116
5 Lattice Dynamics of Molecular Crystals
The optical branch of the vibration normal to the membrane is responsible for the
fluctuation of membrane thickness [38]. The mode with the vanishing wavevector
corresponds to uniform breathing of the volume and is out of consideration under
the assumption of the volume conservation. The physically reasonable length of
wavevector is necessary, accordingly, to describe the thickness fluctuation. It is not
trivial what range of the wavevector is relevant to the thickness fluctuation depending on assumed models. The equipartition law seems not to hold for molecules,
accordingly.
When the membrane is not a molecular bilayer but a single layer, the dispersion relation has only the acoustic branch. The discussion on the deformation as a
whole remains the same as that of bilayers, while one about the thickness fluctuation becomes significantly different. Its absence can be a claim when one believes
the anti-phase displacements of molecules being essential. On the other hand, the
dynamics with a sufficiently short wavelength of the molecular scale may serve as
a microscopic origin of the thickness fluctuation sensed by dull techniques without
atomic resolution. In both cases, the equipartition law does not hold in these cases.
Discussions on membranes often proceed on the area basis. In this respect, it
is noteworthy that the average area of molecular occupancy should correctly count
the number of molecules involved. No special attention is necessary for the singlelayered membrane. In the case of bilayers, the numbers of modes involved in acoustic
and optical branches are exactly half of the total number of molecules. The energy
carried by each branch is the same as one in the single-layer case, in which only the
acoustic branch exists.
References
1. M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford,
1954)
2. D.J. Hooton, Phil. Mag. 3, 49–54 (1958)
3. S. Takeno, Prog. Theor. Phys. 45, 137–173 (1970)
4. A. Nakanishi, T. Matsubara, J. Phys. Soc. Jpn. 39, 1415–1416 (1975)
5. N.M. Plakida, A.V. Belushkin, I. Natkaniec, T. Wasiutynski, Phys. Status Solidi b 118, 129–133
(1983)
6. N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965)
7. M. Toda, J. Phys. Soc. Jpn. 22, 431–436 (1967)
8. S. Takeno, Prog. Theor. Phys. 71, 395–398 (1984)
9. F. Fillaux, C.J. Carlile, Phys. Rev. B 42, 5990–6006 (1990)
10. L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edn. (Butterworth-Heinemann, Oxford, 1982)
11. S. Califano, V. Schettino, N. Neto, Lattice Dynamics of Molecular Crystals. Lecture Notes in
Chemistry, vol. 26 (Springer, New York, 1981)
12. D. Kirin, J. Chem. Phys. 100, 9123–9128 (1994)
13. G.S. Pawley, S.J. Cyvin, J. Chem. Phys. 52, 4073–4077 (1970)
14. K. Saito, T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 61, 679–688 (1988)
15. P. Debye, Ann. Phys. 348, 49–92 (1913)
16. I. Waller, Z. Phys. A 17, 398–408 (1923)
17. M.H. Lemee, L. Toupet, Y. Delugeard, J.C. Messager, H. Cailleau, Acta Cryst. B 43, 466–470
(1987)
5 Lattice Dynamics of Molecular Crystals
The optical branch of the vibration normal to the membrane is responsible for the
fluctuation of membrane thickness [38]. The mode with the vanishing wavevector
corresponds to uniform breathing of the volume and is out of consideration under
the assumption of the volume conservation. The physically reasonable length of
wavevector is necessary, accordingly, to describe the thickness fluctuation. It is not
trivial what range of the wavevector is relevant to the thickness fluctuation depending on assumed models. The equipartition law seems not to hold for molecules,
accordingly.
When the membrane is not a molecular bilayer but a single layer, the dispersion relation has only the acoustic branch. The discussion on the deformation as a
whole remains the same as that of bilayers, while one about the thickness fluctuation becomes significantly different. Its absence can be a claim when one believes
the anti-phase displacements of molecules being essential. On the other hand, the
dynamics with a sufficiently short wavelength of the molecular scale may serve as
a microscopic origin of the thickness fluctuation sensed by dull techniques without
atomic resolution. In both cases, the equipartition law does not hold in these cases.
Discussions on membranes often proceed on the area basis. In this respect, it
is noteworthy that the average area of molecular occupancy should correctly count
the number of molecules involved. No special attention is necessary for the singlelayered membrane. In the case of bilayers, the numbers of modes involved in acoustic
and optical branches are exactly half of the total number of molecules. The energy
carried by each branch is the same as one in the single-layer case, in which only the
acoustic branch exists.
References
1. M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford,
1954)
2. D.J. Hooton, Phil. Mag. 3, 49–54 (1958)
3. S. Takeno, Prog. Theor. Phys. 45, 137–173 (1970)
4. A. Nakanishi, T. Matsubara, J. Phys. Soc. Jpn. 39, 1415–1416 (1975)
5. N.M. Plakida, A.V. Belushkin, I. Natkaniec, T. Wasiutynski, Phys. Status Solidi b 118, 129–133
(1983)
6. N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965)
7. M. Toda, J. Phys. Soc. Jpn. 22, 431–436 (1967)
8. S. Takeno, Prog. Theor. Phys. 71, 395–398 (1984)
9. F. Fillaux, C.J. Carlile, Phys. Rev. B 42, 5990–6006 (1990)
10. L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edn. (Butterworth-Heinemann, Oxford, 1982)
11. S. Califano, V. Schettino, N. Neto, Lattice Dynamics of Molecular Crystals. Lecture Notes in
Chemistry, vol. 26 (Springer, New York, 1981)
12. D. Kirin, J. Chem. Phys. 100, 9123–9128 (1994)
13. G.S. Pawley, S.J. Cyvin, J. Chem. Phys. 52, 4073–4077 (1970)
14. K. Saito, T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 61, 679–688 (1988)
15. P. Debye, Ann. Phys. 348, 49–92 (1913)
16. I. Waller, Z. Phys. A 17, 398–408 (1923)
17. M.H. Lemee, L. Toupet, Y. Delugeard, J.C. Messager, H. Cailleau, Acta Cryst. B 43, 466–470
(1987)