Molecular rotation in crystals

Chapter 9, on entropy and molecular rotation in crystals and liquids, is concerned mostly with statistical mechanics rather than quantum mechanics, but the two appear together in SP 74. Chapter 9 contains one of Pauling s most celebrated papers, SP 73, in which he explains the experimentally measured zero-point entropy of ice as due to water-molecule orientation disorder in the tetrahedrally H-bonded ice structure with asymmetric hydrogen bonds (in which the bonding proton is not at the center of the bond). This concept has proven fully valid, and the disorder phenomenon is now known to affect greatly the physical properties of ice via the... 

Chapter 9. Entropy and Molecular Rotation in Crystals and Liquids 773... 

The lattice energy of a crystal of known structure (atomic positions) is thus calculated by compiling all possible distances between pairs of atoms in different molecules. The method of atom-atom potentials has been employed to investigate phenomena pertaining to static as well as dynamic lattices and the subject has been reviewed by Kitaigorodsky (1973) as well as by Ramdas Thomas (1980). Typical of the problems that have been investigated by this method are defects and planar faults, phase transitions and molecular rotation in crystals. 

We now consider hydrogen transfer reactions between the excited impurity molecules and the neighboring host molecules in crystals. Prass et al. [1988, 1989] and Steidl et al. [1988] studied the abstraction of an hydrogen atom from fluorene by an impurity acridine molecule in its lowest triplet state. The fluorene molecule is oriented in a favorable position for the transfer (Figure 6.18). The radical pair thus formed is deactivated by the reverse transition. H atom abstraction by acridine molecules competes with the radiative deactivation (phosphorescence) of the 3T state, and the temperature dependence of transfer rate constant is inferred from the kinetic measurements in the range 33-143 K. Below 72 K, k(T) is described by Eq. (2.30) with n = 1, while at T>70K the Arrhenius law holds with the apparent activation energy of 0.33 kcal/mol (120 cm-1). The value of a corresponds to the thermal excitation of the symmetric vibration that is observed in the Raman spectrum of the host crystal. The shift in its frequency after deuteration shows that this is a libration i.e., the tunneling is enhanced by hindered molecular rotation in crystal. 

Fig. 12.8. (PPE) profiles for molecular rotation in crystals of some aromatic hydrocarbons. 1 anthracene (elongation index 1.57) 2 pyrene (1.27) 3 naphthalene (1.24) 4 benzene (1.10) 5 cor-onene (1.00)...

E. Posnjak, F.C. Kracek, Molecular Rotation in the Solid State. The Variation of Crystal 1 Structure of Ammonium Nitrate with Temperature , JACS 54, 2766-86 (July 1932) 4) D.P. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

Because of the orientational freedom, plastic crystals usually crystallize in cubic structures (Table 4.2). It is significant that cubic structures are adopted even when the molecular symmetry is incompatible with the cubic crystal symmetry. For example, t-butyl chloride in the plastic crystalline state has a fee structure even though the isolated molecule has a three-fold rotation axis which is incompatible with the cubic structure. Such apparent discrepancies between the lattice symmetry and molecular symmetry provide clear indications of the rotational disorder in the plastic crystalline state. It should, however, be remarked that molecular rotation in plastic crystals is rarely free rather it appears that there is more than one minimum potential energy configuration which allows the molecules to tumble rapidly from one orientation to another, the different orientations being random in the plastic crystal. 

Molecular rotation In a normal crystal every atom occupies a precise mean position, about which it vibrates to a degree depending on the temperature molecules or polyatomic ions have precisely defined orientations as well as precise mean positions. When such a crystal is heated, the amplitude of the thermal vibrations of the atoms increases with the temperature until a point is reached at which the regular structure breaks down, that is, the crystal melts. But in a few types of crystal it appears that notation of molecules or polyatomic... 

There are a few exceptions to the statements of the previous paragraph. The vibrational Raman spectrum of liquid H2 shows rotational fine structure for H2, the rotational levels are widely spaced and intermolecular forces are reasonably small. Certain solids when heated undergo a transition to a solid state in which molecular rotation in the crystal is possible. Solid H2 undergoes such a transition, as shown by the heat-capacity curve see Davidsoriy Section 16-9. 

Figure 12.8 shows the (PPE) curves (see 12.3.3) for the rotation of flat molecules, (in the rigid environment approximation). An elongation index is defined as D /D (see 12.2.3.1), the ratio of the two molecular dimensions in the molecular plane. The rotation is severely hindered if the elongation index exceeds about 1.3. A large molecule such as coronene rotates easily in the crystal, due to its very small elongation index. For these molecules, the ease of molecular rotation in the solid depends more on molecular shape than on details of the crystal structure. 

