Soft X-Ray Emission and Electronic Structure of Alloys D

SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS D. Fabian

To cite this version:

D. Fabian. SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS. Journal de Physique Colloques, 1971, 32 (C4), pp.C4-317-C4-324. 10.1051/jphyscol:1971458. jpa-00214659

HAL Id: jpa-00214659 https://hal.archives-ouvertes.fr/jpa-00214659 Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C4, suppliment au no 10, Tome 32, Octobre 1971, page C4-317

SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS

D. FABIAN (*) Department of Metallurgy, University of Strathclyde, Glasgow, Scotland

R6sum6. - Nous passons en revue les donnks concernant les bandes d'emission des rayons X mous pour les solides, particulikrement dans le cas des alliages d'aluminium, et nous discutons leur interprktation dans le cadre d'un mod6le bande rigide. Les mesures sur les alliages des mktaux nobles et de transition avec I'aluminium sont interprktks en terme d'ktats liks virtuels ou d'6tat lies rksonnants et semblent confirmer le concept de densit6 klectronique locale.

Abstract. -Soft X-ray band-emission data for solid binary solutions are reviewed, with particular reference to aluminium alloys, and their interpretation is discussed in relation to the rigid-band and virtual-bound-state models. The emission spectra only rarely support the rigid-band model. Measurements on alloys of the noble and transition metals with aluminium are interpreted in terms of virtual-bound or resonance-bound states, and appear to support the concept of local electron densities.

