SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS

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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�

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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 subject of much research commencing at about the time of the well-known text by N. F. Mott and H. Jones -

⁽⁽

The theory of the properties of metals and alloys

))

- which appeared in 1936. It is inte- resting to weigh the advances since then : they have been many, and have derived from both experimental and theoretical work.

However, the theory of the solid state is still not well understood and even for pure metals there are considerable difficulties in the interpretation of experimental data ; for alloys in particular a satisfac- tory theory has continually eluded physicists and metallurgists. The questions that arise, when two different kinds of metal atom are alloyed to form a disordered array, are : how much is the band structure affected, and can we still speak of a Brillouin-zone structure once we have destroyed the translational symmetry of the lattice ? In 1936 Sir Nevi11 Mott wrote

<(

The question has not been cleared up yet in a satisfactory way, but it seems fairly certain from the evidence (: evidence associated with the Hume-Rothery rules and their explanation in terms of a rigid-band model) that the break in periodicity may be considered not to affect the zone structure very much n. In 1970 the question has still not been cleared up, nor in fact is it much nearer to solution.

(*) This paper was prepared while the author was on leave of absence at the Laboratorium fiir Festkorperphysik, Eidg.

Technische Hochschule. Ziirich, Suisse.

Soft X-Ray Emission. - Soft X-ray band emission from metals provided the first experimental proof of the fundamental band theory of crystalline solids, and a considerable impetus was given to the early development of the subject, in connection with the electronic structure of metals and alloys, by the work of Farineau [I], Skinner [2] and Cauchois [3] ; lasting from about 1932 t o 1948. As early as 1933 Jones, Mott and Skinner successfully interpreted the intensity distributions observed [4] for the K and L spectra of simple metals (generally meaning metals whose electro- nic structures do not involve d-bands). The theoretical simplicity of that interpretation, which showed for the low-energy region of the emission band that

for K-emission I(E) cc E3I2 (1) and

for L-emission I(E) cc Ell2 (2) has remained as one of the foundation stones on which the field of soft X-ray spectroscopy of solids has continually developed.

The emission of soft X-radiation occurs following the excitation of the solid to a state in which an elec- tron vacancy is produced in an inner atomic-core level ; commonly a K or L level. The intensity of emission, as a function of energy, is usually written in the sim- plified form

I(E) cc F(E) N(E) (3) 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

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C4-318 D. FABIAN

and F(E) expresses the transition probability for an electron making a transition from an occupied valence or conduction level to the vacant core level. It was from a consideration of the factor F(E), and an esti- mation of the respective amounts of atomic s-state and p-state that will combine to form a zero-order wavefunction for electrons in the lattice, that Jones, Mott and Skinner derived the relation ships (1) and (2).

More rigorously eq. (3) should be written

where the integral is over all electron states k on the constant energy surface S in k space. Strictly F(E) cannot be removed from the integral because of its dependence on the direction of k and, as noted for example by Rooke [5], the k-dependence can be particularly sharp where changes of symmetry are involved. However, for alloys and indeed almost generally for all metals, eq. (3) and the relation ships (1) and (2), are accepted as working simplifica- tions. Following Jones, Mott and Skinner [2], the pro- bability factor F(E) is then expressed in terms of the square of the matrix elements for the transition

where v is the frequency of the emitted radiation, and Y, and Yf are wavefunctions for respectively the initial (valence band) and final (core level) states of the electron.

In what few attempts have been made to calculate 'the emission intensity for the component atoms in an alloy, the energy dependence of the transition probabi- lity has been estimated using eq. (5) as a starting point ; see Stott [6], [7]. This is essentially a one-electron description of the system, and while many-electron calculations have met with some success 181, [9], [lo]

in explaining fine-structure in the emission spectra of pure metals, the alloy problem is very much more complex and has not yet been addressed by the many- body theorists.

Electron Model for Alloys.

^-

Theoretical work in the field of alloys is continually expanding, and among the many models that have been proposed are : The rigid-band model first described by Jones [11] in 1934; the localised screening or virtual boundstate model developed chiefly by Friedel [12] and by Anderson [13] since 1952 : and the two-band model proposed by Varley [14] in 1954. The degree to which these models invoke distinct valence bands for the two components of the alloy increases in the order listed. At one extreme, the rigid-band model assumes a common band to which for a binary alloy both solvent and solute atoms contribute their valence electrons, and regards as negligible the incoherent scattering from the random impurity potential which is super-

posed upon the basic periodic potential of the lat- tice. Mott and Jones developed this simplified pic- ture chiefly to account for the Hume Rothery rules for the alloying characteristics of two metals. At the other extreme, we have the two-band model in which the valence electrons are assumed to exist in two separate sets of energy states, each associated with the potential fields of the ions of one component.

