Frequency Moments: Fine Structure Of X-Ray Spectra In Metals And Other Applications

Fine structure of the edges of X-ray lines (absorption, emission, photoemission) in metals is admitted to frequently possess an asymp-totic power law singularity I(!) ! A(! ?! edge) ? on the line threshold due to the analogue of the infrared catastrophe. This happens due to the quasicontinuity of energy spectra of metals. In this paper the leading term of this power law's exponent is combined with rst exact frequency moments of the relevant spectrum. The moments are obtained from an exact explicit expression and the use is made of the results of the power moments' problem on the half-axis to restore approximate line forms beyond the edge asymptotic region: i.e. to nd the so-called generalized power law I(!) = C! ?(!). Applying the method to a model alkali metal we show that the postasymptotic form strongly depends on the long-range asymptotics of the potential of a core hole (where an electron excited by a quantum has been). Later on Tchebychee-Markov inequality is shown which enables us to nd some parameters of spectra with a number of absorption and nonabsorption regions. This method also requires only a few rst moments. It can be applied to spectra for wide frequency regions.


Introduction
Fine structure of the lines of X-ray spectra in metals has been regarded as an exquisite problem of the many-body quantum theory for the last 30 years { since the edge forms of them were found by Mahan and proved analytically by Nozieres and DeDominicis (MND) 1] to be conspicuous.The set of atomic spectral lines of deep electronic levels (not collectivized in the condensed state) for metals is transformed into a set of the following complexes: a threshold (edge) of each line on \intensity-frequency" plots is followed by a curve with the asymptotical form I(!) = C 0 j!j for absorption (! > 0) emission (! < 0)], and for photoemission { P(?!) = c(?!) ?1 ; !> 0; ! is measured from the corresponding thresholds (see gure 1, for example).
This form is called a power law (PL).Obviously cases of < 0 and < 1 correspond to a threshold singularity.The values of the exponents , were proved to depend on the speci c structure of a metal (which includes the atomic core structure).Later on Anderson, Friedel and Hop eld 1] explained the physical core of this essentially many-particle problem, generally considered on the basis of the one-particle approximation.
The model suggested to describe the process implies that an incident quantum interacts with a deep atomic residue level instantaneously, i.e. at t = t 0 = 0 (t { time measured from the moment when \quantum { 180 A.G.Salistra system" interaction began, t 0 { the moment when con guration is believed to settle locally) an electron is assumed to have been already excited and a hole exists where the electron has been.In fact, t 0 is nite, but much smaller than any time parameter substantial for this problem (frequencies !> !plasma ).The period of time when all the interactions with the external eld and the creation of the hole took place is therefore neglected, only their result is taken into account: the potential of the hole (core-hole potential) is V .It is also supposed that this dV=dt = 0: the potential does not change thereafter.Thus, too slow in uences { lattice, surface, dynamic screening (! !plasma ) { are excluded too.Despite these limitations, the model explains experimental data for prethreshold regions adequately.Of course, semiphenomenological nature of V is helpful, V being easily adjusted to t experimental plots.The potential's sudden switching shifts the mean eld of the system and thus perturbs its energy spectrum, mainly, of course, the conduction band.This means that the quantum energy is not absorbed completely by the deep level electron being excited, a part of the energy is taken to perturb the background electron density.Similar ideas apply, for example, to a manyelectron atom, but the latter has no Fermi surface with thermodynamically in nite number of particles on it.The analogue of the infrared catastrophe happens here: one-particle energy spectrum of a metal is quasicontinuous; that is, almost no energy is required to excite a particle on the Fermi surface.Thus, any amount of \spare" energy (not taken to excite a core electron) can create, however, many one-particle excitations, just as it happens to infrared photons in QED.This formal in nity manifests itself here as observable { the conduction band response follows ( t) ?(t { time) instead of e ?t that one may expect for relaxation.The latter corresponds to a nite number of one-particle transitions across a nite energy gap.