The electronic structure of alloys has been the Soft X-Ray Emission. - Soft X-ray band emission subject of much research commencing at about the from metals provided the first experimental proof of time of the well-known text by N. F. Mott and the fundamental band theory of crystalline solids, H. Jones - (( The theory of the properties of metals and a considerable impetus was given to the early and alloys )) - which appeared in 1936. It is inte- development of the subject, in connection with the resting to weigh the advances since then : they have electronic structure of metals and alloys, by the work been many, and have derived from both experimental of Farineau [I], Skinner [2] and Cauchois [3] ; lasting and theoretical work. from about 1932 to 1948. As early as 1933 Jones, Mott However, the theory of the solid state is still not and Skinner successfully interpreted the intensity well understood and even for pure metals there are distributions observed [4] for the K and L spectra of considerable difficulties in the interpretation of simple metals (generally meaning metals whose electro- experimental data ; for alloys in particular a satisfac- nic structures do not involve d-bands). The theoretical tory theory has continually eluded physicists and simplicity of that interpretation, which showed for the metallurgists. The questions that arise, when two low-energy region of the emission band that different kinds of metal atom are alloyed to form a for K-emission I(E) cc E3I2 (1) disordered array, are : how much is the band structure affected, and can we still speak of a Brillouin-zone and structure once we have destroyed the translational for L-emission I(E) cc Ell2 (2) symmetry of the lattice ? In 1936 Sir Nevi11 Mott wrote <( The question has not been cleared up yet in a has remained as one of the foundation stones on which satisfactory way, but it seems fairly certain from the the field of soft X-ray spectroscopy of solids has evidence (: evidence associated with the Hume-Rothery continually developed. rules and their explanation in terms of a rigid-band The emission of soft X-radiation occurs following model) that the break in periodicity may be considered the excitation of the solid to a state in which an elec- not to affect the zone structure very much n. In 1970 tron vacancy is produced in an inner atomic-core level ; the question has still not been cleared up, nor in fact commonly a K or L level. The intensity of emission, is it much nearer to solution. as a function of energy, is usually written in the simplified form (*) This paper was prepared while the author was on leave of I(E) cc F(E) N(E) (3) absence at the Laboratorium fiir Festkorperphysik, Eidg. Technische Hochschule. Ziirich, Suisse. where N(E)is the electronic density of states at energyE,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971458 C4-318 D. FABIAN and F(E) expresses the transition probability for an posed upon the basic periodic potential of the lat- electron making a transition from an occupied valence tice. Mott and Jones developed this simplified pic- or conduction level to the vacant core level. It was ture chiefly to account for the Hume Rothery rules from a consideration of the factor F(E), and an esti- for the alloying characteristics of two metals. At the mation of the respective amounts of atomic s-state other extreme, we have the two-band model in which and p-state that will combine to form a zero-order the valence electrons are assumed to exist in two wavefunction for electrons in the lattice, that Jones, separate sets of energy states, each associated with Mott and Skinner derived the relation ships (1) and (2). the potential fields of the ions of one component. More rigorously eq. (3) should be written Between these two extremes comes the virtual- bound-state model of Friedel and Anderson, some- times called the resonance-bound-state model, which is developed from the concept of localised screening of where the integral is over all electron states k on the the charge surrounding a dissolved atom of different constant energy surface S in k space. Strictly F(E) valency from the solvent metal ; an idea first proposed cannot be removed from the integral because of its by Mott [15] in 1936. The excess charge on a solute dependence on the direction of k and, as noted for atom is effectively shielded by local distortion of the example by Rooke [5], the k-dependence can be conduction band so that the additional electrons or particularly sharp where changes of symmetry are holes introduced by the solute are localised. Friedel involved. However, for alloys and indeed almost shows that this model does not entirely conflict with generally for all metals, eq. (3) and the relation the rigid-band model. The localised states or virtual ships (1) and (2), are accepted as working simplifica- bound states lie within the valence or conduction band, tions. Following Jones, Mott and Skinner [2], the pro- and are different from bound states which can also bability factor F(E) is then expressed in terms of the exist around a completely ionised solute atom and square of the matrix elements for the transition which result from a local well, in the otherwise uniform lattice potential, causing one or more bound states to be formed in the forbidden energy region-normally below the occupied conduction band. In addition to these descriptions is the virtual crystal where v is the frequency of the emitted radiation, model which assumes that the electrons move in a and Y, and Yf are wavefunctions for respectively the crystal potential that is everywhere ordered and the initial (valence band) and final (core level) states of the average of the potentials for the two pure components. electron. This in some ways is similar to the rigid-band model. In what few attempts have been made to calculate Beeby [16] has used an approximation in which the 'the emission intensity for the component atoms in an crystal potential is reduced to the sum of local potentials, alloy, the energy dependence of the transition probabi- and assumes a particular ordered system of localised lity has been estimated using eq. (5) as a starting point ; potentials whose scattering matrix is the average of see Stott [6], [7]. This is essentially a one-electron those from the localised potentials of each constituent. description of the system, and while many-electron More recently Soven, who found [17] that for weak calculations have met with some success 181, [9], [lo] potentials this localised-potential model reduces to in explaining fine-structure in the emission spectra the virtual crystal approximatibn, has introduced a of pure metals, the alloy problem is very much more promising model [IS] based on a Green-function for- complex and has not yet been addressed by the many- malism and suitable for strong local potentials such body theorists. as those found in transition metal alloys. With this formalism, which he first introduced as the coherent Electron Model for Alloys. - Theoretical work in potential method [17] and which has subsequently the field of alloys is continually expanding, and among been examined in some detail by Velicky Kirkpatrick the many models that have been proposed are : The and Ehrenreich [19], Soven shows [IS] that the density rigid-band model first described by Jones [11] in of states localised around a particular constituent 1934; the localised screening or virtual boundstate atom can be substantially different from the average model developed chiefly by Friedel [12] and by density of states so that the local electron density will Anderson [13] since 1952 : and the two-band model be particularly succeptible to the local environment. proposed by Varley [14] in 1954. The degree to which The picture obtained tends to. fall between the localised these models invoke distinct valence bands for the screening or virtual-bound state model of Friedel and two components of the alloy increases in the order Anderson, and the extreme simplification of a two- listed. At one extreme, the rigid-band model assumes a band model used by Varley ; and if the approximations common band to which for a binary alloy both solvent of this one-electron picture are correct then strong and solute atoms contribute their valence electrons, resonance states can result, and move through the and regards as negligible the incoherent scattering band if the impurity potential is increased, in apparent from the random impurity potential which is super- agreement with the findings of Stott [7] whose calcu- SOFT X-RAY EMISSION AND ELE( ZTRONIC STRUCTURE OF ALLOYS C4-319 lations determine the effect of such resonance or vir- respectively giving rise to A1 L2,,-emission and Mg tual-bound states on the soft X-ray emission from L2,,-emission from the alloy. atoms of the solute metal for dilute alloys. For some binary systems, overlap of the two cons- Harrison [20] also considers the problem from the tituent-metal emission bands occurs, with both cove- viewpoint of soft X-ray emission, but examines it in ring similar or overlapping wave-length regions ; but terms of pseudopotential theory. His findings, for for many, as for example with AI-Mg, they are entirely dilute alloys of simple metals, favour what is effectively separate. A measurement of the A1 and Mg K-emission a rigid-band model, while for alloys in which d-bands from Al,Mg2 formed about the first such investigation are involved his approach predicts resonance-bound of an alloy system, reported in 1937 by Farineau [21] states so strongly localised that they would be seen who found that the two emission bands had similar - in terms of soft X-ray transitions - only by the forms, and were quite different from the K-emission impurity atom from which they originate. band for pure Al. The results appeared to support the simple model of sharing of electrons between A1 and Results for Soft X-Ray Emission from Alloys. - Mg atoms. This led Skinner and Johnston [22] in the Soft X-ray emission is particularly suited to examina- following year to interpret their results, for several tion of the local environment of an atom in an alloy, L-emission and K-emission spectra from a series of because transitions can be observed from the imme- alloys of A1 with Cu and Be, also as supporting the diately local conduction or valence band to core simple common-band model. states of the atom. Moreover these transitions may be However, a careful examination of the A1 and Mg examined for both types of constituent-metal atom L2,,-emission bands of a series of Al-Mg alloys by (for a binary alloy), or for either separately. It is Appleton and Curry [23] in 1965 gave results, repro- helpful to examine figure I which illustrates, schema- duced in figure 2, which sharply conflict with the simple

