Progress in electron Compton scattering

ultramicroscopy ELSEVIER

Ultramicroscopy 59 (1995) 241-253

Progress in electron Compton scattering P. Schattschneider, A. Exner Institut fiir Angewandte und Technische Physik, TU Wien, F~tednerHauptstrasse 8-10, tt-1040 Wien, Austria Received 27 July 1994

Abstract

Use of electrons as a probe instead of photons for Compton scattering experiments has been suggested in 1981 by B.G. Williams et al. The advantages, compared to conventional Compton scattering, are evident: a cross section higher by five orders of magnitude, extraordinary spatial resolution, on-line specimen manipulation, and all the additional information that is available in an analytical electron microscope. With PEELS nowadays available, and with the solution of the background problem, electron Compton profiles (CPs) can now be studied. Directional electron CPs of Si could be obtained by use of a two-beam geometry. In the intermediate momentum region, theory fits our results better than the results from conventional y-ray and X-ray experiments. Electron CPs of amorphous allotropes of carbon could be interpreted in terms of a charge transfer between s and p orbitals. In a crystal, the CP contains interference terms from scattering with different momentum transfer. They are caused by momentum correlation (off-diagonal elements of the momentum density in the reciprocal lattice), and can be described by the mixed dynamical form factor. In the two-beam case these terms are proportional to the excitation error. An experiment is suggested in order to obtain the off-diagonal elements of the momentum density.

1. I n t r o d u c t i o n

Compton scattering with T-rays or synchrotron radiation is a well known tool for probing the m o m e n t u m distribution of valence and conduction electrons in the solid. Use of electrons as a probe instead of photons has been suggested in 1981 [1]. The advantages, compared to conventional Compton scattering, are evident: a cross section higher by five orders of magnitude, extraordinary spatial resolution, on-line specimen manipulation, and all the additional information that is available in an analytical electron microscope. The difficulties from strong multiple scattering - not only in the diffuse background, but also

from B r a g g - C o m p t o n channel c o u p l i n g - were unsurmountable at that time, due to poor modelling of the multiple scattering contributions, and to insufficient instrumentation. With parallel-detection electron energy-loss spectrometers (PEELS) nowadays available, and with the solution of the background problem [2,8], electron Compton scattering can now be successfully done in the transmission electron microscope. In the early nineties, we began with Compton experiments on a J E O L 200CX with a G A T A N 666 PEELS. In the experiment, an energy-loss spectrum in the diffraction plane is obtained, at a scattering angle of typically 70 mrad at 200 kV. In terms of atomic physics, this is an energy-scan

0304-3991/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI 0 3 0 4 - 3 9 9 1 ( 9 5 ) 0 0 0 3 2 - 1

P. Schattschneider,A. Exner/ Ultramicroscopy59 (1995)241-253

242

across the Bethe surface at such high energy that the impulse approximation [3] is valid. The inelastic scattering cross section ~Zo-/~E ~ 0 relates then to the Compton profile (CP)

J(pz) = fx(p)x*(p)

2. Directional Compton profiles in silicon d2pxy

(1)

as 02oOE 392

1 ~rZqs J( E / q - q / 2 ) ,

(2)

where we have used atomic units and assumed that the incident wave is plane. Here, E is the energy loss, q is the fixed momentum transfer in the interaction, proportional to the scattering angle, thus establishing the connection to the double differential cross section. X(P) is the wave function of the ground state of the scatterer, in momentum representation, and the coordinate system is chosen such that the z-axis is parallel to q. In many-particle systems, such as in crystals, when there is no wave function, the integrand in Eq. (1) is replaced by the diagonal element of the reduced one-particle density matrix in momentum space [4], I'(plP)=(P

tion times are ten minutes in our experiments, with a momentum resolution of typically 0.08 a.u.

PIP).

(3)

It is only for one-electron systems that the density matrix can be factorized and reduces to

r(plp') =x*(p)x(p').

