Theory of pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures

A derivation is given for the integrated absorption coefficient of pressure-induced pure rotational and vibrational transitions in binary collisions of homonuclear diatomic molecules of the same chemical species. The previously neglected effects of excited vibrational states, mechanical anharmonicity, and vibration-rotation interaction are taken into account to obtain more accurate absorption coefficients at high temperatures. In the region of the fundamental wave number the excited vibrational states make more of a contribution to the absorption than their relative population would lead one to expect. LIST OF SYMBOLS Aij(k) function for molecule k involving integral of normalized associated Legendre functions aṽ linear absorption coefficient for wave number ṽ excluding stimulated emission, m−1 B integrated Einstein coefficient of absorption, m2 J−1 sec−1 Bṽ spectral Einstein coefficient of absorption, m3 J−1 sec−1 Cl1ζ1l2ζ2 expansion coefficient for μ0z, C m c speed of light, m sec−1 Dl1ζ1l2ζ2 expansion coefficient for μ1, C product of two expansion coefficients, units vary e charge of electron, C functions of Q, Δ and I, C2 m5 h Planck's constant, J sec , J sec dipole moment integral, units vary total angular momentum quantum number for initial state of molecule i total angular momentum quantum number for final state of molecule i Kl1ζ1l2ζ2 expansion coefficient for μz, C m k Boltzmann constant, J K−1 functions of J′ and J functions of Q, Δ, C and D, C m quantum number for component of total angular momentum along z axis for initial state of molecule i quantum number for component of total angular momentum along z axis for final state of molecule i mH mass of hydrogen atom, kg mr reduced mass of oscillator, kg N, N1, N2 number of molecules n, n1, n2 number density of molecules, m−3 O branch for molecule probability of molecule i being in state with quantum numbers having values vi, Ji, mi p number of pairs of molecules vibrational overlap integral for molecule k Q branch for molecule scalar quadrupole moment of molecule i, C m2 qXX, qYY, qZZ elements of quadrupole moment tensor, C m2 R intermolecular distance, m internuclear distance of molecule k, m S branch for molecule s integrated absorption coefficient of transition with wave number removed, m−1 T temperature, °K t time, sec V volume, m3 vibrational quantum number for initial state of molecule i vibrational quantum number for final state of molecule i W transition-dependent factor in s, C2 m5 X, Y, Z Cartesian coordinates with origin at the midpoint of the line connecting two nuclei of molecule, with Z axis running along the internuclear axis, m x, y, z Cartesian coordinates fixed in space (Fig. 1), m Xi position coordinates of electrons, m YJm spherical harmonic Zj number of elemental charges on the jth nucleus zi z coordinate of ith electron, m zj z coordinate of jth nucleus, m average polarizability of molecule i, C2 m2 J−1 vibrational matrix element for molecule k, m anisotropy of polarizability of molecule i, C2 m2 J-1 δ(J′, J) Kronecker delta function e force constant for Lennard-Jones potential, J e0 electric permittivity of free space, C2 N−1 m−2 ΘJm normalized associated Legendre function polar angle of molecule i rad μx, μy, μz components of electric dipole moment, C m ∂μz/∂ri evaluated at and , C μz4 μz for configuration 4, C m (μx)ϱ′ϱ, (μz)ϱ′ϱ components of lectric dipole moment matrix elements, C m ṽ photon wave number, m−1 ṽ0 fundamental wave number of vibration, m−1 ϱṽ radiation energy density per unit wave number, J m−2 dτ elements of volume in configuration space, m3 Φ intermolecular potential averaged over orientations, J azimuthal angle of molecule i, rad ψ wave function, units vary vibrational wave function of molecule k, m− Superscripts 0 equilibrium internuclear distance of diatomic molecule ′′ ∂/∂r1 evaluated at ‡ reduced Subscripts e electronic n nuclear ζ v1, v2, J1, J2 collectively ζ′ v′1, v′2, J′1, J′2 collectively ϱ v1, v2, J1, J2, m1, m2 collectively ϱ′ v′1, v′2, J′1, J′2, m′1, m′2 collectively Full-size table Table options View in workspace Download as CSV