Frequency moments and X-ray spectra in metals 181 This results in I / R t ?exp(?i!t)dt / !?1 .Here depends on the core-hole potential and one-particle states density on the Fermi surface (see (3)).
It may be worth mentioning that, though an apparently singular form had been seen on I(!)-plots much earlier, no idea of edge singularity occured to researchers, besides the values of and could be measured only with a synchrotron as a radiation source.One of the issues which are of interest here is the problem of the generalized power law (GPL), i.e. the line form beyond its threshold asymptotical region, where the simple power law fails to hold, but the model is still applicable.Generally GPL is represented as c! (!)?1 for photoemission or (!) for absorption (emission)].
In this work a method of nding the GPL for photoemission spectra is presented (it can be easily adjusted to the absorption-emission case).The method is based on a general exact expression for a spectrum form 2]. Despite being general, explicit, one-particle, independent of wave basis etc., its direct application is hardly possible, only singular asymptotics can be found.However, its form enables us to obtain expressions for frequency moments 3].
In section 2 we shall restore the GPL by making use of the rst four moments and the known edge asymptotics.We shall apply the expression to a model alkali metal and phenomenological core-hole potentials.It will be shown how line forms depend on long range asymptotics of the potentials.
In section 3 another application of the moments method is presented.Unlike the method in section 2, this one allows to nd characteristics of spectra in wide frequency regions (regions of absorption and nonabsorption), but not very suitable for speci c forms of single lines.

The generalized power law
We take the exact explicit expression for absorption (emission) intensity 2] (thereafter atomic units are used): I(!) = Z 1 0 P(t)I 0 (t) exp(?i!t)dt; (1) I P ( ) = RP(!= W ? ); R; W = const: (2) Here the Laplace image of the rst factor in the integrand describes the photoemission intensity as (2), where R and W are xed for a speci c line (transition).That is, the line form for photoemission is described by the Laplace-transform of P(t) with the reversed !-axis and shifted zero.In what follows we shall refer to P(!) as a photoemission spectrum.P(t) and I 0 (t) are built on the matrix elements of H 0 and H, one-particle hamiltonians of the initial and nal states respectively, H = H 0 + V , where V is a oneparticle core-hole potential, I 0 also contains transition matrix elements (e.g.dipole D ik ); the expressions for them are quite cumbersome (in 3] there are also the rst four frequency moments of I and P).
Discussing, for instance, photoemission one uses a perturbation series in V .Then up to the second order in V when t ! 1 Here summation runs over one-particle states, the Nth level divides free and occupied states at T = 0, i.e.E i 6 E F ; E k > E F , is Euler constant.
The values of and F should be calculated numerically: where D is the distance \lower edge of the conduction band { Fermi level", D { \upper edge of the conduction band { Fermi level", (E) is the density of one-particle states (spin variables are included, 1=4 compensates this degree of freedom).Note that in (5) the rst term is O(1) and the second one is O(10 ?2 ), but the latter is very important.Here the frequency is measured from the one-particle threshold.Note that frequency shift P V ii ? is negative (less energy is required to excite a photoelectron).
Thus, the asymptotic singularity at ! !0 is described because e ?ln t = t ? .For absorption (emission) an additional power term arises and changes ! .
A di erence between (1) and ( 2) should be mentioned.Whereas (1) is a general expression for the whole spectrum, at least within the limitations of the model, (2) describes only some vicinity of a speci c threshold, other lines require other constants R and W. This, however, is not substantial because even for (2) other lines require other core-hole potentials.
Rigorous expressions are, though, too complicated for further straightforward analysis.The authors of 1] solved coupled integral equations to obtain a GPL.There is another method connected with the theory of functions: the method of moments.In this way one combines the asymptotics at 0 with the asymptotics at in nity.The striking example of such approach is the formula of Drude-Lorentz from the conduction theory of metals which uses only the static value of conductivity and its zeroth moment.Thus one gets an approximate line form sparing some computational e orts, at least.Besides, a number of results can be obtained directly from the values of the moments.
