thermal and subthermal neutron scattering

Zeitschrift for Physik 210, 434--456 (1968)

Thermal and Subthermal Neutron Scattering*

WOLFGANG MEHRINGER

Received November 11, 1967

Several problems in thermal and subthermal neutron scattering are considered, such as long-wave phonon multiple scattering, interference, reflexion (sec. 1) and inelastic small angle scattering (sec. 2). A method for solving elastic scattering problems is given and applied to the problem of diffraction from a plane slit (sec. 3). These investigations were partly motivated by experimental results, a discussion of which will be found in the appendix. Finally, as of special interest for experiments with extreme subthermal energies, the contribution of phonons to refraction (sec. 4) and inelastic nuclear spin- flip scattering, such as by a ferromagnet, are discussed (sec. 5).

Introduction

Several aspects of thermal and extremely subthermal neutron scattering by nonmagnetic as well as magnetic systems are considered in this paper.

In the first of five sections an application of a method for high- energy scattering by GLAUBER 1 to the scattering of thermal neutrons by modes of lowest frequency sound is given with a discussion of multiple scattering, "conservat ion of momen tum" , interference, and inelastic reflexion.

In section 2 inelastic scattering by imperfections (dislocations) of a crystal, corresponding to the process of bremsstrahlung in quantum electrodynamics, is described. Experimentally a broadening of one- phonon coherent scattering peaks, as in elastic small angle scattering, should be expected.

In section 3 the problem of elastic scattering by a solid is tackled. The integral equation is used, and via summing up forward scattering and by iteration the exact wave function should be constructable, as could be tested by self-consistency.

The investigations of sections 1, 2, and 3 have partly been motivated by experimental results 2, according to which the interference pattern (dashed line in Fig. 1) of two coherent neutron rays should be destroyed (solid line) when a solid block between the rays is cut in two as shown. A discussion of this result will be given in the appendix.

* Summary of work done as dissertation at "Theoretisches Teilinstitut des Physik- departments der Technischen Hochschule Mfinchen" in 1967.

1 GLAUBER, F.: High energy collision theory. Lect. in Theoret. Phys., Boulder 1958.

2 LANDKAMMER, F. J.: Z. Physik 189, 113 (1966).

Thermal and Subthermal Neutron Scattering 435

Trying to find a field of investigation for very slow subthermal neutrons, we came to discuss the problems of sections 4 and 5.

The change in refraction by phonons, i.e. the optical potential, is viewed in section 4. It is crucially dependent on the high-frequency phonon spectrum and should be measurable if thermal expansion could be avoided.

position of counter

l

T neutron intensity

Fig. 1. Schemat ic representa t ion of interference exper iment 2

Finally in section 5 we discuss inelastic spin-flip scattering, as by a nucleus with magnetic moment in a ferromagnet, which will perhaps become a useful method of investigation in addition to nuclear magnetic resonance and the M6ssbauer effect.

A . C o h e r e n t S c a t t e r i n g

1. Long-wave Phonon Scattering The problem of thermal neutron scattering by a phonon system, with

phonon wavelengths comparable to the linear dimensions and with very large phonon occupation number - i.e. macroscopic density fluctua- tions - a theoretical treatment of phases or multiple scattering respectively being wanted, may be satisfactorily handled using a high energy scattering method by GLAUBER ~. Only small changes of momentum have to be regarded, i.e. a linear treatment is a good approximation for a three-dimensional one. Neglecting reflexion, the approximate solution is

( x , t ) = e i (k x - o~ t) q~ (x , t )

x<0: cp(x, 0 = l

( @ v / f ( Y, ) ) (1.1) x>0 : q)(x,t)=exp dy V t - XvY

hk v = (M, = neutron mass) M.

if the potential V(x, t)=0 for x<O.

29 Z. Physik, Bd. 210

436 W. MEHRINOER:

An expansion of q~ in powers of V will lead to a description of multiple scattering similar to a Born series.

For a nonmagnetic system the interaction of a neutron with the particles of the system F = ~ V(x -x i ) will imply V=wp(x, t) i.e. a

(0 coupling to the particle density p with a constant w=2nh2a/M~. The coherent scattering amplitude is a.

We shall neglect all effects connected with the structure of the system as made up of particles. Multiphonon excitation in a single scattering process must be mentioned here, which will, however, be shown to be of no importance. We use an expansion (the system of length L being fixed at x = 0 and x = L ) :

with

p(x, t) = no '~n cos qn x cos(On t - ~n)

n o =average density of particles,

qn =nn/L,

t2n=cq~; c=veloeity of sound.

