ and the polarization P = (R− 1)/(R+ 1).

The measurement of polarization can be done with different degrees of sophistication. We use apolarizer, a flipper and a second polarizer as analyzer. Each of these elements has its imperfections.Sometimes (often in nuclear physics experiments) one needs the exact polarization of the beam atthe target position. Then one has to separate the contributions to the imperfections of each ofthe elements. One has to determine the efficiencies of the polarizer, the analyzer, and the flipperseparately. The polarizing efficiency P can be defined as the degree of polarization attained byan unpolarized beam on passing through the polarizer. The flipper efficiency f can be defined asthe fraction of spins reversed by the flipper .

The properties of a flipper can be given by a matrix representation as

F =

(ε 1− ε

1− ε ε

)(2.6)

the properties of a polarizer correspondingly by the matrix representation(P11 P12

P21 P22

). In this matrix

P11 gives the part of the incident beam which is transmitted without spin change. P12 gives thepart of the beam that is flipped from spin down to spin up. P21 the part flipped from spin up tospin down and P22 is the part of spin down wich remains in spin down when transmitted throughthe polarizer. These properties can be measured on a test set-up which has a polarizer and ananalyzer and a flipper in front of the test object and a flipper behind the test object as shown infigure 2.2. The countrate without flipper currents gives P11, with only flipper1 on we get P12, withonly flipper2 on we get P21 and with both flippers on we get P22. If P21 6= P12 we have a matrix

2.1. π-FLIPPER AND ITS USE 19

polarizer flipper1 test object flipper2 analyzer

Figure 2.2: Measurement of the properties of a polarizer in the position ”test object”. When notest object is present one can measure the flipper efficiencies.

which is not conserving parity or perhaps an error in measurement. With these measurements weget the polarizer efficiency from(

AB

)=

1√2·(P11 + P12

P21 + P22

)=

1√2

(P11 P12

P21 P22

)(11

)(2.7)

as the polarization from an unpolarized beam(11

)P =

P11 + P12 − P21 − P22

P11 + P12 + P21 + P22=A−B

A+B(2.8)

A flipper behind such a polarizer changes the spin of this beam in the following way:(Af

Bf

)=

(ε 1− ε

1− ε ε

)(AB

)=

(Aε+B(1− ε)A(1− ε) +Bε

)(2.9)

with the polarization

Pf =Aε+B(1− ε)−A(1− ε)−Bε

Aε+B(1− ε) +A(1− ε) +Bε=

(1− 2ε)(B −A)A+B

= −(1− 2ε)P (2.10)

With a second flipper (characterized by ε′ instead of ε) one obtains for the flipper matrix of twoflippers(

ε′ 1− ε′

1− ε′ ε′

)(ε 1− ε

1− ε ε

)=

(1− (ε+ ε′ − 2εε′) ε+ ε′ − 2εε′

ε+ ε′ − 2εε′ 1− (ε+ ε′ − 2εε′)

)(2.11)

which has the same form as for one flipper, only with εeff = ε+ ε′ − 2εε′ and using the same stepsas above one obtains for the spinor after a polarizer and the two flippers(

−(A−B)(ε+ ε′ − 2εε′) +A(A−B)(ε+ ε′ − 2εε′) +B

)(2.12)

with the polarization

Pff ′ = P [1− 2(ε+ ε′ − 2εε′)] with (2.13)

P =A−B

A+B(2.14)

The analyzer always measures the spin up component. This yields countrates as

I↑↑ = A (2.15)I↓↑ = Aε+B(1− ε) (2.16)I↑↓ = Aε′ +B(1− ε′) (2.17)I↓↓ = A− (A−B)εeff (2.18)


Combining these measured values one obtains the flipper efficiency:

I↑↑ − I↓↓

I↑↑ − I↑↓=

(A−B)εeff

(A−B)(1− ε)(2.19)

=ε′ + ε− 2εε′

1− ε(2.20)

=2ε− 2ε2

1− ε= 2ε (2.21)

with the assumption that ε = ε′. One can measure these quantities for the polarizer and theanalyzer and in this way come to very precise results. A similar procedure including 4 flippersand three polarizers for precision measurements of the polarization in an experimental set up isdescribed in [8].

2.2 Behaviour of the neutron in a π/2 flipper

Assume that we need to rotate the incident spin of the neutron |+ > by 900 so that afterwards itis normal to the guide field. We can use the same coil as above, but we have to adjust the field sothat it is inclined by 450 in the xz plane. If the field strength is the same as in the last section forthe π-flipper, then the spin describes again half a Larmor precession on the path length inside thecoil. We have a similar set of equations to solve which should now be easy to do. At the exit fromthe coil the neutron spin will then be in the xy-plane and precess around the guide field outsidethe coil.

z

x ...................................

........................................

..................................

.......~Hπ/2

~Hπ/2~Hπ/2

~Hπ/2

z

x ...................................

.........................................

....................................

..z

x ...................................

.........................................

....................................

...z

x ...................................

.........................................

....................................

...

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

......................

........................................................................................

...........................................................

............................................................

............................................~Hπ/2

~Hπ/2

ddrg............................

..............................y

z

neutronflightpath

..............................

...................... HH

coil field450 inclinedto z and x axis neutron spin

directionbefore the flip

spin directionafter the flip

spin precession inside the π/2-flipper coilxz plane

yz plane

.......................................................................................................................................................................................................

..................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................

............................................................................................. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq...................... qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

Figure 2.3: Half a Larmor precession as a π/2 flipper

2.2.1 Adjustment of the currents in a π/2-flipper

Adjustment of the coil first as a π-flipper gives the current for Hπ. The coil field for the π/2 coilmust be set to Hπ/

√(2). Then the correction coil is used to adjust the field so that the polarization

~P appears to be totally depolarized at the analyzer, i.e. that the counting rate is not changed ifthe spin is flipped using the π-coil (switching the current on or off).

2.2. π/2-FLIPPER AND ITS USE 21

Figure 2.4: Measurement of one of the fluctuating components of the precessing spin polarizationusing the π/2 flipper at the entrance of the flipper. The yz-view shows at the right the precessingpolarization with (Px, Py, Pz). Pz, Pρ, Px and Py are drawn. When it enters into the flipper coilit has e.g. the component Px as shown in xz-view of the figure. By the π/2 coil this is rotated inthe z direction as shown in the figure. The spin polarization at the exit of the coil is beginning aprecession around this component as shown on the left side of the figure.

2.2.2 Measurement of precessing spin polarization, CRYOPAD, SSPAD

By using such a π/2-flipper coil it is possible to determine the polarization for the case of a beamwith precessing spin polarization, i.e. a polarization vector which is oriented in another directioncompared to the field and which is therefore precessing in this field. Such a coil rotates the spincomponent which is normal to the initial z-direction and normal to the flight path into the z-direction. There it can be measured using a spin flipper and a spin analyzer (=a mirror thatreflects only one spin) (see fig.2.4). The ρ-component of the polarization can be measured by theuse of a varable guide field in front of the π/2-flipper. This can consist of a simple rectangular coil(see fig.2.5), which makes possible to shift the ρ-direction of the polarization into the x-directionby accelerating the precession without changing its cone angle. The ρ-direction is found when theπ/2-flipped intensity is at a maximum. The π-flipper to measure the polarization has for suchmeasurements to be between the π/2-flipper and the analyzer and this is possible for as manydetectors as one can equip with the unit of fig. 2.5. If at the same position in space one analysesthe polarization in three space directions by rotating Px with a π/2 flipper and Pρ with the variableguide field and a π/2-flipper into the z-direction where the x, the ρ and the z-component Px, Pρ andPz can be determined using a π-flipper and an analyzing mirror one obtains the direction and themodulus of ~P at the position in space which corresponds to the entrance point into the xz-flippercoil of fig.1.5 or of the second coil of fig.2.5. This allows also to measure the full polarization in


""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

""

""

""

""

"""

PPPq

variableguide field

?

π/2-flipper

π-flipper

--

neutron pathto analyzer

--

polarizedneutronbeam toinvestigate

Figure 2.5: Unit to insert be-tween sample and analyzer tomeasure the complete precess-ing spin polarization. Thefirst coil is a variable guidefield to shift the ρ-componentinto the x.direction withoutchanging the cone angle. ρ-component is found when theπ/2-flipped intensity is a max-imum by variation of the cur-rent in coil 1. The third coilmay be superfluous

the case of a precessing spin polarization and to see whether there is somewhere a depolarizationin the scattering process

This is the way how a polarimeter works. Such polarimeters were developed around 1970 inthree different laboratories [9, 10, 11] In this case the rotation of the spin by rotating it using aslowly rotating guide field, (which we describe in the following section), is not sufficient. This is onlysufficient to measure the component of the polarization parallel to the guide field of a scatteredbeam. If one wants to measure the full polarization one has to use additionaly a π/2-flipper, avariable guide field and position a π flipper in front of the analyzer.

During the discussion in a Summerschool in Julich Dr.Schweika from Julich made the remark,that one should insert the π/2-flipper with the varable guide field in front of the sample and adjustit so that at the center of the sample the spin direction points into the x, y, z direction with the fielddirection of the spinturn coil independently having the x, y, z direction. Then one can measure thefull polarization matrix with 9 elements with all detectors equipped with analyzers simultaneouslyand even in time of flight mode of the instrument. This would be the most general way of measuringthis matrix even with an applied field, i.e. without the necessity of a zero field.

The CRYOPAD fig.2.6 is a polarimeter, which enables the measurement of the three com-ponents of the polarization vector after scattering by a magnetic single crystal for each incidentpolarization vector component separately (in total 9 results for one scattering). J. Brown et al.have shown in many interesting papers how this possibility can be used in crystallography to clar-ify magnetic structures just by the analysis of the polarization matrix Pij . This shows that it isinteresting to be able to measure this matrix. In the papers of this group one can find often thestatement that with a guide field at the sample all polarization components normal to the guidefield arising by the magnetic scattering immediately begin to precess and are so unmeasurable.We have seen, that this is not true, they can be measured by adding a π/2-flipper coil somewherebetween the sample and the analyzer and additionaly a piece of guide field with a coil to find themaximum of the polarization in the ρ-direction with ρ =

√x2 + y2 = cylinder coordinate with

guide field direction as cylinder axis. The angle of the precession cone remains namely constant onthe whole way from the sample to the analyzer. The fact that the finite size of the sample smearesthe components normal to the guide field over a certain angular range does not make a problem,as we have seen for a range of 36 deg in section 1.1.2 on page 11. The sample thickness givingsuch a smearing can be estimated with the help of eq.(2.5 on page 17) which gives the path length

2.2. π/2-FLIPPER AND ITS USE 23

Figure 2.6: Construction of the CRYOPAD taken from [37]

for the precession of π with the given fieldstrength and wavelenght. For 36 deg this would allowa pathlength in the sample of 8 mm for 1 mT and λ = 0.15 nm i.e. quite a large sample. Withsuch a simple improvement one can change a 3d-polarization analysis instrument into a sphericalpolarimeter simultaneously avoiding all the uncertainities which arise from the strong (mutuallyinteracting) fields of the adiabatic spin rotators contradicting the condition of no field at the sam-ple position when the Meissner shield becomes superconducting, of mirror images of the field in


superconductors and from the small size of the rotation coils inside the Meissner shields and thatone has no easy access to the interior of this shield in order to be able to control these fields, andall the problems with cryogenics. To distinguish the Px and Py direction at the sample position,one can calibrate the setup by putting a known sample with only a component in x- or y-directionand look for the current in the adjustable guide field that gives then the maximum. Perhaps onecould use a quartz sample inside a π/2-flipper coil to calibrate the instrument. One can call suchan instrument SSPAD = simplified spherical polarization analysis device. It makes possible mea-surements to determine all components of Pij as with the CRYOPAD. One could even think thatthe CRYOPAD could be simplified using only one Meissner shield, avoid the nutator by two suddenxz-coils (see fig.1.5 on page 13) to rotate the spin inside these coils into the wanted directions.

Figure 2.7: Schematic horizontal cross section ofCRYOPAD. (1) axial guide field (150 Oe); (2) non-magnetic rotation unit; (3) adiabatic rotation (250Oe); (4) concentric Meissner shields; (5) vertical su-perconducting solenoids - precession field (up to 100Oe); (6) zero field chamber with variable temperatureinsert; (7) sample position

Figure 2.8: Polarization directions in differ-ent regions:(1) axially polarized incident neu-trons (2) and (3) overlap, spin adiabaticallyrotated. (4) Meissner shields; (5) precessionregion; (6) zero field; (7) sample position

2.3 Spin behaviour in a static magnetic field varying slowly as afunction of z

What happens if we change the direction of the guide field slowly can be investigated by solving theSchrodinger equation for polarized neutrons in the coil of fig.2.9. The neutron flies with constantvelocity along the z-direction. It experiences a rotating field. We can also apply an additional fieldBz in z-direction. The effective field is

~B = (0, 0, Bz) +B1(cosωt,− sinωt, 0) (2.22)

The time dependent Schrodinger equation for this problem is

∂

∂t

(ψ1

ψ2

)=

µ~σ ~B

ih

(ψ1

ψ2

)= − i

2

(ωz ω1e

iωt

ω1e−iωt −ωz

)(ψ1

ψ2

)(2.23)

We transform this equation to a coordinate system that rotates with the frequency the spin sensesfrom the rotating field ω around the z-axis. This is accomplished by(

ψ1

ψ2

)= ei

ωt2

σz

(ϕ1

ϕ2

)(2.24)

2.3. SLOWLY VARYING MAGNETIC FIELD 25

.............................................................................................................................................................................................................................................................................................................................................................................................................................................

.......

.........

............................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................

............................................................................................................................

............................................................................................................................

............................................................................................................................

...............................................................................

............................................................................................................................

............................................................................................................................

............................................................................................................................

............................................................................................................................

...............................................................................

............................................................................................

.........................................................................................

.......................................................

.................................................................

...............................................................................

................................................................................

.....................................................................................

.........................................................................................

.............................................................................................................................................................................

..............................

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

.....................................

.....................................................

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............................

............................................................

....................................

........................................

....................................

........................................

....................................

........................................ ....................................

........................................................

............................................................

....................................

........................................

....................................

........................................

....................................

........................................

....................................

........................................

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............

.......................

........................

........................

..........................

..........................

............................

.............................

..............................

.................................

..............

........................

..........................

.............................

..............................

........................

...........................

........................

..........................

....................... ...............

....................................

........................................

........................

..........................

.............................

.................................

....................

....................................................................................................................................................................

..................................................................................................................................

...........................................................................................

..................................................................

.......................................................................

