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Electronic band structure and properties of α-U
A. Freeman, D. Koelling, T. Watson Yang
To cite this version:
A. Freeman, D. Koelling, T. Watson Yang. Electronic band structure and properties of α-U. Journal
de Physique Colloques, 1979, 40 (C4), pp.C4-134-C4-135. �10.1051/jphyscol:1979442�. �jpa-00218838�
JOURNAL DE PHYSIQUE Colloque C4, supplément au n° 4, Tome 40, avril 1979, page C4-134
Electronic band structure and properties of a-U (*)
A . J. F r e e m a n , D . D . Koelling (+) a n d T. J. W a t s o n Y a n g (f t)
Physics Department, Northwestern University, Evanston, Illinois 60201, U.S.A. and Argonne National Laboratory, Argonne, Illinois 60439, U.S.A.
(*) Argonne National Laboratory, Argonne, Illinois 60439, U.S.A.
(t t) Physics Department, Case Western Reserve University, Cleveland, Ohio, U.S.A.
Résumé. — Utilisant une version relativiste de la méthode LAPW, nous présentons des résultats d'une étude détaillée de la structure de bande électronique ainsi que de quelques propriétés physiques de l'uranium dans la phase orthorhombique. Les densités d'états totales et partielles sont utilisées pour discuter les résultats de photo-émission de Veal et Lam. Les surfaces de Fermi des bandes prédominantes 6 et 7 présentent une symétrie élevée ainsi que quelques particularités topologiques (nesting) qui peuvent être comparées aux mesures actuelles d'effet dHvA d'Arko et Schirber. Le calcul du facteur de forme neutronique en présence d'un champ magnétique est en bon accord avec les mesures de Maglic et al.
Abstract. — We present some results of a detailed study of the electronic band structure and some physical properties of orthorhombic uranium using a relativistic version of the LAPW method. Total and projected (by orbital angular momentum) densities of states are used to discuss the photo-emission data of Veal and Lam.
The Fermi surfaces of the dominant 6th and 7th bands are found to have high symmetry (and some nesting features) of interest to dHvA measurements being pursued by Arko and Schirber. The theoretical magnetic field induced neutron form factor is found to be in good agreement with the measurements of Maglic et al.
T h e a n o m a l o u s physical p r o p e r t i e s of a - U h a v e m a d e it a subject of great i n t e r e s t for e x p e r i m e n t a l a n d theoretical s t u d y . U n f o r t u n a t e l y , its c o m p l e x s t r u c t u r e ( o r t h o r h o m b i c with 4 a t o m s / u n i t cell) a n d t h e resulting lack of s y m m e t r y h a v e m a d e its t h e o r e - tical s t u d y b y c o n v e n t i o n a l e n e r g y b a n d m e t h o d s exceptionally difficult and c o s t l y . T h u s , while w e w e r e able t o s t u d y t h e light a n d h e a v y actinide metals [1] in their cubic (high t e m p e r a t u r e ) structu- r e s , t h e study of a - U a w a i t e d d e v e l o p m e n t of a c o m p u t a t i o n a l s c h e m e w h i c h w a s n o t only rapid a n d efficient b u t w h i c h a v o i d e d t h e so-called asymptote problem w h i c h plagued o u r earlier efforts.
I n this p a p e r w e r e p o r t s o m e results of our relati- vistic e n e r g y b a n d studies o n a - U using a relativistic v e r s i o n of t h e linearized a u g m e n t e d plane w a v e m e t h o d [2-3] ( L A P W ) . This s c h e m e h a s b e e n s u c - cessfully applied t o t h e s t u d y of s o m e actinide c o m p o u n d s , n o t a b l y U G e3 [4], U I r3 [2], a n d U R h3 [5]. W e h a v e d e t e r m i n e d t h e e n e r g y b a n d s t r u c t u r e , density of states ( D O S ) , orbital angular m o m e n t u m p r o j e c t e d D O S (for u s e in analysing t h e X P S e x p e r i m e n t s of Veal a n d L a m [6], detailed F e r m i surface c r o s s - s e c t i o n s in close collaboration w i t h t h e w o r k of A r k o a n d Schirber [7] r e p o r t e d e l s e w h e r e at this c o n f e r e n c e ) , w a v e f u n c t i o n s a n d m a g n e t i c field i n d u c e d spin densities, n e u t r o n m a - gnetic f o r m f a c t o r s (in close collaboration w i t h a n d t o u n d e r s t a n d t h e m e a s u r e m e n t s of Maglic et
al. [8]), a n d generalized susceptibilities, ^ ( q ) (for investigating possible electronically driven p h o n o n anomalies a n d charge d e n s i t y w a v e s ) . B e c a u s e of s p a c e limitations w e a r e able only t o give a brief indication of this e x t e n s i v e w o r k h e r e . A full r e p o r t is being p r e p a r e d for publication e l s e w h e r e .