(1.2)

A simple calculation then leads to

ff/(X, t)=e i(kx-~176176

for x>=L.

In case of 8, =0 the phase of the neutron wave is changed only by refraction:

K 2~z a n o 1 - -k-- = ----k-z--- (1.4)


with K=wa ve number in the system. By expanding (1.3) we get the sum of all multiple scattering terms, i.e. mulfiphonon excitations. If only one phonon is emitted or absorbed, the change of neutron momentum and energy is

lrkl O" I)

(1.5) I~col=G.

For n>> 1, 2, 3 . . . . . however, scattering is severely restricted to v=c, i.e. 16kl =q , - the "VAN HOW" condition - for only then is the amplitude 1In of the one-phonon wave function compensated:

l__n z- ,lim z-]-I (sin Inn(z-1)+0. (1.6)

To point this out more clearly we use a plane wave potential:

L L V(x, t)~e ~(~x-~t) for - ~ - < x < ~ -

I Vdy "~ff(--~x-aO ~/2ei'(a-~)dY �9 - - L / 2

(1.7)

We see that the amplitude in a coherent scattering event is calculated by summing up all phases of the phonon wave which a neutron with ve- locity v would see along its way (a small change of neutron energy being provided). As the " m o m e n t u m " of a phonon with small wave number q is not sharply defined (A Q~ 1/L for any q), though the energy is, a transfer of momentum ~-q O.e. v:#c) becomes possible. Fourier-transform- ing elqY:

1 L / 2

f(Q)= ~--~- _'I J L/ e'q" dy (1.8)

we calculate the integral in (1.7) for x ~ m :

: Q y - 7, -.-ff- y

S dy $ dQf(a) e = 2 n f . (1.9) - o o - o 0

We shall now discuss some applications.

2 9 *

438 W. MEHRINGER :

a) Interference

The LANDKAMMER 2 experiment can be simulated in a simple way if we let a difference in direction of x correspond to the (lateral) phase difference (see Fig. 1) of the two neutron rays in the plane of interference.

(~,) V///////////////A ~////////////A

{~I V///////////////A .,~ V//////////////A

Fig. 2. Simulation of LANDKAMMER'S interference experiment

For simplicity we assume 6.=,:!:0,5.>~ = 0 : O 1 =f2, qh =~o. Provid- ing

and wnoL6 27zhv <1, (1.10)

so that multiple scattering ]6a)l=nf2, n>2 , can be neglected, we get for the average over t

with

( I ~ i (x , t) + I]/ii (x -~- ~, ~)12)t

( 1 = \ ~ ! (r+s) r c o s - - + s C O S r s s ,

(1.11)

1 1 r = - - ; S = - -

C C - - - 1 - - + 1 v /)

The term cos k~ is modulated by cos (O/v)~, which follows from a superposition of the neutron intensities, i.e. interference patterns, belonging to energies ~o, r and co-f2, as two different energies (averaged) do not interfere. As in LANOKa~C~Cn~R'S experiments we distinguish cases ~ and ft.


~) qo x = ~o n . (See Fig. 3.)

V////////////A

Fig. 3. Neutron rays passing a coherent solid block

The interference pattern with F = - � 8 8 chosen is shown in Fig. 4.

4 cq

1

2~/k ,l~v/Q

Fig. 4. Interference pattern for the situation of Fig. 3. s is exaggerated in magnitude

Only lowest orders of interference (hatched in Fig. 4) are observed in experiment; modulation therefore cannot be detected.

t3) Phases ~0t and (PI~ are not correlated. We have to average over ((pt-q~ii) in (i.11) and the modulation term vanishes. A flattening in intensity should be observed when the support of the solid block, cut as in Fig. 2, is subject to random vibrations and intensities are measured over a long period of time. (See Fig. 5.)

(F=-1/4)

r _ _

2~/~

Fig. 5. Interference pattern for the situation of Fig. 2

For a total flattening however, as observed by LANDKAMMER 2, an amplitude 6krlt of the order

w no L 3krit k 2rchv ~ 1 ; (3krltmLano > 1 0 - 3 (1.12)

440 W. M~muNcrR:

would be required (of course a restriction to one-phonon scattering would be no longer possible in this case; likewise a renormalization of the particle current would be necessary). As the amplitude was estimated to have been only 5~10-s . . .10 -4, destruction of the interference pattern by means of the mechanism described is completely out of question: comparing cross sections for one- and two-phonon scattering we see that summing intensities of all orders of multiple scattering would be of no help.

da 0) 1 da (2> - [ w noL5 ~ 2 ~" 102"'10"" (1.13)

\ 2rchv-]

We notice that the same ratio for da(1)/dtr (2) is found by comparing the first und second Born approximation.