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppx

-yz

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

B(z)

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp

pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp

pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppBz = 0

Bz 6= 0

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

zppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppy

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppppppppppcone described

by B(z)if Bz=0 and Bz 6= 0

Figure 2.9: Coil to generate a helical magnetic field wound on the outside of a hollow cylinder.Depending on whether or not we apply a constant field along the z direction, on the right of thepicture the cone is given which describes the magnetic induction B(z) either as a flat circle or as acone

(ψ1

ψ2

)= i

ω

2σze

i ωt2

σz

(ϕ1

ϕ2

)+ ei

ωt2

σz

(ϕ1

ϕ2

)(2.25)

= eiωt2

σz

[iω

2σz

(ϕ1

ϕ2

)+

(ϕ1

ϕ2

)](2.26)

Inserting this in eq.(2.23) yields

eiωt2

σz

[iω

2σz

(ϕ1

ϕ2

)+

(ϕ1

ϕ2

)]= − i

2

(ωz ω1e

iωt

ω1e−iωt −ωz

)ei

ωt2

σz

(ϕ1

ϕ2

)(2.27)

iω

2σz

(ϕ1

ϕ2

)+

(ϕ1

ϕ2

)

= − i2e−i ωt

2σz

(ωz ω1e

iωt

ω1e−iωt −ωz

)ei

ωt2

σz

(ϕ1

ϕ2

)(2.28)

=

(e−i ωt

2 00 ei

ωt2

)(ωz ω1e

iωt

ω1e−iωt −ωz

)(ei

ωt2 0

0 e−i ωt2

)(ϕ1

ϕ2

)(2.29)

=

(e−i ωt

2 00 ei

ωt2

)(ωze

iωt2 ω1e

iωt2

ω1e− iωt

2 −ωze− iωt

2

)(ϕ1

ϕ2

)(2.30)

=

(ωz ω1

ω1 −ωz

)(ϕ1

ϕ2

)(2.31)(

ϕ1

ϕ2

)= − i

2

[(ω 00 −ω

)+

(ωz ω1

ω1 −ωz

)](ϕ1

ϕ2

)(2.32)

= − i2

(ωz + ω ω1

ω1 −ωz − ω

)(ϕ1

ϕ2

)(2.33)


This is a differential equation with constant coefficients which can be integrated by separation ofvariables as shown above eq.(1.92) with the solution

(ϕ1(t)ϕ2(t)

)= e

− i2

(ωz + ω ω1

ω1 −ωz − ω

)t(

ϕ1(0)ϕ2(0)

)(2.34)

.......................................................................................................................................................................................................................................................................................

......................

.........................

.......................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

ω1

ω

W

x”

z”

........

........

.............................................

........................

...............

γ................

................................

................................

................................

................................

................................

................................

................................

................................

................................

................

................

................

................

................

................

................

................

................

................

......

..............................

................................

.................................

...................................

.....................................

.......................................

...........................................

...............................................

......................................................

........

...................................................................................................................................................................................................................................................................................................................................................................

........................

..................

........................................................................

...................

.......

................................

.............................qqqqqqqqqqqqqqqqqqqqqqqqqqS

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq

ω1

ω

ωz

W

x”

z”

qqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq Sqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqBqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

γ

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq α

Figure 2.10: Definition of angles for the interpretation of the solution of the spin in a slowly varyingfield. If a constant field along the z axis is applied the initial spin direction is along the resultantfield direction, as indicated on the right figure. The precession cone has then the apex angle α andis unrolling on the cone described by B. The equation is the same, and only the initial conditionψ1(0) and ψ2(2) will change.

We conjecture that the solution is equivalent to one obtained by a rotation about an axis whichis inclined with respect to the coordinate axis. To investigate this possibility we have to try whetheror not we can write the exponent using the form ~n · ~σ. If we assume (see figure 2.10)

sin γ =ω + ωz

W(2.35)

cos γ =ω1

W(2.36)

W =√ω2

1 + (ω + ωz)2 (2.37)

we can rewrite the matrix in the exponent as(ωz + ω ω1

ω1 −ωz − ω

)= W ·

(sin γ cos γcos γ − sin γ

)(2.38)

= (cos γ, 0, sin γ) · ~σ = ~n · ~σ (2.39)

A rotation by γ about the y-axis yields the solution as a simple precession around the x”-axis ofthe new system with the precession frequency W:(

ϕ′′1(t)ϕ′′2(t)

)= e−i γ

2σy

(ϕ1(t)ϕ2(t)

)(2.40)


= e−i γ2σye

−i Wt2

(sin γ cos γcos γ − sin γ

)ei

γ2σye−i γ

2σy

(ϕ1(0)ϕ2(0)

)(2.41)

= e

−i Wt2

(0 11 0

)(ϕ′′1(0)ϕ′′2(0)

)(2.42)

= e−i Wt2

σx

(ϕ′′1(0)ϕ′′2(0)

)(2.43)

The complete solution with the explicit dependence on the initial spin direction at time 0 isfinally obtained as: (

ψ1(t)ψ2(t)

)= e

i2[−ωtσz−γσy−Wtσx+γσy+ωtσz ]

(ψ1(0)ψ2(0)

)(2.44)

To visualize this solution we have to visualize all the transformations. The initial condition at theright is first transformed to the rotating system, then inclined by the angle γ between the effectivefield direction (direction of W in fig.2.10) and the initial spin direction at time 0. This effectivefield direction depends on the different components of the rotation and the Lamor precession of thefields. This yields an effective precession frequency W. The spin precesses with angular velocity Win this rotating coordinate system around the axis inclined with angle γ. To see this precession inthe laboratory system, we have to transform this precession around the inclined axis back to thelaboratory frame. For this we bring the axis back to the direction inclined by γ and then back tothe fixed laboratory system. In this process and due to the geometrical conditions of the processthe cone precessing with W is unrolling on the surface which is described by the field direction. Itdepends on the different signs on which side of this surface the precession cone is unrolling. Fig.2.11shows the process for the case when ωz = 0.

2.3.1 Behaviour of the spin in a rotating guide field, adiabatic spin rotator

Figure 2.13 indicates the working of the adiabatic xyz coil as shown by the rotation of the spin inthe y-coil. This behaviour as described above plays a role if one changes the direction of the spinby changing the direction of the guide field. The behaviour in fig.2.11 allows one to estimate themaximum deviation of the spin from the field direction. This maximum deviation between fielddirection and spin direction is the angular diameter of the precession cone i.e. 2γ = arccosω1/Wor 2α = 2| arccosω1/W − arctanωz/ω1|.

In this case and in contrast to the sudden-xz coil in a polarimeter, the spin always remainsparallel to the field direction (depending how well these conditions are fulfilled) and it rotates withthis field direction. In figures 4.2 on page 49, 4.4 on page 55, 4.3 on page 54 this is used to rotatethe spin on its way to the sample in three mutually orthogonal directions. After the scatteringprocess the field direction is rotated back to the z direction and it takes with it that componentof the spin, whose direction is not changed by the scattering process which is the cone axis ofthe resulting precessing spin polarization. In this manner one can unambiguously determine thepolarization component parallel to that of the incident polarization of the beam after the scatteringprocess and simultaneously for many different detectors. This is necessary if one wants to use amultidetector instrument with full polarization analysis. The scattering law of the sample andthe kind of scattering (e.g. paramagnetic, nuclear spin incoherent) determine the behaviour ofthe polarization in the scattering process, the depolarization or the change of direction of the


........

........

........

.......................................................................

......................

..........................

.........................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................

......

............................

.................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

..................................................................................................................................................................................

................................................................

......................

..........................

.....................................

..................................................................................................................................................................................

................................................................

................................

............................................ ................ ................ ................

................................

................................

..

........................

.............................................

.................................................................................................................................

.........................................................................................................................................

............................................. ....................

..................................

.................................. ............................

..............

..........................................................................................................

..............

............................

................................. .........................................

..............

ω

W WS S

B1

B2

B3

M

z

................................................................

......................

..........................

....................................

....

........................

...........................................................................................................................................................................

........................................................................................................................................................................................

ω

..............B(t)

W

ω1

.......................... ...... .......................... ...... ...........................................................................................

..............................

................................................................................................................

...........................

.......................

Figure 2.11: Behaviour of the neutronspin 〈~s〉 relative to the field direction ~Bin the general case. The intersection ofthe field direction with the unit sphere de-scribes the curve B1, B2, B3. The inter-section of the spin direction with the unitsphere describes a curve which originatesby the unrolling of the cone with axis WMand with its apex in M. The apex angle is2γ (see fig.2.10), in the conical case it wouldbe 2α.

........

................................................................................

.......................

.............................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................

.......................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................

.............................................

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

.....................

................................................................................................................................... ..............

..........................................................................................................................

pppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp ppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppp

pppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppB(t)

W

ω

ω1

B(t1) B(t2) B(t3)ω

Wunrolling coneppppppppppppppppppppppppppppppp

Figure 2.12: Behaviour of the neutron spin 〈~s〉 rel-ative to the field direction ~B if the frequency ω issmaller than the Larmorfrequency

incident polarization. If you want to investigate the precessing polarization of the beam you needa polarimeter or you have to add a π/2-flipper coil and a section of variable guide field in aninstrument as in figures 4.2 on page 49,4.4 on page 55, 4.3 on page 54.


adiabatic xyz-coil

..........................................................................................................

........................

.......................

.

z

y

polarizeranalyzer

field of the y-coil

arrows are the spins of the neutrons

.....................................

.......................................

..........................................................................................................................................................................

...............................................................................................................................................................

...................................

................................

....

.......................................................................................................................................................................................................................................................................................................................................................................

...........................................

..................................................................................................................................................................................

.......................................................................................

.................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................

.........

................................

....................................

.................................................................................................................................................................

..............................................................................................................................................................................................

..................................

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp ppppp

ppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppp ppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp

ppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppp

ppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp

pppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp pppppppppppppppp ppppppp

ppppppppp ppppppppppppppp.....................................................................................................................................................................

Figure 2.13: Way how the adiabatic xyz coil works, shown by the rotation of the spin in the y-coil.Near the polarizer and analyzer guide fields, which are far reaching fields decreasing with distanceof the field sources, the spin has the z direction. The spin follows the direction of the field whichresults from the superposition of the coil field and the stray field of the analyzers. This is sufficientlystrong and slowly changing that the condition ω/ω1 1 is fulfilled. The figure shows only theeffect of one of three mutually orthogonal coil pairs, which in the case of a cylinder symmetricarrangement of analysers and guide fields work also in the same way. (see figures 4.2 on page 49,4.4on page 55,4.3 on page 54,

Chapter 3

Density matrix formalism todetermine scattering cross sections

3.1 Introduction

In order to indicate how one has to proceed to obtain the necessary information using the densitymatrix formalism, we apply it to a calculation of the ensemble average of the cross section and tothe polarization and show how we have to apply it to the xyz-method .

We first state some rules for Pauli spin matrices, which are very useful in connection withthe density matrix. Then we define the scattering matrix and give some general properties ofthis scattering matrix. We apply these rules and properties to obtain the nuclear scattering crosssection and the nuclear spin incoherent cross section and the dependence of the polarization forthese sorts of scattering. Formulae as by Blume will be derived in a special chapter (chapter 5)after we have applied the transition probability method to determine the magnetic cross section.With the formulae of Blume we are able to discuss how the different contributions can be measuredusing the xyz method. We will see, where one needs a polarimeter to obtain results and whetheror not the instrument D7 without modification is already sufficient for measuring all these crosssections or has to be changed by adding a π/2-flipper and a variable guide field section.

3.2 Compilation of rules for the Pauli spin matrices

3.2.1 some simple rules

First some useful rules for spin matrices:

σα, σβ = 2δαβ (3.1)= σασβ + σβσα (3.2)

[σα, σβ ] = 2iεαβγσγ (3.3)= σασβ − σβσα (3.4)

σ1σ2 = −σ2σ1 = iσ3 (3.5)σασβ = δαβ + iεαβγσγ (3.6)σ†α = σα (3.7)

det(σα) = −1 (3.8)Tr(σα) = 0 (3.9)

31

32 CHAPTER 3. DENSITY MATRIX FORMALISM

~a · ~σ =∑γ

aγσγ =

(a3 a1 − ia2

a1 + ia2 −a3

)(3.10)

Tr(σασβ) = 2δαβ (3.11)

Tr(σασβσγ) = i∑γ′′

εαβγ′′Tr(σγ′′σγ) (3.12)

= 2iεαβγ (3.13)Tr(σασβσγσδ) = 2(δαβδγδ − δαγδβδ + δαδδβγ) (3.14)

(~a~σ) · (~b~σ) = (~a ·~b) + i~σ · (~a×~b) (3.15)

which can be derived as follows

(~a~σ)(~b~σ) = (aασα)(bβσβ) = aαbβσασβ (3.16)

= (12σα, σβ+

12[σα, σβ])aαbβ (3.17)

= (δαβ + iεαβγσγ)aαbβ (3.18)= [(~a ·~b) + i(~a×~b)~σ] (3.19)

(3.20)

Often a very useful rule for the treatment of functions of spin-12 matrices is the following:

• Any function f of the scalar a+ 2~b~σ linear in ~σ

• can be converted into a linear function f(a+ 2~b~σ) = α+ 2β~b~σb

– with α = 12 [f(a+ b) + f(a− b)] and

– β = 12 [f(a+ b)− f(a− b)]

– with the eigenvalues a± b of the operator a+ 2~b~σ

– and the eigenvalues f(a± b) of the operator f(a+ 2~b~σ)

The use of these rules often enables to treat the 2x2-matrices analytically.

3.2.2 Compilation of some more sophisticated rules for Trσ

To average over the magnetic moment if it is spherically symmetric like in a paramagnet one hasto avoid ~Q⊥ and formulate the equations using the normal component in the neutron magneticmoment. To treat the respectiv eqations one needs rules for the operators

Lh = ~Sh · (~σ − (~e · ~σ)~e) (3.21)~Mh = ~Q⊥h = ~Sh − (~e · ~Sh)~e (3.22)

~e =~κ

|~κ|(3.23)

3.2. COMPILATION OF RULES FOR THE PAULI SPIN MATRICES 33

a unit vector in the direction of the scattering vector. For these one can derive the additional rules

Trσ(~σLh) = 2 ~Mh (3.24)Trσ((~P · ~σ)Lh) = 2( ~Mh · ~P ) (3.25)

Trσ((~P · ~σ)~σLh) = 2i( ~Mh × ~P ) (3.26)Trσ((~P · ~σ)Lh~σ) = −2i( ~Mh × ~P ) (3.27)

Trσ(L1L2) = 2( ~M1 · ~M2) (3.28)Trσ(L1~σL2) = −2i( ~M1 × ~M2) (3.29)

Trσ((~P · ~σ)L1L2) = 2i( ~M1 × ~M2)~P (3.30)Trσ((~P · ~σ)L1~σL2) = 2[( ~M1( ~M2 · ~P ) + ( ~M1 · ~P ) ~M2 − ~P ( ~M1 · ~M2] (3.31)

3.2.3 Some properties of the scattering matrix, which links the density matrixof the incident beam to that of the outgoing beam.

For me a long time the density matrix formalism looked like magic because e.g. Lovsey used itbut without showing why it gives physical results. In the following I want to show the physicalinterpretation of the density matrix formalism, not only of the results but of the whole process i.e. Iwant to give the physical reasons why it works. Only then I can trust the results of this formalism.The Pauli spin matrices and the 2 × 2 unit matrix are linearly independent and any 2 × 2 matrixA can be represented as a linear combination

A = λ01 + λ1σx + λ2σy + λ3σz (3.32)

= λ01 + |(λ1, λ2, λ3)|(λ1, λ2, λ3)|(λ1, λ2, λ3)|

σ (3.33)

i.e. each operator in the spinor space has the form

12

(g1 + h~n · ~σ) (3.34)

like the density operator eq.(1.17). The scattering matrix, which describes the scattering by nu-clei with a nuclear spin (Ix, Iy, Iz) must then also have this form and also the scattering matrixdescribing the magnetic scattering and all interference terms must all have this general form.

