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Nonlinear X-ray diffraction. Determination of valence electron charge distributions
Volume 12, number 4
15 farmary 1972
CHEMKAL, PHYSICS LETTERS
NONLINEAR X-RAY DIFFRACTION. DETERMINATION OF VALENCE ELECTRON CHARGE D~~~~UT~O~S I. FFUSJND Depmtmenr of d
Bell
Tefephne
Physics,Bm-Z&artUniversity.
Ramor-Gsn,
Imx.4
Laboratories, Murray Hill,‘New Jersey 07973, USA Received I4 September 197 1
A new, nonlinear X-ray diffraction technique is described which permits the direct experimental det~m~atio~ of the valence electron charge density in a wide variety of covalently bonded materials.
We describe here the theory of a new, nor&near X-ray diffraction technique which permits a direct experimental measurement of the valence eIectron charge d~st~bution in many covalently bonded crystals. Determination of this quantity is a long sought goal of chemical crystallography since its knowledge may be expected to significantly enhance OUTpresent understanding of chemical bonding. Conventional linear X-ray diffraction methods are
capableof measuringonly the total electron chargedensity. From this valenceelectron chargeden&y may be obtained by subtraction of cafculated inner shell electron distributions. Two important limitations of such an approach are the uncertainties of the calculated inner sheIl electron densities - especially that of the’shell immediately below the valence shell, and the necessity of performing very highly accurate measurements throughout a large region of reciprocal space, since the valence electrons gene&y make only a comparatively small contribution to measurable structure factors. In contrast, the method described here permits a determination of diffracted intensities which, ic favorable circumstances, are governed almost completely by the valence electron charge density. This new technique is based on the spontaneous parametric down conversion of a source of X-rays interacting with a suitabiy oriented crystal. In this nonlinear process an input quantum, called the pump, at the irequenw up, decays into two output quanta, called the signal and idler, at the frequencies os and wil respectively, such that the photon energies are conserved, i.e.,
9
(1)
=ClJs+ti.
?Ite theory of spontaneous parametric down conversion has been discussed by Kleinman 111. Although this work is explicitly for the optical region where the phenomenon is well known f2] , its essential features are also applicable to the X-ray region: quantum noise at the signal and idler frequencies produces In the nonlinear cryseaf a fluctuating sum frequency polarization, Pp. which interacts with the input field, Ep, via J
V
dr
s
Pp.dEp. lhe integration
over the crystal volume requires that the photon momenta
be conserved.
Writing the pump, signal, and idler fields as plane waves of the form
and making use of the periodicity of the crystal, the requirement of momentum conservation may be met by 583
CHEMICALPHYSICSLETTERS
Volume 12,.number4 any of the re+xocd
-pW&
IS fanuary 1972
lattice vectors of the crystal, ~~~~~, such that
kp = k, + k, + Q(hkl).
(2)
Eq. (2) is the law of no&near diffraction 133 appropriate to the present discussion. Ifo, is in the X-ray region, eq. (I) requires that one of the output quanta, e.g. the signal, also be an X-ray quantum. The idler frequency however, can, by a suitable choice of orierrtation of Q(W) relative to kp, be made to occur throughorit the remainder of the electromagnetic spectrum. As the idler frecjuency is varied the underlying physical mechanism of the nonlinear interaction is altered, so that different physical properties of the crystal are sampled in different frequency intervals. This unique feature is of key importance. When the idler and-signal frequencies are of the same order, the nonlinear interaction is governed by the total electron charge density {4,5]. When the idler frequency is in the optical region - below the ultraviolet band gap but above the infrared reststr& - the nonlinear int&raction is determined by the redistri’oution of the electronic charge density caused by an optical field [6{ (IThis is essentially a valence clectrvn property but is not yet the valence electron charge densjty itself. What we demonstrate here is that when the idler frequency lies above that of the ultraviolet band gap, but below the lowest absorption edge of the inner she!! electrons, the no&near interaction is dominated by the charge distribution of the vaience electrons. A formal quantum-mechanical expression, convenient to our present purpose, for the coherent, nonlinear response pf a medium to applied electrotiagnetic fields has been given by Armstrong et al. 17). Their treatment neglects focal fiefd effects which, while unwarranted in the optical region, is an adequate approbation here since in the frequency region above the ultraviolet band gap the linear susceptibility of the medium is small. Extracting from the full expression those terms that are of importance for the present combination of frequencies, and explicitly generalizing to a3i M electron system, w-Awrite for the sum ~re~~~ncy p~~r~~~i~n, PI,,
where the subscripts II and b refer to electrons, Vb operates only on functions of the coordinates of electron b, the sum on j runs over all excited states of the system, and the left subscript 1 refers to the projection of the corresprznding vector normal to kP; only this part of PP can interact with a plane wave pump. We now introduce the approximation that the states of the system can be adequately described by a self consistent field, Hartree-Fock wavefunction- Our purpose in this is simply to enable the separation of eq. (3) Into large and small ~ontxibutions corresponding to the division of the M electrons into ZWOgroups, ‘Wence” and inner “core”. Any other approach that makes this same conceptual distinction will lend to the same conc!usiolLS. Within this framework the states Ii> are each a sum of M! electron products lj> =
(l/(.M!yq I d-qq)
- . _$Jfm(rn&f)[ *
where the orbitais #VI are sofutions
W#o’= ) [R+ + Eo]@, and form a complete c i
= 1
t$Ah#q
orthonormd set 584.
