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4f localization in cerium compounds
Journal of the Less-Common Metals, 94 (1983) 17-22
4f LOCALIZATION
IN CERIUM
17
COMPOUNDS*
Summary The 4f is highly localized in all the rare earths except cerium where the degree of localization is reduced and the 4f wavefunction can mix with the extended states of the solid. Hence, the 4f character of cerium varies from one compound to another, e.g. localized in the hydrides and chalcogenides, slightly delocalized in the pnictides and more nearly band like in some intermetallic compounds. Synchrotron radiation photoemission results for cerium compounds with representative behavior are examined.
Intense efforts are presently under way to describe the complex character of the 4f electron in cerium compounds, i.e. to explain its apparent band-like behavior in some systems but its more localized appearance in others. Regrettably, all of the valuable contributions to the field cannot be listed here. The emphasis of this short paper is on photoemission studies of cerium compounds. Additional references can be found in the publications cited in the remainder of this paper. Photoelectron spectroscopy has been used in a large number of studies to determine the energy and hybridization of the 4f Cl-141 and major theoretical efforts have been undertaken [15-221 to explain the fascinating experimental results. Photoelectron spectroscopy directly probes the electronic states and thereby offers insight into 4f localization. It does so, however, by exciting an electron from the ground state and this may alter the energy level picture of the localized correlated states. One ofthe challenges for studies of cerium systems is then to determine the extent to which one-electron concepts can be applied and to evaluate the importance of final state effects [3, 4, E-221. In this paper, we briefly review results for three classes of cerium compounds which exhibit representative 4f behavior: CeS and CeH, for which spectroscopic evidence *Paper presented at the Sixteenth Rare Earth Research Conference, Tallahassee, FL, U.S.A., April B-21,1983. 0022-5088/83/$3.00
The Florida State University,
0 Elsevier Sequoia/Printed
in The Netherlands
18
suggests a non-interacting localized 4f level [2, S] ; CeRh, which appears to have substantial 4f hybridization [2] ; cerium and the cerium pnictides which display two 4f-related features [3--51.
98
7
65
Energy
Fig. 1. Photoemission spectrum for DyAs which shows the dramatic because of photoionization cross-section effects.
4
3
Below
enhancement
2
E,
I O*E,
(eV)
of 4f states
Fig. 2. Comparison of EDCs for CeAs with the calculated one-electron density of states for GdAs to emphasize that low energy photoemission spectra are best used to reveal valence band features. 4f contributions become visible for hv 2 25 eV.
19
The 4f-derived features can be highlighted by several photoemission techniques. First, the hv dependence of the photoionization cross section of valence band states can be examined, recognizing that the Z(1+ 1)/r’ term in the effective potential influences the onset of photoionization and shifts the oscillator strength to higher photon energy for f states than for d-, p- or s-derived states [23]. Hence the 4f-related features can be identified by comparing photoelectron energy distribution curves (EDCs) at low photon energy with those at higher energies. The dramatic cross-section enhancement of the 4f can be seen in Fig. 1 where we show EDCs for DyAs which have been normalized to the emission intensity of the arsenic-derived p states within about 4 eV of l& [Zl]. Structure associated with the dysprosium 4f multiplets appears at binding energies greater than 4 eV and the structure at - 5 eV reflects the ‘F, multiplet state. As shown, the 4f features are scarcely visible at hv = 30 eV but are dominant at hv = 43 eV. For hv <, 25 eV, only the valence band states are seen EDCs for CeAs for 10 eV d hv < 40 eV are compared in Fig. 2 with the density of states resulting from a one-electron band calculation for GdAs C24, 251. As can be seen, the spectral features are related to non4frelated properties of the electronic structure for hv 5 30 eV. Indeed, detailed analysis of the hv dependence of the dispersion of the emission features allows an identification of the critical points in the band structure [24]. A second way to identify the 4f character involves the resonant photoemission process [2,26] (see other papers in these proceedings for discussion of resonant photoemission). Two processes compete quantum mechanically to give the final state, one involving direct excitation of the 4f electrons and the other involving 4d core excitation and decay which results in 4f ejection. In Fig. 3 we display EDCs for CeH, which are arranged so as to show the pre-resonance curve where the 4f emission is suppressed (about 114 eV) and spectra exhibiting resonance enhancement of the 4f emission by more than an order of magnitude. As Figs. l-3 demonstrate, the techniques of cross-section enhancement and resonant enhancement are extremely useful in identifying the 4f character of the ED&. Additional insight can be obtained by comparing EDCs for isomorphic cerium and lanthanum compounds [2] since the resulting difference curves draw attention to the 4f character, as shown for LaH, and CeH, in Fig. 3. In Fig. 3 we show EDCs for CeH,, a compound in which the 4f is isolated and can be readily identified by either cross-section or resonant enhancement or by comparison with LaH,. Analysis of the electronic properties of CeH, and LaH, has shown [Z] that the 4f lies at -2 eV with a full width at half-maximum (FWHM) of 0.75 eV. The states closer to EF are cerium 5d derived, while those centered at -5 eV comprise the so-called bonding band and are hybridized metal-hydrogen states [27]. The results for LaH, show the density of states to be quite low in the region where the 4f is observed [Z]. Similar conclusions have been drawn from studies of CeS, CeSe and CeTe [5, 63, where the 4f lies between chaleogen-derived p states and metal-derived d states (EB = 2.4 eV; FWHM z 0.7 eV). In Fig. 4 we show EDCs for CeRh, taken at hv = 113.7eV (pre-resonance) and a series of energies through resonance. These reveal 4f behavior represen-
20
I
0
8
6
BINDING (a)
4
2
ENERGY
O=EF
8
6 BINDING
(eV) @I
4
2
6
O=E,
ENERGY
4
BINDING
(eV)
(4
I
I 2
b--t. O=Er
ENERGY &)
4
2
BINDING
O=Er
ENERGY (eV)
0-4
Fig. 3. Comparison of results for (a) LaH, and (b) CeH, for photon energies corresponding resonant photoemission to show enhancement of the 4f feature near 2 eV. Fig. 4. (a) Resonant photoemission character.
results for CeRh, and (b) difference curves representing
to
the 4f
tative of systems with apparent 4f hybridization. The difference curves highlight the 4f character of the bands and were obtained by subtracting each EDC from the pre-resonance curve, These results were discussed in ref. 2 as part of a study of several cerium intermetallics which were reported to have varying degrees of 4f occupancy. The spectral results showed (4f)’ character, even for CeRh, which had previously been identified as (4f)O. Our conclusion, based on these spectra and partial yield spectra, was that the 4f was sufficiently extensive to mix, a conclusion supported by recent calculations by Koelling [28]. In Figs. 5 and 6 we show cross-section enhancement EDCs for the cerium pnictides [S] and resonant photoemission EDCs for c+Ce [4] as examples of systems for which more than one 4f-derived spectral feature is observed (see other papers in these proceedings for a discussion of resonant photoemission). For the pnictides, we postulated that the relative intensities of the 0.6 and 3 eV features change by an order of magnitude because of varying screening of the 4f photohole by empty 5d conduction band states. A recent cluster calculation by Fujimori [lS] estimated the effects of screening pnictide ligands on the photoemission spectra. Although the interpretations differ, both emphasize the importance of final state effects in photoemission from narrow band materials. For a-Ce and y-Ce, Wieliczka and coworkers [4] have observed two 4frelated features and have shown that the 2 eV feature increases in relative intensity compared with the structure at Er in the y phase. They concluded that the degree of 4f wavefunction overlap changes at the phase transition and the photohole is better screened in the more compressed a phase. Liu and Ho [15] proposed that d-f Coulomb effects might explain this observation and others have discussed f-f correlation energies as being responsible for inducing the distribution off weight measured experimentally.
21
a 4f ~04
4f
I
-2’
-IIresolution
/
:
:
/
?
: ‘\ :I \
CeP
“7,ky
l.o/Ly
1
0
\I, 0 ?
~ 3
1
1
1 1
!. L :
EINDING
ENERGY
(et!)
-10
-8
Energy
-6
-4
-2
Relative To
E,=o
E,
(eV)
Fig. 5. Photoemission results for cerium pnictides showing two 4f-derived emission features ( - 0.6 and - 3.0 eV) and the importance of screening the photohole (final state effects): ---, hv = 30 eV; --, hv = 50 eV; --, 4f emission, The ratio of the two features (inset) indicates changes in screening. Fig. 6. Photoemission spectra for u-Ce showing 4f-derived features at - 2 eV and EF. These can be highlighted by comparing spectra at 40 and 60 eV or 115 and 120 eV.
The results of the valence band photoemission cited above (and especially through the detailed references) combine with important core level studies, bremsstrahlung isochromat and X-ray absorption studies and theory to give an increasingly precise view of the 4f in a variety of systems [29]. It remains an important challenge to develop a completely consistent framework within which the various results can all be accommodated.
