THE ROLE OF X-RAY EMISSION SPECTROSCOPY IN THE RESEARCH OF THE ELECTRONIC STRUCTURE OF TRANSITION METALS AND THEIR ALLOYS

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THE ROLE OF X-RAY EMISSION SPECTROSCOPY IN THE RESEARCH OF THE ELECTRONIC

STRUCTURE OF TRANSITION METALS AND THEIR ALLOYS

V. Nemoshkalenko

To cite this version:

V. Nemoshkalenko. THE ROLE OF X-RAY EMISSION SPECTROSCOPY IN THE RESEARCH OF THE ELECTRONIC STRUCTURE OF TRANSITION METALS AND THEIR ALLOYS. Journal de Physique Colloques, 1971, 32 (C4), pp.C4-225-C4-229. �10.1051/jphyscol:1971442�. �jpa-00214643�

JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 32, Octobre 1971, page C4-225

THE ROLE OF X-RAY EMISSION SPECTROSCOPY IN THE RESEARCH OF THE ELECTRONIC STRUCTURE

OF TRANSITION METALS AND THEIR ALLOYS

V. V. NEMOSHKALENKO

Physics of Metals Institute AS UkrSSR, Kiev

Due to the improvement of the experimental equip- ment and the use of modern high-speed electronic computers in the field of theoretical calculations of the electronic structure of solids, X-ray emission spectro- scopy has been widely employed within recent years.

A good correspondence between the experimentally certain shape of the aluminium L-band [I] and silicium K and L-bands [2], [3] on one hand and the theoretical calculations of the intensity distribution in the X-ray emission bands conducted on the basis of the general formula

on the other hand makes it possible to hope that the works on deciphering the electronic structure of transition metals and their alloys will be also a success.

It should be immediately noted that the calculations of the electronic structure of solids are rather compli- cated and laborious even in case you use the highest speed electronic computers. Therefore the role of an experiment is especially important and cannot be represented only by the role of the criterion of the correctness of selection of the corresponding theoretical calculation methods especially as they are still far from being perfect and because of the above mentioned difficulty they are confined to a comparatively small number of points in the Brillouin zone which in the main correspond to the high symmetry states. The latter considerably lessens their importance and makes the experiment the only source of information on the electronic structure of solids.

We have already noted [4] that at present there is accumulated the large number of experimental data on the parameters of X-ray emission spectra of tran- sition metals, their alloys and compounds which could serve as a basis of a successive model of their electro- nic structure. But an absence of the single point of view on their interpretation has led to essentially diffe- rent ideas 15-71 about the electronic structure of transi- tion metals even in the limited field of atomic numbers.

In addition, both the experimental data obtained by means of other methods (electronic specific heat, para- magnetic suspectibility, the temperature of transition

to a superconductivity state) and the results of corres- ponding theoretical calculations were used in all cases to prove the model representations developed.

X-ray emission spectroscopy is a unique experimen- tal method making it possible in principle to study all the energy spectrum of solid electrons both collectivized and localized near the atoms of different types. As to the rest of the methods all of their, but theoretical calculations, make it possible to measure a certain parameter directly or indirectly connected with the density of states at the Fermi level N(EF). It is necessary to note that due to a number of specific peculiarities of X-ray spectroscopy (low-energy satellites and self- absorption) it is this field of the spectrum which is measured with the worst error. We could compare the shape of the X-ray emission bands with the dependence of the coefficient of electronic specific heat y on the electron aton1 relation but for this purpose we should satisfy the two requirements : [l] manifestation of all special points of the zone of valency or that of the conductivity in the X-~ay emission spectrum of the atoms studied and [2] validity of the model of the rigid band for the selected area of electron-atom relations.

The first requirement may be evidently satisfied at least in case we use the maximum possible number of spectra with a different symmetry of the inner scanning level and we shall discuss the problem a little bit later.

As to the second level it is absolutely evident that it is accepted in the work [8] and is further used as an important experimental confirmation obtained by another method, of the model representations of the electronic structure of elements of the beginning of I, I1 and 111 long periods developed in this work.

Let us discuss in detail the now available experimen- tal data on the electronic specific heat for the field of electron atom relations from 4 to 6, i. e. just that one which is discussed in the work [6]. As a result of numerous measurements on variour alloys [8-161 it was found that y for binary alloys of the elements of the I and I1 long periods corresponding to the given field of the electron atom relations lie on the curve shown in figure 1. As it has been already absolutely justly noted in the work [6] this curve is a result of measurements taken on a large number of various

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971442

C4-226 V. V. NEMOSHKALENKO

f'feciron-to-Atom Patio. e/a

systems of alloys both between the element of one period and between the elements of various periods.

