ISSN 1063-7834, Physics of the Solid State, 2007, Vol. 49, No. 1, pp. 126–135. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © V.Yu. Trubitsin, E.B. Dolgusheva, 2007, published in Fizika Tverdogo Tela, 2007, Vol. 49, No. 1, pp. 121–130.
126
1. INTRODUCTION
In recent years, the structural transformations in zir-
conium at high temperatures and pressures have been
intensively studied both experimentally and theoreti-
cally. The most attention has been concentrated on the
structural instability of the high-temperature
β
(bcc)
phase at atmospheric pressure and on the specific fea-
tures of the transition from this phase to the
α
(hcp)
phase at a temperature below
T
= 1136 K. It is com-
monly believed that the
β
α
phase transition under
atmospheric pressure is directly related to the softening
of the transverse phonon with wave vector
k
= (1/2)(1,
1, 0) (
N
phonon) experimentally observed in
β
-Zr [1,
2].The main theoretical argument supporting this con-
clusion is the effective potential for the
N
phonon cal-
culated within the “frozen-phonon” model in [3]. It fol-
lows from those calculations that the effective potential
for the
N
phonon in Zr has a double-well form and,
hence, the square of the phonon frequency when calcu-
lated within a harmonic approximation becomes nega-
tive. The fact that the phonon frequency is imaginary
indicates instability of bcc zirconium in the ground
state.
Using perturbation theory for anharmonic effects in
crystals, it was shown in [3] that, at high temperatures,
the
β
phase of zirconium becomes stable with respect to
atomic displacements corresponding to the
N
phonon if
the
N
phonon interacts with other transverse vibrational
modes with wave vectors directed along the [110] axis.
However, using a modified quasi-harmonic approxima-
tion [4], it was shown in [5] that, for this phase to
become stable against
N
-mode displacements, it suf-
fices to take into account the intrinsic anharmonicity of
the
N
mode.
Clearly, for
β
-Zr to be stable, the lattice must be sta-
ble not only against atomic displacements correspond-
ing to the
N
phonon. Calculations performed in [6]
within the frozen-phonon model showed that
β
-Zr is
also unstable with respect to atomic displacements cor-
responding to longitudinal vibrations with wave vector
k
= (2/3)(1, 1, 1) (
L
l
phonon). If the parameters of the
crystal are such that the crystal is above its triple point
(
T
= 973 K,
P
= 5.5 GPa), then these displacements are
involved in the temperature-induced transformation
from the
β
to the
ω
phase (with an AlB
2
-type structure).
At room temperature, the
ω
phase of zirconium has
been experimentally observed at pressures from 2.2 to
30–35 GPa [7, 8]. At zero pressure, the instability of
β
-Zr with respect to atomic displacements of this type
manifests itself in a sharp decrease in the frequency of
longitudinal vibrations, which was determined from the
inelastic neutron scattering spectra in the vicinity of
k
=
(2/3)(1, 1, 1) [1].
At present, there are no theoretical studies explain-
ing the mechanism of stabilization of the high-temper-
ature
β
phase of zirconium with respect to
L
l
-mode dis-
LATTICE DYNAMICS
AND PHASE TRANSITIONS
Effect of Anharmonic Longitudinal and Transverse Vibrations
with Wave Vector k = (2/3)(1, 1, 1) on the Structural Stability
of
b
-Zr under Pressure
V. Yu. Trubitsin and E. B. Dolgusheva
Physicotechnical Institute, Ural Division, Russian Academy of Sciences, ul. Kirova 132, Izhevsk, 426000 Russia
e-mail: tvynew@otf.pti.udm.ru
Received February 21, 2006; in final form, April 24, 2006
Abstract
—The pressure and temperature dependences of the frequencies of a strongly anharmonic longitudi-
nal and a transverse vibrational mode with wave vector
k
= (2/3)(1, 1, 1) in
β
-Zr are studied. The frequencies
are calculated by solving a set of stochastic Langevin differential equations with a thermal bath of the white-
noise type. In the calculations, a two-mode effective potential is used. This potential is found in the framework
of the density-functional theory using the “frozen-phonon” model and assuming that the contribution from the
electron entropy to the free energy is dependent on the atomic displacements. Based on the calculated pressure
and temperature dependences of the spectral density of transverse vibrations, the region of stability of the bcc
phase of zirconium is determined for pressures up to 35 GPa, and the result is in good agreement with the exper-
imental data. The bcc lattice is found to be unstable with respect to atomic displacements characteristic of vibra-
tions with wave vector
k
= (2/3)(1, 1, 1). This instability is of importance not only in the
β
ω
transition
but also in the
β
α
transformation, which is observed to occur at pressures less than 5 GPa.
PACS numbers: 63.20.Ry, 05.10.Gg, 63.20.Kr, 71.15.Nc
DOI:
10.1134/S1063783407010210