Principal values and directions of a deformation tensor
from a set of unit cell parameters
(e.g. thermal expansion, compressibility, compositional deformation, ...)
by A.Filhol, J. Lajzerowicz and M. Thomas
The deformation U of unit cell of a crystalline material under a constraint C is described by a symmetrical secondrank tensor. The constraint can be temperature T, pressure P, composition x, etc., as shown in the nonexhaustive list below:
Isobaric thermal expansion (e.g. see [3]) 

Isothermal compressibility (e.g. see [2]) 

Compositional deformation in an alloy [A1xBx] (e.g. see[4]) Also  Swelling of a monomer single crystal as function of the polymerization rate [5]  Cell deformation of a protein crystal as a function of water content 
With "l" a dimension in the material. 
Principal values and directions of U as a function of C are of interest since they tell us about chemical 'repulsive' interactions in the structure.
When the unit cell is cubic, othorhombic, hexagonal, ..., the principal values (d1,d2,d3) and directions of U are trivial. For low symmetry cells (triclinic, monoclinic) this is less straighforward and a program such as DEFORM is necessary.
The coefficients of the tensor of deformation U can be obtained from the unitcell parameters (three lengths _{} and three angles _{} with i=1 to 3) and their derivatives [1,4]:
These derivatives can be obtained from a set of unit cell parameters as a function of the constraint C. DEFORM performs a polynomial fit for each of the cell parameters not imposed by symmetry. From these polynomials it is straightforward to get both cell parameters and derivatives as a function of C. Then principal values and directions are computed using the above equations.
 Note 1: A polynomial fit is often sufficient but this may not be true in the vicinity of a phase transition. The parameter divergence being exponential then a more appropriate fit function must be used.
 Note 2: It is wise not to use polynomial degrees higher than two. More complex curve shapes are generally a clue to there being a phase transition.
The method implemented in DEFORM is simple and offers an internal coherence check through the extra fit of the variation of the measured cell volume. This "observed" curve is compared to the "calculated" curve, i.e. the curve of cell volumes computed from the fitted cell parameters. Any significative discrepancy between the two indicates that one or more of cellparameter fit is not correct. Again, this is often the clue of a phase transition or of its vicinity.
Thermal expansion of TEA.(TCNQ)_{2} [3]
DEFORM is a old program (1987) to which I will hopefully offer a multiplatform GUI some day !
The original VAX FORTRAN code can be downloaded HERE. [not yet available]
Bibliography
1 A. Filhol (1985) "Evolution comparée en fonction de la température ou de la pression des propriétés physiques et structurales de conducteurs organiques unidimensionnels", PhD, Univ. Bordeaux I, 26 April 1985, nº 835.
2 Room and highpressure neutron structure determination of TTFTCNQ.Thermal expansion and isothermal compressibility
Filhol A., Bravic G., Gaultier J., Chasseau D. and Vettier C. (1981) Acta Crystallographica B 37, 12251236.
3 Structural evolution of the onedimensional organic conductor triethylammonium7,7,8,8tetracyanopquinodimethane (1:2) [TEA(TCNQ)2] in the temperature range 40 to 345 K.
Filhol A. and Thomas M. (1984) Acta Crystallographica B 40, 4459.
4 The tensor of compositional deformation. A new crystallographic way to analyse syncrystallization
Chanh N.B., Clastre J., Gaultier J., Haget Y., Meresse A., Lajzerowicz J., Filhol A. and Thomas M. (1988) Journal of Applied Crystallography 21, 1014.
5 Aimé J.P. (1983) PhD, Univ. of Paris VII, France.
Copyright 2008, Institut LaueLangevin
Last update: 3 Oct 2008, A. Filhol <filhol(at)ill.eu>