+ is The interaction as expressed by phonon creation operators aq simply calculated in a Debye model.

V(r)=wp(r)= 2rch2noa ( h____~l~ M. ~ elq" (aq+a+~) \2 I~q o9 a /

=s ~ e~ qr f q( aq + a +a) (1.14) q

with Nm

#q=--~-- (Nm=mass of the system).

We also can compare the two-phonon scattering cross section term of the second Born approximation with the corresponding term of single scattering. This is known for the particle picture of a crystal 3

d,.(2) ~ ( 2zc no a ,~a VBorn .7_(2) ~ ~ } ~102...106. (1.15) r162 Us ing seat

b) Inelastic Reflexion

Reflexion is neglected in (1.1) and only forward scattering is described. There is however a connection between the elastic and inelastic parts of a solution which can be studied by requiring the series expansion of I~=ei(kx-~ t) for - - L / 2 < x < L / 2 to be a solution of the time-dependent Schr6dinger equation.

About elastic reflexion we might remark that it should exist for a system in any state, i.e. a confinement to solids is not necessary, if only the energy of the neutron or particle scattered is chosen sharp enough

3 See for example: TURCttIN, V. F.: Slow neutrons (1963). Israel Program for Scientific Translations. Davey 1965. - - EaELSTArF, P. A.: Thermal neutron scattering. New York: Academic Press 1965.


(the question about wave-packets is more intricate of course) as in this case the momentum is transferred to the system as a whole.

For brevity we give just the result for the ratio of the cross sections for inelastic and elastic reflexion, where a potential energy of

V=w no(1 +6 e - i(qx+Qt))

for - L / 2 < x <L/2 has been assumed:

d~(r) ~ i n e l

d a~} -

=(W.oa~ 2 \ b y /

i ( a k - q ) - .

Practically, the denominator always has to be averaged over k.

(1.16)

2. Bremsphonons

A phonon may be emitted or absorbed in a process of second order when a neutron is scattered by an imperfect crystal. This is completely analogous to the process of bremsstrahlung, in which a photon is emitted by a charged particle being scattered in an electrostatic potential. The static density of particles in a crystal with dislocations may be described by a function

Po (r) = no + a n i f ) . (2.1)

As is well known, small angle scattering results from ~ n (r) in a process of first order (see Fig. 6).

VY='-~ - ~ e; ek' I an(r) e-*{k-k')'dar /~ k, k' (La)

I I I I ' k i kL ,L _

Vf Fig. 6. Small angle scattering via interaction V f

(2.2)

442 w. MEHRINGER:

The interaction of a charged particle with the field of radiation is replaced in our case by the interaction of a neutron with the phonon field:

p S + V =--~- Z fqc, ck,(aq+a+,) ~ e-'(k-u'-q)'d3r. k, k', q (L 3)

(2.3)

The emission (or absorption) of a "bremsphonon" (q) is then described by the two diagrams of Fig. 7:

I I I I

kf t k i t .t _ 1- _ _

Fig. 7. Diagrams contributing to bremsphonon emission

The corresponding matrix-elements for scattering in the Born approximation are

where (FI VP(EI-Ho+iq) -~ VS +Vf(EI-Ho+iq) -1 V 1' II>

k z Ho=Z O~q(a+ aq+�89 --~-~ct, ck=H~+K,

q

(2.4)

and I F) , I I ) are the final and initial states respectively (eigenstates of Ho).

From this we get the cross section for emission:

~3a - a2n~ ( k ~ - c l q l ) 08q 0 f2q a O k e ( 2 7 0 3 m c z

f 2 13 [ 1 1 �9 ]v~,_,,+ql Iq [ (k'-q)2-k~ +o~ +-(ks+q)2-k~'

2M~ 2M.

2 (2.5)

The phonon energy is designed by ~q in (2.5) and f2q, f2,~ are solid angles of the phonon and neutron emitted. A characteristic difference can be seen: for the case of bremsstrahlung we have ~3 aN I ql-1 for q ~ 0 as there is no factorf~ ,,~ I q l and

, [nq+l~ ~

a s c o n t r a s t e d to o u r c a s e (~3 O" ---)" [ q ].


