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Calculation of high resolution electron and helium energy loss cross sections using surface vibration spectral densities
Surface Science 224 (1989) L983-L988 North-Holland, Amsterdam
L983
SURFACE SCIENCE LETTERS CALCULATION OF H I G H R E S O L U T I O N E L E C T R O N AND H E L I U M ENERGY L O S S C R O S S S E C T I O N S U S I N G SURFACE VIBRATION SPECTRAL D E N S I T I E S P. K N I P P Department of Physics and The James Franck Institute, The University of Chicago, Chicago, 1L 60637, USA
and Burl M. H A L L Department of Physics and The Institute for Surface and Interface Science, University of California, Irvine, CA 92717, USA Received 19 July 1989; accepted for publication 20 September 1989
It is shown that vibrational eigenvectors (outputs of a slab calculation) are not required for the calculation of high resolution electron energy loss spectra (EELS) or helium energy loss spectra. Instead, vibrational spectral densities (outputs of a Green's function calculation) may be input directly to such calculations. This facilitates the treatment of scattering from bulk phonons as well as from surface phonons.
Two theoretical methods exist for the study of phonons in the vicinity of a crystalline surface: slab calculations and Green's function calculations. The former method regards the vibrations of a slab of atoms extending infinitely in two directions (2- and 3%) but consisting of N(_< 100) layers stacked in the 2-direction [1]. A 3N X 3N matrix must be diagonalized to yield the phonon eigenvectors e~n)(Q) and frequencies ~%(Q), where Q is the (two-component) wave vector, l is the layer number, and n = 1, 2 . . . . . 3N. A small number of these phonons are localized to the surface. The Green's function method, on the other hand, considers the vibrations of a crystal which extends semi-infinitely in the t-direction (as opposed to truncating after N layers) [2]. For this reason, the latter approach more aptly describes the crystal in a surface scattering experiment, whose thickness typically exceeds 1 ram. In addition, the Green's function method involves manipulations of matrices which are quite small ( - 10 x 10) when compared to those in slab calculations. Many of these relatively small matrices must be inverted, but the Green's function procedure is usually quicker and more 0039-6028/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L984
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
stable than a slab calculation, which requires the diagonalization of a single large matrix. Despite its apparent analytic and numerical advantages, the Green's function method has been largely neglected because it does not directly yield the phonon eigenvectors and eigenvalues, heretofore irreplaceable inputs to a surface scattering calculation [3]. Instead, it yields the spectral density function
z.,e,, ~ f
~,' (Q)3
*G(Q)]
(1)
n
for an arbitrarily large set of layers l and l'. (In this and all subsequent equations, the subscript a or 13 specifies one of three cartesian directions.) The eigenvectors and eigenvalues of the surface phonons can be trivially extracted from eq. (1) but this is not feasible for most bulk phonons, owing to their continuous frequency bands and to their multiple (roughly threefold) degeneracy. At the outermost layer of a clean surface, bulk phonons and surface phonons contribute to atomic vibrations in comparable amounts [4], as has been seen in helium scattering from Ag(111) [5]. Unlike He atoms, electrons are able to penetrate a few layers before exiting the crystal, thereby further increasing the importance of bulk phonons. This has been seen for EELS from Ni(100) [6], from C u ( l l l ) [7], and from N i ( l l l ) [8]. The main point of this Letter is to show that the spectral density function is in fact more suitable for use in such scattering calculations than the eigenvectors and eigenvalues themselves, for both surface phonons and bulk phonons. The theory for EELS is more advanced than that for helium scattering and will be detailed more thoroughly in this Letter. The adiabatic approximation is used to evaluate the (non-Born) matrix element f(ki, k f) for electron scattering from a crystal with nonequilibrium atomic positions { R }. For scattering from a vibration with wave vector Q = k i - k r , this matrix element is expanded in powers of the displacement u/ [9]:
f ( k i , kf, { R } ) = f ( k i, kf, {R0} ) + Zu~,8~-~, ~, + ½a,81l Y'~,l'l allg'8l"
32f ~UaI~U8
l, q-
. . . .
