semi classic a electron optics

7/31/2019 Semi Classic A Electron Optics

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Semiclassical approximation in electron optics

M. Lenc

Institute of Theoretical Physics and Astrophysics, Masaryk University Brno, Czech Republic

Abstract

The paper gives a brief review of the standard electron optical classical and wave pictures

followed by a detailed analysis of the semiclassical approximation for relativistically

modified Pauli equation. General formulae for the semiclasssical approximation are illustrated

on the simple case of paraxial approximation in rotationally symmetric fields.

Keywords: Electron optics, semiclassical approximation.

PACS: 41.85.-p, 03.65.Sq.

1.Introduction

Standard equations used in electron optics are the result of the sequence of several reasonable

simplification steps. We shall not have to deal with breaking of the electroweak symmetry;

usually considered energies are very low and quantum electrodynamics should be quite

satisfactory theory. But even this is not necessary, as the field intensities under consideration

are very weak. So the fundamental physics begins for us with Dirac equation for electrons in a

given external electromagnetic field. The first simplification step is to consider the electrons

only, i.e. instead of Dirac equation for 4d space time bispinor we have to solve a relativistic

Institute of Theoretical Physics and Astrophysics, Faculty of Sciences, Masaryk University,

Kotl sk 2, CZ-61137 Brno, Czech Republic

e-mail: [email protected]


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form of Pauli equation for 3d space spinor. In the second simplification step the semiclassical

approximation is used, mostly in its zero and first orders (with respect to Planck constant ).

Many interesting results were obtained in that respect in electron optics (the excellent review

is in [1]) and they were generally ignored in quantum mechanical textbooks. In this paper we

will present in detail the results of the semiclassical approximation in the description of the

electron motion in time independent electric and magnetic fields.

2. Classical description

We will start with the usual Lagrange function ([2], 16)

( ) ( ) ( )

12 2

2

2, 1 ,

vL r v m c e v A r e r

c= + (1)

which does not explicitly depend on time, thus the energy is conserved. Here m is the mass

and e the charge of the electron. For the generalized momentum p and Hamilton function we

have

( ) ( )

( ) ( )( ) ( )

12 2

2

1 22

2 2 4 2

, , ,

1

, .

L m vp e A r H r p p v L

v v

c

H r p p e A r c m c e r m c

= = + =

= + + =

(2)

The choice of additive constant for conserved energy is typical for charged particle optics: it

is assumed, that for the electron with the negative charge the electrostatic potential is positive

and it is equal to zero at points where the particle velocity is equal to zero. A specific quantity

called the relativistically corrected potential is defined in particle optics [3]

( ) ( ) ( )* 21 .2e

r r rm c

= (3)

From eq. (2) we have for relativistic Lorentz factor the expression


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( )1 2 2

2

2

11 .

1

e r

m cv

c

= =

(4)

Denoting

( ) ( )1 2*

2p r m e r= (5)

and removing the square root in (2), we can define the characteristic function ( ),p r

( ) ( ) ( )2

21, 0 .2

p r p e A r p rm c

= = (6)

With this choice of the characteristic function the Hamilton equations are

d d, ,

d d

r p

s p s r

= =

(7)

where, by comparison of the expression for p e A in (2) and (7), the parameter ds is

( ) ( )( )1

2 2 2 dd d d .

c ts c t r

= = (8)

From eq. (7) the trajectory equation is obtained

2

2

d d.

d d

r rm c e E B

s s= + (9)

Substitution

p S= (10)

in eq. (6) leads to the Hamilton Jacobi equation

( ) ( ) ( )2

20 .S r e A r p r = (11)

The solution of (11)

( )1 2, ,S S r c c= (12)


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depends on two arbitrary constants (the third constant was used as the energy of the electron),

which we will use for the solution of the continuity equation in the semiclassical

approximation.

For the optical system with the straight (z) axis it is advantageous to separate from (6) the

momentum component zp and consider it as a Hamilton function

( ) ( )( )1

2 222 .z x x y y z

p p p e A p e A e A = (13)

From Hamilton equations

( ) ( ) ( ) ( )dd d, , ,d d d

yz z z zx

x y

d pp p p ppx yz p z x z p d z y

= = = =

(14)

trajectory equations are obtained

( )( )

( )

( )

12 2 2

12 2 2

12 2 2

12 2 2

d1 ,

d 1

d1 .

d

1

x z y

y z x

p x me x y E y B B

z px y

p y me x y E x B B

z p

x y

= + + +

+ +

= + + +

+ +

(15)

3. Wave optical description

Standard coordinate representation

p pi

= (16)

allows us to obtain from (6) time independent Schrdinger equation

( ) ( ) ( ) ( )2

2= .e A r r p r r

i (17)

