X-ray isochromats of NaCl, KCl and KI

Volume 24.A, n u m b e r 4

PHYSICS

X-RAY

ISOCHROMATS

LETTERS

OF

NaCI,

13 F e b r u a r y 1967

KC1

AND

KI

S. B E R G W A L L Institute of Physics, University of Uppsala, Sweden and M. E L A N G O Institute of Physics and Astronomy, Estonian Academy of Sciences, Tartu, Estonian SSR Received 11 J a n u a r y 1967 The intensity distribution n e a r the s h o r t wavelength limit of the continuous X - r a y s p e c t r u m has been studied for NaC1, KC1 and KI by m e a n s of the i s o c h r o m a t method. Information about the density of s t a t e s in the conduction band and the c h a r a c t e r i s t i c energy l o s s e s obtained from the p r e s e n t investigation is in q u a l i t a tive a g r e e m e n t with the t h e o r e t i c a l calculations and experimental values respectively.

K n o w l e d g e of t h e s t r u c t u r e of t h e c o n d u c t i o n b a n d i s i m p o r t a n t f o r t h e s o l u t i o n of a g r e a t n u m b e r of p r o b l e m s in s o l i d s t a t e p h y s i c s . U n f o r t u n a t e l y , in t h e c a s e of i o n i c c r y s t a l s , t h i s k n o w l edge is rather incomplete at present. Complications arise in obtaining information on this subject from ultraviolet and X-ray absorption and r e f l e c t i o n s p e c t r a d u e to t h e o v e r l a p p i n g of s t r u c t u r e s of d i f f e r e n t b a n d s , d i s t u r b i n g e f f e c t s of i n n e r b a n d h o l e s , a n d t h e a c t i o n of s e l e c t i o n r u l e s [ 1 , 2 ] . T h e r e f o r e , i t w o u l d s e e m p r o f i t a b l e to a p p l y t h e X - r a y i s o c h r o m a t m e t h o d to a n i n v e s t i g a t i o n of t h e c o n d u c t i o n b a n d s t r u c t u r e s of i o n i c crystals since this method has already been app l i e d f o r a n a l o g o u s p u r p o s e s in t h e c a s e of s o m e metals and semiconductors [3-5]. T h e a p p a r a t u s a n d m e t h o d of i n v e s t i g a t i o n u s e d in this work are essentially identical with those d e s c r i b e d in r e f s . 6 a n d 7. T h e s p e c t r o m e t e r w a s a d j u s t e d f o r t h e 2 2 8 5 . 0 0 X . U . w a v e l e n g t h of t h e Cr K~ I line. The ionic crystals investigated were e v a p o r a t e d o n t o s o l i d c o p p e r c y l i n d e r s in a c o n ventional evaporation chamber and used as anodes in the X-ray tube. The emission current in the t u b e w a s k e p t s t a b l e a t 10 m A a n d t h e w o r k i n g p r e s s u r e w a s a l w a y s a r o u n d I 0 /I T o r r . T h e i s o c h r o m a t s p r e s e n t e d a r e t h e a d d e d c u r v e s of t h r e e to f o u r i n d e p e n d e n t r e c o r d i n g s . T h e t o t a l c o u n t i n g time at each point was about I000 seconds. T h e r e s u l t s of t h e e x p e r i m e n t s a r e s h o w n in t h e f i g u r e . In a l l c a s e s t h e l o w - e n e r g y s t r u c t u r e of t h e i s o c h r o m a t s s t a r t s w i t h a c o m p a r a t i v e l y s m a l l p e a k f o l l o w e d b y a n u m b e r of m o r e i n t e n s e peaks. 23O

N o C I ~ 6oo

Uo 500

I

I

~; KCL~

I

"

600

50o 5o0

i

i

f

•

40e

~ I

o

12

~ I

,o

2~) z~ v

vo~,~

Fig. 1. I s o c h r o m a t c u r v e s of NaC1, KC1 and KI. The zero voltage c o r r e s p o n d s to the bottom of the conduction band, which is conventionally chosen.