”
of a one-electron Schrtidinger equation
(4)
CHEMICALPHYSICSLETTERS
Volume 12, number 4
15 Ianuw 1972 (7)
Because only one-electron ordinates is diagonalized. ly enables us to write the estimate the contribution
operators appear in the matrix elements of eq. (3), the double sum over eiectron coThis, together with the neglect of valence electron-core electron exchange, immediatenonlinear polarization as a sum of contributions irom core and valence electrons. We of the core electrons by introducing the approximation
where hS1, is the energy of the lowest absorption edge of the core electrons; this leads to an upper: I@nit. The oscillator strengths of the valence electron transitions are concentrated near the ultraviolet band gap, Eg, so that here we may make the approximation
Writing the nonlinear polarization as
(IQ)
IPP = NG(hkl) E,Eiq where N is the number density of unit cells and G(hkl) the vector nonlinear stmcture factor, we have G(hkZ) = -_(ie3/2m2wpw~w,),&
[iri*Q(hti)] [Fv(hkl) - a((Oi/.Rc)zF,(hkl)~].
(11)
Here di = Ej/Ei, F,(hkl), which includes dispersion terms, is the linear structure factor of the valence electrons. and F,(hkZ) the corresponding quantity for the core,electrons. Since (Wi/s2,;’ may be assumed smali, the core contribution enters only as a small correction. When necessary, its magnitude may be estimated from theory?, the known total linear structure factor, or the dependence on Oi of the ( ) bracket in eq. (1 I)_ Eq. (11) is the key result of this letter. Its derivation, however, is strongly dependent upon the adequacy of the approximation made in eq. (9)_ Because of this we re-exnmine the physical implications of transforming For the valence electrons the matrix elements in eq. (3) into the linear structure factor. It may be shown [B] that the matrix elements in eq. (3) arise in perturbation theory as
s
dr exp [i(&--kp)
- r]Sp(r,Ei),
V
where 8p(r,Ei) is the change in the electronic charge density of the medium produced by an applied field [G]. Ihe only approximation made here is that local field effects at the short X-ray wavelengths corresponding to wP and w, may be neglected, If the frequency Wi is much greater than any of the internal modes (hound states) of the medium, the motion of one part of the charge distribution’when being driven by the oscillating field Ei cannot be transmitted to any other part; each region oscillates independently. Since the ratio of electronic charge density f For small Q(IzkT)for the K shell, for example, Fc(hkl)/st2e-c 2/nfC. In this limit, however, a precise estimation of eq. (3) is possible. If the photoionization cross section of the K absorption edge is asaJmed to vary as X3, then Fc(hkl)/~Z$ is replaced by g~/2&, where go is the total K-shell oscillator strength. Typically go = 1.4, so that the K shell contribution is $ as large as
indicated in eq. (3). Similar considerations also hold for the other cores. The matrix elements describing the core confributiun hay be recognized as being simply proportional to the dielectric function l (Q.0) of the core electrons and hence decrease more rapidly with hkl than does the iincar structure factor. Because of this the core contribution which. overall, probably decreases more slawly with increasinghkl thti does the vaIenceelectron structure factor, is not expec’d to becorze very important until large vatues of hk[ are attained. At this point both structure factors have, in any event, been reduc& to very SIX& v+lues.