Multiple scattering leads to a strong background (mainly caused by coupling of the Compton event with quasi-elastic scattering) that exceeds the signal except for very thin specimens (see Fig. 1). Diffraction leads to a superposition of various Compton profiles with different peak energies, each stemming from an excited Bragg reflexion [7]. These difficulties led to electron Compton scattering lying dormant for years. Recently the background problem could be solved by parameterized simulation of combined C o m p t o n - p l a s m o n and Compton-quasi-elastic events [8]. The main contribution comes from a combination of single inelastic events in the energy range of the Compton peak with quasi-elastic (high-angle) scattering. For cylindrically symmetric scattering cross sections, the angle- and energy-dependent background can be modelled by means of a Fourier transform for the energy variable and a Hankel transform of zero order for the angular variable:

fj(O0, p) = f dE f~O d0Fj(E, 0) e-iE~°Jo(OP).

(5)

(4)

The momentum resolution in Compton experiments with T-rays is typically 0.5 a.u. [5], and for synchrotron radiation w o r s e than 0.1 a.u. [6]. These figures are given by the beam and collimator divergence. In electron Compton experiments, a number of instrumental conditions determine the resolution. Besides the performance of the magnetic lenses in the TEM, it is the convergence angle of the incident beam, the camera length, the spectrometer entrance aperture, and the energy resolution of the PEELS. With modern equipment, resolutions of 0.02 a.u. are realistic, and could in principle be pressed down to 0.01 a.u. The main problem with photons is the low intensity demanding data collection over a typical time of an hour, whereas at present typical coUec-

Here, F 1 is the quasi-elastic single-scattering Lenz-type [9] profile, and F z is the inelastic single-scattering distribution, given by a hydrogenic model. J0 is the zero-order Bessel function of the first kind. The Fourier transform of the background can be calculated as [8] b = eal+az(e dlh - 1)(e a2f2 - 1).

(6)

The attenuation thicknesses dl. 2 is the respective thickness of the specimen in units of quasi-elastic and inelastic mean free path lengths. The important point here is that b (and its back transform B ( E , 0)) can be described by only two parameters. So, it is relatively easy to fit the background to the experimental profile. The accuracy of the fit is excellent. An example is given in Fig. 2.

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

0.4

.,,I,,

I.,,1,

,I..~,1,

/ '/

b

.,.I,.,

,,,

f-'\

0.3 "2

"~ 1.5

243

"~ 0.2

.......

/ N

•~

"~ 0.5

o.1

1

0

o 1 O0

500

900

1300

1700

'

600

200

'''1

l'''

1000

''1

'''1

1400

1800

energy loss [eV]

energy loss [eV]

]2,= i~

15

"~

10

.~

1

0.5 5 0

,,

1 O0

500

900

1300

1700

200

I.

,,I,~,I

600

eaaergy loss [eV] 25

, . , I , , o l , , , I , , ,

, . . I . . , I , . , I , . . I .

I

,'

,I,

''I

1000

''.I,'

~ I '''I"''

1400

1800

e n ~ g y loss [eV]

, , , I , , , | , , , l , , , I , , , I , , . I , , , I , , , I , , ,

f

]

2O

1.6

j~ 1.2 0.8

0.4

5

0

0 .,,

100

'"

i . , , i , o , i , . , i . , . i , , . i , , , i , , , i . , ,

500

900

1300

energy loss [eV]

1700

200

I ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 600

1000

energy ~

'''1' 1400

'"

I'''

I'' 1800

[eV]

Fig. 1. Total scattering profiles P(E, O) (curves 1) at scattering angles 44 mrad (a, c, e) and 68 mrad (b, d, f) for thickness D o = 0.15 (a, b), Dp = 0.60 (b, c), D o = 1.20 (e, f), together with the elastic background Pi,e(E, 0). The insets show, the calculated low-loss spectra.