Finally, we will use a unit matrix 0 and the vector of Pauli matrices

0

1 0 0 1 0 1 0 ,

0 1 1 0 0 0 1

x x y y z z x y z

ie e e e e e

i

= = + + = + +

(18)


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to write instead of (17)

( ) ( ) ( )( ) ( )2

2

0 = ,e A r r p r r

i (19)

which adds the spin interaction and leads directly to the Pauli equation

( ) ( ) ( ) ( ) ( ) ( )2

2

0 0 .e A r r e B r r p r r

i = (20)

The wave function ( )r is a two-component spinor

( )( )

( )

1

2

.r

r

r

= (21)

The current density is given by the expression ([4], 115)

( ) ( ) ,2 2

ej A

m i m m + + + += (22)

with Hermitian conjugate spinor

( ) ( ) ( )( )* *1 2 .r r r +

= (23)

Using the names Schrdinger or Pauli for the equations, we have in mind the standard non -

relativistic form of the equations with a relativistic relation between the energy and

momentum.

4. Semiclassical approximation

The basic simplification comes from the semiclassical approximation. We shall write

(21) in the form

( )( )

( )( ) ( )

1

2

exp .f r i

r f r S r f r

= (24)


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Functions ( )f r and ( )S r in (24) are real, so that the Hermitian conjugate spinor to (24) is

( ) ( ) ( )( ) ( ) ( )* *1 2 exp .i

r f r f r f r S r + = (25)

Spinor components are normalized, i.e.

( ) ( )2 2

1 2 1 .f r f r+ = (26)

Substitution of (24) into (20) gives the equations

( ) ( ) ( )

( ) ( ) ( )

2 2 2

1 2

2 2 2

1 2

2 0 ,

2 0 ,

z x y

x y z

p i i e B f f e B i B f f

e B i B f f p i i e B f f

+ =

+ + + + =

(27)

where we have shortened the notation by omitting the arguments. The mechanical momentum

is denoted as

.S e A = (28)

Semiclassical approximation consists in neglecting terms with2

, i.e.

2

.f pf

(29)

Let us suppose that the condition (29) is satisfied. Then we have for the zero order (in )

2 2 0p = (30)

(the Hamilton Jacobi equation). It is the equation (30), for the first time obtained (when

seeking the semiclassical solution of the Dirac equation) by Pauli [5], which is sometimes

questioned (discussion e.g. in [6], [7]). Indeed, for the explanation of particle trajectories in

the Stern Gerlach experiment one would like to have the dependence of the classical (zero

order approximation) trajectory on the spin orientation. It is possible to modify the

approximative procedure in that way, nevertheless we will use the standard approach. Thus

for the first order (in ) we put

( )2 0f = (31)


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(the continuity equation) and then eq. (27) reduces to

( )

( )

1

2

2 20 .

2 2

z x y

x y z

i ie B e B i B

f

fi ie B i B e B

=

+ +

(32)

To solve the Hamilton Jacobi equation (30), any of the standard classical methods can be

used. The probability density 2f can be calculated from (31) as a time independent analog

to the Van Vlecks determinant. For the optical system with the straight (z) axis the natural

form of 2f reads ([8], 2.3)

( )( )

2 2

1 21 22

1 2 1 2 22 2 2 2

1 2

,

, , , , .

x y

S SS SF

x c x cc cf x y z c c

S Spy c y c

=

(33)

F is an arbitrary function of two variables. The proof is simple: after substitution of (33) into

(31) we use two identities, obtained by differentiation of (30) with respect to 1c and 2c .

Combining eq. (32) and its complex conjugate and substituting for from eq. (7) allows us

to write a new equation

d,

d

n en B

s m c= (34)

where n is a unit vector in the spin direction

( ) ( ) ( )* * * * * *

1 2 1 2 1 2 1 2 1 1 2 2

.x y zn f f f f e i f f f f e f f f f e

+

+= = + + + (35)

Comparing (34) with (9) we observe the well known result, that (in the semiclassical

approximation) during the motion in the magnetic field the spin orientation remains parallel to

the velocity (e.g. [9], 41). With the notation


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1

2

cos exp2 2

sin exp2 2

if

fi

=+

(36)

we have from eqs. (35) and (26)

sin cos , sin sin , cos .x y zn n n = = = (37)

The current density (22) becomes

( )2

2 2cos .