For example, for photoemission the average frequency (measured from a one-particle line position { W) and centred moments p c 2 = X i<N;k>N jV ik j 2 ; (8) To restore the line form we shall use the known results of the frequency moments problem on semi-axis (Stiltjes problem) 4].An unknown weight function (z) exists with a number of known power moments with an arbitrary d and a nonnegative (t) that satisfy, of course, the restrictions of ( 14).
The results of the moments problem provide an approximate function to have the xed rst moments automatically.The Nevanlinna function should be chosen in order to satisfy some additional conditions; for this case the power law asymptotic behaviour is the same.
There is an inverse method: we can nd a certain !lin (15) ("lin" stands for linear) from ? !id = I n (! lin ) + O(t); (16) that is, supposing that ( 13) is an equation for an unknown function !lin up to the terms linear in t.Thus, the power law will hold automatically up to the linear terms because, as we have said, ?1=(= !id ]) produces the power law.To write the equation explicitly we just omit all linear terms of higher orders.However, the function !lin obtained from the equation ( 16) has no representation (14).It is, though, its behaviour at z = x + i0 with x > 0 that is of signi cance.We can introduce new ~ and ! and assume i.e. on the positive real half-axis.Integrating ( 14) with (x) from (17), one gets < !(x + i0)].Thus, found !complies with all the restrictions and di ers minimally from ! lin on the positive real semi-axis (its imaginary part simply coincides with it).This function should be put into the imaginary part of (13) divided by ?, hence: (1 ?zG 1= ) ! (1 + z(q 1 p 2 + q 2 )) ( ? 1 ? with G = ?cq 1 (ctg ] ?i) (see that G 1= is real), where p 2 = P 2n (0) and Expressions (18), ( 15), (12) together with the de nitions of the constants q 1 ; q 2 ; p 2 ; G constitute the proposed method of restoring line forms.The moments are to be recalculated to a new centre { threshold, therefore is needed: it, as one can see, coincides with the multiplied by (?1) average frequency measured from the threshold (compare (3) and ( 7)) { in what follows we shall take its absolute value as (it is negative).There are some important inequalities that we have mentioned before: all the determinants k and (1)  k must be positive.This leads, for example to nontrivial p c 3 + 2 p c 2 ?(p c 2 ) 2 > 0; (20) p c 2 p c 4 ?(p c 3 ) 2 ?(p c 2 ) 3 > 0: We purpose to describe the photoemission intensity of an alkali metal which, as it is known, is described by a free electrons' model quite adequately.A hypothetical model metal was chosen with one electron per elementary cell and a spherical rst Brilloin zone.Plane waves are basis (r) = ( ) ?1=2 exp(ikr) ( { volume of the system), the dispersion law = k 2 =2.Obviously, only one parameter is left { electron density or the e ective interelectron radius r s .Let r s = 3:93 (it is r s of natrium).
A core-hole potential has not yet been de ned.In fact, this choice lies beyond the framework of the MND model, the core of the model consisting in the notion of a suddenly arising local perturbation considered invariable thereafter.Thus, one neglects both quick processes of creating the potential (! > max !plasma ; E F ]) and slow relaxation (! E F ). Therefore, V is to be introduced into the model as an external rather phenomenological expression (a set of matrix elements).Rigorous treatment of the \quantummetal" interaction would provide such a formula.However, the treatment is impossible or else no model like the MND one with an intermediary potential would be necessary.