Volume 12, number 4
CHEMICAL PHYSICS LETTERS
15 Jam&y ‘.
Valence- and coreelectron
Table 1 binding energies for various elements
Element
c/v
E&V)
E&V)
6
c
Is/25
14.5
N 0
284 403 536
19.5
7
26
15.5
F
690
32 40 11.3 15 19 22 25.3
16.8 17.3 6.8 6.9 7.3 7.6 8.1
z
8 9 13 14 15 16 17
Al Si ? S Cl
2~i3s
1972
.
77 103 139 168 204
&I&
to electronic mass density is a constant (= e/m) each volume element of charge is given the same acceleration as every other, and hence undergoes the same displacement, 6r(Ei). Thz charge distribution thus executes simple rigid-body motion in response to the driving field and Gp(rJTEj)may be written 6p(r,Ei) = -6f (Ei) ‘rJp(r),
(12)
where p(r) is the unperturbed charge distribution. Eq. (12) then immediately leads back to eq. (11) with the displacement Sr(Ei) given by eGr(Ei)=
~e2/mwi;>iEi.
(13)
We consider that when {IGJiis several times the binding energy of the vaience electrons, eq. (12) is an excellent approximation and the use of eq. (11) is justified. We must now, however, examine the question of whether this condition can be met simultaneously with the additional requirement that Wi/~;t, be small. In table 1 we list the ratio E,/Ev- the core binding energy/the valence electron binding energy - for a number of elements of chemical interest. ‘Ihe binding energies employed, taken from appendix 2 of Siegbahn et al. [9], are relative to the vacuum state. In insulating crystals the core binding energies are otiy slightly changed but the valence electron binding energies relative to the conduction band are usually significantly reduced. Our conclusions based on this table are conservative. A ratio of EJE,,of order ten or greater is most desirable, since Wi can then be chosen to make correction terms only of order 10% or less of F,(hkl). For elements much heavier than Cl, E,/E, appears to be too small to be useful. We emphasize that it is mere important to make the ratio Rwi/E,sufficiently large so as to ensure the validity of eq. (121, than to make ~iWilEc small so as to eliminate F,(M) in eq. (11). Although our method is limited to relatively light elements, the entries in table 1 represent many of the most important covalently-bbnding substances. We turn now to a consichzration of how experiments designed to measure F#kl) may be performed. In essence such experiments proceed 3s does any linear Bragg diffraction measurement. The important differences are the small frequency shift of the diffracted beam (w,) relative to the incoming beam (up), the small, but essential, difference in crystal orientation, and the reduced signal level. We discuss each of these factors in turn. The requirement that hwi/Ev bb sufficiently large can be adequately met, we believe; by making this ratio approtiately qual to three. Since typic&values of E, are lo-20 eV, this places the idler wavelength in the neighborhood of 200 i& With a 1 A pump, the ptimp-signal frequency shift is =?4%, so that only moderate’spectral resolution is required. .@l permissible crystal orientations for a given choice of CyhW), Wi, and oF may be computed with the aid of elementary geometry from eqs. (1) and (2). ‘I&angle between k, and ki, however, is a very important variable, and the’actual crystal orientation should be based upor. this parameter. It was shown in’ref. ,161, fig. 1,
Volume 12, number 4
CHEMICAL PHYSICS LE?TERS
15 Janua.ry 1972
that crienting the cry&l such that k, and ki are antiparallel, yields the narrowest signal spectrum into the iargest solid angle. This is highly desirable since it permits the signal to appear as a pronounced peak on whatever background may be present. For this orientation Kleinman’s function [l] , K[s,i), ciefined by K(s,i)= CIS’(Vs-Vi)l-‘,
(I4)
where s’is a unit vector along k,, and vi is the group velocity at Oj, is, neglecting all dispersion,,K(s,i) ~4. Thd variation of the signal wavelength with angle, 6, about this direction is, again neglecting all d@&sion, /Ahs/h,l z $(hi/hs)b’