Acknowledgments It is a pleasure to acknowledge colleagues who participated in the original work cited here. Synchrotron radiation studies were performed at the Synchrotron Radiation Center, University of Wisconsin-Madison, and were enriched by the staff of that laboratory (supported by National Science Foundation Grant DMR-8020164). This work was supported by National Science Foundation Grant DMR-7821080.
22
References
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
J. M. Lawrence, P. S. Riseborough and R. D. Parks, Rep. Prog. Phys., 44 (1981) 1. D. J. Peterman, J. H. Weaver, M. Croft and D. T. Peterson, Phys. Rev. B, 25 (1982) 5530; 27 (1983) 808, and references cited therein. A. Franciosi, J. H. Weaver, N. Martensson and M. Croft, Phys. Rev. B, 24(1981) 3651. D. Wieliczka, J. H. Weaver, D. W. Lynch and C. G. Olson, Phys. Rev. B, 26(1982) 7056. D. Wieliczka, Ph.D. ~~se~t~tio~, Iowa State University, 1982. W, Gudat, R. Rosei, J. H. Weaver, E. Kaldis and F. Hulliger, SoEid State Commun., 41 (1982) 37; J. Appl. Phys., 52(1981) 2133. M. Croft, A. Franciosi, J. H. Weaver and A. Jayaraman, Phys. Rev. B, 24 (1981) 544. M. Croft, J. H. Weaver, D. J. Peterman and A. Frandiosi, Phys. Reu. Lett., 46 (1981) 1104. N. Martensson, B. Reihl and R. D. Parks, Sotid State Commun., 8(1983) 573. J. W. Allen, S.-J. Oh, I. Lindau, J. M. Lawrence, L. I. Johansson and S. B. M. Hagstrom, Phys. Rev. Lett., 46(1981) 1100; Phys. Rev. B, 26(1982)445. A. Platau, A. Callenas and S.-E. Karlsson, Solid State Commun., 37(1981) 829; Phys. Rev. B, 18 (1978) 3820. L. I. Johansson, J. W. Allen, T. Gustafsson, I. Lindau and S. B. M. Hagstrom, SoZid State Commun., 28 (1978) 53. P. A. Cox, J. K. Lang and Y. Baer, J. Phys. F, lI(l981) 113,121. M. Campagna, G. K. Wertheim and Y. Baer, Top. Appl. Phys., 27(1979) Chap. 4. Y. Baer, H. R. Oh, J. C. Fuggle and L. E. DeLong, Phys. Rev. B, 24 (1981) 5384. S. H. Liu and K. M. Ho, Phys. Rev. B, 26(1981) 7052. A. Fujimori, Phys. Rev. B, 27(1982) 3992; personal communication, 1983. D. D. Koelling, SoZid State Commun., 43 (1982) 274. W. E. Pickett, A. J. Freeman and D. D. Koelling, Phys. Rev. B, 23 (1981) 1266. R. Podloucky and D. Glotzel, Phys. Rev. B, 27 (1983) 3390, and references cited therein. B. Johansson, Philos. Mug., 30(1974) 469; Phys. Rev. B, ll(l975) 1367. E. A. Kmetko and H. H. Hill, J. Phys. F, 5(1975) 1119; 6(1976) 1025. R. M. Martin, Phys. Rev. Lett., 48(1982) 362. J. W. Allen and R. M. Martin, Phys. Rev. Lett., 49(1982) 1106. S. T. Mason, Top. Appl. Phys., 26(1978) Chap. 3. J. H. Weaver and D. Wieliczka, unpublished data, 1983. A. Hasegawa and A. Yanase, J. Phys. SW. Jpn., 42 (1977) 492. W. Gudat and C. Kunz, Phys. Rev. Lett., 29f1972) 169. D. J. Peterman, J. H. Weaver and D. T. Peterson, Phys. Rev. B, 23 (1981) 1692. D. D. Koelling, personal communication, 1983. J. C. Fuggle, M. Campagna, Z. Zolnierek, R. Lasser and A. Platau, Phys. Rev. Lett., 45 (1980) 1597. B. Johansson and N. Martensson, Phys. Rev. B, 24 (1981) 4494. F. U. Hillebre~ht and J. C. Fuggle, Phys. Rev. B, 25 (1982) 3550. G. Crecelius, G. K. Wertheim and D. N. E. Buchanan, Phys. Rev. B, 18 (1978) 6419.