244

P. Schattschneider, A. Exner / Ultram&roscopy 59 (1995) 241-253

Contrary to microanalysis where the background can be extrapolated beyond an ionisation edge, by use of simple formulae Such 'as the power-law fit, the background in Comp~on experiments is much more complicated, and cannot be modelled over an energy range of some 100 eV by simple expressions. Only with the simulation of t h e background became a reliable interpretation of CPs feasible. The B r a g g - C o m p t o n coupling cannot be avoided. It is, however, possible to use a scattering geometry in which most of the subsidiary CPs are suppressed. This is the two-beam case. The higher-order reflexions of the systematic row are faint enough to produce only a few percent of the total intensity. The Compton spectra from the

1000600800 i..........

....... ial

'1

°i 200 0

i

0

500

t000 1500 Energie[eV]

2.q00

i

i

2500

1000

600

direct and from the diffracted beam can be made to have the same peak energy when the momentum transfers from both beams are equal, according to Eq. (2), that is, positioning the spectrometer aperture at the symmetry line between the two (Fig. 3). From crystal symmetry, it follows moreover that the momentum densities projected onto the two scattering vectors are identical, hence the CPs in Eq. (1) are identical. In this scattering geometry only one CP contributes to the measured intensity - see Fig. 4. According to this particular geometry, the CPs are no more obtained in high-symmetry directions as in conventional Compton experiments. Profiles in an arbitrary direction can be calculated by expansion of wave functions into cubic harmonics [10]. The first reliable directional electron Compton profiles (CPs) of a single crystal were obtained in 1993 [2]. The profiles are broader than those from conventional Compton scattering, see Fig. 5. The reason is not known, but it is well possible that coherence effects (see Section 4) are responsible for the broadening. In the Compton literature, the interesting quantity is the anisotropy of momentum density, that is, difference of CPs obtained with the scattering vector in different crystal directions. The difference CPs in Si are comparable to earlier w o r k ( F i g . 6). In the intermediate momentum region, theory [10] fits our results better. The notorious underestimate of amplitudes in the difference profile, a well known effect in conventional Compton work, does not show up in our experiment.

........................ ,................. .................. ................. ,............................ ,!bl, 1 ...................................................... -...................................................... ;......................................... 3. Compton profiles in carbon allotropes

200400 .... ~

...........-

0

0

500

1000 1500 2000 2500 Energie[eV] Fig. 2. Compton spectrum of amorphous carbon. (a) As measured. (b) Same with background according to Eq. (6). Thin lines: simulated spectrum and quasi-elastic background (best fit).

Fig. 7 shows Compton spectra of amorphous carbon (a-C) and amorphous diamond-like carbon (a-D). Since diamond has stronger bonds than a-C, one would expect a broader momentum distribution than in a-C, according to the virial theorem. At first glance, there is no significant difference. Note, however, the strong background contribution, concentrated at the Compton maximum. This may mask faint differences in the

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

245

k0+ g

ko I r I I I //

k

0 =k

// /

=k'

Fig. 3. Scattering geometry for suppression of Bragg-Compton channel coupling. The vectors k, k' are crystallografically equivalent.

momentum distribution t. The total counts in the spectra are 1 x 106 for a-C and 5 x 105 for a-D, quite comparable to X-ray work. Note, however, that the momentum resolution is much better, and that we had a total time of 500 s for collection of one spectrum. Fig. 8 shows CPs obtained from the spectra by background subtraction and rescaling of the abscissa in momentum units. The dashed lines roughly separate the core from the valence contributions. Fig. 9 shows the theoretical CPs of s- and p-electrons, calculated with H a r t r e e - F o c k atomic wave functions [11]. Fig. 10 is the difference (a-D minus a-C) in the momentum density of the two CPs in Fig. 8, normalized to 4 electrons per atom (crosses). In the following we consider a simple atomistic model in order to explain the difference profile. Carbon has one s- and three p-electrons in the

valence shell, and the CP of one atom of carbon is the sum (7)

J(p) =Js(E/q - q/2) + 3Jp(E/q - q/2),

where Js,p are the directional averages of the CPs of a single s- and p-electron. We observe that any hybrid orbital, s p 3, s a y , can be written as a linear combination of Is) and IP) orbitals X = Ea=laiXi • Referring to Eq. (1), the Compton profile of one

l h 000 :

aperture

~



<010>

1 ~ j d h 4 0 0

~T 1 In the present case, a simple power-law fit would have resulted even in a broader Compton profile for a-C.

