Carvalho, F.

Ehret, G.

Glaeser, W.

KITopen

Dezember  1964 K F K  262  Ins t l t u t  f u r  Angewandte  Kr rnph \ - s ik  15 I L ; ~  „ .. * - .  . - The  Phonon Frequent>- Dis t r ibu t ion  of Vanadium W .  G l ä s e r ,  F. Car l - a lho ,  G .  E h r e t  Als Manuskript vervielfältigt. Fü r  diesen Bericht behalten wir uns a l le  Rechte vor.  Gesellschaft  fü r  Kernforschung m. b. H. Kar l s ruhe  KERNFORSCHUNGSZENTRUM KARLSRUHE December 1964 Institut für Angewandte Kernphysik The Phonon Frequency Distribution of Vanadium W. Gläser, F. Carvalho*, and G. Ehret / iC - A paper subrnitted to the IAEA Symposium on the Inelastic Scatter- ing of Neutrons Bombay, India, 15 - 19 December 1964 *On leave of absence from Junta de Energia Nuclear, Portugal Gesellschaft für Kernforschung m.b.H., Karlsruhe I. I n t r o d u c t i o n  The i n v e s t i g a t i o n  of  i n e l a s t i c  s c a t t e r i n g  o f  s low n e u t r o n s  i n  s o l i d s  y i e l d s  v a l u a b l e  i n f o r m a t i o n  on t h e  dynamic p r o p e r t i e s  o f  s u c h  m a n y - p a r t i c l e  s y s t e m s .  I f  t h e  c o h e r e n t  s c a t t e r i n g  i s  dominant ,  s i n g l e  phonons can  be "seen"  a n d ,  as a r e s u l t  of s y s t e m a t i c  meas- urements  o f  phonons i n  c e r t n i n  d i r e c t i o n s  i n  s i n g l e  c r y s t a l s ,  d i s -  p e r s i o n  c u r v e s  can  be c o n s t r u c t e d .  I f  t h e  s c a t t e r i n g  i s  i n c o h e r e n t ,  t h i s  method i s  n o t  p o s s i b l e .  I n  t h i s  c a s e ,  however ,  t h e  f r e q u e n c y  d i s t r i b u t i o n  of  t h e  normal  modes cnn  be deduced i n  a more o r  l e s s  d i r e c t  way from t h e  measured s c a t t e r i n g  d i s t r i b u t i o n .  Comparison of  e x p e r i m e n t a l  nnd t h e o r e t i c n l  f r e q u c n c y  d i s t r i b u t i o n s  o f t e n  sug-  g e s t s  a more r e a l i s t i c  p i c t u r e  of t h e  p r o ~ e r t i e s  of  t h e  s c a t t e r e r .  U n f o r t u n a t e l y ,  o n l y  a  few s u b s t a n c e s  s c a t t e r  e l m o s t  c o m p l e t e l y  i n -  c o h e r e n t l y .  From t h e s e ,  vanndium, w i t h  a c u b i c  body c e n t c r e d  l a t t i c e ,  h a s  a r e l a t i v e  s i m p l e  s t r u c t u r e .  Seve r r i l  worke r s  - 4t have  r e -  p o r t e d  f r e q u e n c y  d i s t r i b u t i o n s  of vanadium o b t a i n e d  by t h e  s c a t t e r -  i n g  of b e r y l l i u m  f i l t e r e d  c o l d  n e u t r o n s .  I n  t h e  p r e s e n t  P a p e r ,  a somewhat d i f f e r e n t  rnethod i s  d e s c r i b e d  i n  which t h e  s c a t t e r i n g  lnw o f  vanadium i s  measured f o r  a r e l a t i v e l y  l a r g e  r a n g e  of  momentum t r a n s f e r  , R Q ,  r.nd one rgy  t r u n s f  e r ,  5 W ( 0  < Q (: 14 ; 0 < Ci w < <2k T ) .  S t a r t i n g  w i t h  the s c a t t e r i n g  law v a l u e s  and  u s i n g  t h e  ß e x t r a p o l a t i o n  t e c h n i q u e  p roposed  by E g e l s t a f f  (5J,values of  t h e  f r e q u e n c y  d i s t r i b u t i o n  f u n c t i o n  hzve been  o b t ü i n e d .  The o n l y  as- s u m p t i o n  made f o r  t h e  pu rpose  of p e r f o r m i n g  t h e  i t e r a t i o n  o f  t h e  ex- t r a p o l a t e d  v a l u e s  i s  t h u t  t h e  mot ions  of  t h c  atoms a r e  harmonic .  