INELASTIC NEUTRON SCATTERING AND LATTICE DYNAMICS OF METALS IN QUASI-ION APPROXIMATION

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INELASTIC NEUTRON SCATTERING AND LATTICE
DYNAMICS OF METALS IN QUASI­ION APPROXIMATION
by
Κ KREBS and K. HÖLZL
1969
Joint Nuclear Research Center
Ispra Establishment ­ Italy
Reactor Physics Department ΛίΤίΑΙ-ί
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Par». Il
INELASTIC NEUTRON SCATTERING AND LATTICE DYNAMICS
OF METALS IN QUASI-ION APPROXIMATION
by K. KREBS and K. HÖLZL
European Atomic Energy Community - EURATOM
Joint Nuclear Research Center - Ispra Establishment (Italy)
Reactor Physics Department
Luxembourg, August 1969 - 142 Pages - 36 Figures - FB 185
On the basis of the pseudo-ion model described in Part I, the
following lattice dynamical quantities have been calculated :
formfactors, effective potentials, phonon densities of state, dynamical
structure factors and effective Debye temperatures.
Results are presented for 10 cubic metals. Listings of the pertinent
FORTRAN-4 programs are given in Appendix I, II and III.
EUR 3621 e
Part. II
INELASTIC NEUTRON SCATTERING AND LATTICE DYNAMICS
OF METALS IN QUASI-ION APPROXIMATION
by K. KREBS and K. HÖLZL
European Atomic Energy Community - EURATOM
Joint Nuclear Research Center - Ispra Establishment (Italy)
Reactor Physics Department
Luxembourg, August 1969 - 142 Pages - 56 Figures - FB 185
On the basis of the pseudo-ion model described in Part I, the
lollovving lattice dynamical quantities have been calculated :
formfactors, effective potentials, phonon densities of state, dynamical
structure factors and effective Debye temperatures.
Results are presented for 10 cubic metals. Listings of the pertinent
FORTRAN-4 programs are given in Appendix I, II and III. EUR 3621 e
Part. Il
EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM
INELASTIC NEUTRON SCATTERING AND LATTICE
DYNAMICS OF METALS IN QUASI ION APPROXIMATION
by
K KREBS and K. HÖLZL
1969
Joint Nuclear Research Center
Ispra Establishment - Italy
Reactor Physics Department ABSTRACT
On the basis of the pseudo-ion model described in Part I. the
following lattice dynamical quantities have been calculated :
formfactors, effective potentials, phonon densities of state, dynamical
structure factors and effective Debye temperatures.
Results are presented for 10 cubic metals. Listings of the pertinent
FORTRAN-4 programs are given in Appendix I. II and III.
KEYWORDS
INELASTIC SCATTERING FORM FACTOR
NEUTRONS PHONONS
LATTICES . DEBYE TEMPERATURE
METALS NUMERICAL
IONS FORTRAN
ATOMIC MODELS Page:
1. INTRODUCTION 5
2. MODEL
3. PROGRAMS 7
H. RESULTS θ
5. CONCLUSION 8
REFERENCES 9
TABLE 1. - Parameters for Li, Na, K, Rb 11
TABLE 2. -s for Cu, Al, Pb2
TABLE 3. - Parameters for Fe, Ni, Pt I3
FIGURE CAPTIONS I4
FIGURES 15
APPENDIX - Program PSAF XI -m SKFD
APPENDIX III - Program CVDWF INELASTIC NEUTRON SCATTERING
AND LATTICE DYNAMICS OP METALS IN QÏÏASI-ION APPROXIMATION
*)· 1. INTRODUCTION
This report contains results of lattice dynamical calculations which were
made on the basis of the pseudo­ion model *■' described in Part I · The
reduction of the experimental inelastic neutron data to pseudo­ion form fac­
tors was done by means of a non­linear least square fitting program, called
PSAF. Using another program, SKFD, frequency distributions and dynamical
structure factors have been computed. These quantities are essential for any
calculation of electrical and thermal resistivities or of thermodynamical
quantities like specific heats, Debye temperatures or Debye­Waller factors.
The computation of the latter quantities is done by a program called CVDWF.
Listings of the mentioned FORTRAN­4 programs are given in the Appendix.
2. MODEL
In this passage we recall briefly our model and the underlying assumptions.
The interaction energy E(K) between any two lattice particles including 2nd
order effects may be written as » '
E(K) = E,. (K) + E. ,(K)
dir ind
,e2Z2 f Gc< *re2Z2(K) Gind(K) (1 __!_,) Q)
κ2 ε(Κ)
Ωο \ K2
The first term is the direct Coulomb interaction energy between two extend­
ed ions with form factors G (Κ), which go to 1 for K­K). The indirect inter­
action via the valence electrons is determined by the form factor
Gind(K) = Gc(K> ­ Gorth(K) (2)
G *v(K) *s tne ^orm factor describingorthogonalization effects,forK­K)it
goes to zero, i.e. G th(K) correspondstoaneutral "charge" distribution.
The dielectric function ε(Κ) is given by
k2
ε(Κ) = 1 +­f D(t) , (3)
Κ
where k = Hk^/wa and t = K/2k_(k„:Fermiwavevector, a : Bohrradius),
c F oFFo
*) Manueoript received on 5 May, 1969. δ
D(t) = 1 in the Thomas-Fermi approximation. In the RPA approximation it
, 5)
is given by
2
D(t) = f(t) = \ + ^-Z~- m \ψζ\ (Ό
4t
3)
Including exchange effects we have
D(t) = (l - ——-= ) f(t) (5)
\ 2 + 6t /
and if we take into account also the fact that the indirect interaction it-
3)
self is modified by exchange effects , we may define an effective dielec­
tric constant using
D(t) = «i>— (6)
" 8VkFil+3t2
The corresponding dielectric function will be called ε (Κ).
Equation (1) may be reduced to the following screened interaction between
pseudo-ions
2 2 G2. (K)
E(K) = ΐϊ|Α. -ξ , (7)
"ο Κ ε(Κ)
where the model form factor G,.(K) = G. , (K). This reduction is possible un­
ti ina
der the following assumptions:
a) the ion cores are small and non-overlapping,
b) the form factors are radially symmetric,
c) the ions behave as rigid particles, i.e. during vibration no moments are
excited.
Since (7) is an interaction between extended particles and since it depends
explicitly on volume, the validity of the Cauchy relations is not required.
G (K) = G (Κ)/ε(Κ) ·*■ 0 for K-K), thus this quantity characterizes a neutral
particle, the so-called pseudo-atom . The pseudo-atom is composed of the
bare particle G(K) and the screening cloud GSCL(K) = - G(K)(1 - 1/ε(Κ)).
If the assumptions a), b) and c) are justified for a certain metal, the po-