1. Model predictions at second order phase transitions
Kraków
September 2003
Model predictions at second order phase transitions
  H.Böhm
Institut für Geowissenschaften der Universität,
55099 Mainz, Germany
J.Kusz
Institute of Physics, University of Silesia,
Katowice, Poland

2. Question
When a structure transforms from a high symmetric high temperature phase to a phase of lower symmetry by a second order phase transition, can we predict models for the phase after the transition ?

3. Answer
Yes
LANDAU theory postulates, that the distortions at a second order phase transition occur according to a single irreducible representation of the group of the wave vector.
That means that we can predict possible deformations, which will occur at the phase transition

4. Example
NbTe4 and TaTe4
Nb, Ta Te
Both iso-structural compounds transfer from a tetragonal structure of high symmetry to a (modulated) structure of lower symmetry through a second order phase transition.
The phase after the transition can be described as a 2 x 2 x 3 superstructure

5. Diffraction pattern
Evidence: additional superstructure (or satellite) reflections occur at the phase transitions:
RT: (h h l) layer c*
[1 1 0]*
[1 1 0]*
[1 1 0]* c* a*
HT: (h h l) layer
The variation of the pair of satellite reflections with temperature:

6. Deformation modes LANDAU theory postulates, that a second order phase
transition occurs according to a single irreducible representation of the group of the wave vector.
SG: P4cc (103)
k=m(0,0,1) 1
41z 2z
43z cx
c-xy cy
cxy t1 t2 -1 t3 t4
(m=1/3) a* c*
There are four one-dimensional (and one two-dimensional) irreducible representations
(Kovalev: Irreducible Representations of Space Groups (Gordon & Breach N.Y.)

7. Deformation modes What are the deformations, if we only consider the
listed one-dimensional representations.
The symmetry in the group of the
wave vector is :
P4cc (103) 1
41z 2z
43z cx
c-xy cy
cxy t1 1
t1 (identical representation) has no change of symmetry:
This does not correspond to a phase transition

8. Deformation modes
What are the deformations, if we consider the representation t2. 1
41z 2z
43z cx
c-xy cy
cxy t2 1 -1
The symmetry with distortions according to t2 is : P4 t2 a1 a2 3 5
Which deformations occur according
to t2 ? 2 6
The two squares perform a
breathing mode 4 8 7
The x-component 1

9. Deformation modes
Which deformations occur according to t2 for the other components ? 1
41z 2z
43z cx
c-xy cy
cxy t2 1 -1
The y-component
The z-component a1 a3 t2 a1 a2 t2 3 5 2 6 5 8 6 4 8 7 7 1 3 2 1
The two squares perform a
libration mode 4

10. Model calculations D B A C
[1 1 0]* c* A C
The phase after the transition can be described as a 2 x 2 x 3 superstructure.

11. Model calculations D B A C A=B C=D The modulation of the mode
(according to t2)
has a period of
3 along c.
A and B are in-phase
C and D are in-phase
but anti-phase to A an B.

12. Results
[1 1 0]* c*
[1 1 0]* c*
RT: (h h l) layer

13. Crystal Chemical Interpretation
3.00
2.87