08- 4 I 11 -Pure metals I --- AhMPI 1, I( 06- A L,2 Mg 7

4r :04-

02-

38 42 46 50 60 64 68 72 Energy E ( eul

FIG. 2. - Mg and A1 L2,3 emission from AI-Mg alloys. Reproduced, with permission, from Appleton and Curry 1231.

rigid-band model. In our picture of this model, figure 1, the common valence band has a width between those for pure A1 and pure Mg and should, if the model is valid, give rise td the same bandwidths for Mg L2,,-emission and A1 L2,,-emission. Appleton and Curry find that these bandwidths are different from one another and, with changing composition, remain about equal to those for the respective pure metal. Independent measurements by Dimond [24] confirm FIG. 1. - Schematic picture for A1 and Mg L-emission from these results ; and closely similar conclusions were adjacent cells in the lattice of an AI-Mg alloy, for the rigid-band reached for Mg-bi alloys by Catterall and Trotter [25]. model. These observations appear to indicate that separate Indicated approximately, above and below, are the radial localised valence bands must be associated with each parts of the free-atom wavefunctions for the L2 and L3 core levels, type of atom. We may picture in figure 3 the situation and for the 3s valence electrons of Al and Mg. that will apply if this is the case. Two densities of states, one for each,type of atom, will predominate tically, the situation approximating Co the rigid-band locally. Some form of two-band model is implied model for individual but adjacent A1 and Mg cells in similar to that proposed by Varley. This has been the lattice of an A1-Mg alloy. We can picture here the advocated also, to explain the soft X-ray emission transitions that occur from the valence band for the results, by Rooke [24] who tentatively explores a two- alloy to either the L2 and L, leve'ls of an A1 atomic band model in terms of Thomas-Fermi localised core, or the L2 and L, level of a Mg atomic core, screening. C4-320 D. FABIAN

