Relationships in Raman intensity theories,in particular the Mayants-Averbukh theory

Brief surveys are given of the Mayants-Averbukh Raman intensity theory, and of the polar tensor Raman intensity theory recently presented by Bogaard and Haines. It was found that these intensity theories in essence are equivalent. In addition, the appearances of the symmetry invariant parameter matrices F_n⁰ of the Mayants-Averbukh theory were derived and tabulated for various symmetries of bond n. These matrices, and a single bond coordinate system, can be used as a convenient alternative to the Mayants-Averbukh treatment of bonds which have some kind of symmetry with respect to the midpoint of the bond. A modification of the Mayants-Averbukh treatment is also suggested. The rotational mode equations of the Mayants-Averbukh theory have been investigated to elecudate the constraints which they impose on Raman intensity theories based on the bond polarizability model. It was found that the valenceoptical theory is in conformity with the rotational modes only if all electrooptical parameters $\text{?α}{}_{\mathrm{ii}}{}^{\mathrm{(n)}}\text{?γ}{}_{\mathrm{p}}$ are neglected, where α_ii⁽ⁿ⁾ (i = 1, 2, 3) are the diagonal components of the polarizability α⁽ⁿ⁾ of bond n, and γ_p is the pth internal angular coordinate. Furthermore, the valence-optical theory was found to be strictly applicable only for cylindrical bond symmetry, C_mv (m ≥ 4). A generalized valence-optical Raman intensity theory, allowing also for non-zero off-diagonal components α_ij⁽ⁿ⁾, was found to be incompatible with the rotational mode equations of the Mayants-Averbukh theory. However, its basic polarizability equation was useful for suggesting a unique interpretation of a set of f parameters (elements of F_n⁰) in terms of components of the anisotropic part of a symmetric bond polarizability.     

Interpretation of the d-parameters,occurring in the Mayants-Averbukh theory of infrared intensity,in terms of the bond dipole moment concept: Equivalence with the valence-optical theory in the zeroth approximation for linear and planar molecules

From a comparison with the valence-optical theory of infrared intensity in the zeroth approximation, it is suggested that the d-parameters (elements of the effective charge tensor D_n⁰ of a bond) occurring in the Mayants-Averbukh theory can be expressed in terms of bond dipole moment components and their derivatives with respect to components of the Cartesian bond displacement vector. With this interpretation, which may be suitable for practical application, the Mayants-Averbukh theory formally appears as a combination of the “special” and the generalized valence-optical theory. In particular, it is found that for linear molecules the Mayants-Averbukh theory is equivalent to the “special” valence-optical theory in the zeroth approximation. For planar molecules the Mayants-Averbukh theory is equivalent to the generalized valence-optical theory (zeroth approximation) for in-plane modes and to the “special” valence-optical theory for out-of-plane modes.     

Quasiclassical mobility for extrinsic semiconductors

A quasiclassical formulation for mobility in extrinsic semiconductors is presented based on scattering from ionized impurity atoms. Quantum theory enters the otherwise classical Chapman-Enskog expansion of the Boltzmann equation through incorporation of the Thomas-Fermi interaction potential together with the Bom approximation for evaluation of scattering integrals. The following expression results for mobility μ_i, (cgs):

$\text{μ}{}_{\mathrm{i}}\text{3}\text{2}\text{?}{}^2\text{n}{}_{\mathrm{s}}\text{e}{}^3\text{m}{}^1\text{2}\text{2}\text{k}{}_{\mathrm{B}}\text{T}\text{2φ}\text{3}\text{2}\text{1}\text{f(γ)}$

,

$\text{f(γ)=[(1+γ)e}{}^{\mathrm{\gamma }}\text{E}{}_1\text{(γ)?1]}$

, where n_s is impurity concentration, m¹ is effective mass, E₁(γ) is the exponential integral, ? is dielectric constant and γ is dimensionless Thomas-Fermi energy. The structure of the dimensional factor in the preceding expression for μ_i agrees with previous expressions for this parameter.     

Rotational analysis of a red A′ 2Σ+-X 2Πi system of CuO

High dispersion rotational analysis of a red CuO band system has led to the identification of an $\text{A′}{}^2\text{Σ}{}^{\mathrm{+}}\text{-X}{}^2\text{Π}{}_{\mathrm{i}}$ transition. The anomalous appearance of the branches is due to a very large spin splitting of the ²Σ upper state. The influence of centrifugal distortion effects on this spin splitting (γ_D and γ_H parameters) is essential for explaining the band structure. A reassignment of electronic symmetries of all the ²Σ states of CuO is proposed.     