Volume 24A, number 4

PHYSICS L E T T E R S

For the interpretation of the observed structure of the isochromats the theoretical calculations of the conduction band structure of KI [8] and the results of the investigation of the characteristic energy losses in alkali halides [9] can be used. According to ref. 8 the bottom of the conduction band of KI is m a d e up of states of s-symmetry and very close to them states of d-symmetry both originating from the K + ions. The states of d-symmetry from the I- ions and the states of p-symmetry from the K + ions can be found a few eV higher. In accordance with the above we suppose that the first peak in the isochromat curves, which appears 2 eV above the bottom of the conduction band of KC1 and KI, and 3 eV for NaCI, c o r r e sponds to the o v e r l a p p e d density of the states of s - and d - s y m m e t r y of the c a t i o n s . This r e s u l t s e e m s to c o n f i r m the g u e s s , that the s t a t e s of d - s y m m e t r y of Na + ought to be found at higher e n e r g i e s than those of K +. The second peak app e a r s at about 7 eV above the bottom of the conduction band in NaC1 and KC1 (not w e l l - r e s o l v e d for the l a t t e r ) , and at about 6 eV in KI. Thus, we may conclude that this peak is due to the anion s t a t e s of d - s y m m e t r y . According to the s e m i e m p i r i c a l e s t i m a t e s in ref. 2, these states m u s t show up at higher e n e r g i e s in the case of the chlor i d e s as c o m p a r e d with the iodides. The third peak, at about 10 eV above the bottom of the conduction band for all t h r e e s a l t s , may be f o r m e d by cation s t a t e s of p - s y m m e t r y , but d i s t u r b i n g effects f r o m c h a r a c t e r i s t i c energy l o s s e s will c e r t a i n l y be p r e s e n t in this region.

13 February 1967

The following peaks in the i s o c h r o m a t c u r v e s a p p e a r at r e g u l a r i n t e r v a l s f r o m the f i r s t two peaks. These i n t e r v a l s (12 eV for NaC1, 14 eV for KCI and 12 eV for KI) a r e in good a g r e e m e n t with the c h a r a c t e r i s t i c e n e r g y loss values (11.75 and 12.70 eV for NaC1, 13.9 and 14.1 eV for KC1 and 11.8 eV for KI) given in ref. 9. It s e e m s therefore r e a s o n a b l e to i n t e r p r e t these peaks as originating f r o m e l e c t r o n s , which have e x p e r i enced single o r m u l t i p l e e n e r g y l o s s e s in the target material. The authors a r e much indebted to P r o f e s s o r P e r Ohlin for providing f a c i l i t i e s for this i n v e s tigation. F i n a n c i a l support f r o m the Swedish N a t u r a l Science R e s e a r c h Council is gratefully acknowledged.

References 1. L.G. Parratt, Hey. Mod. Phys. 31 (1959) 616. 2. J.C. Phillips, Phys.Rev. 136 (1964) A 1705. 3. H. Claus and K.Ulmer, Z.Physik 185 (1965) 139. 4. S.Berg'wall, Z.Physik 193 (1966) 13. 5. S.Bergwall, Z~Physik 187 (1965) 495. 6. P.Ohlin, Diss.Uppsala Universitets .~rsskrift 1941. 7. A. Nilsson, Arkiv Fysik 6 (1953) 513. 8. Y. Onodera, M. Okazaki and T. Inui, Techn.Rep.ISSP, Univ. of Tokyo Ser. A Nr 209 (1966). 9. P.E.Best, Proc. Phys. (London) 79 (1962) 133. M. Creuzberg, Z. Physik 196 (1966) 433.

* * * * *

UNIQUENESS

THEOREM FOR NONLINEAR THERMODYNAMICS*

IRREVERSIBLE

R. E. NETTLETON Sandia Laboratory, Albuquerque, New Mexico Received10 January 1967

Any kinetic equation linear in first and second time derivatives and non-linear in system ordering parameters can be expressed uniquely in Onsager canonical form, linear in the thermodynamic forces. This leads to evaluation of a significant free energy term. It has b e e n p r e v i o u s l y d e m o n s t r a t e d [1] by applying a p r o j e c t i o n o p e r a t o r technique of Zwanztg [2] to the LiouviHe equation that i r r e v e r s i b l e t h e r m o d y n a m i c s can be applied to phenomonolo-

gical equations which exhibit i n e r t i a l effects in * This work was supported by the U. S. Atomic Energy Commission.

231