Fig. 4. Scattering geometry for the symmetric two-beam case. Diffraction plane with incident beam (000) and g = (400). Arrows: scattering vectors Qo, Qg.

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

246

hybrid orbital can be split into a direct and a mixed term

J(pz)

=

p-wave functions, it follows that all mixed terms must vanish, and with coefficients aia / = 0.25 for s p 3 hybrids, Eq. (7) is retained

Ea,a?fx,(p)x? (p) d2pxy + Eaia/fxi(P)Xj*(p )

J = Js + 3Jp.

d2pxy.

(8)

(9)

The same expression holds for amorphous graphite, with sp 2 hybrids and a single p-orbital. Apparently, the CPs of all amorphous carbon allotropes should be identical. Thus, there should not be any difference between the CPs of a-C and

i
In amorphous media, contributions to J from products with uneven parity vanish, due to the directional averaging. From the parity of s- and

...............................i..............................i...............................i................................i............................... :

3

& 2"

1

0

0 . 2

0.5

I

i

i

i

1

i

i

,

I

I

1.5

p~ [a.u.] =

t

~

,

2

I

J

t

i

I

b

2.5

,

,

i

i

<19,3,2>

0.1 0.05

~

o

-0.05 -0.1 -0.15 - 0 . 2

,

0

,

,

,

",

0.5

,

,

.

,

,

,

1

.

.

.

1.5

.

,

,

2

,

,

,

2.5

Pz [a.u.] Fig. 5. (a) Direct comparison of the Compton profile in (19, 3, 2) direction measured by means of electron scattering and ~/-ray scattering. (b) Difference profile A J = J R ~ - J ¢ l between the 7-ray profile and the electron scattering profile for (19, 3, 2) and

(22, 19, 4~.

247

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

a-D. This is in contrast to experiment (see Fig. 10). The reason is t h a t wave function overlap in bonding orbitals changes the charge distribution in the lobes. A n exact theoretical treatment would necessitate calculation of crystal wave functions, as has b e e n done for crystalline Si [10] in the F L A P W approach, and a directional average. For a qualitative reasoning, o n e can expand the unknown wave function into atomic orbitals. Giving equal weight to the three p-type orbitals, we then have for a-D Jd = PdJs + (4 - Pd) Jp,

where Pd = l al 12 is the weight of the s-type wave function in a-D. The difference CP (diamond minus carbon) is then 4-]o

The first factor on the right-hand side is the charge transfer, in the sense of additional charge in the bonds of a-D, as c o m p a r e d to a-C. The dashed line in Fig. 10 is the theoretical difference profile for one electron transferred from Is) to IP) orbitals. The full line is the

(10) ,

0.2

,

,

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

•

. .

0.1

.

.

.

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

.

.

.

!

. . . . . . .

l--electron I

i .............

~l

•

Reed

i

i

i

IJ

O

Sch01ke

= •I

............ i........................- ........... t . . . .

i ..--o ...................... i .............................. :

__~>~_\o

o

.

i ...............................

•

~

.

i ...............................

~'- . ~ , , ~ ...............--.-~ ~

0.05

.

i

a 0.15

!

(11)

= (pd-po)(J,-]p).

,

i ................................ •

0

•

~

...........i..............~...'. ...........~... ~

~...............................

*

=

-0.05

-0.1

-0.15

_~................................................................

i ................................................................................................

-0.2

0

0,4

0.8

1.2

1.6

2

Pz [a.u.] 0.2

J

;

I

I

l

I

t

i

i

J

!

b 0.15

............................... i.............................................................................. - ....

" o.1 "

..../""~'.,4:--.-.-.i-.-~,

.

~:i..........................7~; ........'~; .. ~

'~__7-

...........