This shows there is no chance for using such a process to explain LANDKAMM~R'S interference experiment. Bremsphonons might however be studied by comparing coherent scattering by crystals with different density of dislocations. "Peaks" belonging to a certain phonon energy should be broadened for a high density of dislocations, as follows from the possible transfer of momentum described by nonvanishing V[~_k~+q.

3. Elastic Scattering and Diffraction at a Plane Slit

a) Integral Equation and General Method of Solution

In connection with interference LANDKAMMER 2 investigated diffraction. For checking and improving his results, since the KmCHHOFF'S theory used by him apparently is not a good approximation for treating diffraction of monochromatic neutrons by a " long transparent" medium, we shall tackle the problem by trying to solve the scattering problem directly. We want to describe scattering of a plane wave e i~ r (thermal or subthermal energy) by a target consisting of N nuclei fixed at positions rj and with scattering lengths a t. The target is well approximated by a solid at low temperature. Inserting

27c 1i 2 N V(r) = M , ~' at 6 ( r - r j) (3.1)

j = l

into the integral equation for elastic scattering, we get

e i k ] r j - r ] O (r) = e i k ~ + Z ( - a j) ~ (r j) (3.2)

j [rj--r[

where j = i is omitted in calculating tfl(ri). This is done automatically when ~ is replaced by S" Simple iteration gives ~ as a sum of multiply scattered spherical waves. This will prove to be a good solution if the target length is small (<7~10 -3 cm for thermal neutrons) so that dispersion can be neglected.

For a target of length L, however, with wnoL/hv~>2~ we have to think of another method - we shall require small k in order to avoid Bragg scattering - which is suggested by the small reflectivity as compared to the large value of ~ obtained by summing up slowly changing phases in forward scattering. [This is simply demonstrated for a plane wave e ikx scattered by a slab parallel to the (y, z)-plane using e i~x for

(r j) in (3.2) with K = k - w no/h v.]

444 W. MEttRINGER:

Consider the target "sl iced" perpendicular to k (Fig. 8).

3 4

Fig. 8. "Slicing" of a target

Beginning from the left ( r is calculated for adjoining slices (v =2, 3, 4, ...) using forward scattering only, i.e. if ~kv is known for slices 1, 2 and 3 then ~4 can be calculated by (3.2).

It should be noted that the thickness of slices has to be chosen only smaller than the "dispersion length" 7=2~hv/wno in practice and not "infinitely" small, i.e. ,~2n/k. A better solution - with reflexion included - may be constructed then by reinserting the "forward scattering solution" into (3.2). Repeating this procedure the "exact" solution finally should be obtained as being indicated by consistency, i.e. by finding from calculation the same function ~ as inserted.

b) Application to Diffraction at a Plane Slit

This situation is shown in Fig. 9.

w -----~ J~ - x

position of "screen" (counting apparatus)

Fig. 9. Diffraction at a plane slit with coordinate axes used for calculation

In the experiment of the second reference 2:

2d~(1. . .10) 10 -4 cm

L~(0.1. . .1) cm

R = 5 m .


It will be indicated below that the approximation ~ = e ~K~ in the solid will suffice for widths 2 d of the slit used in experiment.

With (3.2) we get ~ (L) by adding and subtracting the waves which would be scattered if nuclei were in the slit:

8iklrs-rj] e iKxj ~ ( r s ) = e i k X ' e i ( n - k ) 5 - - ( - a)j.Z(~,iO I r~- r j ~ " (3.3)

Naturally it will be difficult to tell how good this approximation is in the dependence on target length L and position r~ as well as on the width d.

We define ~s(p) as the sum of scattered waves of fictitious nuclei in the slit (plus nuclei of the solid blocks) for a slab of "dispersion length" 7 = 2 n / k - K at positions indicated by dots in Fig. 10. A rough calculation gives ~k~/~p <~d/nT.

For L ( > l ) not " too large" our approximation should be " g o o d " for widths d~7 (7~ 10 -2 ... 10 -3 cm for thermal neutrons). Comparing

2d �9 nuclei con t r i bu t i ng

to ~'s to%

F Fig. 10. Estimate of ~s/F/p

our results for 2d~(1 ... 5) 10 -4 cm this to be correct.