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 the charge surrounding a dissolved atom of different valency from the solvent metal ; an idea first proposed by Mott [15] in 1936. The excess charge on a solute atom is effectively shielded by local distortion of the conduction band so that the additional electrons or holes introduced by the solute are localised. Friedel shows that this model does not entirely conflict with the rigid-band model. The localised states or virtual bound states lie within the valence or conduction band, and are different from bound states which can also exist around a completely ionised solute atom and 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 model which assumes that the electrons move in a crystal potential that is everywhere ordered and the average of the potentials for the two pure components.

This in some ways is similar to the rigid-band model.

Beeby [16] has used an approximation in which the crystal potential is reduced to the sum of local potentials, and assumes a particular ordered system of localised potentials whose scattering matrix is the average of those from the localised potentials of each constituent.

More recently Soven, who found [17] that for weak potentials this localised-potential model reduces to the virtual crystal approximatibn, has introduced a promising model [IS] based on a Green-function for- malism and suitable for strong local potentials such as those found in transition metal alloys. With this formalism, which he first introduced as the coherent potential method [17] and which has subsequently been examined in some detail by Velicky Kirkpatrick and Ehrenreich [19], Soven shows [IS] that the density of states localised around a particular constituent atom can be substantially different from the average density of states so that the local electron density will be particularly succeptible to the local environment.

The picture obtained tends to. fall between the localised

screening or virtual-bound state model of Friedel and

Anderson, and the extreme simplification of a two-

band model used by Varley ; and if the approximations

of this one-electron picture are correct then strong

resonance states can result, and move through the

band if the impurity potential is increased, in apparent

agreement with the findings of Stott [7] whose calcu-

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SOFT X-RAY EMISSION AND ELE( ZTRONIC STRUCTURE OF ALLOYS C4-3 19

lations determine the effect of such resonance or vir- tual-bound states on the soft X-ray emission from atoms of the solute metal for dilute alloys.

Harrison [20] also considers the problem from the viewpoint of soft X-ray emission, but examines it in terms of pseudopotential theory. His findings, for dilute alloys of simple metals, favour what is effectively a rigid-band model, while for alloys in which d-bands are involved his approach predicts resonance-bound states so strongly localised that they would be seen - in terms of soft X-ray transitions - only by the impurity atom from which they originate.

Results for Soft X-Ray Emission from Alloys. -

Soft X-ray emission is particularly suited to examina- tion of the local environment of an atom in an alloy, because transitions can be observed from the imme- diately local conduction or valence band to core states of the atom. Moreover these transitions may be examined for both types of constituent-metal atom (for a binary alloy), or for either separately. It is helpful to examine figure I which illustrates, schema-

FIG. 1. - Schematic picture for A1 and Mg L-emission from adjacent cells in the lattice of an AI-Mg alloy, for the rigid-band model.

Indicated approximately, above and below, are the radial parts of the free-atom wavefunctions for the

L2

and L3 core levels,

and for the 3s valence electrons of Al and Mg.

tically, the situation approximating Co the rigid-band model for individual but adjacent A1 and Mg cells in the lattice of an A1-Mg alloy. We can picture here the transitions that occur from the valence band for the alloy to either the L2 and L, leve'ls of an A1 atomic core, or the L2 and L, level of a Mg atomic core,

respectively giving rise to A1 L2,,-emission and Mg L2,,-emission from the alloy.

For some binary systems, overlap of the two cons- tituent-metal emission bands occurs, with both cove- ring similar or overlapping wave-length regions ; but for many, as for example with AI-Mg, they are entirely separate. A measurement of the A1 and Mg K-emission from Al,Mg2 formed about the first such investigation of an alloy system, reported in 1937 by Farineau [21]

who found that the two emission bands had similar forms, and were quite different from the K-emission band for pure Al. The results appeared to support the simple model of sharing of electrons between A1 and Mg atoms. This led Skinner and Johnston [22] in the following year to interpret their results, for several L-emission and K-emission spectra from a series of alloys of A1 with Cu and Be, also as supporting the simple common-band model.