When ignoring multiphonon processes, this series is truncated after two terms, yielding f ( k i, kr, { R } ) -=-fLEED(ki, kf) q- E u l . h l ( k i , t
kf),
(2)
where fLEED is the low-energy electron diffraction matrix element and
A,,,t(ki, k¢)= 3f/3u,~l is calculated in much the same way as fLEED [10,11]. Only the second term in eq. (2) needs consideration when not at a diffraction condition (i.e., Q 4: G), because fLEED vanishes in this case.
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
L985
The inelastic scattering rate is then calculated using Fermi's golden rule:
l
/
where 1I) and I F ) are the initial and final vibrational states of the crystal and hw = hZ(k 2 - k Z ) / 2 m is the electron energy loss (or gain). The large outer brackets ( . . . )i indicate that the last step of the calculation requires a thermodynamic average of all possible initial vibrational states of the crystal. The calculation proceeds along the same line as that for inelastic neutron scattering [12], by Fourier expanding the delta function: ['(ki, kf) =
~,AalA~l,(~_,(IluatlF)fdte-itt~-(Er-EI)/hl(Flu~z'[1)l otflll'
\
aflll'
\
F
I
F
= E A~,A~cf dt e-i~t(u~tufft,(t)l.
1
(3)
a,Bll"
The bracketed quantity is a time-dependent correlation function whose Fourier transform describes the energy-loss dependence of the cross section [13]. In the harmonic approximation of lattice dynamics, this correlation function can be expressed in terms of the spectral density function:
f d, e-i¢ot =
.
f d, e-i'~t~'nn
e(n)•(n')* al ¢:fll'
~f~-~.~' ([at" + a.] × [ a . , ( t ) + at",(t)])
~(n)~(n)*
= f dt e-i~t Y'~ ~al ~BI" n
= Z -e(n)•(n)* -al c:fll' -
N(~o) + 1
o~
(N(~o.) e -i=-t + [N(~o.) + 1] e i="t )
('On
{ N ( ~ . ) 8 ( w + ~o.) + [N(~o.) + 1] $ ( ~ o - ¢o.))
P,~Bu'(Q, o~),
L986
P. Krupp, B.M. Hall / High resolution electron and helium energy loss cross sections
where N(~0) = [ e x p ( h c o / k B T ) - 1] 1. This yields the final result
F(ki,
kf)-
y, A~,(k,, aflll"
U(~0)+l
kr)A¢,,(ki,
kf)p~/~u,(Q, 0~).
(4)
Because p ( ~ ) is an even function, eq. (4) yields b o t h the energy gain a n d energy loss features, whose ratio equals e x p ( h ~ / k B T ). In eq. (4), A / d e p e n d s only on the details of the interaction between the electron a n d the static lattice (e.g., scattering phase shifts, crystal geometry, work function), whereas p ( ~ ) d e p e n d s only on the details of the lattice d y n a m i c s (e.g., force constants, crystal geometry). A similar expression of the EELS cross section in terms of p ( w ) has b e e n previously derived, b u t it relies o n u n p h y s i c a l z - d e p e n d e n c e of the electron wave f u n c t i o n [4]. Accordingly, the analytical f o r m u l a for A / ( k ~ , k f ) f o u n d therein is n o t expected to accurately describe a typical EELS experiment. A t t e n u a t i o n of the electron wave f u n c t i o n b y other miscellaneous inelastic processes causes A~ to a p p r o a c h zero rapidly with increasing l. F o r surface p h o n o n s , p u , ( ~ ) approaches zero rapidly as l or l ' increases, whereas this is obviously not true for b u l k p h o n o n s . I n practice, the series in eq. (4) converges
== t~ _Q
ii ft
•
I !
I ~ 0
15
ii
""
"' 20
* I
.'"
**
I!
"
I~ 25
30
35
energy loss (meV)
Fig. 1. Single-phonon cross section of 170 eV electrons scattered from Ni(111) with an incidence angle of 70 o and an exit angle of 46 °, which correspond to a momentum transfer of Q = M (a high-symmetry point at the boundary of the (two-dimensional) surface Brillouin zone). Only the energy loss spectrum is shown here. The dashed line depicts the scattering from the top layer, the solid line from the top two layers, and the dotted line from the top five layers. Note that destructive interference between the top two layers essentially eliminates the scattering intensity from the 31 meV surface phonon. The lattice dynamical model used is that of masses connected by linear springs with nearest neighbors only. All of the springs have the same strength except for the ones within the top layer, which are softened by 20%.