We have k Ir~-~jl z d

e ,. ano f ... "d~l ~ d ~ e i K C e ~k a ~ e irx'~ N - - j d e j Io-r ' l I r s - r j l R o -d -

with coordinates p(r ~/, 0 for nuclei in the slit. Expanding we get

l p - r s [ ~ R - r (y _~/)2 2R

which leads to e i (K-k)L 1 L a n o e i k R l ~ l ( r s ) = e i k R e i(K-k) - t - ~ - i ( K - k )

with diffracted intensities of 2 shows

(3.4)

~2

- - + 2 R ( 3 . 5 )

~ k " J(ys) 1 f i__~_ l (3.6)

A factor e -pr was added in (3.4)because of the finite extension of the plane wave. The constant fl can be neglected for LANDKAMMER'S experiment. It is useful however for fixing the sign of the root. Requiring

I ik \~

4 4 6 W . MEHRINGER :

(as seen for large fl) we have 1 - i

( - i )~-= ]/~ . ( 3 . 7 )

We still have to calculate

d ik(ys-II) 2 V~RR~ j (ys) = Se 2R dt l= (A+iB). (3.8)

- d

If lysl >>d and kd2/2R< 1 (as for our case) we get

, . ~ ] / S R / k y s 2 ~ s i n ( ~ _ f l _ ) A ~ V k y 2 cos \ ~ - ] (3.9)

~Vky2s sin sin ( ~ - ~ - )

Eq. (3.9) shows that the width d of the slit has almost no influence on the oscillatory behaviour ~sin(kya,/2R), fixing only the amplitude. A useful representation is given by Fm~SNEL'S integrals

k k 2--R- (Ys +'/)2 2-~" (Y* - d)2

COS U COS U

A= ol /---7- du (-r-) ol 1/u k k

2 7 (r~ + a)2 . 2 -~ (Y~- a)z . s i n u s i n u

B= oI au (-r- ) oI V

F d u

(3.10)

- - d u

for Ys (<>) d.

For a comparison with experiment (neglecting absorption by Bragg scattering) we must take into consideration that eikXfor the incoming neutron wave is a good approximation only for the limited region of space near the target. Since it is nearly a spherical wave and the target is at equal distance from entrance slit and screen, a factor �89 has to be multiplied by the plane-wave part of ~ (rs).

ano (1_ e-i(K-1o L)(1 I ~b(r~)12 = + ( K - k ) k (3.11)

As ( K - k ) k = -2roan o we have {...} =0.2. This shows that all material properties (a, no) are contained in the

phase (K--k)L only, being a function of L for given energy or of energy for constant L. This effect then should be detectable for sharp neutron energy, i.e. with

A(K-k)L<2rc or (3.12)

Ak AE k 10-3...10 -1 k - 2 ~ < ~ L[cm]

Thermal and Subthermal Neutro~l Scattering 447

Ivl 2

I t ~-( (K-k)L ,

r

I t l,r , I I I I I I /T '

chosen values: 2d=5l t

k=l.5.108cr R=5m

. . . . I

-50

= - i ( K - k ) g - 1 ,~

I

10 30 50

Fig. 11. Dependence of diffraction pat tern on e - i (K-k)L

Intensities which could have been measured in LANDKAMMER'S diffraction experiments for a width 2 d = 5 . 10 - 4 c m are shown in Fig. 11 for phases e-*{K-k)L=l, i, --1 and --i.

Ir [ r 12 = (0 .5 - 0.4 B) 2 + (0 .4 A) 2

1r (3.13)

I ~(-~ A)z +(O.4 B) 2.

For a broad energy spectrum however, as used in z, ( l~(r , )]2)k, the intensity averaged over energy, must be calculated. Putting rapidly

448 W. MEI-IRINGER:

oscillating terms He- i ( r -k )L equal to zero, we get

< I~(r,)I=>~ - <0.25 - 0.2 (A + B) + (0.2) 2 [(A + B) 2 + 2 (A 2 + B 2) + (A - B) 2] >k. (3.14)

Since A and B are functions which vary slowly with k to replace <A (y~, k)> k by A (y~, <k>) will be a reasonable approximation.

For widths 2 d = 1 #, 5# (and 20#) results are shown in Fig. 12.

O.6O

0.50

0.4o

"~--~ 0.30

Q20

0.10

/ vatues chosen: [

/

I~i2d=20ll .2d=11 a

0 I I r I I I

0 20 40 60 Y,[~]

Fig. 12. Intensity averaged over spectrum as to be compared with experimental results by LANDKAMMER

In the experiment 2 the small maximum at y~=0 was not detected. This should partly be due to the lateral superposition of intensities when the slit is slightly oblique (see appendix).

For 2 d = 1 p the change in intensity from the ease d = 0 is only about 1/6 at most. Therefore diffraction will not explain why the pattern vanishes in an interference experiment.