tration function s(E) invoked by Skinner and Johnston has meaning, it probably implies the concept of separate sets of energy states subsequently assumed in the Varley two-band model. Wavefunction expectation values, as indicated by the free-atom radial components in figure 1, do not assist us to decide between the one-band and two-band models in cases such as Al-Mg, but they do - as we shall see -provide a guide for the interpretation of emission spectra when d-bands are involved. First, however, we should note the importance of the changes of shape in the A1 and Mg L,.,-emission bands (Fig. 2). Appleton and Curry [27]-have pointed out similar FIG.3- - Schematic picture for A1 and Mg L-emission, effects for many alloy sysfems ;namely that the relative from adjacent cells in the lattice of an Al-Mg alloy, when a two- band model applies. intensity is always reduced in the high-energy part of the emission band when alloying occurs with a lower- valency metal. This we can interpret in terms of the The results obtained by Skinner and Johnston for two-band model : in the case of Al-Mg, where elec- Be K-emission from dilute alloys of Be in Cu and A1 trons are transferred from the A1 atoms to Mg atoms, appeared also to indicate that the bandwidths associa- if the electron states associated with an A1 atom are ted with the solute atoms are the same as for the pure distinct from those associated with an Mg atom then solute metal. They interpreted their results, to retain we should expect a decrease in intensity in the upper agreement with the conclusions reached by Farineau part of the A1 band where the deplenishing would and with the rigid-band model, by proposing that while mostly occur. A corresponding increase is observed the separate atoms in fact see the same common in the upper part of the Mg band. Alternatively we band they will only be involved in transitions with can consider the excess charge left on an A1 atom as electrons that can penetrate to the interior core levels drawing off part of its outer electrons and forming of the atom ;the atoms select, so to speak, their own bound states (of s-symmetry) that partly screen the temporary valence electrons >>.Skinner and Johnston upper valence states from the core. introduced into the expression for I(,?), eq. (3), a Clearly there is some evidence for a two-band model sorting function s(E) to give in these AI-Mg results. Certainly there is little to support the rigid-band model, nor the description given by Harrison ; although Neddermeyer [28], who and regarded s(E) as expressing this property of an has recently obtained very similar data for this system, atom core to select its own valence electrons from finds that for concentrated alloys his results do tend the common band, and to be distinct from the proba- towards the rigid-band description. One explanation bility factor F(E). The function F(E) cannot alone be must always be borne in mind : this concerns the ques- responsible for this would provide no explanation of tion of clustering in the alloys which, if it occurs to why the wider constituent-metal emission band (the any marked extent, could result in significant pure- Al-emission bandwidth in the case of Al-Mg alloys) component emission from within the alloy. However is greater than expected for a common rigid band. clustering, although it would account for the constant To provide a rough indication of the extent to which bandwidth, cannot easily explain the considerable core-level and valence-band electrons might penetrate changes of shape that occur in the emissfon bands. from one atom cell to the next, the radial part of the Furthermore, short-range order in Al-Mg is possibly Hartree-Fock free-atom wavefunctions for the 2p more likely than clustering, a view supported by the and 3s electrons of both A1 and Mg have been shown, strong formation of intermetallic phases. approximately, in figure 1. The A1 3p-electrons Turning to the results for other alloy systems, contribute also to the valence band but these have not Curry [29] has given a valuable review of the experi- been included because only s-states can be involved in mental findings which appear mostly to indicate band- transitions to the L, and L,(Z~'/~and 2p3I2) core widths that do not change appreciably on alloying. This states and, although the symmetry of the valence particularly is the case for noble and transition-metal p-states can be affected by the crystal field so that they alloys, in which we find also much evidence for the are to some extent involved, transitions giving rise existence of virtual-bound or resonance-bound state. to L2,,-emission will result predominantly from In figure 4 the results are shown for the A1 L2,,- s-electrons in the band. emission from Al-Ag alloys, reported by Fabian, The sharpness of the L, and L, levels is immediately Lindsay and Watson [30]. A strong peak is caused by apparent, and the extent to which the valence s-states transitions from Ag d-states in the alloy valence band arising from neighbouring atoms can be seen by a to A1 core states. If we examine now the radial part given atomic core can be roughly gauged. If the pene- of the free-atom 4d wavefunction for Ag, which is SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS 04-321

