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Submitted on 1 Jan 1981
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DYNAMICS IN DISPLACIVE PHASE TRANSITIONS
R. Englman
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au n o 12, Tome 4 2 , de'cembre 1981 page c6-755
DYNAMICS IN DISPLACIVE PHASE TRANSITIONS
R. Englman
Israe l Atomic Energy Comission, Soreq Nuclear Research Center, Yavne, Israel
Abstract.- First passage rates of an interacting, anharmonic system of phonons (some of which are unstable, soft) are calculated as function of temperature. The classical equations of motion are amended by ef non- local term representing tunnelling.
1. Introduction.- Studies (ly2) of the approach of large systems to a new configu- ration present pictures of a limited number of degrees of freedom (soft modes) that have several points of stability and are in interaction with other (hard) degrees of freedom or with a random force-field. The theoretical methods utilize Fokker-Planck or master-equations. Experimentally soft mode dynamics have been investigated, e.g. by Raman s~attering.'~) Times of passage to the new equilib-
o 8
rium point(4) vary widely, between 10 -10 times inverse lattice frequencies, (5) and the factors that influence this variation are of interest.
For phonon-physics and in studies of displacive first order transitions (as in Cu or Fe-doped spinels) theories of fluctuation-induced phase-transitions are insufficiently productive since (a) they treat mainly the vicinity of the bifur- cation point, whereas most of the passage time is taken up far from this point and (b) quantum mechanical tunnelling is not included in the classical studies.
2. Description.- Our treatment (which follows on a previous work(6)) is for a monatomic, anharmonic lattice which is unstable in some low-frequency modes. The modes are admixed by qxqxq-type anharmonic terms whose strengths are about .1
in units of zero point motion, the mode mixing operating among adjacent,as well as within, wave-number shells and are critical for the transition. In our computa-
3
tion the Brillouin zone is divided into cells, numbering up to 10
.
The soft modes possess cubic terms (of strengths .02-.06) whose pbysical effect on the passage resembles that of a second minimum (see Fig. 1). In the equations of motion of the soft modes we add to the kinematic velocity a tunnelling termJOURNAL DE PHYSIQUE
where w is the (harmonic) soft mode frequency, M the atomic mass, V the local potential and V the height of the maximum in the potential being a function of
max
the remaining coordinates. The exponential form is exact in the WKB-approximation for a parabolic barrier.
(7)
Initial conditions for the modes at their minima assume a thermal distribution in the velocities squared.3. Results.- First passage rates have been calculated for a monatomic, Debye-type lattice having dynamic properties and mean mass equal to those of MgO. As is well known,(2) the rates depend heavily on the initial conditions and emphasis is placed
on fast passage conditions (leftward initial velocities). Rates shown in Fig. 2
as function of temperature T are those for soft modes occupying about 1.5% of the Brillouin zone. To test the effect of the tunnelling, Eq. (I), calculations were also made for a lattice whose mean mass is reduced by 20, i.e. for a proton lattice with the bonding properties of MgO. Down to 1.2'~ (% .8 cm-l) the effect of
tunnelling on the transition is small but is still distinguishable below 20'~.
4. References
(1) H. Haken, Synergetics (Springer, Berlin, 1978)
(2) R. Kubo et al. (Ed.), Nonlinear Nonequilibrium Statistical Mechanics, Kyoto Seminar in Prog. Theor. Phys. Suppl.
3
(1978)(3) K. Hisano and K. Toda, Solid State Comun.
3,
915 (1978) (4) M.I. Dykman and M.A. Krivoglaz, Physica A m A , 480 (1980)(5) A.D. Bruce, K.A. Miller and
W. Berlinger, Phys. Rev. Letters
42,
185 (1979)(6) R. Englman, Chem. Phys.
2,
227 (1981)L
Mode displacement
(ql
Fig. 1.- Forms of potentials used for soft modes (full curve) and hard modes (dotted curve). The broken curve is a cubic approximation to the soft-mode potential.