First we want to investigate some properties of this scattering matrix. We describe the changeof the spin state by the transformation

χ(f)c′ = Mc′cχ

(i)c (3.35)

Mc′c is a 2 × 2 matrix, which can also change the reference axes. In this case it would be Mc′c =Rc′cMcc where Rc′c changes the reference axes from c to c’ and Mcc is the scattering matrix for thesystem c. The most general form of this Mcc and assuming rotational invariance would then be

Mcc = g1 + hxσx + hyσy + hzσz = g1 + ~h · ~σ (3.36)

Here g and ~h are complex quantities. The components of ~h and ~σ are associated with the referenceaxes c. We assume these to be defined in the center of mass system. In an experiment ~h must berelated to the scattering process. If the ingoing and outgoing wave vectors are ~k and ~k′ we select

~M =(~k − ~k′)|~k − ~k′|

(3.37)


~N =~k × ~k′

|~k × ~k′|(3.38)

~K =~k + ~k′

|~k + ~k′|(3.39)

three unit vectors, which form a Cartesian coordiante system. Taking the axes x,y,z along thevectors ~M , ~N , and ~K, we can write

~h · ~σ = hx( ~M · ~σ) + hy( ~N · ~σ) + hz( ~K · ~σ) (3.40)

The determination of the coefficients would correspond to a full polarimeter measurement measuringfor each of three incident polarizations three components of the outgoing polarization.

In many cases additional symmetries occur such as parity conservation. Parity conservationpostulates ~M → − ~M , ~N → ~N , ~K → − ~K, ~σ → ~σ. The consequence for the scattering processwould then be hx = hz = 0, and writing for hy = h the general matrix is of the form

Mcc = g1 + h( ~N · ~σ) (3.41)

This is just the form we had assumed above for the general form of the scattering matrix. In thisform g and h are related to scattering amplitudes describing the interaction of the incident neutronswith the target nuclei.

The use of polarimetry allows to identify processes in which the parity is not conserved as wedid for example in [35, 36]. It may play a role if there are magnetic helices of only one helicity,because then parity is violated.

In the case of conservation of these symmetries the process of scattering of a beam with spincan be described further by applying the transformation with Mcc in the following way. For largevalues of r, we require that Ψ may be written as the sum of an incoming plane wave in spin stateχ

(i)c and an outgoing spherically scattered wave in spin state χ(f)

c = Mccχ(i)c . This results in the

Born approximation for the case of a beam of particles with spin:

Ψ ≈ eikzχ(i) +eikr

rMcc(ϑ, ϕ)χ(i) as r →∞ (3.42)

We wish now to obtain the procedure for finding the density matrix of the resultant neutronbeam after the scattering process by the scattering matrix of the form found above.

ρoutc = χ(f)

c χ(f)†c (3.43)

= Mccχ(i)c χ(i)†

c M †cc (3.44)

= Mccρ(i)c M †

cc (3.45)

This is the relationship between the outgoing and the incident density matrix in terms of thescattering matrix Mcc. This procedure also holds for beams which are unpolarized or partiallypolarized. The resultant beam is described by a density matrix which is the product of three 2× 2matrices.

If one takes the trace of ρ(out)c one obtains the expectation value of the scattering matrix operator

for the incident beam described by the density matrix ρ(in)c .

dσ

dΩ= Trρ(out)

c = Tr(Mccρ(i)c M †

cc) (3.46)

3.3. APPLICATION: NUCLEAR COHERENT AND INCOHERENT SC. 35

Using ρoutc one can obtain the polarization of the scattered beam (see eq.(1.40 on page 10)

~P =Trσ(σρ(out)

c )

Trσρ(out)c

=Trσ(σMccρ

(in)c M †

cc)

Trσ(Mccρ(in)c M †)

(3.47)

We now want to apply this to a simple example.

3.3 Application of the density matrix formalism to obtain nuclearcoherent and nuclear spin incoherent scattering and their po-larization

3.3.1 Scattering of a polarized beam by a spinless target: quartz measurementto determine the flipping ratio

We now apply these rules first for a spinless target, i.e. to an aggregat of atoms without nuclear spinsand without magnetism. The scattering matrix has then the form of a complex number NNe

iφN

which in kinematic approximation is the result of the Fourier transform of the crystallographicstructure. As is well known, the interaction of the neutron with a nucleus without nuclear spin isdescribed by the expression for the scattering amplitude:

VN (~κ) = VN (~k − ~k′) =2πh2

m0

∑ν,j

bjei(~k−~k′)(~Rν+~dj) (3.48)

~k,~k′ are the wave vector of the incident and outgoing neutrons, ~Rν is the position of the originof the unit cell ν, ~dj is the position of the atom j with scattering length bj in the unit cell ν.Specializations of this general expression for single crystals, powder patterns, materials with shortrange order, liquids and amorphous materials will be derived when we need it to investigate suchmaterials. The behaviour of this scattering amplitude in crystalline materials is the special topicof crystallography and all information about it is gathered in the international X-ray tables. Herewe only need to know the most general form, which this expression can have for the scatteringmatrix. For the case where one has no symmetry center the above sum VN (~κ) will be a complexnumber N with modulus N and phase φN or with real and imaginary parts N (r) = N cosφN andN (i) = N sinφN . But V †

N (~κ)VN (~κ) is always real and positiv. Only the way to calculate this ischanging in noncentro symmetric crystals as then in eq.(3.48) one has also an imaginary part. Onecan also split this into different factors like the lattice factor, the structure amplitude and the shapefunction. But without magnetic interaction terms and nuclear spin terms we have no term withthe form (~σ · ~Op) with an operator ~Op. So the scatterred intensity is obtained by

dσ

dΩ= Trσ(ρN †N) (3.49)

=12Trσ[(1 + ~P · ~σ)N †N ] (3.50)

= N †N = N2 (3.51)

as Trσ(1) = 2 and Trσ(~σ) = 0. For the polarization ~Pout of the outgoing beam with ~Pin as thepolarization of the incident beam we get

~Poutdσ

dΩ= Trσ(ρN †~σN) (3.52)


P(out)β

dσ

dΩ=

12Trσ[(1 + P (in)

α σα)N †σβN ] (3.53)

=12Trσ(N †σβN + P (in)

α σασβN † N) (3.54)

= P (in)α δαβN

†N (3.55)

= P(in)β |N |2 (3.56)

= P(in)β N2 (3.57)

as Trσασβ = δαβTr(1 00 1

)= 2δαβ and Trσ = 0.

This is nothing else but the fact that for pure nuclear coherent scattering the scattered beamhas the differential scattering cross section dσ

dΩ = N †N and the polarization of the scattered beam isthe same as that of the incident beam. If the incident beam is polarized in the x direction then alsothe outgoing beam is polarized in the x direction, and correspondingly for y and z. For quartz wehave such a scattering, which has no nuclear spin incoherent and no magnetic cross section. Fusedquartz is characterized by broad diffuse scattering as it behaves like a frozen liquid. It enables tomeasure the polarization product properties of the analyzers in a broad angular range, which isneeded for the correction of the results of the multidetector for finite flipping ratio.

3.3.2 Correction for finite flipping ratio

Any reliable results of a measurement should not depend on the properties of the measuring equip-ment. But if one measures with bad polarizers and analyzers one gets other results than withgood ones. So one has to be able to correct for this faults. The quality of the equipment can becharterized by the flipping ratio of the scattering by a purely coherent scatterer, and for this we usea quartz sample. The polarization is not a linear function but in the way it is visualized in fig.3.1and 3.2 one can use simple rules of elementary geometry of proportionality to correct the flippedand nonflipped scattering intensities of the sample. From fig.3.2 one sees immediately that

PQuartz =I↑ − I↓I↑ + I↓

(3.58)

=I↑/I↓ − 1I↑/I↓ + 1

(3.59)

=Rquartz − 1Rquartz + 1

(3.60)

Icorr↑↑ + Icorr

↑↓ = I↑↑ + I↑↓ = I (3.61)Icorr↑↑ − Icorr

↑↓I

=I↑↑ − I↑↓I↑ − I↓

(3.62)

Icorr↑↑ − Icorr

↑↓ =I↑↑ − I↑↓PQuartz

(3.63)

From this follows by some simple calculations

Icorr↑↑ = I↑↑ +

1Rquartz − 1

(I↑↑ − I↑↓) (3.64)

Icorr↑↓ = I↑↓ −

1Rquartz − 1

(I↑↑ − I↑↓) (3.65)

which one can also immediately see in fig.3.2 applying simple geometric rules.


PPPPPPPPPPPPPPPPPPPPP

I

0

I2

I↑(P )

I↓(P )

0 0.5 PQuartz 1

PQuartz·I

6

?

6

6I↓

I↑ Figure 3.1: Vizualization forthe intensities I↑ and I↓ forthe polarization with a flip-ping ratio 7

PPPPPPPPPPPPPPPPPPPPP

I

0

I2

0 0.5 PQuartz1

6

6

6

6

I↓

I↑

Psample

HHHH

HHHHHH

HHHH

I↑↑

I↑↓

Icorr↑↑

Icorr↑↓

-

-

Figure 3.2: Vizualization of the backgroundcorrected intensities I↑ and I↓ of the quartzfor the flipping ratio about 4, the measuredand background corrected intensities for thesample and the flipping ratio corrected in-tensities of the sample Icorr

↑↑ without flip, andIcorr↑↓ with flip. The correction assumes that

the quartz should only scatter without flipand relates the measured sample intensitiesto an ideal quartz scatterer

The correction for finite flipping ratio is then done in following way: For the three polarizationdirections x,y,z we measure the scattering of a fused quartz sample with and without spin flip.As we know from the above discussion that this scattering does not change the polarization weuse these scattered intensities without spin flip I↑n and with spin flip I↑n for detector number n(after background correction) to determine the properties of the measuring set-up of this detector.These properties are contained in the product PpolarizerPanalyzerFflipper with Ppolarizer=polarizationefficiency of the polarizer, Panalyzer=polarization efficiency of the analyzer and Fflipper= flipperefficiency (see 2.1.1 on page 18). As a measure for this ”polarization product” we use the flippingratio Rn for each detector n given by

Rn =I↑n

I↓n(3.66)

From this we can determine the polarization product for each detector

(PpolPanalFflipper)n =Rn − 1Rn + 1

(3.67)

Rn serves also for the correction for finite flipping ratio of the measured data. This flipping ratiocontains all polarization losses by depolarizations in the polarizer, the analyzer the guide fields andthe imperfections of the polarizer, the analyzer and the flipper.


The correction for detector n can then be performed by using eq.(3.64) and eq.(3.65). In theliterature you can also find the application of eq.(3.64,3.65) with Rquartz replaced by the flippingratio of the sample. From the above figures you can see immediately the nonesense of such a cor-rection. It would only calculate the intensity that would have been measured without polarizationanalysis but with losses by the transition of the beam through polarizer and analyzer, which youcan have easier by just adding the (backgroud corrected) intensities with and without flip.

3.3.3 Scattering of polarized neutrons by a target with only nuclear spin inco-herence: vanadium calibration

For absolute calibration measurements one normally uses the scattering of a standard scattererwhose cross section and scattering behaviour is well known. Vanadium is a nearly ideal incoherentscatterer, having a coherent scattering cross section of only 0.0184 barn and an incoherent crosssection of 5.187 barn. Because of the incoherence the angular scattering is isotropic, so one knowsthat every detector should get the same scattered intensity if one has real single scattering behaviourand no absorption. But these latter effects can be corrected for. The incoherence of the scatteringof vanadium results from interaction of the neutron with the nuclear spin of vanadium. For nuclearspin incoherent scattering the behaviour of the scattering length is given by two numbers b+ and b−

which are the eigenvalues of the scattering length operator b = b+ 12bNI · σ . As I · σ= J2−I2− 1

4σ2

= J(J + 1)− I(I + 1)− 34 this yields

I · σ = I for J = I +12

(3.68)

I · σ = −(I + 1) for J = I − 12

(3.69)

we obtain

b+ = b+12bNI (3.70)

b− = b− 12bN (I + 1) (3.71)

b =(I + 1)b+ + Ib−

2I + 1(3.72)

bN =2(b+ − b−)

2I + 1(3.73)

As an example that shows that the available values in tables are often questionable: In the tablewe find for vanadium which is 99.5% 51V with I = 7

2 the values b+ = 5.11fm and b− = −7.52fmand this gives b = −0.4156fm and bN = 3.158fm. This should give the known nuclear incoherentcross section of vanadium, but it does not. Perhaps the authors which measured the values usedother definitions of b+ and b−. In fact one can find different definitions in the literature. It is afactor 2 which is lost. What is really measured is only the difference. The values itself are thencalculated using other informations.

We apply the density matrix method now to the case with only the operator B(I · σ) withB = 1

2bN but we assume that the nuclei are not polarized,i.e.

〈Ix〉 = 〈Iy〉 = 〈Iz〉 = 0 (3.74)

〈I2x〉 = 〈I2

y 〉 = 〈I2z 〉 =

13I(I + 1) (3.75)

dσ

dΩ= Tr(ρB†(~σ · ~I†)B(~σ · ~I) (3.76)


=12Tr[(

(1 00 1

)+ σαP

(in)α )(σβI

†β)B†B(σγIγ)] (3.77)

=12Tr[σβσγI

†βIγB

†B + B†BσασβσγP(in)α I†βIγ ] (3.78)

=12B†B[2δβγI

†βIγ + 2iεαβγP

(in)α I†βIγ ] (3.79)

= B†B[I†βIβ + i ~P (in)(~I† × ~I)] (3.80)

= B†B[I(I + 1) + i ~P (in)(~I† × ~I)] (3.81)

Now we have to average over the nuclear spin orientation of ~I. As ~I is an angular momentum, weknow that ~I† = ~I and that ~I × ~I = i~I. This shows that the vector product expression in eq.(3.81)is linear in nuclear spin so that it averages to zero for an unpolarized nuclear target. Therefore weobtain finally

dσ

dΩ= B†BI(I + 1) (3.82)

For the polarization of the beam scattered by unpolarized nuclear spins we obtain

P (out)γ

dσ

dΩ= Tr(ρB†(~σ · ~I†)σγ(~σ · ~I)B) (3.83)

=12Tr[(

(1 00 1

)+ σαP

(in)α )(σβI

†β)σγ(σδIδ)] (3.84)

=12B†BTr[σβσγσδI

†βIδ + σασβσγσδP

(in)α I†βIδ] (3.85)

= B†B[iεβγδI†βIδ + (δαβδγδ − δαγδβδ + δαδδβγ)P (in)

α I†βIγ ] (3.86)

= B†B[−i(~I† × ~I)γ + (~P (in) · ~I†)Iγ + (~P (in) · ~I)I†γ − P (in)γ (~I† · ~I)] (3.87)

= B†B[−i(~I† × ~I)γ − P (in)γ I(I + 1) + (~P (in) · ~I†)Iγ + (~P (in) · ~I)I†γ ] (3.88)

Now we average again over nuclear spin orientations and obtain by using eqs.(3.74-3.75)

(~P (in) · ~I†)Iγ = (P (in)x I†x + P (in)

y I†y + P (in)z I†z)Iγ =

13P (in)

γ I(I + 1) (3.89)

(~P (in) · ~I)I†γ = (P (in)x Ix + P (in)

y Iy + P (in)z Iz)I†γ =

13P (in)

γ I(I + 1) (3.90)

With this we obtain

P (out)γ

dσ

dΩ= B†BP (in)

γ I(I + 1)[13

+13− 1] (3.91)

= −13P (in)

γ I(I + 1)B†B (3.92)

To understand the physical meaning of this result we have to know thatdσ

dΩ= I↑↑γ + I↑↓γ = B†BI(I + 1) (3.93)

P (out)γ

dσ

dΩ=

I↑↑γ − I↑↓γ

I↑↑γ + I↑↓γ

(I↑↑γ + I↑↓γ ) = I↑↑γ − I↑↓γ = −13B†BI(I + 1) (3.94)

for γ = x,y or z. Solving this two equations for the intensity without spin flip I↑↑γ and for theintensity with spin flip I↑↓γ we obtain immediately

I↑↑γ =13|B|2I(I + 1) (3.95)

I↑↓γ =23|B|2I(I + 1) (3.96)


The physical meaning of this is that irrespective of the incident polarization direction (i.e. P in xor y or z direction) the outgoing scattering resulting from unpolarized nuclear spins is always withone third without flip and two third with flip of the neutrons in the polarized beam.