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
L987
J
4"--
I
-40
-20 0 20 energy loss (meV)
4O
Fig. 2. Simulated EELS result showing both energy gains and losses, obtained by convolving the curve in fig. 1 with a 6 meV wide Gaussian. after only a few layers. Fig. 1 illustrates this for N i ( l l l ) . The sharp features at 16 meV (Rayleigh mode) and 31 meV ( " g a p " mode) are due to surface phonons, whereas the two broad bands at 18-27 meV and 34-37 meV are due to bulk phonons. For a direct comparison with experiment, these curves should be convoluted with the particular instrument function - typically Gaussian with a 6 meV linewidth. This is shown in fig. 2. Although we demonstrate the results for a very simple system, this method works just as well for surfaces with adsorbates, relaxation, reconstruction, a n d / o r complicated force constants. The only constraint is that the surface must exhibit a two-dimensional periodicity which is commensurate with the monatomic substrate. Along different lines, Bortolani et al. have calculated the single-phonon cross section for helium scattering from Ag(111), a very smooth surface [5]. It takes the form / ' ( k i, k f ) =
N(~o)+ 1 iC(ki, kf)] 2 ~0
× £ I - i B e ~ " ' ( Q ) + Qe~("(Q)128[~0 - % ( Q ) ] ,
(5)
n
where 2 is parallel to Q, C is a complicated function of ki and kf, and B is the "softness parameter", equalling 2.2 ,~-a for A g ( l l 1). Performing the sum reproduces eq. (4), with A t ( k i, k f ) - 8l.oC(k i, k f ) ( - i B 2 + (2). As for EELS, the single-phonon cross section factorizes into two parts; one part depends only on the interaction between the incident particle and the static lattice, and the other part depends only on the lattice dynamics.
L988
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
In conclusion, a previous EELS calculation is reformulated in terms of correlation functions, rather than vibrational eigenvectors. In the harmonic approximation, these correlation functions can be simply expressed in terms of easily calculable spectral densities. A previous helium scattering calculation is also expressed in terms of spectral density functions. P.K. acknowledges support by the National Science Foundation Materials Research Laboratory at The University of Chicago, and B.M.H. acknowledges support by the US Department of Energy, through Grant No. DE-FG0384ER45083.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
R.F. Allen, G.P. Alldredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1648. J.E. Black, T.S. Rahman and D.L. Mills, Phys. Rev. B 27 (1983) 4072. Y. Chen, Z.Q. Wu, S.Y. Tong and J.E. Black, Surface Sci. 210 (1989) 271. V. Roundy and D.L. Mills, Phys. Rev. B 5 (1972) 1347. V. Bortolani, A. Franchini, F. Nizzoli and G. Santoro, Phys. Rev. Letters 52 (1984) 429. B.M. Hall and D.L. Mills, Phys. Rev. B 34 (1986) 8318. M.H. Mohamed, L.L. Kesmodel, B.M. Hall and D.L. Mills, Phys. Rev. B 37 (1988) 2763. W. Menezes, G. Tisdale, P. Knipp, U. Fano and S.J. Sibener, to be published. H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). C.H. Li, S.Y. Tong and D.L. Mills, Phys. Rev. B 21 (1980) 3057. Mu-Liang Xu, B.M. Hall, S.Y. Tong, M. Rocca, H. Ibach, S. Lehwald and J.E. Black, Phys, Rev. Letters 54 (1985) 1171. L. Van Hove, Phys. Rev. 95 (1954) 249. Although this derivation assumes that the state of the crystal is described by a wave function, eqs. (3) and (4) follow just as easily if one assumes that the crystal is described by a density matrix. To compare directly with experiment, eqs. (3) and (4) must be multiplied by a simple energyand angle-dependent factor. See, e.g., eq. (2.14) of ref. [10].