However, a careful examination of the A1 and Mg L2,,-emission bands of a series of Al-Mg alloys by Appleton and Curry [23] in 1965 gave results, repro- duced in figure 2, which sharply conflict with the simple

FIG. 2. - Mg and A1

L2,3

emission from AI-Mg alloys.

Reproduced, with permission, from Appleton and Curry 1231.

0 8 -

rigid-band model. In our picture of this model, figure 1, the common valence band has a width bet- ween 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 these results ; and closely similar conclusions were reached for Mg-bi alloys by Catterall and Trotter [25].

These observations appear to indicate that separate localised valence bands must be associated with each type of atom. We may picture in figure 3 the situation that will apply if this is the case. Two densities of states, one for each, type of atom, will predominate locally. Some form of two-band model is implied similar to that proposed by Varley. This has been advocated also, t o explain the soft X-ray emission results, by Rooke [24] who tentatively explores a two- band model in terms of Thomas-Fermi localised screening.

1 1

4 -

Pure metals

I

---

^{A h}MPI

0 6 - ^1,I(

A L,2 Mg 7

4r

:

^{0 4 -}

0 2 -

38 4 2 46 50 60 6 4 68 72

Energy E ( eul

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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 emis- sion 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- The results obtained by Skinner and Johnston for

Be K-emission from dilute alloys of Be in Cu and A1 appeared also to indicate that the bandwidths associa- ted with the solute atoms are the same as for the pure solute metal. They interpreted their results, to retain agreement with the conclusions reached by Farineau and with the rigid-band model, by proposing that while the separate atoms in fact see the same common band they will only be involved in transitions with electrons that can penetrate to the interior core levels of the atom ; the atoms select, so to speak, their own temporary valence electrons >>. Skinner and Johnston introduced into the expression for I(,?), eq. (3), a sorting function s(E) to give

and regarded s(E) as expressing this property of an atom core to select its own valence electrons from the common band, and to be distinct from the proba- bility factor F(E). The function F(E) cannot alone be responsible for this would provide no explanation of why the wider constituent-metal emission band (the Al-emission bandwidth in the case of Al-Mg alloys) is greater than expected for a common rigid band.

To provide a rough indication of the extent to which core-level and valence-band electrons might penetrate from one atom cell to the next, the radial part of the Hartree-Fock free-atom wavefunctions for the 2p and 3s electrons of both A1 and Mg have been shown, approximately, in figure 1. The A1 3p-electrons contribute also to the valence band but these have not been included because only s-states can be involved in transitions to the L, and L , ( Z ~ ' / ~ and 2p3I2) core states and, although the symmetry of the valence p-states can be affected by the crystal field so that they are to some extent involved, transitions giving rise to L2,,-emission will result predominantly from s-electrons in the band.

The sharpness of the L, and L, levels is immediately apparent, and the extent to which the valence s-states arising from neighbouring atoms can be seen by a given atomic core can be roughly gauged. If the pene-

valency metal. This we can interpret in terms of the two-band model : in the case of Al-Mg, where elec- trons are transferred from the A1 atoms to Mg atoms, if the electron states associated with an A1 atom are distinct from those associated with an Mg atom then we should expect a decrease in intensity in the upper part of the A1 band where the deplenishing would mostly occur. A corresponding increase is observed in the upper part of the Mg band. Alternatively we can consider the excess charge left on an A1 atom as drawing off part of its outer electrons and forming bound states (of s-symmetry) that partly screen the upper valence states from the core.

Clearly there is some evidence for a two-band model 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 has recently obtained very similar data for this system, finds that for concentrated alloys his results do tend towards the rigid-band description. One explanation must always be borne in mind : this concerns the ques- tion of clustering in the alloys which, if it occurs to any marked extent, could result in significant pure- component emission from within the alloy. However clustering, although it would account for the constant bandwidth, cannot easily explain the considerable changes of shape that occur in the emissfon bands.

Furthermore, short-range order in Al-Mg is possibly more likely than clustering, a view supported by the strong formation of intermetallic phases.

Turning to the results for other alloy systems, Curry [29] has given a valuable review of the experi- mental findings which appear mostly to indicate band- widths that do not change appreciably on alloying. This particularly is the case for noble and transition-metal alloys, in which we find also much evidence for the existence of virtual-bound or resonance-bound state.