These results will be used in the next chapter where we derive formulas for magnetic scatteringand include nuclear scattering without derivation. In the next chapter we wish to derive thecross section for magnetic scattering using the xyz method, and employ the method of tansitionprobability calculation, (this method to calculate the polarization behaviour is complimentary).This method should also be known. In the literature this method is also very often used. Therewe derive the general form of magnetic scattering, which differs from nuclear spin scattering bythe fact that only the component of the magnetic moment which is orthogonal to the scatteringvector contributes to the scattering. This is the first topic of the next chapter. Then we developea method for describing the behaviour of the spin if the quantization axis is changed from thez to the x or y direction. With these methods we calculate the transition probabilities for spinflip and non spin flip scattering for these quantization axes and their relationship to the magneticcorrelation functions Mµν with µ, ν = x, y, z. We apply the results to get formulas for powder andsingle crystal samples which will be used in the following chapters. To demonstrate the advantagesof the density matrix method, we will also use it to first derive the same results. After this we willthen use it to find the behaviour for more general cases and interpret these results by visualizingthe respective behaviour for pure magnetic scattering and for interference terms and including thecase of non centro symmetric systems. These play a role especially for Bragg scattering by morecomplicated spin structures as they were investigated with the CRYOPAD (see chapter 5.6).

3.3.4 Summary about the density matrix formalism

One can see that the density matrix method offers a simple and very effective method to treat spinproblems. One only has to use some simple rules for the calculation of the traces of combinationsof Pauli spin matrices. This strength results from the law we found in eq.(1.35 on page 10) andthe fact that the Pauli spin matrices combined with the 2× 2 unit matrix form a complete systemto express all hermitian 2× 2 matrices. They are all of the form A

(1 00 1

)+B(~σ ~O). To interpret the

results one has to use the relationships of eq.(3.93 on page 39, 3.94). I myself learned these thingswhen I collaborated with Boris Toperverg on diffuse scattering by the roughness in multilayers,where we treated the multilayer system using the pseudo spin 1

2 method.

Bibliography

[1] Sakurai, J.J. Modern Quantum Mechanics Benjamin/Cummings Publishing Company, MenloPark, California 1985

[2] Dirac, P.A.M. Spinors in Hilbert Space Plenum Press, New York and London 1974

[3] Goldstein, H. Klassische Mechanik Akademische Verlagsgesellschaft, Frankfurt am Main 1972

[4] Scharpf, O., J. Appl. Cryst. 11, 631, (1978)

[5] Scharpf, O., Strothmann, H., Physica Scripta T24, 58, (1988)

[6] Mezei, F. Journal de Physique 45, (1984), C3-223

[7] Mezei, F. in Imaging Processes and Coherence in Physics Schlenker, M. et al. editors, SpringerVerlag, Heidelberg (1980) p.282.

[8] Serebrov, A.P., Aldushchenkov, A.V., Lasakov, M.S., Kuznetsov, I.A., Stepanenko, I.V. Nu-clear Instruments and Methods A 357, (1995), 503

[9] Reckveldt, M.Th., J. de Physique 32, (1971), C579

[10] Mezei, F. Z.Physik 255, (1972), 146

[11] Okorokov, A.I., Runov, V.V., Volkov, V.I., Gukasov, A.G.: JETP 69, (1975), 590

[12] Sakurai, J.J. Modern Quantum Mechanics Benjamin/Cummings Publishing Company, MenloPark, 1985

[13] Capellmann,H., J.Magn.Magn.Materials.28,250(1982)

[14] Emery,V.J.: Phys.Rev.Lett.58,2794(1987)

[15] Lovesey,S.W.:Theory of neutron scattering from condensed matter. Vol. 2: Polarization effectsand magnetic scattering. Oxford: Clarendon Press 1986

[16] Capellmann,H., Vieira,V.: Phys.Rev.B25,3333(1982)

[17] Brink, D.M., Satchler, G.R. Angular momentum Clarendon Press, Oxford 1971

[18] Brussaard, P.J., Tolhoek, H.A. Physica 23, (1957) 955

[19] Brenig, W. Annalen der Physik 5, (1996) 123

[20] Zhou, C., Schulz, H.J. Phys. Rev.B 52, (1995) R11557

41

42 BIBLIOGRAPHY

[21] Aharonov, Y., Albert, D.Z., Casher, A., Vaidman, L. Physics Letters A 124, (1987) 199

[22] Aharonov, Y., Albert, D.Z., Vaidman, L. Phys. Rev. Lett. 60, (1988) 1351

[23] Busch, P. Physics Letters A 130, (1988) 323

[24] Golub, R., Gahler, R. Physics Letters A 136, (1989) 178

[25] Golub, R., Gahler, R. Nuclear Instrum. and Methods A284, (1989) 188

[26] Rose, M.E. Elementary theory of angular momentum John Wiley & sons, New York 1967 (5.thedition)

[27] Schertler, K., Thoma, M.H., Annalen der Physik 5, 103, (1996)

[28] Edmonds, A.R. Angular momentum in quantum mechanics Princeton University Press, Prince-ton, New Jersey 1974

[29] Robson, B.A. The theory of polarization phenomena Clarendon Press, Oxford 1974

[30] Abragam, A., Winter, J.M., Phys.Rev. Letters 1 (1958), 374

[31] Donally, B.L., Clapp, Th., Sawyer, W., Schultz, M. Phys. Rev. Letters 12, 502

[32] Clausnitzer, G., Fick, D. Nuclear Instruments and Methods 47 (1967), 171

[33] Beurtey, R. in Polarized Targets and Ion Sources Proceedings of the International Conferenceon Polarized Targets and Ion sources, Saclay, France (1966), Edited by La Direction de laPhysique d’Etudes Nucleaires de Saclay, Boite postale Nr.2, 91 Gif-sur-Yvette

[34] DeKieviet, M., Dubbers, D., Schmidt, Ch., Scholz, D., Spinola, U. Phys. Rev. Letters 75(1995), 1919

[35] Heckel, B., Ramsey, N.F., Green, K., Greene, G.L., Gahler, R., Scharpf, O., Forte, M., Dress,W., Miller, P.D., Golub, R., Byrne, J., Pendlebury, J.M, Physics Letters B, 119, 298, (1982)

[36] Heckel, B., Forte, M., Scharpf, O., Green, K., Greene, G.L., Ramsey, N.F., Byrne, J., Pendle-bury, J.M., Phys. Rev. C 29, 2389, (1984)

[37] F.Tasset, Physica B, 156 & 157, 627 (1989)

Chapter 4

Theory of the XYZ-difference methodusing polarized neutrons for theseparation of coherent, spinincoherent and magnetic scatteringcross sections and employing amultidetector: Requirements for suchan instrument.

4.1 Abstract

Equations are derived for the polarization analysis using a multidetector to separate coherent, spinincoherent and magnetic scattering using three dimensional analysis or the xyz-difference method.Then the conditions for the magnetic guide fields to rotate the polarization into the three directionsx,y and z are discussed on the basis of the existing instrument D7 at the ILL in Grenoble.

4.2 Introduction

Experiments with polarized neutrons and especially with polarization analysis always suffer fromintensity problems. The application of multidetectors similar to experiments without polarizationanalysis can help to overcome such problems. However in the ‖-⊥ difference method [1] the need torotate the polarization into the direction parallel to the scattering vector prevents its application ina multidetector. We want to derive equations which show how a polarization analysis experimentfor paramagnetic and antiferromagnetic scatterers and a multidetector instrument has to be carriedout. Then we want to demonstrate how the using method can be implemented in a real instrument.We demonstrate this using the example of the existing instrument D7 at the ILL in Grenoble.

0 first published as O.Scharpf, H.Capellmann, in phys. stat. sol. (a) 135, (1993), 359

43

44 CHAPTER 4. THEORY OF XYZ-DIFFERENCE METHOD

4.3 XYZ-Difference method

First we re-derive the ‖-⊥-method and afterwards we generalize it to the xyz-difference method,which can be used for the simultaneous measurements of a multidetector polarization analysisexperiment.

The double differential neutron scattering cross section is given by the relation(∂2σ

∂Ω∂E′

)ss′

=(mn

2πh2

)2

· k′

k

∑λλ′

pλ| < k′s′λ′|V |ksλ > |2δ(hω + Eλ − Eλ′) (4.1)(∂2σ

∂Ω∂E′

)=

(mn

2πh2

)2

· k′

k

∑λλ′

pλTrσ(ρ|〈k′λ′|V |kλ〉|2)δ(hω + Eλ − Eλ′) (4.2)

where quantum numbers without prime correspond to the initial states, quantum numbers withprimes correspond to final states, k,s are the quantum numbers for the neutron, λ are the quantumnumbers of the target. V is the interaction between neutron and target.

Eq.(4.2) we get by factorizing the spin part 〈s′|, |s〉 and write it as the density matrix ρ =∑|s〉〈s|∑

ss′

|〈k′s′λ′|V |ksλ〉|2 =∑ss′

|〈s′|〈k′λ′|V |kλ〉|s〉|2 = Trσ(ρ|〈k′λ′|V |kλ〉|2)

Eq.(4.2) does not give the transition probability s to s’ directly but by Trσ(ρσ|〈k′λ′|V |kλ〉|2) asshown in 3.47 on page 35.

For magnetic scattering the magnetization density of the electrons Me(r) interacts with thedipole field of the neutron. This density of the magnetization of the electrons has an orbital partML(r) and a spin part MS(r)

Me(r) = ML(r) + MS(r) = ML(r)− gµBS(r) (4.3)

The dipole field of the neutron Bn(r) is

Bn(r) = ∇r ×(

∇r ×µn

|rn − r|

)(4.4)

with µn = γµN , γ = −1.9132, µN = nuclear magneton, g=gyromagnetic ratio of the electron,µB = eh

2mec = Bohr magneton.So one finally has (index e corresponds to electron, index n to neutron respectively)

Ve =1

4πµBMe(r)Bn(r) (4.5)

The matrix element of 1|rn−r| relative to rn is

< k′| 1rn − r

|k >= eiκ · r∫eiκ·(rn−r)

|rn − r|d3rn =

eiκ · r

κ2(4.6)

with κ = k − k′. So we get with e = κ/|κ| unit vector in direction of κ

< k′|Bn(r)|k >=1κ2κ× (µn × κ)eiκ · r = γµNµBe

iκ · r(σn − (e · σn)e) (4.7)

and< k′|Me(r) ·Bn(r)|k >= Me(r) ·

κ

κ× (µn ×

κ

κ)eiκ · r = Me(r) · µn⊥e

iκ · r (4.8)

4.3. XYZ-DIFFERENCE METHOD 45

withµn⊥ =

κ

κ× (µn ×

κ

κ) = µn(σn − (e · σn)e) (4.9)

For a crystal with magnetic atoms with |Mν | Bohr magnetons µB in the direction Mν/|Mν | atthe positions (xν , yν , zν) and the scattering vector in the direction (h,k,l) gathering from (4.1) and(4.5) all factors as 4π mn

2πh2µBµNγ = γ e2

2mec2= γ r0

2 = 0.2695 · 10−12cm this results in the magneticstructure amplitude

Fmaghkl = 0.2695 · 10−12 ·

∑magn. atoms ν

in unit cell

Mνfν(hkl)e2πi(hxν+kyν+lzν)[cm] = (A1, A2, A3) (4.10)

Mν measured in units of µB, fν(hkl) is the form factor of the magnetic atom ν, r0 = e2

mec2is the

classical electron radius.

Fmag⊥hkl = (A1, A2, A3)−

[(ha ,

kb

lc) · (A1, A2, A3)](h

a ,kb

lc)

k2

b2+ k2

b2+ l2

c2

Figure 4.1: Geometry of equation (4.11) andthe equations (4.27 on page 48) et seq.. x andy axes must be in the scattering plane. Thesedirections are determined by the field directionof the adiabatic spin turner coil pair, which isselected as x direction and correspondingly thatfor the y direction.

For the following we choose the axis such that the scattering vector κ = k − k′ lies in thex,y plane. For the multidetector instrument to be discussed different detectors have angles α withrespect to the x-axis (determined by the direction of the field in the spin turner coils, see fig.4.1)i.e. κκ = (cosα, sinα, 0). Therefore

µn⊥ =κ

κ2× (µn × κ) = (µx sin2 α− µy sinα cosα, µy cos2 α− µx sinα cosα, µz) (4.11)

and

Me · µn⊥ = Mx(µx sin2 α− µy sinα cosα) +My(µy cos2 α− µx sinα cosα) +Mzµz

= µx(Mx sin2 α−My sinα cosα) + µy(My cos2 α−Mx sinα cosα) + µzMz

= µn ·Me⊥. (4.12)

Me(r) is the magnetization density of the electron and µn = −γ eh2mncσn = −γµNσn is the magnetic

moment of the neutron, σn are the Pauli spin matrices for the neutron spin.For the presentation of the xyz difference method it will be convenient to introduce raising or

lowering operators with respect to all three axes:

xµ±=

12(µy ± iµz)

yµ±= 1

2(µz ± iµx)zµ±=

12(µx ± iµy) (4.13)


Eigenstates to µν = −γµNσν ; ν =x,y,z are indexed as |ν↑> and |

ν↓>. E.g. σx|

x↑>= |

x↑>;

σx|x↓>= −|

x↓>; σy|

y

↑>= |y

↑> etc. with |x↑>= 1√

2|

z↑> +|

z↓>, |

x↓>= i√

2−|

z↑> +|

z↓>,

|y

↑>= 1√2|

z↑> +i|

z↓>, |

y

↓>= 1√2i|

z↑> +|

z↓>.

A flip operation is possible for polarizations in all three directions x, y or z. It corresponds tothe fact that for the spin up/down at the polarizer the spin at the analyzer is down/up. This spin

flip is described by operations asxσ+ |

x↓>= |

x↑> and

xσ− |

x↑>= |

x↓> etc. for the other directions,

yielding

<x↑ | x

σ+ |x↓>= 1 <

x↓ | x

σ+ |x↓>= 0 <

x↓ | x

σ− |x↑>= 1 (4.14)

etc. This admits a generalization by allowing operations using mixtures of eigenstates as e.g.