But t h e n  t h e  s e p a r a t i o n  of rnulti-phonon p r o c e s s e s  i s  s t r a i g h t f o r -  ward nnd no p r e v i o u s  e v n l u n t i o n  of  t h e  Debye-Waller f a c t o r  i s  n e c e s s a r y .  11. E x ~ e r i m e n t a l  Arrangement and P r o c e d u r e  k beam f r o n  t h e  K a r l s r u h e  r o t a t i n g - c r y s t a l  t i m e - o f - f l i g h t  s p e c t r o -  me te r  d e s c r i b e d  e n r l i e r  i n  d e t n i l  fi] p r o v i d e d  t h e  i n c i d e n t  mono- e n e r g e t i c  n e u t r o n s  i n  a n  ene rgy  r a n g e  of  18 t o  80 mev. P r imary  ene rgy  r e s o l u t i o n  was 5 % a t  18 mcV and t i m e  r e s o l u t i o n  e b o u t  20 p s e c / m .  A s k e t c h  of t h e  nppurn tus  i s  shown i n  F i g .  I .  The d i s t a n c e  sample-  - d e t e c t o r  was 2 m .  Nine d e t e c t o r s  n t  s c a t t e r i n g  a n g l e s  betwcen 20 and 0 140 were used  s i m u l t n n e o u c l y .  A t  p r e s e n t ,  two d i f f e r e n t  t y p e s  of  detectors are  in use, namely ~e'-counters, 1 inch diimeter, forming 6 banks with an cffective aren of 155 Cm2, nnd Li F-znS scintillctors, 5 inch in diameter. For dnta ncquisition nnd reduction, a multiple input data acquisition System ( M I D A S )  with a Control Data 160-k cornputer ( 8 ~  core memory) as centrzl unit is employed / T ] .  This computer hnndles simultaneoucly on-line four different renctor beam experiments. The eventsrecorded by the time-of-flight detectors of the scattering experiment are fed into n multiparameter coding unit which digitizes the experiment number, the particular detector num- ber and the neutron time-of-flight. This unit is connected with the computer by means of ccntrzl buffer stntion. Z The scattering sample was n 7 X 4.5 cm vanadium plntc with a thick- ness of 0.2 Cm. The transmission of the szmple was 85 % for incident neutrons of the smnllest cnergy 2nd sbout 90 % for the highest energy used. For each of the thrce incident energies (about 18, 40 2nd 80 meV) n run of approximately 50 hours duration was cnrried out. This corresponds to a few hundred counts in the significant inelastic channels. For background elimination sinilar runs without the sample were made. Because of the rotational symmetry of the crystal nono- chromator, background contributions from fast neutrons and uncor- related slow neutrons are constant in time and are, therefore, easily corrected. 111. Data Processing and Results --- -The quantity to be directly calculated from the rneasured scatter- ing distributions is the scattering law S ( a , ß ) ,  defined as follows: where is the double differential neutron scattering cross-section, the free atom cross-section of the scattering nucleus, the Boltzmann constant, the absolute temperature of the scatterer, the ratio of the mass of the scattering nucleus to the mass of the neutron, the element of solid angle hbwhich the neutrons are scat- tered, E. and  E  t h e  i n c i d e n t  and s c a t t e r e d  n e u t r o n  e n e r g i e s ,  and  t h e  s c a t t e r i n g  a n g l e .  F o r  i n c o h e r e n t l y  s c a t t e r i n g  s o l i d s  a d i r e c t  r e l a t i o n  can  be e s t a b -  l i s h e d ,  i n  t h e  harmonic a p p r o x i m a t i o n ,  between t h e  phonon f r e q u e n c y  d i s t r i b u t i o n  of t h e  c r y s t a l  l a t t i c e  P(ß) and t h e  l i m i t i n g  v a l u e  of  S ( ~ , ß ) / c x  f o r  U S O :  I n  a f i r s t  s t e p  a t l c a l l i b r a t i o n  programi1 was r u n  f o r  t h e  pu rpose  of  c a l c u l a t i n g  t h e  d e t e c t o r  e f f i c i e n c y  a s  a  f u n c t i o n  of  ene rgy .  