FIG. 5. - Schematic picture for A1 L2.3-emission from an Ag-A1 alloy. The radial part of the free-atom wavefunctions for the 4 d states of Ag have been indicated approximately so that the overlap with the A1 Lz and L3 levels can be gauged. FIG.4. - A1 L2,s-emission spectra obtained for a series of A1-Ag alloys. Lindsay, are shown in figure 6. We observe that a The prominent peak that develops at - 65 eV is interpreted strong peak occurs in the A1 L,,,-emission, which we in terms of transition from the Ag d-band to A1 core levels. attribute to transitions from resonance-bound d-states of the copper atoms to neighbouring A1 core states. indicated approximately in figure 5, we can say that The peak appears also to move to lower energies, and such transitions are not altogether unexpected ; they to broaden, as the concentration of Cu is increased ; probably arise from resonance states involving the bearing in mind that the 20 %-Cu alloy is two-phase, Ag d-band. In addition to the A1 L,,,-emission, transitions giving rise to Ag N,,,-emission have been indicated in figure 5 ; however, this emission band for the alloys was of too low an intensity to be observed. The N, and N3(4 p) levels are widely separated for Ag (13.0 eV compared to -- 0.03 eV for the L, and L, of Al) ; and we note that the expected overlap betwee adjacent cells for these levels, indicated by their approxi- mate free-atom wavefunctions, appears to agree with the large widths observed for these core states in X-ray photoemission from Ag metal [31], usually regarded as due chiefly to lifetime broadening. Our interpretation in terms of Ag d-states for these spectra, described by Marshall et al. 1321, requires hybridization of the d-states with s-states in the valence band, and is supported by measurements on Al-Zn AI -CU Alloys and AI-Cu alloys. We find no such resonance peak in I the A1 L,,3-emission from Al-Zn, which is entirely to be expected since the d-bands in Zn lie below the 4 s so that little or no hybridization with s-states occurs. But with Cu, where the d-band lies right in the middle of the valence band, a strong resonance-bound state results. 74 70 65 60 The measurements for Al-Cu alloys by Watson and FIG. 6. - Al L-emission spectra for AI-Cu alloys (see text). C4-322 D. FABIAN containing a tetragonal 8-phase and a fcc a-A1 phase. Cu, the position of the strong peak is less easily corre- This may correspond to the features described by lated with the energy at which the d-band for the pure Stott [7]. For these alloys the M2,,-emission band transition or noble metal is expected to occur. However from Cu overlaps the A1 L2,,-emission, and an esti- in all these cases strong intermetallic bonding occurs, mate of this band for the 80 %-Cu alloy has been in which the d-electrons will play a large part ; this made from the measured M2,,-emission for pure would cause the d-states to appear at lower energy, copper. and also possibly make the transitions more likely. Considerable support for the interpretation that The results are probably best explained in terms of a these are localised d-states being seen by the sol- form of two-band model and local electron-state vent metal, comes from the work of Harrison and densities, which Soven [18] shows can be expected to Curry [33], who observe the same strong peak in A1 differ considerably from the average density of states L2,,-emission from all alloys of the noble-metals with for the alloy. The theoretical concepts differ only in aluminium, and from the work of Williams et al. [34] formalism from the localised virtual or resonance who observe it in Al-Au alloys. Harrison and Curry states of Anderson and Friedel ; the theory for these also examine the overlap of the radial functions R,,(r) probably only applies for dilute alloys, but there seems for Cu and r3RZp(r) for Al, centred on adjacent A1 and to be no reason why strongly localised states would not Cu atoms, and conclude that since the product of these occur in concentrated alloys. The formation of reso- is involved in the matrix element for the transition nance-bound states for noble and transition metals from Cu 3d to A1 2p that this transition is likely to is also well formulated by Harrison [20] ; although he occur. suggests that they would not be seen by the solvent- Harrison and Curry [33] conclude also that resonance metal atoms for an alloy in which Cu for example is bound states are being observed in the A1 L-emis- the solute, which is not in agreement with our obser- sion from transition-metal alloys with aluminium, an vations. observation supported by the results obtained for Localised states have been reported also in the A1-Pd alloys by Watson and Kapoor, in conjunction interpretation of optical data for Ag-Pd alloys, by with Nemoshkalenko [35], shown in figure 7. In the Karlsson, Myers and WalldBn [36] ; and in the interpretation of photoemission data for Cu-Ni, by Seib and Spicer [37], and Ag-Pd, by Norris and Nils- son [38]. Norris and Myers (to be published) show also that the experimental data for Ag-rich Ag-Pd alloys are best explained in terms of resonance-bound states and localised electron densities.