L983
SURFACE SCIENCE LETTERS CALCULATION OF H I G H R E S O L U T I O N E L E C T R O N AND H E L I U M ENERGY L O S S C R O S S S E C T I O N S U S I N G SURFACE VIBRATION SPECTRAL D E N S I T I E S P. K N I P P Department of Physics and The James Franck Institute, The University of Chicago, Chicago, 1L 60637, USA
and Burl M. H A L L Department of Physics and The Institute for Surface and Interface Science, University of California, Irvine, CA 92717, USA Received 19 July 1989; accepted for publication 20 September 1989
It is shown that vibrational eigenvectors (outputs of a slab calculation) are not required for the calculation of high resolution electron energy loss spectra (EELS) or helium energy loss spectra. Instead, vibrational spectral densities (outputs of a Green's function calculation) may be input directly to such calculations. This facilitates the treatment of scattering from bulk phonons as well as from surface phonons.
Two theoretical methods exist for the study of phonons in the vicinity of a crystalline surface: slab calculations and Green's function calculations. The former method regards the vibrations of a slab of atoms extending infinitely in two directions (2- and 3%) but consisting of N(_< 100) layers stacked in the 2-direction [1]. A 3N X 3N matrix must be diagonalized to yield the phonon eigenvectors e~n)(Q) and frequencies ~%(Q), where Q is the (two-component) wave vector, l is the layer number, and n = 1, 2 . . . . . 3N. A small number of these phonons are localized to the surface. The Green's function method, on the other hand, considers the vibrations of a crystal which extends semi-infinitely in the t-direction (as opposed to truncating after N layers) [2]. For this reason, the latter approach more aptly describes the crystal in a surface scattering experiment, whose thickness typically exceeds 1 ram. In addition, the Green's function method involves manipulations of matrices which are quite small ( - 10 x 10) when compared to those in slab calculations. Many of these relatively small matrices must be inverted, but the Green's function procedure is usually quicker and more 0039-6028/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L984
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
stable than a slab calculation, which requires the diagonalization of a single large matrix. Despite its apparent analytic and numerical advantages, the Green's function method has been largely neglected because it does not directly yield the phonon eigenvectors and eigenvalues, heretofore irreplaceable inputs to a surface scattering calculation [3]. Instead, it yields the spectral density function
z.,e,, ~ f
~,' (Q)3
*G(Q)]
(1)
n
for an arbitrarily large set of layers l and l'. (In this and all subsequent equations, the subscript a or 13 specifies one of three cartesian directions.) The eigenvectors and eigenvalues of the surface phonons can be trivially extracted from eq. (1) but this is not feasible for most bulk phonons, owing to their continuous frequency bands and to their multiple (roughly threefold) degeneracy. At the outermost layer of a clean surface, bulk phonons and surface phonons contribute to atomic vibrations in comparable amounts [4], as has been seen in helium scattering from Ag(111) [5]. Unlike He atoms, electrons are able to penetrate a few layers before exiting the crystal, thereby further increasing the importance of bulk phonons. This has been seen for EELS from Ni(100) [6], from C u ( l l l ) [7], and from N i ( l l l ) [8]. The main point of this Letter is to show that the spectral density function is in fact more suitable for use in such scattering calculations than the eigenvectors and eigenvalues themselves, for both surface phonons and bulk phonons. The theory for EELS is more advanced than that for helium scattering and will be detailed more thoroughly in this Letter. The adiabatic approximation is used to evaluate the (non-Born) matrix element f(ki, k f) for electron scattering from a crystal with nonequilibrium atomic positions { R }. For scattering from a vibration with wave vector Q = k i - k r , this matrix element is expanded in powers of the displacement u/ [9]:
f ( k i , kf, { R } ) = f ( k i, kf, {R0} ) + Zu~,8~-~, ~, + ½a,81l Y'~,l'l allg'8l"
32f ~UaI~U8
l, q-
. . . .