4. Optical Potential

How neutron refraction is influenced by phonons is a problem which our Debye model handles smoothly. We have to solve the scattering problem (see (2.4) and (1.14)).

[Hs+K,(r)+ V(r)] kg=-Evk 7'. (4.1)

The system and the neutron are in states Iv) and e ~k" (energies Wv and ek, W,+e~ =E~k) before scattering. We take as usual 4

~ = Z [P>f ; u(') (4.2) It

leading to (K,--ek)f~ k= --<v] V(r) [ 7J> --- --U**f~k(r) (4.3)

for the "elastic par t" f~ 4.

In the second equation of (4.3) the optical potential was defined. If, as in our case, f~k(r) represents the dominating part of 7 ~, then a

distorted-wave treatment of the problem will be good:

n ~ '= E(Ho+ VvO+(V(r ) - V~k)] ~'=Ev ~ (4.4)

ku= [ v>f~*-~ 1 (V-Urn) ~P. E~k-(Ho + Uvk) + i ~l

U~k then can be calculated iteratively. Up to second order we have

u f k(r) (4.5) [ 1 (V(r) - V,k)] [v> f:U(r). ~<vl V(r) t l + Evk--(Ho + Uvk)+ i tl

Replacing U~k and f : k in the second term on the right by the result of the first term: U~ ~ =s and f(o)~k= e ~ K, with

h 2 K 2

2M, ~-s=~k (4.6)

we obtain

Uo,> k~ s +'S U<~> k=s +'S rT(r) ,~<~> kT • U~ ~)

<no> + 1 = s + s 2 ~ , f - o f o ti2 (4.7)

Q . o _ho)a_ - (Q2_2QK)+iq 2M.

+ t? <%> ]" li c o o - - ~ - ~ (Q2 + 2 Q K) + i tl ]

4 See for example: Wu, T.Y., and U. OnMtrRA: Quantum theory of scattering. London: Prentice Hall 1962.

450 W. MEHRINGER :

A diagram of the second term of (4.7) is shown in Fig. 13.

Fig. 13. Self energy of a neutron in a phonon field

The imaginary part of U<~>k describes damping of a plane wave e IK" by one-phonon scattering.

We shall only discuss the real part as contributing to refraction. Both terms ~ (no) (referring to primary emission or absorption) just cancel for phonons of small Q when Q2 < QK and nQ >> 1, as for the situation of section 1).

For the case of very slow subthermal neutrons (k~106), however, scattering from a system in thermal equilibrium at high temperature gives a "measurable" effect: the problem is, of course, how to avoid or compensate for the change of U by thermal expansion (A U ~ 10 -2 s for AT,~ 103 ~ The essential contribution comes from phonons of large Q. Neglecting QK we get U independent of K. For a rough estimate ~oQ = c Q is used. We need not take care of the singularity in the absorption term when calculating the principle value integral if c is chosen large enough. This shows however that in an actual calculation great care has to be taken since the sign of 6 U(<~>) k depends in a sensitive way on the high- frequency energy spectrum. As to using very slow neutrons, the essential point lies in

M, 6K = - - ~ r g - 6 U. (4.8)

Provided c is large, i.e.

hqo 2M n c

- - = ~ < 1 (qD = Debye wavenumber) (4.9)

we get for the part of fi U<~>k depending on temperature (with (nQ) kT/hcQ)

s 2 1+7 _ 2 7 n o m c ~2 h3

(4.10)

= S . n2 ha -b "4"'" �9 nomc

~5 U (T) is seento be very small. For T g 103 ~ u ( r ) g ( 1 0 -4 ... 10-8)s.


For smallest K (hZK2/2M,,~s) we get 3 K g 10 -z ... 10 z "measurable". The "emiss ion" term contribution -,~ 1 of (4.7) is independent of temperature:

3U(~)_ - s M~qo 7 - 2 + l n ( l + 7 ) 2rcZ h2no m

(4.11) sM~2c [7 3 ~4 ] = - s I T - q - + . . . . j 0-Ss [ 1

n o i n

so that 13 V (1) - M n c 2

(~ U (T) ~ k T (4.12)

B. Incoherent Scattering

5. Inelastic Nuclear Spin-flip Scattering

The energy of nuclei with spin and magnetic moment is split in strong magnetic (as in a ferromagnet) or electric fields. The energy corresponding to unit change of the nuclear magnetic quantum number is taken up by a neutron in a spin-flip scattering process.