(15)
Taking as typical values consistent with:the idler attenuation Ah,/& = low3 and Ai/Xs= 200, we find for the maximum permissible full angle 26,, 2: OS”, so that the solid angle into which an acceptable signal is emitted is x 10m4sr. A% We turn now to a computation of the magnitude of the signal output. The radiated signal power is, of course, proportional to N2G(hkZ)-G*(I~kT), where N is the number density of unit cells. In addition, this quantity must be summed over all interacting states of polarization of the pump, signal, and idler fields [lo]. Using the facts that the crystal orientation differs only slighhtlyfrom Bragg’s angle, Bg, and wp = ws = wx, we find for an unpolarized pump,
With P, the signal power, Pp the input pump power, and f? the effective crystal length, we may, following ECleinman [I] , write
where Q is the fine-structure constant and ro the classical electron radius. Since the frequency shift of the signa is, in general, of the same order as the Compton shift, we must also concern ourselves with the radiated Compton power, PG. For this we have [ 1 i] ?JARgPp
= Nr; [Z- Z(W)] [;(l + cos22QB)] ,
(18)
where Z- Z(hkl) is the effective number of electrons that contribute to the Compton scattering [i2] . We evahate eqs. (17) and (18) for the (002) reflection of LiF. From the work of Stewart [13] we may infer that F,(OO2) = 4.4 for the F-ion. Choosing X, = 1.54 .& (Cu Ka), and Xi = 200 A, and neglecting the Li+ ion, we have for eq. (13) PsiAapp = 3.9 X 10-g. From ref. [ 121 we fiid Z- Z(OO2)for LiF = 5.0, then PC/As2&?Pp = 179 X iUm4. For these parameters the Cumpton shift, Ak, is A& = 72 X 10-j A. while the signal shift is A,-$, = 11.9 X
10-j A, so that the signaloverlapsthe Comptonpeak, In generalthe spectralwidth of the Comptott
profit
is of
order Ah while that of the signal is determined by AS2,, eq. (IS). For AQs = 104sr, the signaI spectral width is 1 less than Fg the Compton width, so that the signal appears as a very distinct peak on the side of the Compton profde. The situation in this regard appears veiy similar to that which exists in the case of X-ray Raman scattering as reported by Das Gupta [14]. Measurement of F,(hkl) may be seen to be very similar in degree of difficulty to these experiments. Since a number of substances of special interest here have already been studied by Das Gupta, and since it is apparent that significant-improvements in instrumentation are possible, we anticipate that measurements of the valence electron charge distribution are capable of being routinely performed for a wide variety of materials. 587
:
: Volums i2,number _ ‘. ‘,
..Refltetices”.
CHEMiCAL PHYSICS LETTERS
4 ,,
.,
,,:. .,
..,
:
.‘. (2) .f3j
RA ?Jyeqand S,E, I@ris,Phys, 1; Freund, Phys. Rev. Letters 21
l! &iary
1972
.
:_’
.:--.~-f~] D_A. Kfeinman. Phys_ Rev. 174 (1968)‘1027.
”
..
.’
Rev,168 (1168) 1064.
. ..
(1968) 1404. [4f I. Freund and B.F. Levine, Phys. Rev. Letters 23 (1969) 654. [5] I. Freund and RF. Levine, Opt. Commun. 3 (1971) 101. ’ [6] 1. Freivtd and B.F. Levine, Phys. Rev. Letters 25 (1970) 1241; P:hI. J$enberger and S.L. McCall, Phys. Rev. 3A (1971) 1145. [?] 3.A. Annnrong, N. BIoembergen, J. Ducuing and P.S. Fe&an, Phys. Rev. 127 (1962) 1918,eqs. (2.8)~(2.10). [a] I. Freund and B.F. Levine, to be published, f93 K. sjegbahn et al.; ESCA (Atmquist and W&sells, Uppsala, 1967) appendix 2. IlO] 1. Freund and B.F..Levine, Phys. Letters 31A (1970) 456. [ 11 J R.W. James, OpticaJ,principles of the diffraction of X-rays (Cornell Univ. Press, Ithaca, 1965) ch, 3, 1121 ~&I&X& tiblesfor X-ray aystilogiaphy, Vol. 3 (Kynoch Press, ~~~~~, 1965) section 3.4, ‘[13] R,.F. Stewart, J.Chem. Phys. 53 (1970) 205. [14] .K. Das Gupta, Phys. Rev. Letters 3 (1959) 38; Phys. Rev. 128 (1962) 2181.