In this discussion at attempt will be made to describe in greater detail the structure and motion for a larger number of condis crystals. A special effort will be made to point-out the differences between condis crystals on the one hand, and liquid and plastic crystals on the other. It seems reasonable, and has been illustrated on several examples, that molecules with dynamic, conformational disorder in the liquid state show such conformational disorder also in the liquid crystalline and plastic crystalline states The major need in distinguishing condis crystals from other mesophases is thus the identification of translational motion and positional disorder of the molecular centers of gravity in the case of liquid crystals, and of molecular rotation in the case of plastic crystals. 

This volume contains data on the geometric parameters (intemuclear distances, bond angles, dihedral angles of internal rotation etc.) of free polyatomic molecules including free radicals and molecular ions. (For the diatomic stmctures measured by high-resolution spectroscopy, see [1], and for molecular stmctures in crystals, see [2].)... 

Janik and co-workers (1969) carried out incoherent inelastic neutron scattering studies on solid methane. They studied the lattice vibrations and molecular rotation in this solid near the melting point. Ito (1964) reported the lattice vibrational Raman spectrum of solid methyl iodide. Durig, Craven, and Bragin (1970) studied the low-frequency vibrations in the solid phases of two classes of molecules (CH3)jMCl and (CH3)3MBr where M was C, Si, and Ge. The infrared spectra of solid CCI4, benzene, and CS2, pure and activated by the impurities I3 and HCl, were studied by Munier and Hadni (1968). Colombo (1968) described the low-frequency Raman spectrum of single crystals of imidazole. 

The entropy of fusion of a normal crystal can be considered as the sum of the increase in disorder due to the breakup of the crystal lattice, the configurational contribution, and that due to greater fieedom for internal motions. The configurational contributions predominate. They can be considered as the sum of the translational disorder and the effect of onset of molecular rotation. In this light a transition from a normal crystal to a plastic crystal introduces some of the disorder normally associated with melting. The sum of the entropy of transition to a plastic crystal and the subsequent entropy of fusion is within the range of the entropy of fusion of normal crystals. Molecular rotation in plastic crystals are not usually free. Thus the entropy of transition is somewhat less than what would be observed for a transition to free rotation. 

Fig. 1. Examples of temperature dependence of the rate constant for the reactions in which the low-temperature rate-constant limit has been observed 1. hydrogen transfer in the excited singlet state of the molecule represented by (6.16) 2. molecular reorientation in methane crystal 3. internal rotation of CHj group in radical (6.25) 4. inversion of radical (6.40) 5. hydrogen transfer in halved molecule (6.16) 6. isomerization of molecule (6.17) in excited triplet state 7. tautomerization in the ground state of 7-azoindole dimer (6.1) 8. polymerization of formaldehyde in reaction (6.44) 9. limiting stage (6.45) of (a) chain hydrobromination, (b) chlorination and (c) bromination of ethylene 10. isomerization of radical (6.18) 11. abstraction of H atom by methyl radical from methanol matrix [reaction (6.19)] 12. radical pair isomerization in dimethylglyoxime crystals [Toriyama et al. 1977].

Press, W., 1981, Single-Particle Rotations in Molecular Crystals. Springer Tracts in Modem Physics, Vol. 92 (Springer, Berlin). Punnkinen, M., 1980, Phys. Rev. B 21, 54. 

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

Smectic phases are more highly ordered than nematic phases, and with an ordering of the molecules into layers. There are a number of different smectic phases which reflect differing degree of ordering. Crystal smectic phases are characterised by the appearance of inter-layer structural correlations and may in some cases be accompanied by a loss of molecular rotational freedom. 

Oxygen in the solid state consists of 02 molecules. From 24 K to 43.6 K they are packed as in a-F2. Under pressure (5.5 GPa) this packing is also observed at room temperature. Below 24 K the molecules are slightly tilted against the hexagonal layer. From 43.6 K up to the melting point (54.8 K) the molecules rotate in the crystal as in /3-F2. Under pressure oxygen becomes metallic at approximately 100 GPa, but it remains molecular. 

Electrons of still lower energy have been called subvibrational (Mozumder and Magee, 1967). These electrons are hot (epithermal) and must still lose energy to become thermal with energy (3/2)kBT — 0.0375 eV at T = 300 K. Subvibrational electrons are characterized not by forbiddenness of intramolecular vibrational excitation, but by their low cross section. Three avenues of energy loss of subvibrational electrons have been considered (1) elastic collision, (2) excitation of rotation (free or hindered), and (3) excitation of inter-molecular vibration (including, in crystals, lattice vibrations). 

---