Abstract:
The algebraic expressions for the anharmonic contributions to
the Debye-Waller factor up to 0(A ) and 0 L% ) £ where ^
is the scattering wave-vector] have been derived in a form suitable
for cubic metals with small ion cores where the interatomic potential
extends to many neighbours. This has been achieved in terms of
various wave-vector dependent tensors, following the work of Shukla
and Taylor (1974) on the cubic anharmonic Helmholtz free energy.
The contribution to the various wave-vector dependent tensors from
the coulomb and the electron-ion terms in the interatomic metallic
potential has been obtained by the Ewald procedure. All the restricted
multiple whole B r i l l o u i n zone (B.Z.) sums are reduced to single whole
B.Z. sums by using the plane wave representation of the delta function.
These single whole B.Z. sums are further reduced to the •%??
portion of the B.Z. following Shukla and Wilk (1974) and Shukla and
Taylor (1974).
Numerical calculations have been performed for sodium where the
Born-Mayer term in the interatomic potential has been neglected
because i t is small £ Vosko (1964)3 • *n o^er to compare our calculated
results with the experimental results of Dawton (1937), we
have also calculated the r a t io of the intensities at different temperatures
for the lowest five reflections (110), (200), (220), (310)
and (400) . Our calculated quasi-harmonic results agree reasonably
well with the experimental results at temperatures (T) of the order of
the Debye temperature ( 0 ). For T » © ^ 9 our calculated anharmonic
results are found to be in good agreement with the experimental results.The anomalous terms in the Debye-Waller factor are found not to be
negligible for certain reflections even for T ^ ©^ . At temperature
T yy Op 9 where the temperature is of the order of the
melting temperature (Xm) » "the anomalous terms are found to be
important almost for all the f i ve reflections.