<z↓ |σx|

y

↑>=1√2

<z↑ |σy|

y

↑>= 1√2

<y

↓ |σz|x↑>= −1 + i

2(4.15)

etc. Such relationships are needed for the generalized polarization analysis [18] where one mea-sures the x, y or z component of the polarization P’ after scattering for the incident polarizationP = (1, 0, 0), (0, 1, 0) or (0, 0, 1), respectively. The density matrix method does not need all thisspecializations, it is representation independent and yields all combinations, also those containingmixtures of eigenstates in one go, as we will see in chapter 5 on page 61.

4.3.1 The ‖–⊥ method

4.3.1.1 Magnetic part of the neutron cross section

For the ‖-⊥ method the polarization is rotated successively parallel and perpendicular to the scat-tering vector and by taking the difference of these two measurements one separates the magneticscattering from the other sorts of scattering. This method uses a single detector and the rotationof κ parallel to the x-axis can be achieved by setting α = 0 in eq.(4.11) or (4.12). For the xyzdifference method this will be generalized to arbitrary angles at a later stage. For α = 0

Me · µn⊥ = µyMy + µzMz = −γµN (σnyMey + σnzMez) (4.16)

DefiningxM± (r),

y

M± (r) andzM± (r) analogically to the

νµ± operators, eq.(4.16) can also be

written asMe · µn⊥ = (

xM+

xµ− +

xM−

xµ+) = −γµN (

xM+

xσ− +

xM−

xσ+) (4.17)

Remembering that Me(r) is still a function of r and thereby also Mx(r), My(r), Mz(r) andνM± (r)

one obtains for the magnetic part of the function Γ‖ss′(r, ω):

(γµN )2Γ‖ss′(r, ω) =

∑λλ′

pλ| < λ′s′|Me · µn⊥|λs > |2δ(hω + Eλ − Eλ′) (4.18)

=∑λλ′

pλ| < λ′s′|xM+

xµ− +

xM−

xµ+ |λs > |2δ(hω + Eλ − Eλ′)

= (γµN )2∑λλ′

pλ| < λ′s′|xM+

xσ− +

xM−

xσ+ |λs > |2δ(hω + Eλ − Eλ′)

A.Spin flip scattering.

Let us now investigate the spin flip neutron scattering cross section for the polarization parallel tothe scattering vector i.e.

4.3. XYZ-DIFFERENCE METHOD 47

P ‖ κ ‖ ex = (1, 0, 0)

If we can neglect spin orbit coupling theνσ± operators can be applied separately and so one obtains

from (4.18)

Γ‖↑↓(r, ω) =

∑λλ′

pλ| < λ′|xM+ |λ > |2δ(hω + Eλ − Eλ′)

=∫

xM+ (r, t)

xM− (0, 0) eiωtdt (4.19)

= Myy(r, ω) +Mzz(r, ω) + i(Mzy(r, ω)−Myz(r, ω))and similarly

Γ‖↓↑(r, ω) = Myy(r, ω) +Mzz(r, ω)− i(Mzy(r, ω)−Myz(r, ω))

The double brackets indicate thermal (i.e.∑

λ pλ . . .) and quantum mechanical average (i.e.∑

λ′ <λ|M+|λ′ >< λ′|exp(itEλ′)M−exp(−itEλ)|λ >). In the following, intermediate steps will be omittedand we will always write immediately the time Fourier transformMxx(r, ω) etc. of the correspondingcorrelation functions, or for brevity . . ., especially in the longer formulae in eq.(4.27) et seq.and where no special abbreviations are in use. The factor i in (4.19 on page 47) signals a phaseshift of π

2 . It disappears when multiplied by a similar factor i resulting from the Fourier transformof an odd part of a correlation function, as e.g. Mzy −Myz for a helical structure. The resultingcross section is of course a real expression.

For polarization normal to the scattering vector i.e.

P ⊥ κ ‖ ez = (0, 0, 1)

Γ⊥↑↓(r, ω) =

∑λλ′

pλ| < λ′|My|λ > |2δ(hω + Eλ − Eλ′)

= Myy(r, ω) = Γ⊥↓↑(r, ω) (4.20)

By taking the difference (as mentioned above) one obtains:

Γ‖↑↓(r, ω)− Γ⊥

↑↓(r, ω) = Mzz(r, ω) + i(Mzy(r, ω)−Myz(r, ω))

Γ‖↓↑(r, ω)− Γ⊥

↓↑(r, ω) = Mzz(r, ω)− i(Mzy(r, ω)−Myz(r, ω)) (4.21)

B.Non-spin flip scattering

For P = Px ‖ κ: in the non-spinflip scattering cross section one obtains analogously

Γ‖↑↑(r, ω) = 0 (4.22)

for P ‖ ez ⊥ κ

Γ⊥↑↑(r, ω) = Mzz(r, ω) (4.23)

and the difference is hereΓ⊥↑↑(r, ω)− Γ‖

↑↑(r, ω) = Mzz(r, ω) (4.24)

Γiss′(r, ω) derived above are functions which can be used to express the double differential cross

section by combining eq.(4.1),(4.5) and including the factor eiκ · r from eq.(4.8) to perform thespatial Fourier transform, taking into account the fact that Me(r) is a function of r. This yields


the magnetic part (indicated by the signˆ ) of the dynamical scattering function Siss′(κ, ω) as a

function of κ and ω in addition to a form factor f(κ)2 which describes the Fourier transform of themagnetization density. For the sake of simplicity these processes of the spatial Fourier transformhave not been included above, and they will also not be included in the following derivations. Thisis done in order to avoid the essential physics being hidden behind pure formalism, which is assumedto be known and which is not essential for the aims of this paper. It will be assumed as performedwhen one finally needs the dynamical structure factor. Fourier transforming Γi

ss′(r, ω) as derivedabove to determine Si

ss′(κ, ω), one obtains the double differential cross section

∂2σ

∂ω∂Ω

∣∣∣∣∣i

ss′

=(r0γ

2

)2 k′

kSi

ss′(κ, ω)f2(κ) (4.25)

for i=‖,⊥. The differential cross section is obtained from the integral over ω defined as

dσ

dΩ

∣∣∣∣iss′

=(r0γ

2

)2 ∫ k′(ω)k

Siss′(κ, ω)f2(κ)dω (4.26)

with i=‖, ⊥.

4.4 General case of multidetectors

For the case of a multidetector instrument like that in fig.4.2 it is not possible to fulfil simultaneouslythe condition of the polarization parallel to the scattering vector for all detectors. In this casethe method described above has to be modified in the following manner: we assume the generalgeometry of fig.4.2, eqs.(4.11) and (4.12) and discuss again the various cross sections. However, forthe case of multidetectors the expressions become longer:

4.4.1 Spin flip cross sections (magnetic part only)

The polarization is defined as a vector with P = (Px, Py, Pz) = Pxex +Pyey +Pzez the magnitudeof which describes the degree of polarization according to |P| = n↑−n↓

|n↑+n↓| . Polarization in x-direction(which can be achieved by putting the field in the spin rotation coil of fig.4.2 in the direction to betaken as x-direction) is then designated by Px = 1 or P=(1,0,0) or P ‖ ex.

4.4.1.1 Polarization in x-direction

Because <x↓ |σx|

x↑>= 0 the only remaining contribution in eq.(4.12) is

(γµN )2Γx↑↓(r, ω) =

∑λλ′

pλ| < λ′x↓ |µy(My cos2 α−Mx sinα cosα) +

µzMz|λx↑> |2δ(hω + Eλ − Eλ′) (4.27)

which withxA±= My cos2 α−Mx sinα cosα± iMz can be written as

Γx↑↓(r, ω) =

∑λλ′

pλ| < λ′x↓ |(x

σ+

xA− +

xσ−

xA+)|λ

x↑> |2δ(hω + Eλ − Eλ′)

=∑λλ′

pλ| < λ′|xA+ |λ > |2δ(hω + Eλ − Eλ′) (4.28)

4.4. GENERAL CASE OF MULTIDETECTORS 49

Figure 4.2: Multipurpose instrument D7 at the Institute Laue Langevin in Grenoble. At the rightone bank is shown in the non-analyzing version with analyzers removed by the pneumatic elevatorsand the collimator inserted instead of the guide fields in front of the analyzers on the other banks.The neutrons arrive from the monochromator through the beryllium filter, the polarizer, the flipperand the chopper to the sample in the cryostat. From there they are scattered and arrive throughthe guide fields at the analysers and detectors. With the spin turner coils the polarization of theneutrons can be rotated in the x, y or z direction.

As in (4.19) <x↓ | x

σ− |x↑>= 1 and all other terms involving

xσ+ are zero. If we can neglect spin orbit

coupling we can evaluate the above matrix element with respect to the spin variables. This resultsin

Γx↑↓(r, ω)

= xA−

xA+

= (My cos2 α−Mx sinα cosα)(My cos2 α−Mx sinα cosα) + MzMz −iMz(My cos2 α−Mx sinα cosα) +i (My cos2 α−Mx sinα cosα)Mz

= MyMy cos4 α+ MxMx sin2 α cos2 α+ MzMz −(MxMy + MyMx ) sinα cos3 α− i[ (MzMy −MyMz ) cos2 α+(MxMz −MzMx ) sinα cosα] (4.29)


(Remember that the correlation functions MxMx etc. are always to be understood asMxx(r, ω) etc. (see remarks eq.(4.19))). In determining Γx

↓↑ we obtain xA+

xA− and thereby

the same expression as in (4.29). However –i is replaced by +i in the phase shifted part.

4.4.1.2 Polarization in y direction

With Py = 1 or P = (0, 1, 0) and

y

A±= Mz ± i(Mx sin2 α−My sinα cosα)) (4.30)

Γy↑↓(r, ω) =

y

A−y

A+= MzMz + MxMx sin4 α+ MyMy sin2 α cos2 α

−(MxMy + MyMx ) sin3 α cosα− i[(MxMz −MzMx ) sin2 α+ (MzMy −MyMz ) sinα cosα] (4.31)

Also here for Γy↓↑ we obtain the same expression as in (4.31) but instead of –i we have to write +i.

See also (4.40).

4.4.1.3 Polarization in z direction

(γµN )2Γz↑↓(r, ω) =

∑λλ′

pλ| < λ′z↑ |µx(Mx sin2 α−My sinα cosα)

+µy(My cos2 α−Mx sinα cosα|λz↓> |2δ(hω + Eλ − Eλ′) (4.32)

and withzA±= (Mx sin2 α−My sinα cosα)± i(My cos2 α−Mx sinα cosα) (4.33)

this yields

Γz↑↓(r, ω) =

zA−

zA+

= MxMx (sin4 α+ sin2 α cos2 α)+ MyMy cos2 α+(MxMy + MyMx ) sinα cosα) (4.34)

4.4.2 Non flip cross section for the magnetic part only

Now we calculate the NON FLIP functions Γν↑↑ for the magnetic part only:

P = Px

(γµN )2Γx↑↑(r, ω) =

∑λλ′

pλ| < λ′x↑ |µx(Mx sin2 α

−My sinα cosα)|λx↑> |2δ(hω + Eλ − Eλ′) (4.35)

= (γµN )2MxMx sin4 α+ MyMy sin2 α cos2 α−(MxMy + MyMx ) sin3 α cosα (4.36)

4.5. APPLICATION TO A POWDER MEASUREMENT 51

P = Py

Γy↑↑(r, ω) = MyMy cos4 α+ MxMx sin2 α cos2 α

−(MyMx + MxMy ) sinα cos3 α (4.37)

P = Pz

Γz↑↑(r, ω) =MzMz (4.38)

These results for Γνss′ can be combined in the following matrix equations for the phase shifted

and non phase shifted parts: For the non phase shifted part one obtains

Γx↑↓(r, ω)

Γy↑↓(r, ω)

Γz↑↓(r, ω)

Γx↑↑(r, ω)

Γy↑↑(r, ω)

Γz↑↑(r, ω)

=

sin2 α cos2 α cos4 α 1 − sinα cos3 αsin4 α sin2 α cos2 α 1 − sin3 α cosαsin2 α cos2 α 0 − sinα cosαsin4 α sin2 α cos2 α 0 − sin3 α cosαsin2 α cos2 α cos4 α 0 − sinα cos3 α0 0 1 0

Mxx(r, ω)Myy(r, ω)Mzz(r, ω)Mxy(r, ω) +Myx(r, ω)

(4.39)

and for the phase shifted partΓx↑↓(r, ω)

Γx↓↑(r, ω)

Γy↑↓(r, ω)

Γy↓↑(r, ω)

=

− sinα cosα − cos2 αsinα cosα cos2 αsin2 α − cosα sinα− sin2 α cosα sinα

(Mzx(r, ω)−Mxz(r, ω)Mzy(r, ω)−Myz(r, ω)

)(4.40)

For the elastic case these equations, combined with equation (8), correspond to the followingequation given for the special case of P ‖ κ for spin flip scattering by Moon, Riste, Koehler [8]:

dσ±∓

dΩ=∑i,j

pip∗je

iK·(ri−rj)[S⊥i · S⊥j ∓ iz · (S⊥i × S∗⊥j] (4.41)

However, 4.39 and 4.40 are also applicable to inelastic scattering and for the geometry of the threeorthogonal directions x,y,z, with x and y in the scattering plane as this is needed for the applicationof a multidetector with time of flight analysis.

4.5 Application to a powder measurement

For a powder with a collinear magnetization the correlation functions MxMx=MyMy =MzMz . All mixed ones as MxMz , MxMy ,MzMy are zero. Then magneticcross sections become much simpler: the phase shifted parts (equ.(4.40)) are all zero and in matrix(4.39) only the first three columns remain for describing the spinflip and the non spin flip part ofthe respective magnetic cross sections. We now define

∂2σνss′

∂Ω∂ω=k′

k

(r0γ

2

)2

f2(κ)Sνss′(κ, ω) (4.42)

etc. corresponding to (4.25)-(4.26), with Sνss′(κ, ω) being the space Fourier transform of the func-

tions Γνss′(r, ω) and ν=x,y,z; s,s’=|

ν↑> or |

ν↓>. From eq.(4.39), and

∂2σparam

∂Ω∂ω=k′

k

23

(r0γ

2

)2

f2(κ)M(κ, ω) (4.43)


∂2σspinincoh

∂Ω∂ω=k′

kB2Sincoh(κ, ω) (4.44)

∂2σcoh

∂Ω∂ω=k′

kb2Scoh(κ, ω) (4.45)

Then we gather all contributions for the respective scattering and obtain six cross sections eachmeasured with the three polarisation directions in x, y and z direction, with and without neutronspin flip respectively. These include all kinds of scattering i.e. coherent, spin incoherent, isotopeincoherent and magnetic scattering.