4f LOCALIZATION
IN CERIUM
17
COMPOUNDS*
Summary The 4f is highly localized in all the rare earths except cerium where the degree of localization is reduced and the 4f wavefunction can mix with the extended states of the solid. Hence, the 4f character of cerium varies from one compound to another, e.g. localized in the hydrides and chalcogenides, slightly delocalized in the pnictides and more nearly band like in some intermetallic compounds. Synchrotron radiation photoemission results for cerium compounds with representative behavior are examined.
Intense efforts are presently under way to describe the complex character of the 4f electron in cerium compounds, i.e. to explain its apparent band-like behavior in some systems but its more localized appearance in others. Regrettably, all of the valuable contributions to the field cannot be listed here. The emphasis of this short paper is on photoemission studies of cerium compounds. Additional references can be found in the publications cited in the remainder of this paper. Photoelectron spectroscopy has been used in a large number of studies to determine the energy and hybridization of the 4f Cl-141 and major theoretical efforts have been undertaken [15-221 to explain the fascinating experimental results. Photoelectron spectroscopy directly probes the electronic states and thereby offers insight into 4f localization. It does so, however, by exciting an electron from the ground state and this may alter the energy level picture of the localized correlated states. One ofthe challenges for studies of cerium systems is then to determine the extent to which one-electron concepts can be applied and to evaluate the importance of final state effects [3, 4, E-221. In this paper, we briefly review results for three classes of cerium compounds which exhibit representative 4f behavior: CeS and CeH, for which spectroscopic evidence *Paper presented at the Sixteenth Rare Earth Research Conference, Tallahassee, FL, U.S.A., April B-21,1983. 0022-5088/83/$3.00
The Florida State University,
0 Elsevier Sequoia/Printed
in The Netherlands
18
suggests a non-interacting localized 4f level [2, S] ; CeRh, which appears to have substantial 4f hybridization [2] ; cerium and the cerium pnictides which display two 4f-related features [3--51.
98
7
65
Energy
Fig. 1. Photoemission spectrum for DyAs which shows the dramatic because of photoionization cross-section effects.
4
3
Below
enhancement
2
E,
I O*E,
(eV)
of 4f states
Fig. 2. Comparison of EDCs for CeAs with the calculated one-electron density of states for GdAs to emphasize that low energy photoemission spectra are best used to reveal valence band features. 4f contributions become visible for hv 2 25 eV.
19
The 4f-derived features can be highlighted by several photoemission techniques. First, the hv dependence of the photoionization cross section of valence band states can be examined, recognizing that the Z(1+ 1)/r’ term in the effective potential influences the onset of photoionization and shifts the oscillator strength to higher photon energy for f states than for d-, p- or s-derived states [23]. Hence the 4f-related features can be identified by comparing photoelectron energy distribution curves (EDCs) at low photon energy with those at higher energies. The dramatic cross-section enhancement of the 4f can be seen in Fig. 1 where we show EDCs for DyAs which have been normalized to the emission intensity of the arsenic-derived p states within about 4 eV of l& [Zl]. Structure associated with the dysprosium 4f multiplets appears at binding energies greater than 4 eV and the structure at - 5 eV reflects the ‘F, multiplet state. As shown, the 4f features are scarcely visible at hv = 30 eV but are dominant at hv = 43 eV. For hv <, 25 eV, only the valence band states are seen EDCs for CeAs for 10 eV d hv < 40 eV are compared in Fig. 2 with the density of states resulting from a one-electron band calculation for GdAs C24, 251. As can be seen, the spectral features are related to non4frelated properties of the electronic structure for hv 5 30 eV. Indeed, detailed analysis of the hv dependence of the dispersion of the emission features allows an identification of the critical points in the band structure [24]. A second way to identify the 4f character involves the resonant photoemission process [2,26] (see other papers in these proceedings for discussion of resonant photoemission). Two processes compete quantum mechanically to give the final state, one involving direct excitation of the 4f electrons and the other involving 4d core excitation and decay which results in 4f ejection. In Fig. 3 we display EDCs for CeH, which are arranged so as to show the pre-resonance curve where the 4f emission is suppressed (about 114 eV) and spectra exhibiting resonance enhancement of the 4f