Here e f f e c t s  of  sample t h i c k n e s s ,  mnin ly  a b s o r p t i o n ,  a r e  t a k e n  i n t o  a c c o u n t .  The c a l c u l a t i o n  i s  based  on t h e  P l a c z e k  heavy-mass ex- p a n s i o n  (81. A f u n c t i o n  of  t h e  t y p e  p roposed  by Harris e t  a l .  h a s  been  f i t t e d  t o  t h e  c a l c u l a t e d  e f f i c i e n c y  v a l u e s .  The r e s u l t i n g  p a r a m e t e r s  were f e d  i n t o  t h e  computer  t o g e t h e r  w i t h  t h e  raw d a t a  from sample - in  and sample-out  r u n s  and  a second c a l c u l a t i o n  was s t a r t e d  w i t h  t h e  " s c a t t e r i n g  lzw program' ' ,  which d e l i v e r s  t h e  W ,  ß,S(ct,ß), S(a,ß)/crc and  t h e  s t a t i s t i c a l  e r r o r s  of t h e  measurement f o r  e a c h  t i m e  c h a n n e l .  I n  t h i s  program t h e  b u i l t - i n  p o s s i b i l i t y  of  smooth ing  t h e  raw-data  i n  s e l e c t e d  t ime-channe l  i n t e r v a l s  h a s  been  t a k e n  t o  advan tage  when a d v i s a b l e ,  namely f o r  t h e  background d a t a  and t h e  sample - in  d a t a  a t  t h e  l o w e s t  e n e r g y .  A t y p i c a l  r e s u l t  f o r  ~ ( ~ , ß > / c u  vs .  t ime-channe l  i s  shown i n  F i g .  2. The c a l c u l a t e d  S(o<,ß)/rx v a l u e s  a r e  t h e n  p l o t t e d  a g a i n s t  w a n d  e x t r a p o l a t e d  t o  M =  0 f o r  ß v a l u e s  at 0.05 i n t e r v a l s  ( F i g .  3 ) .  I n  t h e  f irst  e x t r a p o l a t i o n  t h e  r e s o l u t i o n  e f f e c t s  t h a t  t e n d  t o  i n -  c r e a s e  t h e  g r a d i e n t  of  t h e  S/oc c u r v e s ,  e s p e c i a l l y  a t  l o w  W , have n o t  been  c o r r e c t e d  f o r .  Then t h e  LEAP program 001 h a s  been  r u n  w i t h  t h o s e  f i r s t  e x p r a p o l a t e d  v a l u e s ,  assuming t h a t  t h e  g r n d i e n t  of t h e  LEAP o u t p u t  c u r v e s  would be a p p r o x i m a t e l y  c o r r e c t .  Based on t h i s  g r a d i e n t  a c o r r e c t i o n  f o r  t h e  e l a s t i c  peak  r e s o l u t i o n  w a s  performed u s i n g  t h e  a s sumpt ion  t h a t  t h c  peek  shape  and a l s o  t h e  ene rgy  r e s o l u t i o n  f o r  a  g i v e n  t ime-channe l  2nd a  g i v e n  i n c i d e n t  ene rgy  a r e  i ndependen t  of  t h e  s c a t t e r i n g  z n g l e .  It was f u r t h c r  assurned t h a t  t h e  &-dependence of t h e  e l a s t i c  am- p l i t u d e  i n  a g i v e n  channe l  i s  d e s c r i b e d  by t h e  Debye-Waller f a c -  t o r .  S h i s  r e s o l u t i o n  c o r r e c t i o n  d e t e r m i n e s  ~2 p a r a l l e l  d i s p l a c e m e n t  of t h e  c u r v c s  c a l c u l a t e d  w i t h  LEAP which i s  of s i g n i f i c n n c e  o n l y  f o r  low ß v a l u e s .  With t h e  new e x t r a p o l a t e d  v z l u e s  a cecond LEAP c a l c u l a t i o n  was made and t h e  r e s u l t s  were a g u i n  compared w i t h  t h e  mezsured d a t z ;  t h i s  p r o c e d u r e  w a s  r e p e a t e d  u n t i l  t h e  s l o p e  of t h e  L - l i n e s  ( F i g . 