When ignoring multiphonon processes, this series is truncated after two terms, yielding f ( k i, kr, { R } ) -=-fLEED(ki, kf) q- E u l . h l ( k i , t
kf),
(2)
where fLEED is the low-energy electron diffraction matrix element and
A,,,t(ki, k¢)= 3f/3u,~l is calculated in much the same way as fLEED [10,11]. Only the second term in eq. (2) needs consideration when not at a diffraction condition (i.e., Q 4: G), because fLEED vanishes in this case.
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
L985
The inelastic scattering rate is then calculated using Fermi's golden rule:
l
/
where 1I) and I F ) are the initial and final vibrational states of the crystal and hw = hZ(k 2 - k Z ) / 2 m is the electron energy loss (or gain). The large outer brackets ( . . . )i indicate that the last step of the calculation requires a thermodynamic average of all possible initial vibrational states of the crystal. The calculation proceeds along the same line as that for inelastic neutron scattering [12], by Fourier expanding the delta function: ['(ki, kf) =
~,AalA~l,(~_,(IluatlF)fdte-itt~-(Er-EI)/hl(Flu~z'[1)l otflll'
\
aflll'
\
F
I
F
= E A~,A~cf dt e-i~t(u~tufft,(t)l.
1
(3)
a,Bll"
The bracketed quantity is a time-dependent correlation function whose Fourier transform describes the energy-loss dependence of the cross section [13]. In the harmonic approximation of lattice dynamics, this correlation function can be expressed in terms of the spectral density function:
f d, e-i¢ot =
.
f d, e-i'~t~'nn
e(n)•(n')* al ¢:fll'
~f~-~.~' ([at" + a.] × [ a . , ( t ) + at",(t)])
~(n)~(n)*
= f dt e-i~t Y'~ ~al ~BI" n
= Z -e(n)•(n)* -al c:fll' -
N(~o) + 1
o~
(N(~o.) e -i=-t + [N(~o.) + 1] e i="t )
('On
{ N ( ~ . ) 8 ( w + ~o.) + [N(~o.) + 1] $ ( ~ o - ¢o.))
P,~Bu'(Q, o~),
L986
P. Krupp, B.M. Hall / High resolution electron and helium energy loss cross sections
where N(~0) = [ e x p ( h c o / k B T ) - 1] 1. This yields the final result
F(ki,
kf)-
y, A~,(k,, aflll"
U(~0)+l
kr)A¢,,(ki,
kf)p~/~u,(Q, 0~).
(4)
Because p ( ~ ) is an even function, eq. (4) yields b o t h the energy gain a n d energy loss features, whose ratio equals e x p ( h ~ / k B T ). In eq. (4), A / d e p e n d s only on the details of the interaction between the electron a n d the static lattice (e.g., scattering phase shifts, crystal geometry, work function), whereas p ( ~ ) d e p e n d s only on the details of the lattice d y n a m i c s (e.g., force constants, crystal geometry). A similar expression of the EELS cross section in terms of p ( w ) has b e e n previously derived, b u t it relies o n u n p h y s i c a l z - d e p e n d e n c e of the electron wave f u n c t i o n [4]. Accordingly, the analytical f o r m u l a for A / ( k ~ , k f ) f o u n d therein is n o t expected to accurately describe a typical EELS experiment. A t t e n u a t i o n of the electron wave f u n c t i o n b y other miscellaneous inelastic processes causes A~ to a p p r o a c h zero rapidly with increasing l. F o r surface p h o n o n s , p u , ( ~ ) approaches zero rapidly as l or l ' increases, whereas this is obviously not true for b u l k p h o n o n s . I n practice, the series in eq. (4) converges
== t~ _Q
ii ft
•
I !
I ~ 0
15
ii
""
"' 20
* I
.'"
**
I!
"
I~ 25
30
35
energy loss (meV)
Fig. 1. Single-phonon cross section of 170 eV electrons scattered from Ni(111) with an incidence angle of 70 o and an exit angle of 46 °, which correspond to a momentum transfer of Q = M (a high-symmetry point at the boundary of the (two-dimensional) surface Brillouin zone). Only the energy loss spectrum is shown here. The dashed line depicts the scattering from the top layer, the solid line from the top two layers, and the dotted line from the top five layers. Note that destructive interference between the top two layers essentially eliminates the scattering intensity from the 31 meV surface phonon. The lattice dynamical model used is that of masses connected by linear springs with nearest neighbors only. All of the springs have the same strength except for the ones within the top layer, which are softened by 20%.