These cross sections read

∂2σx↑↓

∂Ω∂ω=

12∂2σparam

∂Ω∂ω(cos2 α+ 1) +

23∂2σspin

incoh

∂Ω∂ω(4.46)

∂2σy↑↓

∂Ω∂ω=

12∂2σparam

∂Ω∂ω(sin2 α+ 1) +

23∂2σspin

incoh

∂Ω∂ω(4.47)

∂2σz↑↓

∂Ω∂ω=

12∂2σparam

∂Ω∂ω+

23∂2σspin

incoh

∂Ω∂ω(4.48)

∂2σx↑↑

∂Ω∂ω=

12∂2σparam

∂Ω∂ωsin2 α+

13∂2σspin

incoh

∂Ω∂ω+∂2σcoh

∂Ω∂ω+∂2σisotop

incoh

∂Ω∂ω(4.49)

∂2σy↑↑

∂Ω∂ω=

12∂2σparam

∂Ω∂ωcos2 α+

13∂2σspin

incoh



incoh

∂Ω∂ω(4.50)

∂2σz↑↑

∂Ω∂ω=

12∂2σparam

∂Ω∂ω+

13∂2σspin

incoh



incoh

∂Ω∂ω(4.51)

By combining the measured cross sections one can separate the different contributions fromone another. Thus the two following combinations yield the paramagnetic cross section with noadditional assumption other than that of a powder sample, namely that the correlation functionsMiMi for i=x,y,z are all equal and all the mixed correlation functions are zero:

∂2σparam

∂Ω∂ω= 2

(∂2σx

↑↓∂Ω∂ω

+∂2σy

↑↓∂Ω∂ω

− 2∂2σz

↑↓∂Ω∂ω

)(4.52)

∂2σparam

∂Ω∂ω= 2

(2∂2σz

↑↑∂Ω∂ω

−∂2σx

↑↑∂Ω∂ω

−∂2σy

↑↑∂Ω∂ω

)(4.53)

Correspondingly one obtains the ∂2σspinincoh/(∂Ω∂ω) by the following combination:

∂2σspinincoh

∂Ω∂ω=

32

(3∂2σz

↑↓∂Ω∂ω

−∂2σx

↑↓∂Ω∂ω

−∂2σy

↑↓∂Ω∂ω

)(4.54)

as can be checked by combining eq.(4.46),(4.47) and (4.48). The ∂2σcoh/(∂Ω∂ω) can be obtainedby subtracting the cross sections obtained in a similar manner using ∂2σz

↑↑/(∂Ω∂ω):

∂2σcoh


incoh

∂Ω∂ω=∂2σz

↑↑∂Ω∂ω

− 12∂2σparam

∂Ω∂ω− 1

3∂2σspin

incoh

∂Ω∂ω(4.55)

Coherent scattering cannot be distinguished from isotope incoherent scattering. These relationshipsare given in [2, 9, 16, 20] without derivation.

4.6. DERIVATION AND LIMITS OF VALIDITY OF P′ = −κ(κ ·P) 53

4.6 Derivation and limits of validity of P′ = −κ(κ ·P)

With equation (4.39) we can now derive the equation, already given by Halpern and Johnson [11]:

P ′ = −κ

κ(κ

κ· P ) (4.56)

with P=polarization of the incident beam and P’=polarization of the beam scattered by a para-magnetic sample. It is often cited [5, 14, 15] with more or less convincing or even misleadingdemonstrations. Its main content is: (i) the scattered beam is polarized in the direction of thescattering vector (but see [5]), (ii) the spin is flipped and (iii) the intensity decreases with thecosine squared of the angle of rotation of the analysis. This is analogous to the law of Malus inoptics. With our method of demonstration we can see its range of validity. If κκ = (cosα, sinα, 0)and the incident polarization is P = (Px, 0, 0) and one determines P′ = P′

x i.e. the polarization inx direction, application of (4.56) yields P ′

x = −κxκxPx = −Px cos2 α and correspondingly for Py

(4.56) yields P ′y = −Py sin2 α and for Pz always P ′

z = 0.Using eqs.(4.39) and (4.40) it is possible to determine this polarization of the beam scattered

by a paramagnetic or antiferromagnetic sample with disappearing interference term . This is doneusing the definition of polarization i.e. the equation

P ′ν =

Γν↑↑ − Γν

↑↓Γν↑↑ + Γν

↑↓Pν (4.57)

For non-collinear structures one cannot expect eq.(4.56) to be valid, as this can even give rise totwo different results for ↑↓ or ↓↑ flips. For collinear structures one obtains for the general case

P ′x = Px

(1− 2 cos2 α)(Mxx sin2 α+Myy cos2 α− (Mxy +Myx) sinα cosα)−Mzz

Mxx sin2 α+Myy cos2 α− (Mxy +Myx) sinα cosα+Mzz

P ′y = Py

(1− 2 sin2 α)(Mxx sin2 α+Myy cos2 α− (Mxy +Myx) sinα cosα)−Mzz

Mxx sin2 α+Myy cos2 α− (Mxy +Myx) sinα cosα+Mzz

P ′z = Pz

Mzz −Mxx sin2 α−Myy cos2 α+ (Mxy +Myx) sinα cosαMzz +Mxx sin2 α+Myy cos2 α− (Mxy +Myx) sinα cosα

(4.58)

These equations are also valid for a single crystal, but they are more general than eq.(4.56). Iffor example the magnetization of the single crystal is in the z-direction, eq.(4.58) yields P ′

z = Pz,P ′

x = −Px, P ′y = −Py independent of α. Eq.(4.56) cannot comprise this result. Eq.(4.56) is only

valid for a collinear powder if Mxx = Myy = Mzz and without interference term . For this case weobtain from (4.58)

P ′x = −Px cos2 α P ′

y = −Py sin2 α P ′z = 0 (4.59)

These are exactly the results of eq.(4.56).For the case of paramagnetic and antiferromagnetic powder samples eq.(4.56) is very useful.

Its content is, that the polarization of the beam scattered by a paramagnetic or antiferromagneticsample has the direction of the scattering vector. In this case the direction of the polarizationis very easy to determine experimentally from the three components of the xyz method usingtan2 α = (Γy

↑↓−Γz↑↓)/(Γ

x↑↓−Γz

↑↓) with Γν↑↓ given in eq.(4.39) and it is proportional to the respective

scattered intensity. This can then immediately indicate whether or not the scattering vector hasthe direction of an elastic scattering vector or that of an inelastic one. This can be done using anintegral measurement without time of flight analysis. This was first shown by Maleev [12] and usedby Mezei [13].


4.7 Instrumental

Fig.4.2 gives a perspective view of a multidetector instrument adapted to the xyz-difference methodof polarization analysis for three different wavelengths λ=0.31 nm, 0.48 nm or 0.57 nm. The differentchapters of this book describe the different possibilities of the instrument and so together resultin a general description of the instrument. In this chapter we are interested in that part of theinstrument which is necessary to guarantee the working of the rotation of the polarization intothe x, y and z directions for these wavelengths. This comprises essentially everything that mayinfluence the field direction. Different sample environments such as cryostats or furnaces have toavoid fields at the sample position by bifilar heater winding. The chopper disks for the time offlight have to avoid rotating metallic parts in the polarized beam, which would depolarize the beamby eddy currents. In the disks of the chopper the aluminium in the windows therefore has to beremoved, and not only the absorbing gadolinium layer.

Figure 4.3: (a) Field pattern of one detector bank in a vertical section. Some lines (dotted) ofconstant field strength are given near the points where the field changes considerably. The arrowsshow the field direction in the centre point of the arrow line (zero of vertical scale: plane ofscattering, zero of horizontal scale: instrument axis). (b) Field strength in the scattering plane onthe way from the sample centre to the centre of the detector without current in the spin turn coil.

Fig.4.3a shows a vertical section through one detector bank. The detectors are located on banks,each with 16 3He-detectors of 10 cm in height and 5 cm in diameter, with 2 atm gas pressure, andat a distance of r=150 cm away from the sample. Fig.4.3b shows the field in the scattering planeon the way from the polarizer to the sample (dotted) and from the sample to the analyzer (solid).If all analyzers are in the measuring position, then the field direction in the scattering plane iseverywhere perpendicular to this plane and has the same direction. The magnetic backwards fluxnecessary to achieve div B = 0 is always located inside the permanent magnets only. In the guidefield in front of the polarizer a Mezei flipper with a Brookhaven correction coil around it [8] servesas flipper. A monitor behind the sample and the chopper together with an analyzer serves formonitoring the flipping ratio and the sample transmission.

The instrument comprises 4 such banks of 16 detectors each. Every second detector, i.e. 32

4.7. INSTRUMENTAL 55

detectors, is equipped with a supermirror analyzer of 5 cm x 10 cm cross section as described in[3, 4] where the alignment and the achieved transmission and flipping ratios are also described asa function of wavelength. Fig.4.4 shows the contours of constant field strength in the scatteringplane in the surroundings of one bank with 16 detectors and the associated guide fields.

Figure 4.4: Magnetic field of one bank in the scattering plane. The field direction is perpendicularto the figure plane. The dotted lines are lines of constant field strength. The solid lines describethe lay-out of the guide field set-up, as given in fig.4.3 in a vertical cut.

Behind the chopper and centred at the sample position there is a spin rotator. This consists ofa cube of three coil pairs to define the polarization direction of the beam at the sample position.The field strength is large enough to enable an adiabatic rotation of the polarization direction intoeither the x,y or z direction [6, 7] but not too strong, so that the field at the sample does not changethe sample magnetization essentially. Fig.4.5 shows the field of this coil in the scattering plane inthe absence of other sources of magnetization. This field should show rotational symmetry aroundthe coil axis.

With four detector banks in the instrument the fields of these four banks are linearly superposedso that for example in the centre of the instrument there is a region of 4×0.25 mT = 1.0 mT resultingfrom all four banks (see figs.4.3 and 4.4). Therefore for polarization analysis measurements that donot need spin rotation there is no need for an additional guide field at the sample position, and onecan even remove the xyz-spin-turner coil cube to increase the accessible angular range. For the caseof the xyz-difference measurement, and for measurements with the polarization in x or y direction,one has to adjust the z-coil-current to compensate this z-field in the instrument. Otherwise thefield would have also a component in the z-direction and consequently would not point exactly inthe x- or the y-direction. All these conditions have to be kept under surveillance, because theycan change with time, e.g. by changing the environment of the instrument, which is now kept asiron-free as possible.

This spin rotation has to take place on the flight path from the polarizer guide fields to thesample, and from the sample to the analyzer guide fields. For the scattering process in the samplewhich we call the xyz-difference method and which was treated theoretically in the first part ofthis paper, we need at the sample position three different, well defined polarization directions


Px, Py, Pz. After the scattering process and on the way to the analyzers there is again a spinreorientation process which rotates the component of the neutron spin, which was brought into thex or y direction for the scattering process, respectively, back to the z direction. This component isthan transmitted by the analyzers. No rotation is required for the z component.

Figure 4.5: Field in and around the spin turn coil. The dotted lines are lines of constant fieldstrength in mT for a current of 7.7 Ampere. The arrows in the lower left quarter show the fielddirection in this part of the plane at the center of the arrows. Solid thick lines: position of twocircular coils with the coil axis z’ in the figure plane through the zero point of the abscissa.

In the spin turner coils the spin of the neutrons has to follow the field direction. At the polarizerand the analyzer the guide fields are always high, roughly 10 mT at the polarizer and 40 mT atthe analyzer with the fields being oriented along the vertical direction (z-direction). If the spinturner coils are excited for the horizontal x or y direction, then the vertical field in front of thebank decreases with the distance from the bank and is exactly compensated at the sample positionby the z-coils. This combines with the field in the x direction, field strength of which behaves asshown in the contour plots of fig.4.5. This figure indicates the field direction (arrows) and the linesof constant field strength (dotted) of one coil pair of the spin turn coils for a current of 7.7 Ampere.(0,0) is the centre of the sample on the scattering plane and on the axis of the instrument. As thecoils are circular the figure is valid for the whole space. It can be applied for all three coil pairs andcan be used to describe the total resulting field of all three coil pairs just by linear superposition ofthe figure with z’ in x or y or z direction respectively. From the field patterns in figs.4.3 to 4.5 onecan obtain the exact rotation of the guide field as a function of the distance from the centre of thespin turner coil for each detector. Examples of this are given in fig.4.6 for a neutron path throughthe centre of a spin turner coil and for a neutron path just behind the edge of such a coil. In orderto allow the neutron spin orientation to follow the field direction one has to fulfil the condition ofadiabaticity i.e. the change of the field direction has to be slow enough. Fig.4.6 shows that thefield direction is changing by 620 = 1.082 rad on the way from the centre to the entrance into the

4.7. INSTRUMENTAL 57

guide field for the analyzer in the range of r = 25 cm to 45 cm. Fig.4.6b shows that the rotation isindeed homogeneous over this distance of 20 cm and is not taking place over the whole path. Theneutron with a velocity v(m/s)=3956/λ(A) feels this as a rotation of |ω| = 1.082·v

s . The conditionof adiabaticity requires |ω| |ωL| with ωL = −γH being the Larmor frequency. It is known[6, 7] that the maximum deviation of the neutron spin direction from the magnetic field directionis given by the angle 2δ = 2 arctan ω

ωL. In our case ωL = −2π · 29164 H

mT = 183242.8 HmT [s−1] and

for λ = 4.8 A we have ω = 1.081 39560.2·4.8 = 4459.2 [s−1]. This corresponds to 2δ = 2.220 for a field

of 1.255 mT as rotating guide field, which is the minimum field in the region of the rotation offig.4.6 b. On the way from the polarizer to the sample one gets exactly the same deviation. Forthe wavelength of 5.7 A this deviation is 1.870 and for 3.1 A it is 3.40. On the way through thecorner of the coil the behaviour is a little more complicated as seen in fig.4.6 c) and d). There onecan observe two rotations of the field, one in ϕ and one in ϑ. With the same sort of estimate onefinds there a maximum deviation for a wavelength of 4.8 A of 2.20 for the ϕ– rotation and 0.740

for the ϑ–rotation. The behaviour of the neutron spin in a conically rotating guide field is treatedfurther in chapter 2 and in [19].

Figure 4.6: a) and c) rectangular, b) and d) spherical field components of the guide field on theway of the neutron for two characteristic neutron pathes, a) and b) through the center of the coils,c) and d) near the corner of the coils. In b) and d) the ordinate scale is mT for the field |H| anddegree for the angles ϑ, ϕ.

The error introduced into the measured magnetic cross section by such a deviation of the


neutron spin direction can be estimated by using the law derived above of the polarization directionfor paramagnetic scattering (eq.(4.56)). From this one obtains a maximum error of 0.15% for themeasured intensities corresponding to the deviation of 2.20 at the sample position and another0.15% for the deviation of 2.20 on the way from sample to analyzers. By analogy to the law ofMalus for optical polarization the intensities are proportional to cos2(2δ) with 2δ being the angleof rotation of the polarization. Because of this small maximum error and also due to the size ofthe detectors of 10 cm on a radius of 1.5 m the angle ϑ is only determined to within 3.80. It is notpractical to try to diminish this angle of 2δ = 2.20.

A quantitative test is described in chapter 10 and in [20] of cross section measurements usingthis instrument and a sample whose cross section can be calculated exactly for different κ andω. As this test cross section is only slightly dependent on the scattering vector this measurementrepresents another test for the fulfilment of the adiabaticity condition for all the neutron flightpaths through the fields of the spin rotation coils.

A remark is in order concerning the difference of ↑↓ and ↓↑ flip measurements, which play arole only in Px or Py or P‖ measurements with spin flip. For the instrument described above thiscan be measured by inverting the current in the x or y coils. This process inverts the initial spinon the sample from ↑ to ↓ and if the sample flips the set-up of the instrument is able to measurethe ↑↓ and the ↓↑ flip cross sections separately.