Clearly there are two possibilities for measuring those rather small energy differences ( ~ 10 -6 eV in a ferromagnet). First by using neutrons of a certain thermal energy and measuring energies before and after scattering by making use of Bragg reflexion and Doppler effect - this is being done at the moment. Another possibility which we shall consider in more detail consists of using extremely subthermal neutrons with an energy of about the difference of spin energies and measuring the reaction threshold.

It need not be said that, though there are nearly no complications considered in our description of the scattering process, the method may prove to be of valuable help in addition to the methods of recoilless emission and nuclear magnetic resonance.

Using the Born approximation as usual, we have to keep in mind that large corrections due to the absorption cross section ~ v -1 at very low subthermal energies may become necessary: experiments might fail when the scattered neutron flux becomes too small. To make the whole thing simple we neglect correlations between (uncorrelated) states IL M)=II1M1) rI2M2) ... [INMN) of the nuclei and states In) of the " la t t ice" for describing states I v ) of the total system.

[v)=ll, M) In). (5.1)

30 Z. Physik, Bd. 210

4 5 2 W . MEHRINGER:

For a transfer r of momentum and co of energy to the system the cross section for spin-flip 5 is then described by

a2cr(~) k(~) Ij=~ O~2O--------~ = k ~oP(Vo) (n le i" '~ lno>

II, M ) ~ 2 (5.2) �9 (I , MI (+1 2 b j s l j I~> 6(Ev-Evo-co)

where I+ >, I - ) are spin states of the neutron describing polarization in the direction of the crystal field: sz I - ) = +�89 Coherent terms vanish; after summing over final states we have the result:

~2a(- ~) k(:~) N 0aa-----/--- k Zp(vo)Z I<nl e~K'Jlno>[ 2

n, vO j = 1

. b2(ij o - o ++_Mj)( i i+Uj+l)6(E_E.o+A(f f~) co ) (5.3) with

A(~:)--w o ~ - - ~ ( ~ s T- 1 ) - - E ( ~ # )

as the energy difference between spin states of nucleus (j). For A ~ 1 0 -6 eV thermal weighting gives a constant value for all

p(M ~ n r >10 -2 ~ To simplify further we assume our system to be a Bravais lattice with

isotopes s (fraction p~, energy levels being split up by As as for a magnetic field) distributed statistically. Then

02 a(~)O0 a~ = ~ p~ b2 ~2 is( is+ 1) ~ ) ( 1 r co) $

and k(:~)

Zp( o)Z I< l e'~'~lno>[Z6(E.-E.o+a, -co) (5.4) 71,710 j = l

earl be calculated by just replacing co by co-T-A~ in the incoherent structure function a Silo(r, co) (harmonic approximation). With k(:~)/k= (1 -(co/Eo)) ~ (Eo is the neutron energy before scattering) we get in lowest order when ] n) = ] no) (omitting index s)

~(o=~)=e-Z WN ( 1 - co ~ ~ 6 (co-T-A) (5.5) \ ~o!

so that

d~2 ~ IT- . (5.6)

Taking A > 0 (which generally is the ease) we could hope to find a characteristic step for da(-)/dO at threshold energy Eo =A (Fig. 14).

s HALPERN, O., and M. H. JOHNSON: Phys. Rev. 55, 898 (1939),


Two questions have to be considered for a real experiment - in supplement to our Born-approximation result. They arise from the existence of two effective magnetic fields in a ferromagnet differing in direction, magnitude and local extension. As the magnetic moment of a neutron is correlated to its spin by g, <0, it has to pass a potential

(I+&/Eo) 1/2

1

(1-a/Eo) 1/2 / ~

A ~ EO Fig. 14. Energy dependence of cross sections

Fig. 15. Change of k by scattering in a ferromagnet

barrier due to contact interaction with polarized Is-electrons after having changed its state of spin from I - ) to I + ) . This goes via tun- nelling as the linear extension L of the barrier is small and k(-)L~ 1 for k(-)2/2M, is of the order of A; to detect the deviations from da(-)/df2~ (1-A/Eo) ~ for Eo immediately above the reaction threshold may be a difficult experimental task.

Apart from this small effect, the average potential - as given by the coherent forward scattering amplitude of both nuclear and magnetic scattering - which is seen by a neutron after a spin flip process will be relevant to the part of da(-)/df2 which can be observed experimentally (Fig. 15, 16, 17 and 18).