The high resolution infrared spectrum of the ν1 band of HOCl

The ν₁ bands of HO³⁵Cl and HO³⁷Cl have been recorded. Both the A- and B-type rotational transitions of these hybrid bands have been completely assigned, and spectroscopic constants have been obtained for both the ground and upper state. The ratio of the electric dipole moment derivatives $\text{(}\text{?μ}{}_{\mathrm{a}}\text{?Q}{}_1\text{)}\text{(}\text{?μ}{}_{\mathrm{b}}\text{?Q}{}_1$ has been found to be 0.985 ± 0.05 for ν₁.     

Nuclear moments of the low abundant natural isotope 176Lu and hyperfine anomalies in the lutetium isotopes

The atomic beam magnetic resonance method combined with laser-induced state- and isotopeselective detection of metastable atoms has been used to investigate the hyperfine structure of the ²D ground multiplet in ¹⁷⁵Lu and ¹⁷⁶Lu. The analysis of the data yields not only accurate values for the hyperfine interaction constants, the nuclear magnetic dipole moment of ¹⁷⁵Lu, and the electronic g_J, factors, but also the first directly measured value of the nuclear magnetic dipole moment of the low abundant isotope ${}^{176}\text{Lu}\text{: μ}{}_{\mathrm{I}}\text{(}{}^{176}\text{Lu) = 3.1692 (45)μ}{}_{\mathrm{<mtext>N</mtext>}}$ (corrected for diamagnetic shielding). The spectroscopic quadrupole moment of ¹⁷⁶Lu was calculated from the ratio of the B-factors and the quadrupole moment of ${}^{175}\text{Lu}\text{: Q}{}_{\mathrm{<mtext>s</mtext>}}\text{(}{}^{176}\text{Lu}\text{) = 4.92 (3)}\text{b}$. Moreover, the magnetic hyperfine anomalies for the isotopic chain ^{175,176,176m,177}Lu were determined. A quadrupole hyperfine anomaly between ¹⁷⁵Lu and ¹⁷⁶Lu was not found when comparing the ratio of the B-factors in the states ${}^2\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{and}{}^2\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$. From a comparison of the quadrupole moment of ¹⁷⁵Lu obtained from the hyperfine structure data and the quadrupole moment measured in muonic lutetium atoms semi-empirical Sternheimer shielding factors could be estimated.     

Dipole moment function and vibration-rotation matrix elements of HCl35 and DCl35

The infrared intensity measurements and molecular beam electric resonance dipole moment measurements for HCl and DCl have been reviewed. A method not previously exploited is used to determine infrared matrix elements from the electric resonance dipole moment measurements. A ‘best’ set of matrix element values was selected for HCl and from these the M_i-coefficients of a polynomial dipole moment approximation were determined; M₀ = 1.0935±0.0007 D, $\text{M}{}_1\text{= 0.947±0.023 D/}\text{A}\text{?}$, $\text{M}{}_2\text{= 0.015±0.041 D/}\text{A}\text{?}{}^2$, $\text{M}{}_3\text{= -0.814± 0.116 D/}\text{A}\text{?}{}^3$. Calculations using this dipole moment function for both HCl and DCl are shown to give good agreement with available band strength and vibration-rotation interaction factor measurements. RKR potentials are also calculated for both molecules.     

Resolution of the discrepancies concerning the optical and microwave values for B0 and D0 of the X 3Σg− state of O2

The discrepancies concerning the optical and microwave values of B₀ and D₀ for the X${}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state of O₂ have been removed by a nonlinear least-squares fit to all of the lines of the O₂, $\text{b}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}\text{-X}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ Red Atmospheric bands recorded by Babcock and Herzberg (Astrophys. J., 108, 167, 1948). The resulting values for B₀″ and D₀″ are in excellent agreement with the Raman and microwave values. Improved values are determined for B₁″, D₁″, γ₁″ (spin-rotation), and ?₁″ (spin-spin). Both γ_v″ and ?_v″ increase in magnitude from v″ = 0 to v″ = 1. Improved Dunham Y_i0 and Y_i1 expansion coefficients are determined for the $\text{b}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ state, from which the Rydberg-Klein-Rees potential is constructed.     

Backbending behaviour of rotational bands in 163,165,167Yb

Several rotational bands in ^163,165,167Yb are observed in (HI,xnγe^?) experiments. The $\text{i}{}_{\mathrm{<mtext>13</mtext><mtext>2</mtext>}}$ and $\text{3}\text{2}{}^{\mathrm{?}}\text{[521]}$ bands do not backbend, whereas the $\text{5}\text{2}{}^{\mathrm{?}}\text{[523]}$ bands do, indicating additional processes besides the rotational alignment of one $\text{i}{}_{\mathrm{<mtext>13</mtext><mtext>2</mtext>}}$ neutron pair that are responsible for the backbending.     