T h e t h r e e lattice p a r a m e t e r s a n d t h e a t o m i c posi- tional p a r a m e t e r , u, w e r e o b t a i n e d b y extrapolating t h e t e m p e r a t u r e d e p e n d e n t X-ray m e a s u r e m e n t s of B a r r e t t et al. [9] t o t h e t e m p e r a t u r e s a s s u m i n g t h a t t h e s y s t e m w a s cooled u n d e r p r e s s u r e in o r d e r t o avoid t h e anomalies a s s o c i a t e d with transitions at T < 43 K . U s i n g t h e s e p a r a m e t e r s ( a = 5.360 A , b = 11.086 A , c = 9.328 A , a n d /x = 0.102) our m o - del w a r p e d muffin tin potential w a s c o n s t r u c t e d in t h e s t a n d a r d overlapping c h a r g e d e n s i t y model using t h e K o h n - S h a m - G a s p a r ( a = 2/3) e x c h a n g e . I n our calculation w e u s e t h e f3 d2 s1 configuration. Diffe- r e n t e s t i m a t e s including a pseudo-self-consistent m o d e l (as w a s u s e d earlier for U G e3) t o d e t e r m i n e an i m p r o v e d c h o i c e of configuration, indicate t h a t a configuration in w h i c h roughly 0.25 t o 0.5 electron a r e t r a n s f e r r e d from t h e f t o t h e d levels is m o r e a p p r o p r i a t e . T h e calculated b a n d s t r u c t u r e w a s fit- ted with a F o u r i e r series a n d u s e d in o u r d e t e r m i n a - tions of t h e F e r m i s u r f a c e , D O S a n d * ( q ) .
Figure 1 s h o w s t h e / = 3 angular m o m e n t u m p r o - j e c t e d D O S . W e see a g o o d deal of s t r u c t u r e in t h e t o t a l D O S (not s h o w n ) arising from t h e lower sym- m e t r y a n d t h e hybridized set of s, p , d, a n d f b a n d s . T h e r e is a rapidly increasing total D O S j u s t a b o v e EF
(*) Supported by the U.S. NSF, the AFOSR, and the DOE.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979442
ELECTRONIC BAND STRUCTURE AND PROPERTIES OF a-U C4-135
ALPHR-URANIUM
1
p
t NERGY l R Y 1I
Fig. 1. -The I = 3 projected DOS.
which has relevance to specific heat and susceptibili- ty determinations in alloys of uranium and to optical absorption measurements. The 1 = 3 DOS correlates well with the XPS data of Veal and Lam [6] whereas the total DOS does not. This result is consistent with the fact that the higher angular momentum (more localized) f orbitals have much larger transition matrix elements. The large amount of 1 = 3 DOS above E , indicates that this is the cause of the buildup in the total DOS in this same energy region.
One might be tempted to attribute the occupied f-character (i.e. the f-DOS below E,) to tailing caused by hybridization with the s, p, and d states.
Our calculated Fermi surface cross-sections are presented and discussed elsewhere in these procee- dings. In the remaining space we present the magne- tic field induced spin densities decomposed by 1- value. The Fourier transform of these spin densities make the dominant contribution to the neutron magnetic form factor. Figure 2 compares the free ion f state charge density (broken curves) with the 1 = 3 decomposed radial density at (or close to) the Fermi energy, 1.135 Ry. All densities are normalized to unity over the r space shown (solid curves) so that they must be multiplied by their occupation. (The f contribution is roughly 80 % with the majority of the remainder comprised of d-character.) We find that the two 1 = 0 densities agree very closely, the 1 = 1 densities are slightly shifted for large r, and the 1 = 2 densities deviate only near the sphere radius. The major difference is in the 1 = 3 densities with the band result showing the large radial expansion expected for itinerant electrons in the lower half of the band. It is this expansion which is responsible for the contracted values of the magnetic neutron form factor seen by Maglic et al. [8].
Fig. 2. - Comparison of the free ion f state charge density with the 1 = 3 decomposed radial density at (or closed to) E,.
Although hybridization is present in this region of
energy, many of the wave functions show nearly Our results indicate that the f-electrons in a - U are pure f-character and so hybridization alone is not not only quite itinerant in nature (and hence treata- responsible for the observed occupied f states but ble by a band model), but are dominantly present in involved the formation of a true f-band. the experimental data.
References
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[2] KOELLING, D. D., PTOC. of the Second Intern. Conf. on the Electronic Structure of the Actinides, J. Mulak, W. Suski and R. Troc (Ossolineum, Wroclaw, Poland) ; KOELLING, D. D. and ARBMAN, G . O., J. Phys. F 5 (1975)
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[3] ANDERSEN, 0 . K . , Phys. Rev. B 12 (1975) 3060.
[4] &KO, A. J. and KOELLING, D. D., Phys. Rev. B 17 (1978) 3104.
[5] ARKO, A. J. and KOELLING, D. D. (private communication).
[6] VEAL, B. W. and LAM, D. J . , Phys. Rev. B 10 (1974) 4902.
[7] SCHIRBER, J. E., ARKO, A. J. and FISCHER, E. S., Solid State Commun. 17 (1975) 553.
[8] MAGLIC, R. C., LANDER, G. H., MUELLER, M. H. and KLEB, R., Phys. Rev. B 17 (1978) 308.
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