15 farmary 1972
CHEMKAL, PHYSICS LETTERS
NONLINEAR X-RAY DIFFRACTION. DETERMINATION OF VALENCE ELECTRON CHARGE D~~~~UT~O~S I. FFUSJND Depmtmenr of d
Bell
Tefephne
Physics,Bm-Z&artUniversity.
Ramor-Gsn,
Imx.4
Laboratories, Murray Hill,‘New Jersey 07973, USA Received I4 September 197 1
A new, nonlinear X-ray diffraction technique is described which permits the direct experimental det~m~atio~ of the valence electron charge density in a wide variety of covalently bonded materials.
We describe here the theory of a new, nor&near X-ray diffraction technique which permits a direct experimental measurement of the valence eIectron charge d~st~bution in many covalently bonded crystals. Determination of this quantity is a long sought goal of chemical crystallography since its knowledge may be expected to significantly enhance OUTpresent understanding of chemical bonding. Conventional linear X-ray diffraction methods are
capableof measuringonly the total electron chargedensity. From this valenceelectron chargeden&y may be obtained by subtraction of cafculated inner shell electron distributions. Two important limitations of such an approach are the uncertainties of the calculated inner sheIl electron densities - especially that of the’shell immediately below the valence shell, and the necessity of performing very highly accurate measurements throughout a large region of reciprocal space, since the valence electrons gene&y make only a comparatively small contribution to measurable structure factors. In contrast, the method described here permits a determination of diffracted intensities which, ic favorable circumstances, are governed almost completely by the valence electron charge density. This new technique is based on the spontaneous parametric down conversion of a source of X-rays interacting with a suitabiy oriented crystal. In this nonlinear process an input quantum, called the pump, at the irequenw up, decays into two output quanta, called the signal and idler, at the frequencies os and wil respectively, such that the photon energies are conserved, i.e.,
9
(1)
=ClJs+ti.
?Ite theory of spontaneous parametric down conversion has been discussed by Kleinman 111. Although this work is explicitly for the optical region where the phenomenon is well known f2] , its essential features are also applicable to the X-ray region: quantum noise at the signal and idler frequencies produces In the nonlinear cryseaf a fluctuating sum frequency polarization, Pp. which interacts with the input field, Ep, via J
V
dr
s
Pp.dEp. lhe integration
over the crystal volume requires that the photon momenta
be conserved.
Writing the pump, signal, and idler fields as plane waves of the form
and making use of the periodicity of the crystal, the requirement of momentum conservation may be met by 583
CHEMICALPHYSICSLETTERS
Volume 12,.number4 any of the re+xocd
-pW&
IS fanuary 1972
lattice vectors of the crystal, ~~~~~, such that
kp = k, + k, + Q(hkl).
(2)
Eq. (2) is the law of no&near diffraction 133 appropriate to the present discussion. Ifo, is in the X-ray region, eq. (I) requires that one of the output quanta, e.g. the signal, also be an X-ray quantum. The idler frequency however, can, by a suitable choice of orierrtation of Q(W) relative to kp, be made to occur throughorit the remainder of the electromagnetic spectrum. As the idler frecjuency is varied the underlying physical mechanism of the nonlinear interaction is altered, so that different physical properties of the crystal are sampled in different frequency intervals. This unique feature is of key importance. When the idler and-signal frequencies are of the same order, the nonlinear interaction is governed by the total electron charge density {4,5]. When the idler frequency is in the optical region - below the ultraviolet band gap but above the infrared reststr& - the nonlinear int&raction is determined by the redistri’oution of the electronic charge density caused by an optical field [6{ (IThis is essentially a valence clectrvn property but is not yet the valence electron charge densjty itself. What we demonstrate here is that when the idler frequency lies above that of the ultraviolet band gap, but below the lowest absorption edge of the inner she!! electrons, the no&near interaction is dominated by the charge distribution of the vaience electrons. A formal quantum-mechanical expression, convenient to our present purpose, for the coherent, nonlinear response pf a medium to applied electrotiagnetic fields has been given by Armstrong et al. 17). Their treatment neglects focal fiefd effects which, while unwarranted in the optical region, is an adequate approbation here since in the frequency region above the ultraviolet band gap the linear susceptibility of the medium is small. Extracting from the full expression those terms that are of importance for the present combination of frequencies, and explicitly generalizing to a3i M electron system, w-Awrite for the sum ~re~~~ncy p~~r~~~i~n, PI,,
where the subscripts II and b refer to electrons, Vb operates only on functions of the coordinates of electron b, the sum on j runs over all excited states of the system, and the left subscript 1 refers to the projection of the corresprznding vector normal to kP; only this part of PP can interact with a plane wave pump. We now introduce the approximation that the states of the system can be adequately described by a self consistent field, Hartree-Fock wavefunction- Our purpose in this is simply to enable the separation of eq. (3) Into large and small ~ontxibutions corresponding to the division of the M electrons into ZWOgroups, ‘Wence” and inner “core”. Any other approach that makes this same conceptual distinction will lend to the same conc!usiolLS. Within this framework the states Ii> are each a sum of M! electron products lj> =
(l/(.M!yq I d-qq)
- . _$Jfm(rn&f)[ *
where the orbitais #VI are sofutions
W#o’= ) [R+ + Eo]@, and form a complete c i
= 1
t$Ah#q
orthonormd set 584.