Conclusion. - It is clear from the large number of electron models for alloys that have been proposed and developed that considerable uncertainty exists ; the more recent theoretical evidence tends to favour localised electron densities, and to point to the importance of the local environment of the impurity atom. Experimentally we have, for a bulk metal 'or alloy, only a few direct methods for studying the electronic density of states over the whole valence or conduction band ;these, in the order in which they have developed historically, are soft X-ray spectroscopy, ultra-violet photoelectron spectroscopy, and X-ray 1. AI-Pd Alloys photoelectron spectroscopy which gives the deepest energy probe and permits therefore examination also of the core levels. Of these soft X-ray spectroscopy clearly provides a valuable tool with which to examine local electron densities, although much improved theory is needed for proper interpretation ;probably many-electron calculations and correct assessment of Fro. 7. - A1 L-emission spectra for A1-Pd alloys (see text). the effect of the local core hole will bring the necessary refinement. measurements by Watson and Kapoor a strong resonance peak is again observed and probably moves Acknowledgements. -The author is indebted to the through the band with increasing concentration of U. K. Science Research Council for a grant in support Pd. They find similar results for the alloys of Nb, V of this research. It is a pleasure also to thank : Pro- and Co with Al. In these cases, and in those of Pd and fessor E. C. Ellwood for continued support and SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS C4-323

encouragement ; Professor G. Busch for his interest, tions to the manuscript ; and Drs. L. M. Watson, and for a visiting appointment at E. T. H. Ziirich ; G. M. Lindsay and Q. Kapoor for assistance and for Dr. H. Chr. Siegmann for discussion, and for sugges- communication of their experimental results.

