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8. Curie's Principle

The crystal symmetry depends upon the state of the crystal. If, due to some external influence, there is a change in the state of the crystal, there may also be a change in the crystal symmetry. The symmetry of a given state of a crystal may be determined using the Curie principle from the symmetry of the crystal free of any external influence and from the symmetry of the external influence.

According to Curie when various natural phenomena are piled upon each other forming a system, the dissymmetries are added up leaving only those elements which separately, in each phenomenon regarded in itself, were present. By dissymmetry Curie meant the sum of the absent symmetry elements.⁸ Curie's principle in itself may be formulated in the physics of crystals as follows: the symmetry group of a crystal under an external influence ($\~{K}$) is given by the greatest common subgroup of the symmetry group of the crystal without the influence (K) and of the symmetry group of the external influence (G) considering also the mutual position of the symmetry elements of these groups:⁹

$$\~{K} = K \cap G$$ (8.1)

Curie's principle expressed in other words: a crystal under an external influence will exhibit only those symmetry elements that are common to the crystal without the influence and the influence without the crystal .⁵

As an example let us investigate the change of symmetry in the ADP (ammonium dihydrogen phosphate) crystals in an electric field of various directions. The ground state symmetry of these crystals is $\=42m$, i.e. it has a fourfold inversion axis (which contains in itself also a two-fold rotation axis). The fourfold inversion axis lies in the line of intersection of two mutually perpendicular planes of symmetry. Two diad axes are perpendicular to the fourfold inversion axis and at 45$^{\circ}$ to the planes of symmetry. This is demonstrated in axionometric and stereographic representation respectively in Figs. 2 and 3. The symmetry of the electric field is $\infty {m}$, i.e. it corresponds to the symmetry of a cone, which has an infinite-fold rotation axis containing every possible rotation axes of lower symmetry including also the twofold axes, further on the infinite-fold rotation axis lies in the line of intersection of an infinite number of mirror planes.

 

Figure 2: The decrease of the symmetry of an ADP crystal in an electric field of the [001] direction.

 

Figure 3: The change of the symmetry of an ADP crystal in electric fields of various directions.

First let us investigate the case when the electric field points in the [001] direction, which means that the vector of the electric field is parallel with the fourfold inversion axis of the crystal. The symmetry elements of the electric field do not include the inversion axis, therefore according to the Curie principle the resulting symmetry elements do not contain this axis. However, it should be observed that the fourfold inversion axis contains also a twofold rotation axis which is a symmetry element of the electric field, consequently the symmetry elements of the crystal in an electric field of the direction [001] will also contain this twofold axis. Of the basic symmetry elements the two mirror planes are also symmetry elements of the electric field, thus they are conserved in the crystal too. The twofold rotation axes perpendicular to the line of intersection of the mirror planes do not belong to the symmetry elements of the electric field, consequently they will disappear. Summing up the common symmetry elements of the electric field and the crystal in this field we have two mirror planes perpendicular to each other and a two-fold rotation axis in the line of intersection of the mirror planes. Thus the symmetry of the ADP crystal in the electric field of the [001] direction is reduced to the symmetry of the orthorhombic mm2 point group (Figs. 2 and 3a). If the electric field acts in the [$\=100$] direction, i.e. along a twofold rotation axis perpendicular to the fourfold inversion axis no mirror plane of the crystal coincides with the mirror planes of the electric field which results in the disappearance of the mirror planes. Further on also the fourfold inversion axis (together with the twofold rotation axis connected with the inversion axis), and also from the two other twofold axes one axis will disappear, since the electric field has only in one direction a rotation symmetry. This way only the twofold rotation axis (along which the electric field is effective) remains conserved with the result that the symmetry of the crystal is reduced to the monoclinic point group 2 (as depicted in Fig. 3b). If the electric field influences the crystal in the [$\=1\=10$] direction, i.e. in one mirror plane, the common symmetry element will be one mirror plane: the crystal symmetry is reduced to the point group m of the monoclinic system (Fig. 3c). Finally if the electric field points in an arbitrary [hkl] direction, different from the directions already discussed, no symmetry element of the crystal and the electric field coincides. Consequently no symmetry element is preserved. In this case the symmetry of the crystal is reduced to the trivial point group 1 of the triclinic system (Fig. 3d).

It follows from the foregoing that the originally optically uniaxial ADP crystal will under the influence of an electric field behave like an optically biaxial crystal.

It should be noted that the Curie principle constitutes only a special case of the great principle of the superposition of the symmetry groups. A detailed discussion of this subject, however, would go beyond the scope of this paper, one can refer to the book of Shubnikov and Koptsik.¹⁰

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