-0.2

i

t

HF1

i ...................... o------i ............................................................................. • Reed

o.o5

. . . . . . . . . .

- ............................................................................................

-0.15

I

FLAPW

"

0

o

SchOlke

.

Z

~.............................................................

i

0.4

0.8

1:2

1.6

2

Pz [a.u.] Fig. 6. (a) Comparison of the differential Compton profile AJ = J19,a,2-J22,]9,4 obtained by means of electron scattering and y-ray (Reed) X-ray (Schiilke) scattering, respectively. (b) Photon results compared to Hartree-Fock (HF1) and Full Potential Linearized Augmented Plane Wave (FLAPW) calculations [10].

248

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253 600

A possible combination of electron Compton scattering in amorphous materials with ( e - , 2 e - ) experiments [12] is currently under discussion. The idea is to construct a synthetic CP from e-, 2 e - data and compare it with a real CP.

Compton spectrum, simulated spectrum and simulated background i i t

(a) 500

400

-e

300

4. Coherence effects 200

100

0

500 '

1o'o 0

15o 0

2000 '

2500

energy loss [eV]

600

Compton spectrum, simulated spectrum and simulated background i = t ,

(b)

In a crystal, the probe electron is a coherent superposition of plane waves. As a consequence, there are well defined phase relations between the direct beam and the Bragg-reflected beams. In the two-beam case with symmetric scattering geometry (Fig. 4), two identical Compton spectra are superimposed, belonging to the 0 and

500

4O0

Compton profile and linear model for core contributions = i i i

300

(a) 300 250 200

',, 200

100

"~, ~N~..... 150

0

I 500

[ I 1CO0 1500 energy loss [eV]

/ 2000

2500

Fig. 7. Compton profiles of (a) a-C, and (b) a-D with best fit (full line) and simulated background (dashed).

100

50

-3

difference profile, smoothed with a Gaussian of F W H M given by the momentum resolution of the experiment. As can be seen, charge is transferred to much higher momentum than predicted by the simple atomic model. This is plausible because the Fermi momentum is different in the two allotropes: changes should then occur in the vicinity of the Fermi momenta. The non-applicability of the atomic model implies that solid-state effects are significant in amorphous diamondlike carbon. Only the subsidiary structure at 0.8 a.u. is probably caused by the Is) ---> ]p) charge transfer. According to the virial theorem, the results show that the ground state energy in a-D is lower than a transfer from s- to p-orbitals would allow.

-2

-1

0 momentum [a.u.]

1

2

Compton profile and linear model for core contributions

200 180

'

'

-2

-1

~

'

'

0

1

2

160 140 120

t0ol 80 i

5o! 40 20 0 -3

momentum [a.u.]

Fig, 8. As Fig. 7 after background subtraction.

(b)

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

g beams. This superposition of intensities is a valid assumption only when the transmitted (0) and t h e reflected ( g ) beams are incoherent to each other. When there are phase relations, an interference term in the Compton cross section should occur. The effect is analogous to the interference term in the double-slit experiment. Expanding the wave function of the incident electron into Bloch waves a/ta(r ) =

Eu(ej)

249

vacuo. We assume a plane wave for the detected electron, ~ b ( r ) = eikb~. The transition matrix element for the inelastic interaction is 1 < b l V l a ) = ~u~J)-~-~-?
ei(k¢Y)+g) r,

j,g

J,g

e~ej'r" li),

~:'jg

(12)

Qje = k(J) + g - k b .

with k (y) = k a + ~,(J), where g are reciprocal lattice vectors, j is the Bloch wave index, and y(Y) is the deviation of the jth dispersion surface from the wave vector in

la> = I~F~> ® [i>, [b> = [ ~ b ) ® If> are the initial, respectively the final states of the combined system, and l i), If) the initial, respectively final

0.9

I

'B i g g s . S ' 'B i g g s . p . . . . . 0.8

0.7

0.6

0.5

0.4

0.3

t /

x

0.2

0.I

-3

-2

-I

0

1

2

Fig. 9. Theoretical Compton profiles of s-electrons (full line) and p-electrons (dashed) in atomic carbon.