We have to stress that here the line forms were correctly restored up to linear in !terms.This does not mean that the farther line form is wrong, it just means that it is unreliable.It contains peaks near .We cannot state that the peaks are to be made smooth by further approximations, though this also happens, but the measure of smoothening requires research.The value of the linear term in Q 4 , for example, is of the order 10 2 , which means that restoring holds up to ! < 0:01Ry 0:1E F .Observing that the GPL is connected with S(E; E 0 ) R R jV kk 0 j 2 dodo 0 (E) (E 0 ) at least to the second order in V , we see that for a semielliptical or rectangular zone and V = const (as in 1]) S(E; E 0 ) has but the only maximum at E = E 0 = E F .But for other V and (E) this may not hold, and, if S possesses a maximum (even a local one) at E = E 1 ; E 0 = E 2 ; E 1 ; E 2 6 = E F , then a peak at ! ' E 1 ; E 2 is possible.Partial con rmation of this statement was obtained by restoring the line form for a rectangular and semielliptical state density and constant V .We did not show this in gures, so that there are not too many lines in them, but (!)-plots (see further) for this case are much smoother than those for other cases (though, peaks arise here too, but far from the threshold: E F ).
Discussing the results on the background of experimental data is rather di cult because, though the data are not scarce, their precision is far from what is needed for any comparison.Singular exponents are obtained, but any deviations from power laws are not.So one can only compare theoretical predictions with numerical models or argue whether an experimental value corresponds to a model because the model is wrong or because the value was measured at frequencies where a curve follows not the power law but the GPL.For example, the authors of 1] claim that = 0:20 for natrium was obtained 5] too far from its threshold where the GPL goes lower than the power law (i.e.(!) grows for !=EF < 1).To prove this they solved integral equations numerically for the contact type potential V kK = ?V (V { a positive constant) and semielliptical (E) / (E 2 F ?E 2 ) 1=2 or the rectangular state density: N(E) = const(0 < E < 2E F ); N(E) = 0(E < 0; E > 2E F ).
Our results show that the signi cance of the GPL's deviation strongly depends on the long-range asymptotics of potentials.We have got di erent types of 's behaviour for the two potentials.(!) was calculated as We concluded that the longer is the range of the potential (the more prominent the peak of its k-transform is in k = 0), the closer it follows the corresponding power law.We also discussed a test example of the contact type ?V (i.e. the transform of the overlocalized A (r)) with = 0:2: V = const; c = 1:774; (26) = 0:0215; p c 2 = 1:264 10 ?3 ; p c 3 = 1:063 10 ?4 ; p c 4 = 9:07 10 ?5 : which produces (!) that lies even further from the power law than the short range potential's one.Dependencies (!) are presented in gure 2. Such conclusion is not as trivial as it can appear.The extremal long range case should be just an overall shift of the mean eld, the corresponding spectral line is a -like peak, nothing like a power law can occur here.That is obvious from the physical sense of the phenomenon: the singular form is created by multiple electron excitations across the Fermi surface, but those can happen only if matrix elements V kk 0 with E(k) < E F , E(k 0 ) > E F (not diagonal in occupation number) are nonzero, which is not the case for (k?k 0 ) { a matrix element of V (r) = const.On the other hand, the power law cannot certainly hold forever, the line has nite moments, and the power law does not.
Thus, we state that, researchers 1] being right, the measure of GPL deviation and uncertainty of experimental values of substantially depend on the long range asymptotics of the core-hole potential.
And, certainly, we obtained a prethreshold line form.Note that calculating moments is much easier than solving integral equations numerically; for some cases (contact-type included) almost all the moments can be found analytically.The method appears applicable at least near the threshold (! E F ).Some remarks should be made here concerning the form of (!) farther from the threshold.We cannot claim that nonlinear parts of the curves in gure 2 do not result from our method faults.We have, obviously, d 2 =d! 2 < 0. However, this holds only for !> 10 ?3 .For ! < 10 ?3 , in fact, d 2 =d! 2 > 0; we do not present it in the gure, because there is no appropriate scale to show this slight curving, even a logarithmic one is not enough.This partially coincides with the results 1] for the mentioned contact-type ?V : (!)-curves have d 2 =d ln ! 2 > 0 .However, there the authors do not mention if the stronger d 2 =d! 2 > 0 holds.At ! ' 10 ?3 the trend changes (for stronger potentials it may happen farther).It must happen just because d 2 =d! 2 > 0 cannot hold too far from the edge, the frequency moments being nite, so (!) must become negative sooner or later, probably somewhere near .