In figure 4 the results are shown for the A1 L2,,-

emission from Al-Ag alloys, reported by Fabian,

Lindsay and Watson [30]. A strong peak is caused by

transitions from Ag d-states in the alloy valence band

to A1 core states. If we examine now the radial part

of the free-atom 4d wavefunction for Ag, which is

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SOFT X-RAY EMISSION AND ELECTRONIC STRUCTURE OF ALLOYS 04-321

FIG. 4. - A1 L2,s-emission spectra obtained for a series of A1-Ag alloys.

The prominent peak that develops at - 65 eV is interpreted in terms of transition from the Ag d-band to A1 core levels.

indicated approximately in figure 5, we can say that such transitions are not altogether unexpected ; they probably arise from resonance states involving the 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 adja- cent 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 and AI-Cu alloys. We find no such resonance peak in 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.

The measurements for Al-Cu alloys by Watson and

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.

Lindsay, are shown in figure 6. We observe that a strong peak occurs in the A1 L,,,-emission, which we attribute to transitions from resonance-bound d-states of the copper atoms to neighbouring A1 core states.

The peak appears also to move to lower energies, and to broaden, as the concentration of Cu is increased ; bearing in mind that the 20 %-Cu alloy is two-phase,

AI -CU Alloys

I

74 70 65 60

FIG. 6. - Al L-emission spectra for AI-Cu alloys (see text).

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C4-322 D. FABIAN

containing a tetragonal 8-phase and a fcc a-A1 phase.

This may correspond to the features described by Stott [7]. For these alloys the M2,,-emission band from Cu overlaps the A1 L2,,-emission, and an esti- mate of this band for the 80 %-Cu alloy has been made from the measured M2,,-emission for pure copper.

Considerable support for the interpretation that these are localised d-states being seen by the sol- vent metal, comes from the work of Harrison and Curry [33], who observe the same strong peak in A1 L2,,-emission from all alloys of the noble-metals with aluminium, and from the work of Williams et al. [34]

who observe it in Al-Au alloys. Harrison and Curry also examine the overlap of the radial functions R,,(r) for Cu and r 3 RZp(r) for Al, centred on adjacent A1 and Cu atoms, and conclude that since the product of these is involved in the matrix element for the transition from Cu 3d to A1 2p that this transition is likely to occur.

Harrison and Curry [33] conclude also that resonance bound states are being observed in the A1 L-emis- sion from transition-metal alloys with aluminium, an observation supported by the results obtained for A1-Pd alloys by Watson and Kapoor, in conjunction with Nemoshkalenko [35], shown in figure 7. In the

1 . AI-Pd Alloys

Fro. 7. - A1 L-emission spectra for A1-Pd alloys (see text).

measurements by Watson and Kapoor a strong resonance peak is again observed and probably moves through the band with increasing concentration of Pd. They find similar results for the alloys of Nb, V and Co with Al. In these cases, and in those of Pd and

Cu, the position of the strong peak is less easily corre- lated with the energy at which the d-band for the pure transition or noble metal is expected to occur. However in all these cases strong intermetallic bonding occurs, in which the d-electrons will play a large part ; this would cause the d-states to appear at lower energy, and also possibly make the transitions more likely.

The results are probably best explained in terms of a form of two-band model and local electron-state densities, which Soven [18] shows can be expected to differ considerably from the average density of states for the alloy. The theoretical concepts differ only in formalism from the localised virtual or resonance states of Anderson and Friedel ; the theory for these probably only applies for dilute alloys, but there seems to be no reason why strongly localised states would not occur in concentrated alloys. The formation of reso- nance-bound states for noble and transition metals is also well formulated by Harrison [20] ; although he suggests that they would not be seen by the solvent- metal atoms for an alloy in which Cu for example is the solute, which is not in agreement with our obser- vations.

Localised states have been reported also in the interpretation of optical data for Ag-Pd alloys, by Karlsson, Myers and WalldBn [36] ; and in the inter- pretation 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 impor- tance 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 elec- tronic 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 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 the effect of the local core hole will bring the necessary refinement.

Acknowledgements. -The author is indebted to the

U. K. Science Research Council for a grant in support

of this research. It is a pleasure also to thank : Pro-

fessor E. C . Ellwood for continued support and

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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.

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DISCUSSION Mr. JORGENSEN. - In view of the concern expressed

by several of our colleagues about small numerical values of intensities of cc crossed transitions

)),

I am wondering whether the shoulder a t 65 eV in elemental aluminium becoming a maximum in the various alloys with magnesium, copper and silver is a transition from a 3s band becoming almost filled in the alloys.