”
of a one-electron Schrtidinger equation
(4)
CHEMICALPHYSICSLETTERS
Volume 12, number 4
15 Ianuw 1972 (7)
Because only one-electron ordinates is diagonalized. ly enables us to write the estimate the contribution
operators appear in the matrix elements of eq. (3), the double sum over eiectron coThis, together with the neglect of valence electron-core electron exchange, immediatenonlinear polarization as a sum of contributions irom core and valence electrons. We of the core electrons by introducing the approximation
where hS1, is the energy of the lowest absorption edge of the core electrons; this leads to an upper: I@nit. The oscillator strengths of the valence electron transitions are concentrated near the ultraviolet band gap, Eg, so that here we may make the approximation
Writing the nonlinear polarization as
(IQ)
IPP = NG(hkl) E,Eiq where N is the number density of unit cells and G(hkl) the vector nonlinear stmcture factor, we have G(hkZ) = -_(ie3/2m2wpw~w,),&
[iri*Q(hti)] [Fv(hkl) - a((Oi/.Rc)zF,(hkl)~].
(11)
Here di = Ej/Ei, F,(hkl), which includes dispersion terms, is the linear structure factor of the valence electrons. and F,(hkZ) the corresponding quantity for the core,electrons. Since (Wi/s2,;’ may be assumed smali, the core contribution enters only as a small correction. When necessary, its magnitude may be estimated from theory?, the known total linear structure factor, or the dependence on Oi of the ( ) bracket in eq. (1 I)_ Eq. (11) is the key result of this letter. Its derivation, however, is strongly dependent upon the adequacy of the approximation made in eq. (9)_ Because of this we re-exnmine the physical implications of transforming For the valence electrons the matrix elements in eq. (3) into the linear structure factor. It may be shown [B] that the matrix elements in eq. (3) arise in perturbation theory as
s
dr exp [i(&--kp)
- r]Sp(r,Ei),
V
where 8p(r,Ei) is the change in the electronic charge density of the medium produced by an applied field [G]. Ihe only approximation made here is that local field effects at the short X-ray wavelengths corresponding to wP and w, may be neglected, If the frequency Wi is much greater than any of the internal modes (hound states) of the medium, the motion of one part of the charge distribution’when being driven by the oscillating field Ei cannot be transmitted to any other part; each region oscillates independently. Since the ratio of electronic charge density f For small Q(IzkT)for the K shell, for example, Fc(hkl)/st2e-c 2/nfC. In this limit, however, a precise estimation of eq. (3) is possible. If the photoionization cross section of the K absorption edge is asaJmed to vary as X3, then Fc(hkl)/~Z$ is replaced by g~/2&, where go is the total K-shell oscillator strength. Typically go = 1.4, so that the K shell contribution is $ as large as
indicated in eq. (3). Similar considerations also hold for the other cores. The matrix elements describing the core confributiun hay be recognized as being simply proportional to the dielectric function l (Q.0) of the core electrons and hence decrease more rapidly with hkl than does the iincar structure factor. Because of this the core contribution which. overall, probably decreases more slawly with increasinghkl thti does the vaIenceelectron structure factor, is not expec’d to becorze very important until large vatues of hk[ are attained. At this point both structure factors have, in any event, been reduc& to very SIX& v+lues.