Finally the procedure of the xyz-method corresponds to part of a measurement which constitutesa generalization of the neutron polarization analysis [18]. Here, however, only the main diagonalelements of the general matrix, namely Pxx, Pyy, Pzz i.e. the scattering with polarization in xdirection and analyzed in x direction (Pxx) is measured and similarly with y and z components.The full generalised polarization analysis would also measure all other components, i.e. it wouldbe able to measure 9 components of the scattered polarization with and without spin flip, whichneeds correspondingly more measuring time. This would only increase the information about thesample in cases where it has a noncentrosymmetric structure or more generally if there is no parityconservation and no time inversion symmetry, as we have seen in 3.2.3 on page 33. The instrumentD7 can handle such questions for as many detectors as can be equipped with a unit like in fig.2.5 onpage 22 (see 2.2.2 on page 21). The present instrument is equipped with 32 analyzers, it can handlelarge samples and simultaneously afford time of flight analysis of energy changes of the scatteredbeams in the range of up to 30 meV energy gain [17]. To combine these possibilities with at leastsome features of the generalized polarization analysis is a very interesting and challenging task,especially for a multidetector instrument. For symmetry determinations in complicated magneticstructures it may be necessary to measure all components as J.Brown et al. has shown in manyinteresting papers. Derivation of the respective formulae will show which minimal possibility willbe sufficient to provide the maximum possible information. D7 is already a very complicatedinstrument, if one wishes to use all its possibilities. These can certainly still be enhanced, forexample by including this generalization of the spin analysis. Another possible improvement wouldbe spin echo, but this would only be worthwhile if the instrument is moved to an end position of aguide where it could gain a factor ten in intensity.

Bibliography

[1] K.R.A.Ziebeck and P.J.Brown, J.Phys.F 10, 2015 (1980)

[2] O.Schaerpf, Proceedings Series: Neutron Scattering in the Nineties, Int.Atomic Energy AgencyVienna, 85(1985)

[3] O.Schaerpf, Physica B 156&157, 631 (1989)

[4] O.Schaerpf, Physica B 156&157, 639 (1989)

[5] W.G.Williams, Polarized Neutrons, Clarendon Press, Oxford 1988

[6] Ch.Schwink, O.Scharpf, Z.Phys.B 21, 305 (1975)

[7] O.Schaerpf, in ”Neutron Spin Echo” ed.F.Mezei, p.32, Lecture Notes in Physics vol.128,Springer-Verlag, Berlin 1980

[8] R.Pynn, Rev. Sci. Instrum. 55, 837 (1984)

[9] O.Schaerpf, H.Capellmann, Th.Brueckel, A.Comberg and H.Pasing Z.Phys.B 78, 345 (1990)

[10] R.M.Moon, T.Riste and W.C.Koehler, Phys.Rev. 181, 920 (1969)

[11] O.Halpern, M.H.Johnson, Phys.Rev. 55, 898 (1939)

[12] S.V.Maleev, Sov.Phys. JETP Lett. 2, 338 (1966)

[13] F.Mezei, A.P.Murani, J.Magn.Magn.Mat. 14, 211 (1979)

[14] Y.A.Izyumov, R.P.Ozerov, Magnetic Neutron Diffraction, Plenum Press, New York 1970, p.175

[15] S.W.Lovesey, Theory of Neutron Scattering from Condensed Matter, vol.2, p.167, ClarendonPress, Oxford 1986

[16] O.Scharpf and H.Capellmann, Z.Phys.B 80, 253 (1990)

[17] O.Scharpf, Physica B in press. (1992)

[18] M.Blume, Phys.Rev. 130, 1670 (1963)

[19] O.Scharpf, J.Appl.Cryst. 11, 631 (1978)

[20] O.Scharpf, T.Chattopadhyay, H.W.Weber, O.B.Huyn, D.K. Finnemore, phys.stat.sol.(b) inpress.

59

60 BIBLIOGRAPHY

Chapter 5

Density matrix formalism to treatmagnetic scattering more completely

The study of magnetic structures by neutrons shows a very rich world of possible moment arrange-ments and can not be treated here. To enter deeper into this question one should study such bookslike [1] or [2]. There are not only ferromagnetic and antiferromagnetic types of ordering, but alsoferrimagnetic ones. These are known as collinear structures. There exist many kinds of noncollinearstructures. Another group is treated under the name of weak ferromagnetism. There are also mag-netic structures which cannot be described by means of a unit cell: the helicoidal structures. Inthis chapter we want to apply the density matrix method

1. to see that it is much simpler and simultaneously more powerful than the transition probabilitymethod

2. to learn how to get from its formulas the intensities, which we need to get scattering crosssections

3. to include then also the nuclear magnetic interference terms

4. to include also the spin orbit interaction.

5. to discuss in a more general way the possibilities of the general results which one gets by theformulas first derived by Maleev [3] and by Blume [4, 5] .

5.1 General formulas for magnetic scattering only

The thermal and quantum mechanical average can here be included by the density matrix formalism.One could again formulate the whole process as functions of ~r and t or ω as in the last chapter toget the most general form also useful for inelastic scattering. But to show the essential process it isalso here clearer to avoid to write all the variables necessary to show the full process which resultsin the correlation functions. We use eq.(4.8 on page 44) and 4.9 on page 45 for the interactionpotentials for magnetic scattering. This we know from 4.12 on page 45 can be written

µn~σn · ~Q⊥ = µn~Q · (~σn − ~e(~e · ~σn)) = µn~σn( ~Q− ~e(~e · ~Q)) with ~e = ~κ/κ and ~Q = ~Me (5.1)

So the magnetic interaction is described by an expression of the form ~σ · ~Q⊥ which includes themagnetic form factors of the different magnetic atoms and is a function of ~r and ω. We want first

61

62 CHAPTER 5. DENSITY MATRIX FORMALISM FOR MAGNETIC SCATTERING

to derive the magnetic part only. With the form 5.1 we can use the more sophisticated rules of 3.24on page 33 with

L1 = ~Q† · (~σ − ~e(~σ · ~e)) (5.2)L2 = ~Q · (~σ − ~e(~σ · ~e)) (5.3)~M1 = ~Q† − ~e( ~Q† · ~e) = ~Q†

⊥ (5.4)~M2 = ~Q− ~e( ~Q · ~e) = ~Q⊥ (5.5)

5.1.1 I↑↑ + I↑↓

For the total scattered intensity I↑↑ + I↑↓ we get:

dσ

dΩ= Trσ(ρ(~σ · ~Q†

⊥)(~σ · ~Q⊥)) = Trσ(ρL1L2) =12Trσ(1 + ~P · ~σ)(L1L2) (5.6)

= ( ~Q†⊥ · ~Q⊥) + i ~P · ( ~Q†

⊥ × ~Q⊥) (5.7)

5.1.2 I↑↑ − I↑↓

For the polarization of the scattered beam we get:

~P outγ

dσ

dΩ= I↑↑γ − I↑↓γ (5.8)

= Trσ[ρ(~σ · ~Q†⊥)~σ(~σ · ~Q⊥)] = Trσ[ρL1σL2] =

12Trσ[(1 + (~σ · ~P ))1~σL2] (5.9)

= −i( ~Q†⊥ × ~Q⊥)γ + [ ~Q†

⊥γ(Q⊥ · ~P ) + ( ~Q†⊥ · ~P )Q⊥γ − ( ~Q†

⊥ · ~Q⊥)Pγ (5.10)

5.1.3 Express ~Q⊥ by (Mx, My, Mz) and α

Figure 5.1: The geometry of the scattering vec-tor and the magnetization vector to express ~Q⊥ bythe components of ~M and α. With the projectionof ~M into the xy plane ~Mρ = (Mx,My, 0) we get√M2

x +M2y cos(~κ, ~Mρ) = Mx cosα + My sinα. A

unit vector normal to ~κ is (sinα,− cosα, 0). It isQ2⊥ = M2

x +M2y +M2

z − (Mx cosα +My sinα)2 =(Mx sinα−My cosα)2 +M2

z .

Fig.5.1 gives the geometry of the scattering vector and the function ~Q⊥ in relation to theadiabatic spinturner coils x, y, and z. With the help of this figure one finds that

~Q⊥ = (0, 0,Mz) + (Mx sinα−My cosα)(sinα,− cosα, 0) = (A sinα,−A cosα,Mz) (5.11)

5.1. GENERAL FORMULAS FOR MAGNETIC SCATTERING ONLY 63

5.1.4 Derivation of the correlation functions Γνss′(~r, ω)

To obtain the scattering and polarization for the three components P outx , P out

y , and P outz as scatter-

ing with and without flip with ~P in = (1, 0, 0), ~P in = (0, 1, 0) and ~P in = (0, 0, 1) we insert these and~Q⊥ = (Mx sin2 α −My sinα cosα,−Mx sinα cosα +My cos2 α,Mz) in eqs.(5.7,5.10). We must re-member that products of components of ~Q⊥ in these equations correspond to correlation functions,if one includes all variables summations and transforms in the case, which includes inelastic scat-tering. If we forget this and insert ~Q⊥ and the corresponding incident polarizations, we obtain theequations for elastic scattering.

First we evaluate the vector product ~Q†⊥ × ~Q⊥. With

A = Mx sinα−My cosα (5.12)~Q⊥ = (A sinα,−A cosα,Mz) (5.13)

and assuming that A and Mz can be complex, we get

i( ~Q†⊥ × ~Q⊥) =

∣∣∣∣∣∣∣~e1 ~e2 ~e3

A∗ sinα −A∗ cosα M∗z

A sinα −A cosα Mz

∣∣∣∣∣∣∣ (5.14)

= i((−A∗Mz +M∗zA) cosα,−(A∗Mz −M∗

zA) sinα, (−A∗A+A∗A) sinα cosα)(5.15)

= −(2 cosα=(AM∗z ), 2 sinα=(AM∗

z ), 0) (5.16)= −2=(AM∗

z )(cosα, sinα, 0) (5.17)= −2=(MxM

∗z sinα−MyM

∗z cosα)(cosα, sinα, 0) (5.18)

=(MxM∗z ) and =(MyM

∗z ) can only contribute something if Mx or My or Mz or all are complex and

the real and imaginary part do not have the same direction.With this and eq.(5.7) we obtain for the elastic scattering:

dσ

dΩ= I↑↑ + I↑↓ (5.19)

= ~Q†⊥ · ~Q⊥+ i ~P ( ~Q †⊥ × ~Q⊥) (5.20)

= |A|2 + |Mz|2 − 2=(MxM∗z sinα−MyM

∗z cosα)(Px cosα+ Py sinα) (5.21)

= |Mx|2 sin2 α+ |My|2 cos2 α− 2<(M∗xMy) sinα cosα+ |Mz|2

−2=(MxM∗z sinα−MyM

∗z cosα)(Px cosα+ Py sinα) (5.22)

Using eq.(5.10) and P outν ‖P in

ν we get

P outν

dσ

dΩ= I↑↑ν − I↑↓ν (5.23)

= −i( ~Q†⊥ × ~Q⊥)ν +Q⊥νQ

†⊥ν +Q†

⊥νQ⊥ν − (Q†⊥xQ⊥x +Q†

⊥yQ⊥y +Q†⊥zQ⊥z)

(5.24)

P outx

dσ

dΩ= 2=(MxM

∗z sinα cosα−MyM

∗z cos2 α) +

+2AA∗ sin2 α− (AA∗ sin2 α+AA∗ cos2 α+MzM∗z ) (5.25)

= 2=(MxM∗z cosα sinα−MyM

∗z cos2 α) + |A|2(sin2 α− cos2 α)− |Mz|2 (5.26)

P outy

dσ

dΩ= 2=(MxM

∗z sin2 α−MyM

∗z cosα sinα)− |A|2(sin2 α− cos2 α)− |Mz|2 (5.27)


P outz

dσ

dΩ= MzM

∗z +M∗

zMz − (AA∗ sin2 α+AA∗ cos2 α+MzM∗z ) (5.28)

= |Mz|2 − |A|2 (5.29)

with

|A|2 = |Mx sinα−My cosα|2 = |Mx|2 sin2 α+ |My|2 cos2 α− 2<(M∗xMy) sinα cosα (5.30)

Combining these results to get I↑↑x , I↑↑y ,I↑↑z , I↑↓x ,I↑↓y , I↑↓z gives

I↑↑x = |A|2 sin2 α (5.31)= M∗

xMx sin4 α+M∗yMy cos2 α sin2 α− 2ReM∗

xMy sin3 α cosα (5.32)

I↑↑y = |A|2 cos2 α (5.33)

= |Mx|2 sin2 α cos2 α+ |My|2 cos4 α− 2<M∗xMy sinα cos3 α (5.34)

I↑↑z = |Mz|2 (5.35)I↑↓x = |A|2 cos2 α+ |Mz|2 − 2=(M∗

xMz sinα−M∗yMz cosα) cosα (5.36)

= |Mx|2 sin2 α cos2 α+ |My|2 cos4 α− 2<M∗xMy sinα cos3 α+ |Mz|2 (5.37)

−2=M∗xMz sinα cosα+M∗

yMz cos2 α (5.38)

I↑↓y = |A|2 sin2 α+ |Mz|2 − 2=(M∗xMz sinα−M∗

yMz cosα) sinα (5.39)

= |Mx|2 sin4 α+ |My|2 cos2 α sin2 α− 2<M∗xMy sin3 α cosα+ |Mz|2 (5.40)

−2=M∗xMz sin2 α+M∗

yMz cosα sinα (5.41)

I↑↓z = 0 (5.42)

This corresponds to what we found in eqs.(4.39 on page 51) and (4.40 on page 51). The resulteq.(5.31) can immediately be seen by adding 5.21 to 5.26, the result eq.(5.36) by subtracting 5.26from 5.21 with Px = 1 and the others respectively.

5.2 Nuclear-magnetic interference terms

We have investigated the nuclear scattering resulting from N (see 3.3 on page 35) and the magneticscattering resulting from ~σ · ~Q⊥ (see 5.1 on page 61). If both are present then we have to investigatethe scattering matrix resulting from (N +~σ · ~Q⊥)(N † +~σ · ~Q⊥). We have then still to calculate thetraces of the interference terms

N~σ · ~Q†⊥ + ~σ · ~Q⊥N

† (5.43)

We get for the total scattering from the nuclear magnetic interference term

dσ

dΩ= Trσ[ρ(N~σ · ~Q†

⊥ + ~σ · ~Q⊥N†)] (5.44)

=12Trσ[(1 + ~σ · ~P )(N~σ · ~Q†

⊥ +N †~σ · ~Q⊥)]

=12Trσ[NσαQ

†⊥α +N †σαQ⊥α +NσαPασβQ⊥β +N †σαPασβQ⊥β] (5.45)

=12[2δαβNQ

†⊥β + 2δαβN

†PαQ⊥β ] (5.46)

= NPαQ†⊥α +N †PαQ⊥α (5.47)

= 2<(N ~P · ~Q†⊥) (5.48)

5.2. NUCLEAR-MAGNETIC INTERFERENCE TERMS 65

and for the polarization

P outν

dσ

dΩ=

12Trσ[(1 + ~σ · ~P )(Nσν(~σ · ~Q†

⊥) +N †σν(~σ · ~Q†⊥))] (5.49)

=12[NσνσαQ

†⊥α +N †σνσαQ⊥α + σαPαNσνσβQ

†⊥β + σαPαN

†σνσβQ⊥β ]

=12(2δναQ

†⊥αN + 2δναN

†Q⊥α + 2iεανβPαQ†⊥βN + 2iN †εανβPαQ⊥β) (5.50)

= Q†⊥νN +Q⊥νN

† + iN(~P × ~Q†⊥)ν − iN †(~P × ~Q⊥)ν (5.51)

= 2<(Q†⊥νN)− 2=(N † · (~P × ~Q⊥)ν) (5.52)

One can use this again as above to determine the intensities of the scattering without spin flip andwith spin flip for the incident and analyzed polarization in x, y, z direction:

I(x)↑↑ = 2<(NA∗) sinα (5.53)

I(y)↑↑ = −2<(NA∗) cosα (5.54)

I(z)↑↑ = 2<(NM∗

z ) (5.55)

I(x)↑↓ = 0 (5.56)

I(y)↑↓ = 0 (5.57)

I(z)↑↓ = 0 (5.58)

This only gives a contribution if there is a resulting magnetic operator Q⊥, which does notaverage to zero or for magnetic Bragg peaks coinciding with nuclear ones. In the case of ferro-or ferri- magnetism one can force this by an applied magnetic field. For antiferromagnetism thisterm averages to zero if there are equal numbers of spins with one direction and of the oppositedirection in the same lattice plane. If this is not the case this term does not average to zero. Fornone centro symmetric structures these terms are often very helpful to determining the magneticsymmetry. They can also produce none diagonal contributions to the polarization vector.