Fig. 15 shows the change of k during a single spin-flip process. We indicate the scattering nucleus by a dot, the direction and extension of the local magnetic field (1 s) by BK and a circle around the dot. A neutron has wave number k in vacuum and k' in the crystal before, k (-)~ and k (-) after scattering. B indicates the effective (macroscopic) magnetic field in the crystal contributing to refraction. The reaction threshold

30*

454 w . MEHmNGER :

then is fixed by k~ -)1 =0 , i.e. refract ion index zero, leading to

k} -)2 _ 2rcano #,,B (5.7) 2 M,, M ,

where the magnet ic m o m e n t of the neu t ron is designed by p, > 0. The neut ron therefore mus t have an energy of at least

k~-)z Eos=A - I - - - - (5.8) 2M.

so that it can lose energy A in a spin-flip process. Eq. (5.7) shows that k~ -)2 can be either positive or negative, i.e. the threshold of the process m a y be accessible outside or only inside the crystal. In the first case

do.(-) ds

!

A k''~$ (-) 2 2M n

Fig. 16. Cross section for k(s-F> 0

~ E 0

Eos i

Fig. 17. Reaction at threshold for kCs-)Z> 0. The solid line represents the effective potential seen by a neutron before and after spin-flip. The total neutron energy is

represented by a dashed line

however, there is no possibility of measur ing A directly. I t m a y be calculated f rom E o ~ and k~-)2 or be found graphically by extrapolat ing da(-)/df2 as indicated by dashes in Fig. 16. A measur ing of A is possible for k~-)2 < 0, however.

Finally, we state the result for ~(-v-)(r, o9) if one p h o n o n is emit ted or absorbed.

~ - ~ ) ( r , o ) = e - z w [ 1 _ o ~ ~ \ Eo] (5.9)

~q~ [(n~z+ l)5(co'~A_~q~)+n~zfi(ogT_A+ogqz)]. 4, .~ COS 2


By ~ (co-A + co~ z) we have a description of how a p h o n o n makes spin flip possible for E o < A by giving par t of its energy to the nucleus and the rest to the neutron. The " s t e p " at Eo=A of dG(-)/df2 will no t be " s m e a r e d " since after summat ion we get d~r,/df2~ 1/V-E o for all kinds of incoherent or coherent p h o n o n and m a g n o n processes.

Eo s I - - - - - - D - - - - - - - - 1

EO=A

t t

E

Fig. 18b. Reaction above threshold allowing measurement of A

Fig. 18a. Reaction at threshold for k(s-) 2 <0

A p p e n d i x

Geometric Optics in LANDKAMMER'S 2 Interference Experiments Our results in sections 1, 2 and 3 failed to give an explanation of LANDKAMMER'S

interference experiments. For those of his experiments, however, where the solid blocks were rotated, geo-

metry gives the clue. He intented to confirm the thesis that neutrons which had passed through two separated solids would show no interference.

Fig. 19. Geometry of rotated blocks

Maximum intensity results from interference of rays (0). For a length L ~ 1 cm and rotation angle d (0~0,5 ~ when the slit is situated between rays (0) as indicated in Fig. 19, i.e. with one ray (1) crossing the slit, no interference pattern could be observed.

For a correct calculation of the difference in phase between (1) and (1') it would be necessary to project s on (1').

Assuming 2d~ 5 �9 10 -4 cm we have

s ~ 2 d ~ 1 0 - l c m . 8 9

A difference in phase of about 20 wavelengths (/L~ 52~) results. If neutrons of definite energy had been used, there might have been some hope for finding an interference

456 W. MEHRINGER: Thermal and Subthermal Neutron Scattering

pattern ignoring other difficulties such as the high reflectivity at an angle near total re flexion.

For a thermal energy spectrum, however, as used by LANDKAMMER, the relation A k ~ 1/k for the change of k in matter explains why we have to expect an averaged intensity. After further rotation the pattern of interference can be observed when both rays cross the slit.

The failure of those experiments in which the silt was laterally displaced might equally be explained by a small rotation angle due to insufficient adjustment. This seems to have been shown by a partly improved version of this experiment (Dr. KAL~JS, TH Munich), where a slight change of adjustment would induce a radical change of the distorted pattern.

Acknowledgements. For many helpful discussions the author is indebted to Profes- sor Dr. BReNIC, Professor Dr. WILD, Dr. OBERMAm and to Dr. KALES. For special help he would like to thank Mrs. Dr. ROOERSON.

Dr. W. MERRIN~R Bundesforschungsanstalt f~r Lebensmittelfrischhaltung 7500 Karlsruhe, Engesserstr. 20

thermal and subthermal neutron scattering

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