We could expect y-values to be of random character, but the picture is different and all the y-values are confined to the limits of the same curve showing maxi- mum in the field of

-

experimental date obtained as a result of this work are given in figure 2. It turned out that we may expect the density of states at the Fermi level of BCC titanium to be 2.5 times as high as a corresponding value of HCP titanium. Such an unexpected result, naturally, was to

Eiectron-to-atom ratio , e/a

be confirmed and we've got the confirmation as a result of theoretical calculations of the shape of N(E)-curves of elements of the I long period carried out in similar conditions [18]. The closing circumstance is extremely important since in this case we are interested in kinetics of variation of the shape of N(E)-curve caused by atomic number variations. The lover curve in figure 2 shows N(E,)-values for Ti, Cr and V taken from the present work. The same theoretical calcula- tions confirmed the difference in the density of states at the Fermi level of HCP and BCC titanium which has been discovered as a result of experiments. An estima- tion based on the theoretically calculated N(E)-curves gave us the value of N(EF) BCC : N(EF) HCP = 2.1 which quite satisfactorily falls in line with the experi- mentally found (- 2.5).

But in spite of the qualitative agreement of the upper and lower curves in figure 2, the experimentally found values still remain somewaht higher than the estimated one. This divergence can be reduced if we take into consideration that somewhat overrated value of the constant of the electron-phonon interaction was evidently used in the work [17]. In addition, as it has been noted by Weber and Snow [18], they have used

THE ROLE OF X-RAY EMISSION SP 'ECTROSCOPY IN THE RESEARCH C4-227 rather a big stop value when plotting a histograms of

N(E)-curves and this can reduce the seeming height of any sufficiently sharp peak on the N(E)-curve, that in the Fermi level vicinity included. Taking into consideration only these two factors should bring experimentally found and theoretically calculated N(E)-curves closer.

Completing the analysis of the bahaviour of y in the field of electron atom relations from 4 to 6 we may note that the nature of the maximum which is to be found on the preliminarily plotted experimental curves y(e/a) in the field of e/a = 4.6-4.7 is not connected with the variations of the density of states at the Fermi level but wholly depends on decomposition of the BCC solid solution and precipitation of HCP o-phase from it. As to BCC structures the density of states for them at the Fermi level steadily decreases with the increase of the electron atom relation from 4 to 6 and does not reveal a structure which the calculated N(E)-curves are charac- teristic of this circumstance fully eliminates the ques- tion of the c< band rigidity ⁾⁾ for this field of electron atom relations. Simultaneously, figure 2 shows that in future we can also use the data on the electronic specific heat to compare them with the results of theoretical calculations and parameters of X-ray emission bands but in a way different from that des- cribed in the work [6]. Figure 2 is really a convincing evidence of the fact that there is quite a satisfactory correlation between the curves y(e/a) and N(EF) for different 2. In a similar way we could find correlation between y(e/a) and intensity of inflection points of the high energy edge of the X-ray-emission bands. At the same time as we have already mentioned before it is this parameter which is most difficult to be determined due to the X-ray spectrum data. In addition, we should take into consideration the influence of the probability of transition cc summarise ⁾⁾ the intensity of inflection points of high energy branches at least of K- and L-bands. But the present state of the theory of X-ray spectra does not answer the question how it can be done. Therefore comparisons of the kind may be only of preliminary qualitative character. In addi- tion, we should again emphasize the fact that the experimentally determined electronic specific heat coefficient y is connected with the density of states at the Fermi level in terms of :

where 1 is a constant of electron-phonon interaction.

Unfortunately, at present it cannot be determined accurately enough. True, in his recent work [19]

McMillan has found a formula connecting the tempe- rature of transition to the superconductivity state T, with 1 :

T, = --- 9 exp 1.04(1 + ¹⁾

1.45 1 - ,u*(l + ^{0.62 A) (3)}

where p* is a constant of electron-electron interaction

which as McMillan expects has the value of 0.13 for all transition metals. Using the formula (3) he has determined the values of A for Ti and V which appeared to be equal corresponding to 0.38 and 0.60. Unfortu- nately, neither chromiam nor manganese are super- conductors and the formula (3) cannot be used to determine ;1 for these elements. And this practically means that the values of available at present cannot be used to determine N(EF) -- f ( 2 ) in a sufficiently broad field of atomic numbers. In addition we consider valid the ideas expressed in the works [20], [21] that the electrons on the Fermi surface cannot be considered as free electrons but rather as some elementary exci- tations (cc quasi-particles ))) and the N(EF) function determined due to the values of y would give the density of states of these quasi-particles depending upon the element atomic number.

Thus, the experimental data available at present unequivocally show that the rigid band model cannot be used even for elements of the begining of the I, I1 and I11 long periods and the existing regularity of the variation of N(EF) which is of monotonously decreesing character in BCC solid solutions of these elements in the field of electron atom relations from 4 to 6 should be considered as an electronic feature of BCC solid solutions of this type. Decrease in the elec- tron atom relation < 4 is accompanied by a spasmodic reduction of y which is connected with the variation in the crystal structure type. In the light of everything we have just discussed we understand the lack of coordination in the interpretation of the shape of zirconium spectrum M which took place in the work [22] when an attempt was made to use the y(z)-curve as a full N(E)-curve of the transition metal of the I1 long period.

