2.4 Formation Versus Collapse of Order
51
the molecular orientation is ordered uniaxially along the a axis, the equivalence
between the a axis and others is lost. That is, the symmetry lowers.
The two views on an intermediate state are useful and necessary to understand a
variety of molecular systems. They are, however, not equally possible. Generally, the
discussion starting from the ordered state is easier to perform because relevant degrees
of freedom are, even not specified, but restricted by the ordered state. The third law of
thermodynamics is quite suitable for this view. Microscopic (spin) models of phase
transitions are primarily within this view. Detailed investigation on the ordered states
often leads to correct predictions about a phase transition to disordered state(s). A
comprehensive discussion along this line will be given for the melting process of
molecular crystals in Chap. 6. It is, however, emphasized that the question of why the
specific crystal structure is the most stable is out of consideration from the beginning.
The view starting from highly symmetric (disordered) states is much more difficult than the way mentioned in the previous paragraph. In contrast to the previous
case, a wide variety of orders are potentially possible. In principle, a search for all
possibilities and comparisons among them is necessary. Such a question to the most
stable crystal structure for the ensemble of rigid spherical molecules is an example.
Proper consideration of the effect of temperature is also necessary. For example,
the face-centered cubic packing and the hexagonal closest packing are known as
closest packings of spheres with the stacking of hexagonally close-packed layers of
· · ·ABCABC· · · and · · ·ABAB· · ·. It is clear that they and random stacking of A,
B, C are equally stable at the absolute zero as far as we consider interactions only
between the contacting neighbors.
References
1. P. Ehrenfest, Proc. R. Neth. Acad. Arts Sci. 36, 153–157 (1933)
2. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn. (Butterworth-Heinemann, Oxford,
1980)
3. J.-C. Tolédano, P. Tolédano, The Landau Theory of Phase Transitions (World Scientific, Singapore, 1987)
4. W. Lenz, Phys. Z. 21, 613–615 (1920)
5. E. Ising, Z. Phys. 31, 253–258 (1925)
6. L. Onsager, Phys. Rev. 65, 117–149 (1944)
7. A.N. Kolmogorov, Izv. Akad. Nauk SSSR, Ser. Mat. 3 355–359 (1937)
8. W.A. Johnson, R.F. Mehl, Trans. AIME 135, 416–459 (1939)
9. M. Avrami, J. Chem. Phys. 7, 1103–1112 (1939)
10. M. Avrami, J. Chem. Phys. 8, 212–224 (1940)
11. M. Avrami, J. Chem. Phys. 9, 177–183 (1941)
12. M. Tatsumi, T. Matsuo, H. Suga, S. Seki, J. Phy. Chem. Solids 39, 427–434 (1978)
13. H. Horner, C.M. Verma, Phys. Rev. Lett. 20, 845–846 (1968)
14. K. Saito, A. Sato, A. Bhattacharjee, M. Sorai, Solid State Commun. 120, 129–132 (2001)
15. J. Thoen, H. Marynissen, W. Van Dael, Phys. Rev. Lett. 52, 204–207 (1984)
16. P.G. de Genne, Phys. Lett. A 30, 454–455 (1969)
17. H.T. Stokes, D.M. Hatch, Isotropy Subgroups pf the 230 Crystallographic Space Groups (World
Scientific, Singapore, 1988)
51
the molecular orientation is ordered uniaxially along the a axis, the equivalence
between the a axis and others is lost. That is, the symmetry lowers.
The two views on an intermediate state are useful and necessary to understand a
variety of molecular systems. They are, however, not equally possible. Generally, the
discussion starting from the ordered state is easier to perform because relevant degrees
of freedom are, even not specified, but restricted by the ordered state. The third law of
thermodynamics is quite suitable for this view. Microscopic (spin) models of phase
transitions are primarily within this view. Detailed investigation on the ordered states
often leads to correct predictions about a phase transition to disordered state(s). A
comprehensive discussion along this line will be given for the melting process of
molecular crystals in Chap. 6. It is, however, emphasized that the question of why the
specific crystal structure is the most stable is out of consideration from the beginning.
The view starting from highly symmetric (disordered) states is much more difficult than the way mentioned in the previous paragraph. In contrast to the previous
case, a wide variety of orders are potentially possible. In principle, a search for all
possibilities and comparisons among them is necessary. Such a question to the most
stable crystal structure for the ensemble of rigid spherical molecules is an example.
Proper consideration of the effect of temperature is also necessary. For example,
the face-centered cubic packing and the hexagonal closest packing are known as
closest packings of spheres with the stacking of hexagonally close-packed layers of
· · ·ABCABC· · · and · · ·ABAB· · ·. It is clear that they and random stacking of A,
B, C are equally stable at the absolute zero as far as we consider interactions only
between the contacting neighbors.
References
1. P. Ehrenfest, Proc. R. Neth. Acad. Arts Sci. 36, 153–157 (1933)
2. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn. (Butterworth-Heinemann, Oxford,
1980)
3. J.-C. Tolédano, P. Tolédano, The Landau Theory of Phase Transitions (World Scientific, Singapore, 1987)
4. W. Lenz, Phys. Z. 21, 613–615 (1920)
5. E. Ising, Z. Phys. 31, 253–258 (1925)
6. L. Onsager, Phys. Rev. 65, 117–149 (1944)
7. A.N. Kolmogorov, Izv. Akad. Nauk SSSR, Ser. Mat. 3 355–359 (1937)
8. W.A. Johnson, R.F. Mehl, Trans. AIME 135, 416–459 (1939)
9. M. Avrami, J. Chem. Phys. 7, 1103–1112 (1939)
10. M. Avrami, J. Chem. Phys. 8, 212–224 (1940)
11. M. Avrami, J. Chem. Phys. 9, 177–183 (1941)
12. M. Tatsumi, T. Matsuo, H. Suga, S. Seki, J. Phy. Chem. Solids 39, 427–434 (1978)
13. H. Horner, C.M. Verma, Phys. Rev. Lett. 20, 845–846 (1968)
14. K. Saito, A. Sato, A. Bhattacharjee, M. Sorai, Solid State Commun. 120, 129–132 (2001)
15. J. Thoen, H. Marynissen, W. Van Dael, Phys. Rev. Lett. 52, 204–207 (1984)
16. P.G. de Genne, Phys. Lett. A 30, 454–455 (1969)
17. H.T. Stokes, D.M. Hatch, Isotropy Subgroups pf the 230 Crystallographic Space Groups (World
Scientific, Singapore, 1988)