Volume 12, number 4
CHEMICAL PHYSICS LETTERS
15 Jam&y ‘.
Valence- and coreelectron
Table 1 binding energies for various elements
Element
c/v
E&V)
E&V)
6
c
Is/25
14.5
N 0
284 403 536
19.5
7
26
15.5
F
690
32 40 11.3 15 19 22 25.3
16.8 17.3 6.8 6.9 7.3 7.6 8.1
z
8 9 13 14 15 16 17
Al Si ? S Cl
2~i3s
1972
.
77 103 139 168 204
&I&
to electronic mass density is a constant (= e/m) each volume element of charge is given the same acceleration as every other, and hence undergoes the same displacement, 6r(Ei). Thz charge distribution thus executes simple rigid-body motion in response to the driving field and Gp(rJTEj)may be written 6p(r,Ei) = -6f (Ei) ‘rJp(r),
(12)
where p(r) is the unperturbed charge distribution. Eq. (12) then immediately leads back to eq. (11) with the displacement Sr(Ei) given by eGr(Ei)=
~e2/mwi;>iEi.
(13)
We consider that when {IGJiis several times the binding energy of the vaience electrons, eq. (12) is an excellent approximation and the use of eq. (11) is justified. We must now, however, examine the question of whether this condition can be met simultaneously with the additional requirement that Wi/~;t, be small. In table 1 we list the ratio E,/Ev- the core binding energy/the valence electron binding energy - for a number of elements of chemical interest. ‘Ihe binding energies employed, taken from appendix 2 of Siegbahn et al. [9], are relative to the vacuum state. In insulating crystals the core binding energies are otiy slightly changed but the valence electron binding energies relative to the conduction band are usually significantly reduced. Our conclusions based on this table are conservative. A ratio of EJE,,of order ten or greater is most desirable, since Wi can then be chosen to make correction terms only of order 10% or less of F,(hkl). For elements much heavier than Cl, E,/E, appears to be too small to be useful. We emphasize that it is mere important to make the ratio Rwi/E,sufficiently large so as to ensure the validity of eq. (121, than to make ~iWilEc small so as to eliminate F,(M) in eq. (11). Although our method is limited to relatively light elements, the entries in table 1 represent many of the most important covalently-bbnding substances. We turn now to a consichzration of how experiments designed to measure F#kl) may be performed. In essence such experiments proceed 3s does any linear Bragg diffraction measurement. The important differences are the small frequency shift of the diffracted beam (w,) relative to the incoming beam (up), the small, but essential, difference in crystal orientation, and the reduced signal level. We discuss each of these factors in turn. The requirement that hwi/Ev bb sufficiently large can be adequately met, we believe; by making this ratio approtiately qual to three. Since typic&values of E, are lo-20 eV, this places the idler wavelength in the neighborhood of 200 i& With a 1 A pump, the ptimp-signal frequency shift is =?4%, so that only moderate’spectral resolution is required. .@l permissible crystal orientations for a given choice of CyhW), Wi, and oF may be computed with the aid of elementary geometry from eqs. (1) and (2). ‘I&angle between k, and ki, however, is a very important variable, and the’actual crystal orientation should be based upor. this parameter. It was shown in’ref. ,161, fig. 1,
Volume 12, number 4
CHEMICAL PHYSICS LE?TERS
15 Janua.ry 1972
that crienting the cry&l such that k, and ki are antiparallel, yields the narrowest signal spectrum into the iargest solid angle. This is highly desirable since it permits the signal to appear as a pronounced peak on whatever background may be present. For this orientation Kleinman’s function [l] , K[s,i), ciefined by K(s,i)= CIS’(Vs-Vi)l-‘,
(I4)
where s’is a unit vector along k,, and vi is the group velocity at Oj, is, neglecting all dispersion,,K(s,i) ~4. Thd variation of the signal wavelength with angle, 6, about this direction is, again neglecting all d@&sion, /Ahs/h,l z $(hi/hs)b’