3 )  2nd t h e i r  l i m i t  n t  a =  0 s t - , b i l i z e d  themse lves .  Three  r u n s  were made, P(ß) b e i n g  no rma l i zed  t o  oce  a t  e v e r y  r u n .  The v z i r i a t i o n s  of t h e  c a l c u l a t e d  Debye-Waller f u c t o r  2nd of  t h e  z r e a  under  t h e  i n p u t - p ( ß )  d u r i n g  t h e  whole p r o c e s s  were n o t  g r e u t e r  t h z n  10 %. F i g s .  4 and 5 show t h t  o b t a i n e d  S ) = 2nd P ( R )  c u r v e s .  I V .  D i s c u s s i o n  of r e s u l t s  I n  F i g .  6 t h e  f i r m 1  f r c q u e n c y  d i s t r i b u t i o n  P ( v )  e x t r z c t e d  f r o n  t h e  s c a t t e r i n g  1zw measurements  i s  compared w i t h  some of  t h e  p r e -  v i o u s l y  r e p o r t e d  neasu remen t s  u s i n g  t h e  b e r y l l i u m - f i l t e r  t e c h n i q u e .  The areU under  e a c h  c u r v c  i s  norma l i zed  t o  u n i t y .  Roughly a l l o f  them show t h e  Same shape  b u t  i n  d e t z i l s  t h s r e  a r e  d i s a g r e e n e n t s .  The c u r v e  3 i s  c o r r e c t e d  f o r  t h e  d i s t r i b u t i o n  of  t h e  i n c i d e n t  n e u t r o n s  B]. The d e c r e a s e  i n  peak  h e i g h t  i n  o u r  c u r v e  nny be p z r t i a l l y  due t o  same c x t r a  s c n t t e r i n g  u t  h i g h e r  e n e r g i e s ;  7, de- c r e a s e  may be c ~ ~ u s e d  :l lso by r e s o l u t i o n  e f f e c t s  i n  t h e  i n e l z s t i c  spec t rum.  But n n o t h e r  p o s s i b l e  e x p l m a t i o n  i s  t h e  e f f e c t  of mul- t i p l e  s c n t t e r i n g ;  i n  o u r  rnensurement .~ we uscd much t h i n n e r  snmples  t h a n  t h c  o t h e r  worke r s  d i d .  12 - 1 Thepresen t  cu rve  h a s  a small bump a t  3 = 2*L+910 s e c  . The ?X- i s t e n c e  of s u c h  a pe-k h a s  been p r e d t c t e d  p r e v i o u s l y  a t  a b o u t  I2 - 1 V = 2-10 s e c  und a t t r i b u t v d  t o  t h e  Kohn e f f e c t  fil]. Although vanadium h a s  2. h i g h  t r m s i t i o n  t e m p e r a t u r e  f o r  t h e  s u p e r -  - conduc t ing  s t n t e  i n d i c a t i n g  e s t r o n g  e l ec t ron -phonon  i n t e r a c t i o n  which f n v o u r s  a s t r o n g  Kohn e f f s c t ,  s u c h  a  p e a k , i n  p r i n c i p l e , c a n  be e x p e c t e d  a l s o  on  t h e  b a s i s  of  Born-von Ksrman-theory. Using c a rnodel with noncentral nearest neighbour and central next neigh- bour interaction Singh and Bowers L121 calculated a frequency spec- trum showing three peaks. Unfortunately they did not use the correct elastic constants. After the elastic constants of vanadium had been measured by Alers 1131, calculations using the Born-von Karman and the de Launay models have been made 1141. None of the models, how- ever, gives a satisfactory representation of the measured frequency distributions. From the LEAP calculations the Debye-Waller coefficient X =  2 W/a is found to be 4.22 - + 10 %. In the Debye appsoximation a Debye temperature 8 - 354 + 15'~ can be deduced from this value. Start- 0 - - ing from the measured frequency distribution we calcnlated the specific heat as a function of temperature. The results are well described by the above Debye temperature and are,within the ex- perimental errors,in agreement with the low temperature specific heat measurements by Corak et al. (151. Acknowledgernents The authors wish to thank Prof. K. H. Beckurts who has initiated the inelastic neutron scattering work in our laboratory for his continuing support. Dr. P. A, Egelstaff (Harwell) has made the LEAP program available prior to publication. Thanks for invaluable technicel assistance are due to Dip1.