After all, at least a-third of the conduction electrons must be 3p or other non-3s if an analysis in terms of atomic orbitals has any meaning at all. It is then conceivable that the sharp band a t 71 eV (moving to slightly higher energy in the silver alloy) corresponds to orbitals containing a small amount of s-character.

Professor Azaroff most legitimately emphasizes that the beginning of an absorption edge may not always be described as the beginning of the Fermi sea.

A most striking counter-example was found in 1941

by Professor Cauchois demonstrating that the empty

orbitals in metallic gold accessible to excitation of

3d electrons apparently have much higher energy

than those accessible t o 2p, whereas this discrepancy

is very small in metallic thorium. The explanation is

that the angular kinetic energy 1(1 + 1)/2 r 2 keeps high

I-values from the conduction band at high energy,

except in the case of the relatively low-lying empty 5 f

orbitals in thorium. Since g-like orbitals also should

have high energy, transitions from 4f or 5f to the

conduction band are expected to give an impression of

higher Fermi levels. This effect may be modified by

empty 5d and 6d orbitals. Another important pheno-

menon is that f orbitals with small average radius

have a much smaller efectron affinity than ionization

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energy (Bull. Soc. Chim. France 1968, 4745) whereas conduction bands have identical electron affinity and ionization energy. Hence it is perfectly conceivable that an electron added to a lanthanide metal go in the conduction band, though the ionization of a 4f electron needs considerably more energy.

Mr. ULMER. - I think that at the present state of affairs we have a lot of different densities of states and it is an important point to relate these different densi- ties of states to the pertaining experiment from which they were gained. Then many of the different densities of states are reasonable physical quantities. Another problem is to relate these densities of states (depending upon the peculiar experiment) to well defined cal- culated band structure densities of states.

Mr. MANNE. -The density of states is a well defined quantity within the accepted theoretical model.

However, what is measured experimentally is not the density of states but a transition probability which, in one way or another, relates to the density of states.

X-ray photoelectron spectroscopy (ESCA) offers a solution to the problem of relating different X-ray emission spectra to each other and to the Fermi level.

ESCA relates the initial states of the X-ray emission to the Fermi level and the final states can then be related to the Fermi level by subtraction of the transi- tion energies from the ESCA electron binding energy.

The problem of ESCA chemical shifts does not appear in this procedure since one deals only with measure- ments on the same material.

Mr. AZAROFF. - The matter of different density of states curves and the earlier question about matching the Fermi levels in alloys must be considered carefully to avoid semantic difficulties. To take X-ray absorption as an example, what we know is that the onset of absorption marks the beginning of empty (allowed) energy levels. If the collective electron model applies, then this should be at the Fermi level. But suppose that the collective electron model does not work for alloys, or that transition probabities determine the first allowed transition to lie at a different energy, then the onset of absorption will simply work that energy for the

absorbing atom. This way it is possible that the onset of absorption may be different in two kinds of atoms.

Mr. FABIAN (Question to Mr. ULMER). - Perhaps Professor Ulmer or one of his collaborators at Karl- sruhe would comment on the location of the Fermi edge in their isochromat spectra of alloys.

Mr ULMER (after a question of Mr. FABIAN). -We are at the time investigating a possibility of fixing the Fermi edge at our measured isochromats by taking isochromats for different temperatures of the same sample (as has turned out) is a better reference point for the Fermi edge then our former method, which located the Fermi edge at a point at half the height of the isochromat maximum (Ohlin maximum).

Mr. NIKIFOROV (question). - Have you made any calculations for the transition probability of your cross transitions ? We have performed such calcula- tions for some transition metals. It apeared to be about 100 times less than we wanted it to be.

Mr. FABIAN (answer). - No, we have not perfor- med any theoretical calculations concerning the pro- bability of these cross transitions. I believe this to be a much more complicated matrix element to calculate than is often assumed.

Mr. KUNZ (question). - With respect to the posi- tioning of the Fermi levels I would like to comment that there is at least one different approach : Bea- ghehole and Erbach obtained a good understanding of their optical measurements in dilute noble metal alloys by making the adjustment at the bottom of the Fermi sea. These authors, aswell, could not give a strict theoretical foundation of their procedure. This only demonstrates how open these questions still are.