2

fj S e A f f n

m m = + (38)

The solution of (34) could help us to find the spinor (36). The other way to solve (32) is to use

the substitution

( )

( )( ) ( ) ( )( )

0

11

22

exp, , , d .

exp 2

s

z

s

i gf eB x y z

i gf m c

= =

(39)

Denoting

( ) ( )exp 2 ,2 x yi e

B i B im c =

(40)

one obtains instead of eq. (32) either

*1 22 1

d d0 , 0

d d

g gg g

s s+ = = (41)

or an equivalent system of the second order equations

* *1 21 22 2

d d0 , 0 .d d

d g d gg gs d s s d s

+ = + =

(42)

The change of the parameterization from an interval s to the coordinate z means the

substitution

( )1

2 2 2 2

d d .

x y

m cs z

p

(43)


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Coming back to eq. (33) let us mention that choosing 1 0c x= and 2 0c y= and a suitable form

of the function F

( )

( )( )

0

12

0

1

2z

p r

F i r = (44)

leads to the expression

( )( )

( ) ( )( )

1 22 2

0 00

0 1 2 2 2

0

0 0

, .2

z z

S S

x x x yp rf r r i

S Sr r

y x y y

=

(45)

The z components of the mechanical momentum are given by

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 22 2

2

1 22 2

2

0 0 0 0

0 0

,

.

z x y

z x y

S Sr p r e A r e A r

x y

S Sr p r e A r e A r

x y

=

= + +

(46)

Glaser and Schiske [10] obtained the amplitude f in the form (45) for the first time. It is not

difficult to prove, that

( )( )

0 0

0

0

0

1lim , lim .

2r r r r

p rf r r

r r =

(47)

Because (47) holds, the wave function (24) with the amplitude (45) is a good approximation

for the Green function of eq. (17). Knowing the Green function, we can use the semiclassical

approximation even in the situations, where the diffraction takes place, simply by using in the

diffraction integral a semiclassical wave function and the semiclassical Green function,

respectively. Nice example for the calculation of the interference pattern produced by an

electrostatic biprism was given in [11].


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5. Paraxial approximation for rotationally symmetric fields

Rotationally symmetric fields are characterized by the axial electrostatic potential distribution

( )z and the magnetic flux density distribution ( )B z , respectively. The paraxial solution of

eq. (14) is given in terms of the coordinates of the initial point ( )0 0 0 0w z w x i y= = + in the

plane 0z = and the final point ( )w z w x i y= = + in the plane z =

( )( ){ }

( )( ) ( ){ } ( ) ( ) ( ) ( ) 0

expexp .

a a b b a

a

iw r i z w r z r r z r w

r z

= + (48)

Here ( ) ( )0 00 , 1a ar z r z= = and ( ) ( )0 01 , 0b br z r z= = are the two independent solutions of the

paraxial equation

( )( ) ( )

( )

( )( )( )

( )( )( )

1 12 22 22 2 2 2 222

2

ddd0

d d 4 2 d

p z m c p z m cr z e B zp z r z

z z p z p z z

+ ++ + = (49)

and ( )z is the rotation angle

( )( )

( )0

d .2

z

z

Bez

p

= (50)

Our equation (49) describing paraxial trajectory has an unusual look, nevertheless the

substitution ( )1

* 22p m e= leads directly to the well-known form (e.g. [3], 15.1,

( )1

22e m= )

2 2

* *0 .

2 4

Br r r

+ + + = (51)

Thus in the paraxial approximation the solution of the Hamilton Jacobi equation (30) (point

- characteristic) is


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( ) ( ) ( )( )

( )( ) ( )

( )

( )( )

( )( )

( )( ) ( ) ( )

0

2 2 2 2

0 0 0 0

0 0 0 0 0

1 1, =

2 2

1cos sin

z

a b

a az

a

r z r zS r r p d p z x y p z x y

r z r z

p z z x x y y z y x x yr z

+ + + +

+ +

(52)

and the solution of the continuity equation (31) (with the same choice of the arbitrary function

Fas in (45) ) is

( )( )

( )

( )

( )

12

0 0

0, .2 a

p z p zf r r i

r z p z= (53)

The condition for the applicability of the semiclassical approximation (29) takes the form

( )2

12 2

2 22

1 22

dd

2d d2 .

2 2

a

a

r pp

e B pz z

p pr p

++ + (54)

It is always the case in electron optics for the field dependent terms, as we are comparing the

axial density of the optical power of the magnetic and electrostatic lens on the left hand side

with the square of the wave number on the right hand side. It seems, that the condition is not

satisfied in the vicinity of the plane(s)i

z z= , where ( ) 0a ir z = . This is caused by the above-

mentioned fact that our solution is rather the Green function than a regular solution. Denoting

by M and AM the magnification and the angular magnification, we have

( ) ( ) ( )

( ) ( ) ( )( )

( )0

0

, ,

, d .2

i

a i A i b i

z

i i A i i

z

r z z M z z r z z M

Bep z z p M M p z z z

p

= = (55)

Then it holds (i

z z= )


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1 2 22 22

0 00

12

0 0

1lim exp

2 2

,

i iA r r

A r r

M p M pM x yi x y

i M M M

M x yx y

M M M

+ =

(56)

with cos sin , cos sinr i i r i ix x y y y x = + = . Another important choice of constants in the

solution of the paraxial form of the Hamilton Jacobi equation (30) is 1 0xc p= and 2 0yc p=