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253

250

state of the crystal. For the double differential cross section

(14), consists of direct terms (Qj~= Q/g,) and interference terms (Q~g # Q~,g,). Expression (14) can now be evaluated for the two-beam case described above, In the impulse approximation [3], and under neglect of the small quantities T o) in the scattering vectors [14], we eventually obtain

~2 o-

aE a~2 = (271")4 E I I z 6(Ei _ Ef -4-E ) b

(13) we obtain [14]

a2o~

020 -

l

aE a-----~= 4 E u~J)* uq'). ,q2 ~." j,g j' ,g'

~ j g ~ j ' g'

4

aE a~

S(Qj~, Qy.,E).

-~oS(Qo, Qo, E)

=

2 K gQloQ [g2 W 2 ( ~K o

(14) Here we have used the mixed dynamic form factor [13] 1

S(K, K', E) := -2--~f~_ (ilpK(t)p_x,

XS(Q0, Qe, E ) ,

pK is the time-dependent electron density operator. For K=K', this expression reduces to the usual dynamical form factor for inelastic scattering. The differential scattering cross section, Eq.

(15)

differences of the Compton profiles compared to the atomic model i + ++ , ! ~-+ ,

j

o.4L

+

/

.

+++ +

T~-~'~

/

+

!i

+ ~

..+~, ~ q ~ . ~

i

",+

:~

+

o

sin6)]3

where we have used 6 = [T (1) - T(2)]d, and w = sff~ is the dimensionless deviation parameter as de. fined in the dynamical theory of electron diffraction. K 0 is the number of atoms in the unit cell and K~ is a factor depending on the symmetry of the lattice. For g = (h, k, l), and in silicon, K~ = K 0 cos[¼7r(h + k + l)]. The first term in (15) rep-

dt.

li) e let

1

".......

+ +

~*

~+. + % % 2

i

~- ~

::

+.+'

+ ++ ++

-~/~+ * ~

+++4~_ +

~

* +

*'., 4

! .......

/ + ~:-~. ++" " - ~+~.+,+~ ~ ~*\~ +~+" / *S-d'- + ~ +~-1~ I~ .+,~+ +.:~ +. 4++, t.~.~ + + ~

[4+'~'1

++

1"~1~+

+ -3

~-

"

÷~.+

I~

._

,+Xb~+

+

÷i~+'l+:

%

?

+

++

++T

+

÷

+

', -2

-1

0 momentum [a.u,]

1

2

3

Fig. 10. difference of normalized Compton profiles of a-D and a-C. Full line: smoothed with the effective momentum resolution function (~p(FWHM) = 0.08 a.u.), dashed line: difference Js - J o according to Fig. 9.

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253 resents the incoherent scattering from the transmitted (0) and the Bragg-reflected beams (g), it will be termed "direct" in the following; the second term arises from the coherent superposi-

0.012

I

I

I

251

tion of amplitudes from (0) and (g), we will refer to it as 'indirect'. The interference term vanishes in the exact Bragg case (w = 0). By tilting the crystal out of

I

I

I

I

I

(a)

0.01

0.008

0.006

0.004 H 0.002

0

-0.002 400

0.012

I

I

I

500

600

700

I

double I

I

I

differential I I

Ill ~ ~

0.01

I

800 900 I000 e n e r g y [eV]

I

1200

section

cross I

I

llO0

I

I

I

(b)

x%k

~0.008

~o.oo~ H O.O04

0.002

-

-

O/

~00

I

soo

~00

7o0

s00 900 e n e r g y [eV]

i000

noo

1200

1300

Fig. 11. (a) Direct term of the CP in Si (full line) and interference term for the (220) Bragg beam (dashed). The expected difference CP is twice the latter. Note scale change. (b) Total valence CPs in the (220) case, with excitation errrors w = 1 (dashed) and w = - 1 (full line).