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
L987
J
4"--
I
-40
-20 0 20 energy loss (meV)
4O
Fig. 2. Simulated EELS result showing both energy gains and losses, obtained by convolving the curve in fig. 1 with a 6 meV wide Gaussian. after only a few layers. Fig. 1 illustrates this for N i ( l l l ) . The sharp features at 16 meV (Rayleigh mode) and 31 meV ( " g a p " mode) are due to surface phonons, whereas the two broad bands at 18-27 meV and 34-37 meV are due to bulk phonons. For a direct comparison with experiment, these curves should be convoluted with the particular instrument function - typically Gaussian with a 6 meV linewidth. This is shown in fig. 2. Although we demonstrate the results for a very simple system, this method works just as well for surfaces with adsorbates, relaxation, reconstruction, a n d / o r complicated force constants. The only constraint is that the surface must exhibit a two-dimensional periodicity which is commensurate with the monatomic substrate. Along different lines, Bortolani et al. have calculated the single-phonon cross section for helium scattering from Ag(111), a very smooth surface [5]. It takes the form / ' ( k i, k f ) =
N(~o)+ 1 iC(ki, kf)] 2 ~0
× £ I - i B e ~ " ' ( Q ) + Qe~("(Q)128[~0 - % ( Q ) ] ,
(5)
n
where 2 is parallel to Q, C is a complicated function of ki and kf, and B is the "softness parameter", equalling 2.2 ,~-a for A g ( l l 1). Performing the sum reproduces eq. (4), with A t ( k i, k f ) - 8l.oC(k i, k f ) ( - i B 2 + (2). As for EELS, the single-phonon cross section factorizes into two parts; one part depends only on the interaction between the incident particle and the static lattice, and the other part depends only on the lattice dynamics.
L988
P. Knipp, B.M. Hall / High resolution electron and helium energy loss cross sections
In conclusion, a previous EELS calculation is reformulated in terms of correlation functions, rather than vibrational eigenvectors. In the harmonic approximation, these correlation functions can be simply expressed in terms of easily calculable spectral densities. A previous helium scattering calculation is also expressed in terms of spectral density functions. P.K. acknowledges support by the National Science Foundation Materials Research Laboratory at The University of Chicago, and B.M.H. acknowledges support by the US Department of Energy, through Grant No. DE-FG0384ER45083.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
R.F. Allen, G.P. Alldredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1648. J.E. Black, T.S. Rahman and D.L. Mills, Phys. Rev. B 27 (1983) 4072. Y. Chen, Z.Q. Wu, S.Y. Tong and J.E. Black, Surface Sci. 210 (1989) 271. V. Roundy and D.L. Mills, Phys. Rev. B 5 (1972) 1347. V. Bortolani, A. Franchini, F. Nizzoli and G. Santoro, Phys. Rev. Letters 52 (1984) 429. B.M. Hall and D.L. Mills, Phys. Rev. B 34 (1986) 8318. M.H. Mohamed, L.L. Kesmodel, B.M. Hall and D.L. Mills, Phys. Rev. B 37 (1988) 2763. W. Menezes, G. Tisdale, P. Knipp, U. Fano and S.J. Sibener, to be published. H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). C.H. Li, S.Y. Tong and D.L. Mills, Phys. Rev. B 21 (1980) 3057. Mu-Liang Xu, B.M. Hall, S.Y. Tong, M. Rocca, H. Ibach, S. Lehwald and J.E. Black, Phys, Rev. Letters 54 (1985) 1171. L. Van Hove, Phys. Rev. 95 (1954) 249. Although this derivation assumes that the state of the crystal is described by a wave function, eqs. (3) and (4) follow just as easily if one assumes that the crystal is described by a density matrix. To compare directly with experiment, eqs. (3) and (4) must be multiplied by a simple energyand angle-dependent factor. See, e.g., eq. (2.14) of ref. [10].