emission by more than an order of magnitude. As Figs. l-3 demonstrate, the techniques of cross-section enhancement and resonant enhancement are extremely useful in identifying the 4f character of the ED&. Additional insight can be obtained by comparing EDCs for isomorphic cerium and lanthanum compounds [2] since the resulting difference curves draw attention to the 4f character, as shown for LaH, and CeH, in Fig. 3. In Fig. 3 we show EDCs for CeH,, a compound in which the 4f is isolated and can be readily identified by either cross-section or resonant enhancement or by comparison with LaH,. Analysis of the electronic properties of CeH, and LaH, has shown [Z] that the 4f lies at -2 eV with a full width at half-maximum (FWHM) of 0.75 eV. The states closer to EF are cerium 5d derived, while those centered at -5 eV comprise the so-called bonding band and are hybridized metal-hydrogen states [27]. The results for LaH, show the density of states to be quite low in the region where the 4f is observed [Z]. Similar conclusions have been drawn from studies of CeS, CeSe and CeTe [5, 63, where the 4f lies between chaleogen-derived p states and metal-derived d states (EB = 2.4 eV; FWHM z 0.7 eV). In Fig. 4 we show EDCs for CeRh, taken at hv = 113.7eV (pre-resonance) and a series of energies through resonance. These reveal 4f behavior represen-
20
I
0
8
6
BINDING (a)
4
2
ENERGY
O=EF
8
6 BINDING
(eV) @I
4
2
6
O=E,
ENERGY
4
BINDING
(eV)
(4
I
I 2
b--t. O=Er
ENERGY &)
4
2
BINDING
O=Er
ENERGY (eV)
0-4
Fig. 3. Comparison of results for (a) LaH, and (b) CeH, for photon energies corresponding resonant photoemission to show enhancement of the 4f feature near 2 eV. Fig. 4. (a) Resonant photoemission character.
results for CeRh, and (b) difference curves representing
to
the 4f
tative of systems with apparent 4f hybridization. The difference curves highlight the 4f character of the bands and were obtained by subtracting each EDC from the pre-resonance curve, These results were discussed in ref. 2 as part of a study of several cerium intermetallics which were reported to have varying degrees of 4f occupancy. The spectral results showed (4f)’ character, even for CeRh, which had previously been identified as (4f)O. Our conclusion, based on these spectra and partial yield spectra, was that the 4f was sufficiently extensive to mix, a conclusion supported by recent calculations by Koelling [28]. In Figs. 5 and 6 we show cross-section enhancement EDCs for the cerium pnictides [S] and resonant photoemission EDCs for c+Ce [4] as examples of systems for which more than one 4f-derived spectral feature is observed (see other papers in these proceedings for a discussion of resonant photoemission). For the pnictides, we postulated that the relative intensities of the 0.6 and 3 eV features change by an order of magnitude because of varying screening of the 4f photohole by empty 5d conduction band states. A recent cluster calculation by Fujimori [lS] estimated the effects of screening pnictide ligands on the photoemission spectra. Although the interpretations differ, both emphasize the importance of final state effects in photoemission from narrow band materials. For a-Ce and y-Ce, Wieliczka and coworkers [4] have observed two 4frelated features and have shown that the 2 eV feature increases in relative intensity compared with the structure at Er in the y phase. They concluded that the degree of 4f wavefunction overlap changes at the phase transition and the photohole is better screened in the more compressed a phase. Liu and Ho [15] proposed that d-f Coulomb effects might explain this observation and others have discussed f-f correlation energies as being responsible for inducing the distribution off weight measured experimentally.
21
a 4f ~04
4f
I
-2’
-IIresolution
/
:
:
/
?
: ‘\ :I \
CeP
“7,ky
l.o/Ly
1
0
\I, 0 ?
~ 3
1
1
1 1
!. L :
EINDING
ENERGY
(et!)
-10
-8
Energy
-6
-4
-2
Relative To
E,=o
E,
(eV)
Fig. 5. Photoemission results for cerium pnictides showing two 4f-derived emission features ( - 0.6 and - 3.0 eV) and the importance of screening the photohole (final state effects): ---, hv = 30 eV; --, hv = 50 eV; --, 4f emission, The ratio of the two features (inset) indicates changes in screening. Fig. 6. Photoemission spectra for u-Ce showing 4f-derived features at - 2 eV and EF. These can be highlighted by comparing spectra at 40 and 60 eV or 115 and 120 eV.
The results of the valence band photoemission cited above (and especially through the detailed references) combine with important core level studies, bremsstrahlung isochromat and X-ray absorption studies and theory to give an increasingly precise view of the 4f in a variety of systems [29]. It remains an important challenge to develop a completely consistent framework within which the various results can all be accommodated.