P. Schattsehneider,A. Exner/ Ultramieroscopy59 (1995)241-253

252

the Bragg position to w ~- _ 1, the difference CP should reveal the interference term. The mixed dynamical form factor S relates to the m o m e n t u m density as [13]

S(Qo, Qg, E)

= fr(o-00ro-Q,)

that the contribution to the interference term from t h e s-electrons is equal in shape to the direct term,, and should be difficult to observe, The p-electrons, however, contribute with a smaller profile. A calculated difference CP is shown in Fig. 11. Although the effect is subtle it should be possible to observe. This would allow one to obtain the off-diagonal momentum density, via Eq. (18) in the described diffraction setup in the electron microscope. Experiments are under way 3

(16) where the m o m e n t u m density matrix F, as defined in Eq. (3), is an extension of the m o m e n t u m density to many-electron systems. For one-electron systems, we have

F ( p - Q o I p - Qg) = x * ( p - Q o ) x ( p - Q,).

(17) With the definition of the mean scattering vector 1 q = ~(Qo + Qg) and in the coordinate system as

chosen for Eq. (2) - Pz parallel to q - Eq. (16) can be rewritten as

S(pz;g,E)=f

dpxdpyF(plp+g),

5. Conclusions Electron Compton scattering on solids has evolved from a rather exotic idea to an experimental method that can compete with synchrotron work. In two examples, (momentum anisotropy in Si, and momentum distribution in carbon allotropes) we showed that sensible information can be obtained from the experiment. As a third possibility, we suggest an experiment that should yield information on the non-diagonal momentum density in a solid.

(18)

with pz = E / q - q / 2 + g2/8q. It can be easily seen that the mixed dynamical form factor is a projection integral of the non-diagonal m o m e n t u m density matrix on a plane perpendicular to the mean scattering vector q. Note that the symmetry point of the CP (Pz = 0) defines the peak energy in the Compton spectrum. The small term g2/8q shifts the peak position to slightly lower energy than one would expect naively. Expression (18) has been derived by Schfilke and Mourikis [4] for what they call coherent inelastic X-ray scattering 2 Simulations for Si with atomic orbitals show

2 It should be mentioned that these authors find, due to an error in their derivation, a vanishing shift of the indirect profile with respect to the direct one, a fact that is negligible in most cases with synchrotron radiation.

Acknowledgements The experiments of P. Jonas, the calculations of the Si-CPs by C. Blaas, J. Redinger and R. Podloucky, and the expertise of H. Kohl as to the mixed dynamical form factor were of utmost importance. M. Nelhiebe! calculated the indirect term for Si. I.E. McCarthy provided the ultrathin specimen of a-D. This work was sponsored by the Hochschul jubil~iumsstiftung der Stadt Wien.

References [1] B.G. Williams, M.P. Parkinson, C.J. Eckhardt and J.M. Thomas, Chem. Phys. Lett. 78 (1981) 434.

3 A first experiment with Si showed a CP, but the intensity was too low to see a significant difference. Better specimens (thin and unbent over an area of some micrometers) are necessary in order to have sufficient intensity.

P. Schattschneider, A. Exner / Ultramicroscopy 59 (1995) 241-253 [2] P. Jonas and P. Schattschneider, Microsc. Microanal. Microstruct. 4 (1993) 63. [3] P.M. Platzman and N. Tzoar, Phys. Rev. 139 (1965) A410. [4] W. Schiilke and S. Mourikis, Acta Cryst. A 42 (1986) 86. [5] G. Loupias and J. Chomiler, Z. Phys. D (At. Mol. Clusters) 2 (1986) 297. [6] J.R. Schmitz, H. Schulte-Schrepping, J. Degenhardt, S. Mourikis and W. Schiilke, The HARWI-Compton spectrometer: Test of crystal-dispersive analysis using a Ge200-strip-detector, HASYLAB Jahresbericht (1990) p. 613. [7] B.G. Williams, M.K. Uppal and R.D. Brydson, Proc. Roy. Soc. (London) A 409 (1987) 161.

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