References

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DISCUSSION

Mr. JORGENSEN.- In view of the concern expressed A most striking counter-example was found in 1941 by several of our colleagues about small numerical by Professor Cauchois demonstrating that the empty values of intensities of cc crossed transitions )), I am orbitals in metallic gold accessible to excitation of wondering whether the shoulder at 65 eV in elemental 3d electrons apparently have much higher energy aluminium becoming a maximum in the various than those accessible to 2p, whereas this discrepancy alloys with magnesium, copper and silver is a transition is very small in metallic thorium. The explanation is from a 3s band becoming almost filled in the alloys. that the angular kinetic energy 1(1 + 1)/2 r2 keeps high After all, at least a-third of the conduction electrons I-values from the conduction band at high energy, must be 3p or other non-3s if an analysis in terms of except in the case of the relatively low-lying empty 5 f atomic orbitals has any meaning at all. It is then orbitals in thorium. Since g-like orbitals also should conceivable that the sharp band at 71 eV (moving to have high energy, transitions from 4f or 5f to the slightly higher energy in the silver alloy) corresponds to conduction band are expected to give an impression of orbitals containing a small amount of s-character. higher Fermi levels. This effect may be modified by Professor Azaroff most legitimately emphasizes empty 5d and 6d orbitals. Another important pheno- that the beginning of an absorption edge may not menon is that f orbitals with small average radius always be described as the beginning of the Fermi sea. have a much smaller efectron affinity than ionization energy (Bull. Soc. Chim. France 1968, 4745) whereas absorbing atom. This way it is possible that the onset conduction bands have identical electron affinity and of absorption may be different in two kinds of atoms. ionization energy. Hence it is perfectly conceivable that an electron added to a lanthanide metal go in the Mr. FABIAN(Question to Mr. ULMER).- Perhaps conduction band, though the ionization of a 4f Professor Ulmer or one of his collaborators at Karl- electron needs considerably more energy. sruhe would comment on the location of the Fermi edge in their isochromat spectra of alloys. Mr. ULMER.- I think that at the present state of affairs we have a lot of different densities of states and Mr ULMER(after a question of Mr. FABIAN).-We it is an important point to relate these different densi- are at the time investigating a possibility of fixing the ties of states to the pertaining experiment from which Fermi edge at our measured isochromats by taking they were gained. Then many of the different densities isochromats for different temperatures of the same of states are reasonable physical quantities. Another sample (as has turned out) is a better reference point problem is to relate these densities of states (depending for the Fermi edge then our former method, which upon the peculiar experiment) to well defined cal- located the Fermi edge at a point at half the height of culated band structure densities of states. the isochromat maximum (Ohlin maximum). Mr. MANNE.-The density of states is a well defined Mr. NIKIFOROV(question). - Have you made quantity within the accepted theoretical model. any calculations for the transition probability of your However, what is measured experimentally is not the cross transitions ? We have performed such calcula- density of states but a transition probability which, tions for some transition metals. It apeared to be in one way or another, relates to the density of states. about 100 times less than we wanted it to be. X-ray photoelectron spectroscopy (ESCA) offers a solution to the problem of relating different X-ray Mr. FABIAN(answer). - No, we have not perfor- emission spectra to each other and to the Fermi level. med any theoretical calculations concerning the pro- ESCA relates the initial states of the X-ray emission bability of these cross transitions. I believe this to be a to the Fermi level and the final states can then be much more complicated matrix element to calculate related to the Fermi level by subtraction of the transi- than is often assumed. tion energies from the ESCA electron binding energy. The problem of ESCA chemical shifts does not appear Mr. KUNZ(question). - With respect to the posi- in this procedure since one deals only with measure- tioning of the Fermi levels I would like to comment ments on the same material. that there is at least one different approach : Bea- ghehole and Erbach obtained a good understanding Mr. AZAROFF.- The matter of different density of their optical measurements in dilute noble metal of states curves and the earlier question about matching alloys by making the adjustment at the bottom of the the Fermi levels in alloys must be considered carefully Fermi sea. These authors, aswell, could not give a to avoid semantic difficulties. To take X-ray absorption strict theoretical foundation of their procedure. This as an example, what we know is that the onset of only demonstrates how open these questions still are. absorption marks the beginning of empty (allowed) energy levels. If the collective electron model applies, Mr. FABIAN(Answer). - The reason for localising then this should be at the Fermi level. But suppose that the Fermi energy of the two constituent atoms at the the collective electron model does not work for alloys, same energy, is probably only thermodynamic intui- or that transition probabities determine the first allowed tion. I find it very difficult to conceive the electrons transition to lie at a different energy, then the onset of in << adjacent )> atomic cells as having anything diffe- absorption will simply work that energy for the rent from equal chemical potentials.