Acknowledgments It is a pleasure to acknowledge colleagues who participated in the original work cited here. Synchrotron radiation studies were performed at the Synchrotron Radiation Center, University of Wisconsin-Madison, and were enriched by the staff of that laboratory (supported by National Science Foundation Grant DMR-8020164). This work was supported by National Science Foundation Grant DMR-7821080.
22
References
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
J. M. Lawrence, P. S. Riseborough and R. D. Parks, Rep. Prog. Phys., 44 (1981) 1. D. J. Peterman, J. H. Weaver, M. Croft and D. T. Peterson, Phys. Rev. B, 25 (1982) 5530; 27 (1983) 808, and references cited therein. A. Franciosi, J. H. Weaver, N. Martensson and M. Croft, Phys. Rev. B, 24(1981) 3651. D. Wieliczka, J. H. Weaver, D. W. Lynch and C. G. Olson, Phys. Rev. B, 26(1982) 7056. D. Wieliczka, Ph.D. ~~se~t~tio~, Iowa State University, 1982. W, Gudat, R. Rosei, J. H. Weaver, E. Kaldis and F. Hulliger, SoEid State Commun., 41 (1982) 37; J. Appl. Phys., 52(1981) 2133. M. Croft, A. Franciosi, J. H. Weaver and A. Jayaraman, Phys. Rev. B, 24 (1981) 544. M. Croft, J. H. Weaver, D. J. Peterman and A. Frandiosi, Phys. Reu. Lett., 46 (1981) 1104. N. Martensson, B. Reihl and R. D. Parks, Sotid State Commun., 8(1983) 573. J. W. Allen, S.-J. Oh, I. Lindau, J. M. Lawrence, L. I. Johansson and S. B. M. Hagstrom, Phys. Rev. Lett., 46(1981) 1100; Phys. Rev. B, 26(1982)445. A. Platau, A. Callenas and S.-E. Karlsson, Solid State Commun., 37(1981) 829; Phys. Rev. B, 18 (1978) 3820. L. I. Johansson, J. W. Allen, T. Gustafsson, I. Lindau and S. B. M. Hagstrom, SoZid State Commun., 28 (1978) 53. P. A. Cox, J. K. Lang and Y. Baer, J. Phys. F, lI(l981) 113,121. M. Campagna, G. K. Wertheim and Y. Baer, Top. Appl. Phys., 27(1979) Chap. 4. Y. Baer, H. R. Oh, J. C. Fuggle and L. E. DeLong, Phys. Rev. B, 24 (1981) 5384. S. H. Liu and K. M. Ho, Phys. Rev. B, 26(1981) 7052. A. Fujimori, Phys. Rev. B, 27(1982) 3992; personal communication, 1983. D. D. Koelling, SoZid State Commun., 43 (1982) 274. W. E. Pickett, A. J. Freeman and D. D. Koelling, Phys. Rev. B, 23 (1981) 1266. R. Podloucky and D. Glotzel, Phys. Rev. B, 27 (1983) 3390, and references cited therein. B. Johansson, Philos. Mug., 30(1974) 469; Phys. Rev. B, ll(l975) 1367. E. A. Kmetko and H. H. Hill, J. Phys. F, 5(1975) 1119; 6(1976) 1025. R. M. Martin, Phys. Rev. Lett., 48(1982) 362. J. W. Allen and R. M. Martin, Phys. Rev. Lett., 49(1982) 1106. S. T. Mason, Top. Appl. Phys., 26(1978) Chap. 3. J. H. Weaver and D. Wieliczka, unpublished data, 1983. A. Hasegawa and A. Yanase, J. Phys. SW. Jpn., 42 (1977) 492. W. Gudat and C. Kunz, Phys. Rev. Lett., 29f1972) 169. D. J. Peterman, J. H. Weaver and D. T. Peterson, Phys. Rev. B, 23 (1981) 1692. D. D. Koelling, personal communication, 1983. J. C. Fuggle, M. Campagna, Z. Zolnierek, R. Lasser and A. Platau, Phys. Rev. Lett., 45 (1980) 1597. B. Johansson and N. Martensson, Phys. Rev. B, 24 (1981) 4494. F. U. Hillebre~ht and J. C. Fuggle, Phys. Rev. B, 25 (1982) 3550. G. Crecelius, G. K. Wertheim and D. N. E. Buchanan, Phys. Rev. B, 18 (1978) 6419.