The nuclear-magnetic interference term is the term that makes possible the form factor deter-mination, the use of polarizing crystals i.e. a large field of applications not treated here. If theincident beam is not polarized a magnetic field applied to order the magnetic moments can givea polarization in the direction of ~Q⊥ but in general not completely polarized as it is on top of anot polarized intensity. To see this we use the above derived equations and exclude non centrosymmetric terms. We obtain then

I↑↑ + I↑↓ = N †N + (~P (in) · ~Q†⊥)N + (~P (in) · ~Q⊥)N † + ~Q†

⊥ · ~Q⊥ (5.59)

I↑↑ − I↑↓ = ~P (in)N †N +N ~Q†⊥ +N † ~Q⊥ + (~P (in) · ~Q†

⊥) ~Q⊥

+(~P (in) · ~Q⊥) ~Q†⊥ − ~P (in)( ~Q†

⊥ · ~Q⊥) (5.60)

With unpolarized incident beam ~P (in) = 0 these equations simplify strongly and one gets for theout going polarization

P (out) =I↑↑ − I↑↓

I↑↑ + I↑↓=NQ† +N †Q

N †N +Q†Q(5.61)

If for any direction N = ±Q⊥ then

P (out) =N(±N †) +N †(±N)

N †N +N †N= ±1 (5.62)


If this is not the case then one has a polarization on top of an unpolarized background.If the crystal is antiferromagnetic the domains may hinder that this condition is fulfilled in the

whole crystal. The same is valid if one would like to polarize with the helical structure.The reader interested in these topics (form factor determination) should consult [6] or [7].

5.3 Visualization of the full formalism of the density matrix methodfor the 3d PA

Taking together all results of the scattering of polarized neutrons described by the density matrixmethod in chapter 3.3 on page 35, 3.3.3 on page 38, 5.1 on page 61 we get the following equations:

dσ

dΩ= N †N + (~P (in) · ~Q†

⊥)N + (~P (in) · ~Q⊥)N † + ~Q†⊥ · ~Q⊥

+i ~P (in) · ( ~Q†⊥ × ~Q⊥) +B†B[~I† · ~I + i ~P (in) · (~I† × ~I)] (5.63)

~P (out) dσ

dΩ= ~P (in)N †N +N ~Q†

⊥ +N † ~Q⊥ + iN(~P (in) × ~Q†⊥)− iN †(~P (in) × ~Q⊥)

+[(~P (in) · ~Q†⊥) ~Q⊥ + (~P (in) · ~Q⊥) ~Q†

⊥ − ~P (in)( ~Q†⊥ · ~Q⊥)]− i( ~Q†

⊥ × ~Q⊥)

+B†B[−i(~I† × ~I) + (~P (in) · ~I†)~I + (~P (in) · ~I)~I† − ~P (in)~I† · ~I] (5.64)

Now we remember (see 3.94 on page 39) that for the measurement of the polarization in ν-direction(with ν = x, y, z) of the scattered beam

dσ

dΩ= I↑↑ν + I↑↓ν (5.65)

P (out)ν

dσ

dΩ= I↑↑ν − I↑↓ν (5.66)

and that for unpolarized nuclear spins the above equations simplify to

dσ

dΩ= N †N + (~P (in) · ~Q†

⊥)N + (~P (in) · ~Q⊥)N † + ~Q†⊥ · ~Q⊥

+i ~P (in) · ( ~Q†⊥ × ~Q⊥) +B†BI(I + 1) (5.67)

~P (out) dσ

dΩ= ~P (in)N †N +N ~Q†

⊥ +N † ~Q⊥ + iN(~P (in) × ~Q†⊥)− iN †(~P (in) × ~Q⊥)

+[(~P (in) · ~Q†⊥) ~Q⊥ + (~P (in) · ~Q⊥) ~Q†

⊥ − ~P (in)( ~Q†⊥ · ~Q⊥)]

−i( ~Q†⊥ × ~Q⊥)− 1

3B†B ~P (in)I(I + 1) (5.68)

This allows us to determine the scattered intensity with polarization of the incident beam in νdirection and analyzed with the polarization in the same ν direction with ν = x, y, z. This gives

I↑↑xx = N †N +NQ†⊥x +N †Q⊥x +Q†

⊥xQ⊥x +13B†BI(I + 1) (5.69)

I↑↓xx = Q†⊥yQ⊥y +Q†

⊥zQ⊥z + i( ~Q†⊥ × ~Q⊥)x +

23B†BI(I + 1) (5.70)

I↑↑yy = N †N +NQ†⊥y +N †Q⊥y +Q†

⊥yQ⊥y +13B†BI(I + 1) (5.71)

5.4. APPLICATION TO PARAMAGNETIC POWDER SCATTERING 67

I↑↓yy = Q†⊥xQ⊥x +Q†

⊥zQ⊥z + i( ~Q†⊥ × ~Q⊥)y +

23B†BI(I + 1) (5.72)

I↑↑zz = N †N +NQ†⊥z +N †Q⊥z +Q†

⊥zQ⊥z +13B†BI(I + 1) (5.73)

I↑↓zz = Q†⊥xQ⊥x +Q†

⊥yQ⊥y + i( ~Q†⊥ × ~Q⊥)z +

23B†BI(I + 1) (5.74)

This shows again the already well known behaviour that only those components of ~Q⊥ which areperpendicular to the spin of the neutrons can contribute to the scattering with spin flip. And thisis also the case for the nuclear spins. The term with the cross product give only a contribution ifthe <( ~Q⊥) has another direction than the =( ~Q⊥) and not the same size.

More about these things you can find in neutronpol.pdf section 4.4 on the same site.

5.4 Application to paramagnetic powder scattering

Here we want to give the density matrix method derivation of the equation ~P ′ = −~κ(~κ · ~P )/κ2 andvisualize what it says. Practically the latter is already done in chapter 3.7.

5.4.1 Determination of the polarization of paramagnetic scattering using thedensity matrix formalism

To determine the polarization of the paramagnetic scattering we use the fact that

~Me · ~µn⊥ = ~µ · ~Me⊥ (5.75)

already derived in 4.12 on page 45. With ~Si the operator of the electron magnetization includingall factors and sums as in eq.(4.10 on page 45), ~Mi = ~Si− (~e · ~Si)~e its normal component to the unitvector ~e in the direction of the scattering vector, Li = ~Si · (~σ − (~e · ~σ)~e) describing the interactionof the neutron with the electron magnetization the polarization of the neutron beam scattered byLi is

~P out =Trσ(ρL†~σL)Trσ(ρL†L)

(5.76)

=12Trσ(L†~σL+ (~P · ~σ)L†~σL)

12Trσ(L†L+ (~P · ~σ)L†L)

(5.77)

=−i( ~M † × ~M) + ~M †( ~M · ~P ) + ( ~M † · ~P ) ~M − ~P ( ~M † · ~M)

( ~M † · ~M) + i( ~M † × ~M)~P(5.78)

the last line by applying the rules eq.(3.24 on page 33) seq.. For a paramagnet without helicalstructures i.e. without noncentrosymmetric structure the cross product in the nominator anddenominator disappear and we get

~P out =~M †( ~M · ~P ) + ( ~M † · ~P ) ~M − ~P ( ~M † · ~M)

( ~M † · ~M)(5.79)

Now we have to average ~M † and ~M and the products containing them over the random orientationsof the atomic spins ~S†,~S. For random orientation we have

〈(~S† · ~S)〉 = S(S + 1) (5.80)

http://82.135.31.182/neutronpol.pdf#section.4.4


= 〈S†xSx〉+ 〈S†ySy〉+ 〈S†zSz〉 (5.81)

〈S†νSν〉 =13S(S + 1) with ν=x,y,z (5.82)

〈Sx〉 = 〈Sy〉 = 〈Sz〉 = 〈S†ν〉 = 0 (5.83)

With these the average of the first product is

〈(~S† − (~e · ~S†)~e)[(~S · ~P )− (~e · ~S)(~e · ~P )]〉 = 〈~S†(~S · ~P )〉 − 〈~S†(~e · ~S)(~e · ~P )〉−〈(~e · ~S†)~e(~S · ~P )〉+ 〈(~e · ~S†)~e(~e · ~S)(~e · ~P )〉

=13S(S + 1)[~P − (~e · ~P )~e− (~e · ~P )~e+ (~e · ~P )~e]

=13S(S + 1)[~P − (~e · ~P )~e] (5.84)

We used for this

〈~S†(~S · ~P )〉 = 〈(S†x, S†y, S†z)(SxPx + SyPy + SzPz)〉 (5.85)

=13S(S + 1)(Px, Py, Pz) (5.86)

= 〈(~S† · ~P )~S〉 (5.87)

〈~S†(~e · ~S)〉 =13S(S + 1)(ex, ey, ez) (5.88)

〈(~e · ~S†)(~S · ~P )〉 =13S(S + 1)(~e · ~P ) (5.89)

〈(~e · ~S†)(~e · ~S)〉 =13S(S + 1) (5.90)

The average of the second expression gives the same. The third expression and the denominatorcan be averaged in one go

〈( ~M † · ~M)〉 = 〈(~S† − (~e · ~S†)~e) · (~S − (~e · ~S)~e)〉= 〈(~S† · ~S)− (~S† · ~e)(~e · ~S)− (~e · ~S†)(~e · ~S) + (~e · ~S†)(~e · ~S)e2〉

= 〈(~S† · ~S)− (~S† · ~e)(~e · ~S)〉 = S(S + 1)− 13S(S + 1)

=23S(S + 1) (5.91)

So we get for ~P outparamagnetic finally

~P outparamagnet =

13S(S + 1)[ ~

P − (~e · ~P )~e+ ~P − (~e · ~P )~e− 2~P ]23S(S + 1)

= −(~e · ~P )~e (5.92)

Bibliography

[1] Yu.A. Izyumov, R.P. Ozerov, Magnetic Neutron Diffraction Plenum Press, New York (1970)

[2] A.P. Cracknell, Magnetism in Crystalline Materials Pergamon Press, Oxford, New York (1975)

[3] S.V.Maleev, JETP 13, (1961), 860

[4] M.Blume, Phys.Rev. 130, (1963), 1670 Phys.Rev. 133, (1964), A1366

[5] R.I.Schermer, M.Blume Phys. Rev. 166, (1968), 554

[6] J.Schweizer Interpretation of magnetization densities 501-519 Interpretation of the spin den-sities in metals and alloys 479-499 Proceedings of the Nato Advanced Study Institute, editorP.Becker Summer School, Arles 1978 Plenum Press, New York (1980)

[7] P.J.Brown Magnetic neutron scattering Proceedings of the Nato Advanced Study Institute, ed-itor P.Becker Summer School, Arles 1978 Plenum Press, New York (1980) 255-285

69

Index

‖–⊥ method, 44∂2σ/∂ωΩ, 42π/2-flipper

current set-up, 18π-flipper, 15

use of, 15π/2-flipper, 18

use of, 18

adiabatic xyz-coil, 25in D7, 25

angular momentumcommutation relations, 3

Blume formulae, 29, 59Born approximation

for particles with spin, 32

CRYOPAD, 20

density matrix, 5incident beam, 31link between in and out, 31outgoing beam, 31parity conservation, 32powerful relationship, 8useful relationships, 9

density matrix formalism, 29density operator, 6

ensemble average of operator, 7expectation value of operator, 7

flipperstrom, 15

flipper efficiency, 16matrix representation, 16

flipping ratio, 16correction, 34monitoring, 52why to correct, 34

interference term, 31, 38, 51, 59inclusion of, 62

magneticstructure amplitude, 43

magnetizationorbital part, 42spin part, 42

mirror polarizeras potential well, 6

non spin flip scatteringxyz method, 48

non-spin flip scatterin‖–⊥ method, 45

orbitcommutation relations, 3quantum mechanical description, 3

Pauli spin formalism, 4Pauli spin matrix

compilation of rules, 29polarimeter, 26

CRYOPAD, 20historical, 20

polarimetryand parity conservation, 32

polarization, 3, 4, 7–10, 25, 33, 37⊥ scattering vector, 45completely polarized, 6equally distributed spins over a ϕ range, 9incompletely polarized, 6mirrors, 5precessing, 18

measurement, 19Stern-Gerlach, 5

polarization measurement, 16polarizer properties

matrix representation, 16measurement, 16

70

INDEX 71

polarizing efficiency, 16precessing polarization, 18precession cone

unrolling, 25

rotating guide field, 25

scatteringby nuclear spin only, 36by spinless target, 33

quartz, 33nuclear spin

vanadium, 36target with spin only, 36unpolarized nuclear spin, 36

scattering matrix, 31Schrodinger equation

time dependent, 13in constant B-field, 14in slowly varying field, 22

separationof coh.,mag.,inc.scatt., 41using xyz method, 41

slowly varying magnetic field, 22spin

adiabatic xyz-coil, 25commutation relations, 3helical magnetic field

visualization, 25in rotating guide field, 25in slowly varying field, 4in strong inhomogeneities, 4quantum mechanical desciption, 3slowly varying magnetic field, 22space dependent components, 4sudden transition, 15with direction (ϑ, ϕ), 9

spin flip scattering‖–⊥ method, 44polarization in x, 46polarization in y, 48polarization in z, 48xyz method, 46

spin in helical field, 22spin orbit, 59spin orbit coupling, 45, 47spinor

examples, 13spinor space

rotation about x, 12rotation about ~n, 13rotation about y, 12rotation about z, 12rotation in, 10

SSPAD, 13, 19calibration for sample position, 22fig. of coils, 20

Stern-Gerlach, 4, 5

transition matrix element, 42

visualizability, 4visualization

constant spinor, 4

xyz coil, 25xyz method, 29, 38

density matrix, 29elastic case, 49lowering operators, 43multidetectors, 46raising operators, 43

xyz-rotation, 26xz-coil

sudden transition, 9use to rotate spin, 10

polarization elements - p.otto sch¤rpf sj

Documents