-Ing. H. Ripfel, H. D. Jünger and the staff of the Karlsruhe IBM 7070. References Stewart, A. T. and Brockhouse, Bi P?., Rev. Plod. Ph*. - 30 (19581, 250. Eisenhauer, C. M., et al., Phys. Rev. - 109 (1958), 1046. Semljanov, M. G., et al., ItInelastic Scattering of Neutrons in Solids and ~irquids"', Vol. 11, IAEA, Vienna ( 1 9 6 3 ) ~  125. Haas, R. et al., "Inelastic Scattering of Neutrons in Solids 2nd Liquids", Vol. 11, IAEA, Vienna (1963) 145. Egelstaff, P. A., "Inelastic Scattering of Neutrons in Solids and Liquids" , IAEA, Vienna (1961 ) 25, Carvalho, F. et ~ l . ,  Study Group Meeting on Research Reactor Experimental Sechniques, Bucharest, October 26 - 31, (1964). Krüger, G. 2nd Dimrnler G., Proc. EANDC Coni. on the Autornatic Acyuisition 2nd Beduction of Nucl. Datz, Karlsruhe (1964) 149. Placzek, G., Phys. Rcv. - 86 (1952) 377. Harris, E., Duffil, C., "Inelastic Scattering of Neutrons in Solids 2nd Liquids", Vol. I, IAEA, Vienna (196j) 171. Mc Lztchie, R., Progr,zrn description LEAP, AERE Harwell (1962). Pelnh, I,, "Inelactic Scattsring oS Neutrons in Solics 2nd Liquids", Vol. 11, I A E A ,  Vienna (1963) 155. Singh, D. X. 2nd Bowers, !!. L., Phys. REV. - 116 (19.59) 279. Alers, G. A., Phys. Rev, 119 (1960) 1532. Hendricks, J. a . ,  Phys. Rev. 130 (1963) 1377. Corak, \!I. S. et al„ Phys, Bev. 102 (1956) 656. Legends t o  f i g u r e s :  F i g .  1 :  Schemat ic  s k e t c h  of t h e  r o t a t i n g - c r y s t a l  t i m e - o f - f l i g h t  s p e c t r o m e t e r  a t  t h e  FR2 r e a c t o r .  F i g .  2 :  T y p i c a l  p l o t  of  S(&,ß)/b. v a l u e s  o b t a i n e d  w i t h  t h e  " s c ü t -  t e r i n g - l a w  program" F i g ,  3 :  T y p i c a l  c u r v e s  of l o g  (S/oc) vs.U; L i n d i c a t e s  t h e  c u r v e s  from t h e  f i n a l  LEAP c a l c u l a t i o n ;  R 80 and  H 40 t h e  c u r v e s  c a l c u l a t e d  on t h e  b a s i s  of  L and  t h e  r e so lu t ' $on  c o r r e c -  t i o n  s k e t c h e d  i n  t h e  t e x t ,  f o r  i n c i d e n t  e n e r g i e s  of 89 and 40  meV. F i g  . 4: The v a l u e s  of ( S / I &  V S .  ß. F i g .  5: P(ß)  deduced from t h e  measu renen t s .  F i g .  6 :  A compar ison  of s e v e r a l  e x p e r i m e n t a l  f r e q u e n c y  d i s t r i -  b u t i o n s  for vanadium. The number on t h e  c u r v e  g i v e s  t h e  r e f e r e n c e .  Curve 4 i s  t h e  r e s u l t  o f  t h e  p r e s e n t  work. F I G .  1 REACTOR REACTOR- BEAM COLLl- M A T O R  MONITOR 1 FIG. 2 f 1 .o J n a g r i 9  n $$E fiE81@1 @8na~m TIME CHANNEL I I I 102 < - - - - - - 10 -, - - - - - - 1 A A FIG. 3 0 0 " A A  ß positive and negative Oa '\, \ \ \ . n o "'._ = 20 meV '\ 0 0 Y. --- --- m . '. - - * . --- -.. - - - I -  b---A--b- 0 W.-.* 8 = 40 meV -;--(L- -0 --A--*- - *- -- -.-R40 - L A ~-i-----* = 8o mev 0-• 0 p10,2 A - - - - - SI  U FIG. 4 

The phonon frequency distribution of vanadium

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The phonon frequency distribution of vanadium

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