Nonbonded (exp−6) potential parameters for C···C, C···H, and H···H interactions were derived from the crystal structures and properties of aromatic hydrocarbons. The exponent of the C···C repulsion term was taken from a calculation based on the interplanar spacing and compressibility of graphite. The exponent of the H···H repulsion was taken from a quantum‐mechanical calculation of the repulsion between two hydrogen molecules. The coefficients of the attractive and repulsive terms were fitted by weighted least squares to 77 observational equations involving the geometrical crystal structures, elastic constants, and sublimation energies of nine aromatic hydrocarbons. It was found necessary to externally estimate the H···H repulsion coefficient to obtain an appropriate partition between the C···C, C···H, and H···H energy terms. The geometrical mean combining law was applied for the attractive and repulsive coefficients.The resulting nonbonded potential parameters, when used to minimize the lattice energy by a steepest descent procedure, were found to reproduce the crystal structures of benzene, napthalene, and anthracene to within about 0.05 A in the carbon‐atom positions and within about 1% in the lattice constants. No assumption of anisotropic forces based on π‐electron polarizabilities was necessary to reproduce herringbone‐type packing. The position of the minimum in the best C···H nonbonded potential was less than expected, while the minimum in the H···H potential was larger than expected. The minima positions were found to be sensitive to the H···H repulsion coefficient. The correlation coefficients between the potential parameters found by this method were found to be large.

Nonbonded Potential Parameters Derived from Crystalline Aromatic Hydrocarbons

The ``elastic constants'' of a crystal are referred to a Cartesian coordinate system which has a definite relation to the crystal axes. In determining these constants from sound‐velocity measurements at least as many measurements must be made as there are constants to be determined and in general more than one propagation direction will be used. Thus, in order to relate the measured velocities to the elastic properties of the crystal one needs to solve the equation of motion for the direction under consideration, and to know the transformation equations that relate the elastic properties along the propagation direction to the desired elastic constants. In this paper a perturbation solution to the equation of motion is presented as well as the transformation equations for triclinic and all higher symmetry crystal classes. Also presented are simplifications that will be of use in designing experiments and reducing the data. Several mistakes in previously published work are corrected.

Determination of Elastic Constants from Sound‐Velocity Measurements in Crystals of General Symmetry

The lattice dynamics of solid CO2 has been treated throughout the Brillouin zone. The molecular librational degrees of freedom have been treated by the use of Eulerian displacement coordinates. The potential model assumes the molecule represented by two centers of interaction (diatomic model) and each molecular pair interaction term is written as a sum of 6–12 terms between interaction centers on different molecules. The three parameters of the potential are obtained by fitting to the five optic modes, from the crystal energy, and by invoking the condition of zero uniform stress. It is found that the interaction between translational and librational degrees of freedom is large inside the Brillouin zone. The low temperature specific heat was calculated and found to agree well with experimental data. In an appendix, the applicability of the classical harmonic oscillator treatment to small molecular solids is discussed, and it is concluded that this approach is justified for solid CO2.

Intermolecular Potential and Lattice Dynamics of the CO2 Crystal

The complete vibrational spectra of crystalline C6H6 and C6D6 have been calculated for the different pressure‐induced solid phases recently determined at 293 K up to 25 GPa, and compared to Raman scattering data. The normal coordinate analysis has been carried out by using intermolecular Buckingham‐type atom–atom interactions and the intramolecular force field of the free molecule. Results of such frequency calculations are compared to experimental values at ambient pressure. The variation of the relevant crystalline parameters is discussed to construct a model and calculate the vibrational frequencies under pressure. The quantitative fit of the frequency shift of the Raman active modes under pressure demonstrates the necessity of including different C–C and C–H (C–D) bond compressibilities within the benzene molecule. Such intramolecular distance variations which allow to estimate the frequency corrections for the totally symmetric (a1g) breathing modes, have been determined from the observed pressure‐fr...

High pressure solid phases of benzene. II. Calculations of the vibration frequencies and evolution of the bonds in C6H6 and C6D6 up to 20 GPa

On the basis of experimental data on the structure and elasticity tensor of naphthalene crystals at low temperatures a more accurate fitting of the parameters of the C-C, C-H, and H-H potential curves was made. The potential curves involved have the form φ − (A/r6)+ Be−αr and the following numerical parameters: Acc= 421 (kcal/mol)A6, Bcc = 7.16 × 104 kcal/mol, αcc = 3.68 A−1; ACH = 118 (kcal/mol) A6, BCH = 1.86 A 104 kcal/mol, ACH = 3.94 A−1;AHH = 29(kcal/mol) A6, BHH × 0.49 × 104kcal/mol,αHH = 4.29 A−1.



Auf der Grundlage von experimentellen Werten der Struktur und dey Elastizitatstensors von Naphthalenkristallen bei niedrigen Temperaturen wurde eine genauere Anpassung der Parameter der C-C,C-H- und H-H-Potentialkurven durchgefuhrt. Die Potentialkurven haben die Form e − (A/r6)+ Be−αr und die folgenden numerical parameters: Acc= 421 (kcal/mol) A6, Bcc = 7.16×104 kcal/mol, αcc = 3.68 A−1; ACH = 118 (kcal/mol) A6, BCH = 1.86×104 kcal/mol, ACH = 3,94 A−1;AHH = 29(kcal/mol) A6, BHH = 0,49×104kcal/mol,αHH = 4,29 A−1.

Optimal CC, CH, and HH Potential Curves for the Naphthalene Crystal

The elastic coefficients of benzene are determined from sound velocities measured with the use of an ultrasonic pulse technique over a temperature range of 170° to 250°K. At 250°K the results are (in units of 1010 dyn/cm2): c11 = 6.14; c22 = 6.56; c33 = 5.83; c44 = 1.97; c55 = 3.78; c66 = 1.53; c12 = 3.52; c13 = 4.01; c23 = 3.90. These elastic coefficients generally increase as the temperature is lowered. The observed absorption coefficients are relatively large, thus confirming data reported by L. N. Liebermann. However, it was found that the absorption is not isotropic and that the absorption for shear waves is comparable with that for longitudinal waves. Some of the experimental techniques used are described.

Elastic Constants of Single‐Crystal Benzene

: Large single crystals of benzene (C6H6) were aligned by an optical method The aligned crystals were pulsed with ultrasonic waves at a frequency of approximately 10 megacycles per second using quartz crystal transducers The velocity of the ultrasonic waves was measured by comparing the echo transit time with that of simultaneously generated waves moving through a variable length path in a medium in which the velocity of sound was known precisely Velocities of longitudinal and transverse waves along each of the principal axes of the orthorhombic crystal were used to compute the main diagonal terms of the C sub ij matrix The off-diagonal terms were then determined by velocity measurements along arbitrarily selected directions Velocity measurements were made and the elastic constants computed over the temperature range 170 K to 250 K Absorption in the crystal was measured over the same temperature range for each of the directions used for velocity measurements The absorption was determined to be aelotropic

https://calhoun.nps.edu/bitstream/10945/11975/1/determinationofe00hese.pdf

Donal W. Elliott

Papers

Elastic Constants of Single‐Crystal Benzene

Determination of the elastic constants of benzene