(15)
Taking as typical values consistent with:the idler attenuation Ah,/& = low3 and Ai/Xs= 200, we find for the maximum permissible full angle 26,, 2: OS”, so that the solid angle into which an acceptable signal is emitted is x 10m4sr. A% We turn now to a computation of the magnitude of the signal output. The radiated signal power is, of course, proportional to N2G(hkZ)-G*(I~kT), where N is the number density of unit cells. In addition, this quantity must be summed over all interacting states of polarization of the pump, signal, and idler fields [lo]. Using the facts that the crystal orientation differs only slighhtlyfrom Bragg’s angle, Bg, and wp = ws = wx, we find for an unpolarized pump,
With P, the signal power, Pp the input pump power, and f? the effective crystal length, we may, following ECleinman [I] , write
where Q is the fine-structure constant and ro the classical electron radius. Since the frequency shift of the signa is, in general, of the same order as the Compton shift, we must also concern ourselves with the radiated Compton power, PG. For this we have [ 1 i] ?JARgPp
= Nr; [Z- Z(W)] [;(l + cos22QB)] ,
(18)
where Z- Z(hkl) is the effective number of electrons that contribute to the Compton scattering [i2] . We evahate eqs. (17) and (18) for the (002) reflection of LiF. From the work of Stewart [13] we may infer that F,(OO2) = 4.4 for the F-ion. Choosing X, = 1.54 .& (Cu Ka), and Xi = 200 A, and neglecting the Li+ ion, we have for eq. (13) PsiAapp = 3.9 X 10-g. From ref. [ 121 we fiid Z- Z(OO2)for LiF = 5.0, then PC/As2&?Pp = 179 X iUm4. For these parameters the Cumpton shift, Ak, is A& = 72 X 10-j A. while the signal shift is A,-$, = 11.9 X
10-j A, so that the signaloverlapsthe Comptonpeak, In generalthe spectralwidth of the Comptott
profit
is of
order Ah while that of the signal is determined by AS2,, eq. (IS). For AQs = 104sr, the signaI spectral width is 1 less than Fg the Compton width, so that the signal appears as a very distinct peak on the side of the Compton profde. The situation in this regard appears veiy similar to that which exists in the case of X-ray Raman scattering as reported by Das Gupta [14]. Measurement of F,(hkl) may be seen to be very similar in degree of difficulty to these experiments. Since a number of substances of special interest here have already been studied by Das Gupta, and since it is apparent that significant-improvements in instrumentation are possible, we anticipate that measurements of the valence electron charge distribution are capable of being routinely performed for a wide variety of materials. 587
:
: Volums i2,number _ ‘. ‘,
..Refltetices”.
CHEMiCAL PHYSICS LETTERS
4 ,,
.,
,,:. .,
..,
:
.‘. (2) .f3j
RA ?Jyeqand S,E, I@ris,Phys, 1; Freund, Phys. Rev. Letters 21
l! &iary
1972
.
:_’
.:--.~-f~] D_A. Kfeinman. Phys_ Rev. 174 (1968)‘1027.
”
..
.’
Rev,168 (1168) 1064.
. ..
(1968) 1404. [4f I. Freund and B.F. Levine, Phys. Rev. Letters 23 (1969) 654. [5] I. Freund and RF. Levine, Opt. Commun. 3 (1971) 101. ’ [6] 1. Freivtd and B.F. Levine, Phys. Rev. Letters 25 (1970) 1241; P:hI. J$enberger and S.L. McCall, Phys. Rev. 3A (1971) 1145. [?] 3.A. Annnrong, N. BIoembergen, J. Ducuing and P.S. Fe&an, Phys. Rev. 127 (1962) 1918,eqs. (2.8)~(2.10). [a] I. Freund and B.F. Levine, to be published, f93 K. sjegbahn et al.; ESCA (Atmquist and W&sells, Uppsala, 1967) appendix 2. IlO] 1. Freund and B.F..Levine, Phys. Letters 31A (1970) 456. [ 11 J R.W. James, OpticaJ,principles of the diffraction of X-rays (Cornell Univ. Press, Ithaca, 1965) ch, 3, 1121 ~&I&X& tiblesfor X-ray aystilogiaphy, Vol. 3 (Kynoch Press, ~~~~~, 1965) section 3.4, ‘[13] R,.F. Stewart, J.Chem. Phys. 53 (1970) 205. [14] .K. Das Gupta, Phys. Rev. Letters 3 (1959) 38; Phys. Rev. 128 (1962) 2181.