Thus, since the data concerving the electronic spectific heat cannot be used for decoding the intensity distribution in X-ray K-, L-, M-, 0-bands of elements of the I, I1 and I11 long periods, we shall discuss the problem whether it is possible to compare them, with the theoretically calculated N(E)-curves. We should immediately note that any attempts, those made in the work [6] included, of the simplification of the formula (1) for the intensity distribution in the X-ray emission band cannot be considered to have a sufficient foun- dation. In fact, in the case of K-spectra according to the rules of selection there would be a transition elec- trons described in the terms of p-type wave functions from the external energy band. The quadrupole transi- tions may be neglected, at least for the elements of the begining of the period having in mind their extremely small probability [23]. Then the intensity for K-band may be represented in the terms of

where 0 in the second integral means that integration

C4-228 V. V. NEMOSHKALENKO is confined to the field of the distribution of the avew

function of the innershell electron. On the other have the particle density of states of p-symmetry may be represented in the terms of :

We can see that the difference between equations (4) and (5) lies in their end members : in the first case we have the square of the average a@), while in the second case we have the average of the square of a@). There are no mathematical methods by means of which we could transpose one equation to another. In the works 11-31 it has been convincingly shown to what great extent the shape of the X-ray emission band may differ from the N(E)-curve. And in case we have decided to use X-ray spectrum data as a criterion of correctness of the corresponding theoretical methods of the estima- tion of the electronic structure of solids, the estimation should be by all means carried on until we obtain corresponding I(E)-curves which can be compared only to the experiment and only in case you would take into consideration all the factors noted in the work [24]. It is just necessary to note you should be very careful when correcting X-ray spectra having in mind the instrument distortions and the inner level width. In any case you should by no means make corrections in high order provided they are not optimum. In fact it has been shown in the work 1261 that on unjustified increase of the order of correction by the authors of the work [6] with the aim to obtain a more distinct thin structure led in a number of cases to the appearance of a new thin structure which is not present on the experimental curves. An attempt to achieve a <( better coincidence ¹⁾ between the experiment and the calculations at the expense of an unjustified increase of the order of correction is harmful and senseless in the light of equations (4) and (5) which uniquivocally show that such a coincidence even cannot occui .

At the same time the analysis of the X-ray emission spectra of all series makes it possible to measure the energy position of special points of the zone and only these parameters can be immediately compared to the results of the corresponding theoretical calculations.

The X-ray spectrum method is an important physical method of investigation of the zone structure of a substance which is especially applicable to the alloys and chemical compounds where the rigid band model is known to be unfit and the theoretical calculations are additionally complicated by a number of factors in comparison with the calculations of the electronic structure of pure elements. The importance of the methdd lies not only in the fact that the latter makes it possible tostudy all the energy spectrum of electrons as a whole but also in the fact of its selectivity relative to the states of various symmetry in the outer energy band and the possibility to investigate the energy

spectrum of electrons in the vicinity of each component of a complex alloy or a chemical compound. But of special significance in this case is the experiment purity and authenticity. The latter can be verified only by means of comparing the results obtained in different experimental conditions. Unfortunately, the situation in X-ray emission spectroscopy of transition metals of I and 11 long periods is such that when the results concerning K- and L-spectra of the I long period and M-spectra of the I1 long period obtained in diffe- rent laboratories differ from each other, the difference lying in some elements of the thin structure 141, [6], 171, [22], [27]. The note made in the work [6] that the spectra represented in the latter are obtained in better experimental conditions than the spectra given in other works cannot be considered convincity. As a matter of fact K-spectra of the elements of the I long period were obtained in the works [4], 161, [27], the resolving power of the instrument used being the same (-- 0.2 eV), which generally speaking, is of no decisive significance in case of a considerable width of the inner level ( N 0.8 eV). At present only the first low-energy maximum of K-bands of the elements of the I long period and M-bands of the elements of the I1 long period which are well described in the works of different authors may be considered to be reasonably well established. As to the second low-energy maxi- mum its presence on the photomicrograms cannot be considered similarly authentic. Really, let us see what the photomicrometric curves of the KP,.,-band of vanadium obtained as a result of photomicrometry of areas at different levels look like. According t o figure 3 the second low-energy maximum is observed

T H E ROLE OF X-RAY EMISSION SPECTROSCOPY IN THE RESEARCH (3-229

not on all the areas and even in the cases where some fracture of the line representing the intensity of the low-energy branch of the KP,.,-band occurs, the energy position of the inflection point varies from one spec- trum area to another. The same thing means that it would not appear on the averaged curve and it is a result of the averaging of a large number of spectra which is to be considered the most authentic shape of the intensity curve as it has been already done, for

instance, in the work [4]. As to some discrepancy between t h e energy position of the first low-energy maximum of KP, .,-bands of the elements of the I long period and M-bands of the elements of the I1 long period on one hand and special points of the zone on the other it is probablly caused by incompletness and imperfectness of the calculations available at present as it has been properly noted in the work [6].

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