( ) ( )0 0 0 0 0 0 0 0 0 0, , ; , , ; ,x y x yW r p p z S r x y z p x p y= + + (57)

with ( )0 0 0 0 0, , ;x yx x r p p z= and ( )0 0 0 0 0, , ;x yy y r p p z= expressed from the equations

0 0

0 , 0 .W W

x y

= =

(58)

The function W is called mixed - characteristic. The semicolon in the argument of the

functions in eqs. (57) and (58) underlines that 0z should be understood as a parameter rather

than a variable. It holds for all expressions parameterized by the axial coordinate z , but we

use the more precise notation only in this part of the paper. A simple calculation gives

( ) ( ) ( )( )

( )( )

( )

( )

( )( )

( )( )( ) ( )( )

0

2 2 2 2

0 0 0 0 0

0

0 0 0 0

1 1, , ; =

2 2

1cos sin

z

b a

x y x y

b bz

x y x y

b

r z r zW r p p z p d p z x y p p

r z p z r z

z x p y p z y p x pr z

+ + +

+ + +

(59)

for the mixed - characteristic and

( )( )

( )( )

12

0

0 0 0

1, , ;x y

b

p zg r p p z

r z p z= (60)

for the amplitude. The function F used in the calculation of (60) is the same as given in eq.

(44) multiplied by 2 i . Than one can obtain the same expressions (59) and (60) as a result

of the Fourier transform


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( ) ( ){ }

( ) ( ){ }

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

, , ; exp , , ;

, , ; exp , , ; d d .

x y x y

x y

g r p p z iW r p p z

f r x y z i S r x y z p x p y x y= + +

(61)

A brief and lucid introduction to the application of the semiclassical wave functions in the

transfer function theory is given by Hawkes [12].

Finally, let us mention the paraxial form of the spinor equations (39), (40) and (41). We have

( )( )

( )( )

( ) ( )

( )( ){ }

0

*dd , exp 2 ,

2 d 4z

B B wee i

p i p

= = = = (62)

and further

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

0

0 0

0

0

0 0

0

1 1 0 2

*

1 0 2 0 1

*

2 2 0 1

* *

2 0 1 0 2

d

d d d ,

d

d d d .

z

z

z

z

z z

z

z

z

z

z

z z

z

g z g z g

g z g z g

g z g z g

g z g z g

=

=

= +

= +

(63)

In the paraxial approximation we take only the first two terms. Naturally, in a homogeneous

magnetic field the only effect is the change of the relative phase of the spinor components

representing the rotation of the spin components in the x y plane

( )

( )

( ) ( ){ }

( ) ( ){ }1 01

2 2 0

exp.

exp

g z i zf z

f z g z i z

= (64)


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6. Conclusions

In the paper we have fully described the results of semiclassical approximation in the form

suitable for electron optics. Our attention was restricted to the motion of the single electron in

a given time independent electromagnetic field.

References

[1] P. W. Hawkes and E. Kasper, Principles of Electron Optics (Vol. III.), Academic Press,

London 1996.

[2] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Butterworth

Heinemann, Oxford 2000.

[3] P. W. Hawkes and E. Kasper, Principles of Electron Optics (Vol. I.), Academic Press,

London 1989.

[4] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non relativistic Theory),

Pergamon Press, Oxford 1991.

[5] W. Pauli, Diracs Wellengleichung des Elektrons und geometrische Optik, Helv. Phys.

Acta 5 (1932), 179 199.

[6] S.I. Rubinow, J.B. Keller, Asymptotic Solution of the Dirac Equation, Phys. Rev. 131

(1963), 2789 2796.

[7] Yu.A. Kravtsov, O.N. Na da and A.A. Fuki, Waves in weakly anisotropic 3D

inhomogeneous media: quasi isotropic approximation of geometrical optics, Uspekhi

Fizicheskikh Nauk39 (1996), 129 154.

[8] M. Lenc, Immersion objective lenses in electron optics (PhD Thesis), TU Delft, Delft

1992.

[9] V.B. Berestetskii, E.M. Lifshitz and L.M. Pitaevskii, Quantum Electrodynamics,

Butterworth Heinemann, Oxford 1999.


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[10] W. Glaser and P. Schiske, Elektronenoptische Abbildungen auf Grund der

Wellenmechanik. II, Ann. Der Physik (6. Folge) 12 (1953), 267 280.

[11] J. Komrska and B. Vlachov, Justification of the model for electron interference

produced by an electrostatic biprism, Optica Acta 20 (1973), 207 215.

[12] P.W. Hawkes, Image processing based on the linear theory of image formation, in

Computer Processing of Electron Microscope Images (ed. P.W. Hawkes), Springer, Berlin &

New York 1980, 1 33.

semi classic a electron optics

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