X-band ESR studies of forbidden transitions of europium ions in CaF2: Eu crystals

The characteristics of allowed and forbidden transitions of Eu²⁺ ions in CaF₂ single crystals have been investigated at room temperature. The Angular dependences of these transition intensities were theoretically calculated with the use of perturbation theory, and compared with the experimental values. The experimental results can be explained by taking into account crystal field effects $\text{(}\text{1}\text{60}\text{) b}{}_4\text{[O}{}^0{}_4\text{+ 5O}{}^4{}_4\text{]}$, and a hyper fine interaction A_iS · I_i in the spin Hamiltonian.     

The irreducible Raman scattering tensor operator for the 32 crystallographic point groups and its application to the resonance and electronic Raman effect

The irreducible components of the Raman scattering tensor operator $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ under the symmetry of a general point group are calculated. The unitary transformations U(Γ_γΓ_ksΓ_k′s′, ρσ) from the Cartesian $\text{α}\text{?}{}_{\mathrm{\rho \sigma }}$ and spherical $\text{α}\text{?}{}_{\mathrm{Q}}{}^{\mathrm{K}}$ components, respectively, to the irreducible components $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ for the 32 crystallographic point groups are collected in tables. As an example the unitary transformation U(Γ_γΓ_ksΓ_k′s′, ρσ) is used to discuss the behavior of the scattering tensor in a resonance Raman experiment. With the help of the general formalism the scattering tensor for electronic Raman transitions of transition metal ions is calculated. As an example the scattering tensors of electronic Raman transitions within the ⁵T₂ state of the high-spin trigonal distorted octahedral Fe²⁺ are calculated and the refinement of the selection rules is discussed.     

Light composite fermions in a dynamical subquark model of pregauge interactions

The masses of composite leptons and quarks are discussed in a “dynamical subquark model of pregauge interactions”. In this model, the leptons and quarks are made of a spinor and scalar subquark with equal mass, M, and the gauge bosons and Higgs scalar of the SU(3)_c×SU(2)_L×U(1)_Y model are made of a subquark-antisubquark pair. The SU(2)_L×U(1)_Y symmetry is spontaneously broken by the composite Higgs scalar and the (scalar) subquark mass parameter is in turn bounded as $\text{M > 5.4}\text{TeV}\text{(=2π(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{where}\text{G}{}_{\mathrm{<mtext>F</mtext>}}$ is the Fermi coupling constant). The spontaneously generated mass of a lepton or quark, m_i⁽ⁿ⁾ (i = 1, 2; n = 1 ～ N_g), is calculated to be: $\text{m}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{× (4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)α}{}_{\mathrm{<mtext>e.m.</mtext>}}\text{(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/3}\text{6}\text{(=0.35r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{(4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)}\text{GeV}\text{),}\text{where}\text{r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}$ are the parameters satisfying that 0 ? r_i⁽ⁿ⁾ ? 1 and Σ (r_i⁽ⁿ⁾)² = 1;N_g is the total number of generations of the leptons and quarks; α_e.m. is the fine structure constant. The appearance of light composite fermions is related to a specific mechanism of generating global chiral symmetries of the leptons and quarks. Global symmetries of scalar subquarks yield chiral symmetries of the leptons and quarks. Our model turns out to satisfy 't Hooft's anomaly conditions on massless composite fermions.     

Reorientational motion of the OH− ion in cubic sodium hydroxide

The rotational motion of the OH^? ion was studied in cubic NaOH at 575 K with quasielastic incoherent neutron scattering. The data are compared to two simple models yielding values for the radius of rotation R, the translational mean square displacement 〈u²〉_H, the rotational jump rate τ^?1 and the rotational diffusion coefficient D_R. The following parameter values are obtained: (a) rotational jump model: $\text{R = 0.95}\text{A}\text{?}$, $\text{〈u}{}^2\text{〉}{}_{\mathrm{H}}\text{= 0.052}\text{A}\text{?}{}^2\text{, τ}{}^{\mathrm{?1}}\text{= 2}\text{meV}$, (b) rotational diffusion model: $\text{R = 0.99}\text{A}\text{?}\text{, 〈u}{}^2\text{〉}{}_{\mathrm{H}}\text{= 0.046}\text{A}\text{?}{}^2\text{, D}{}_{\mathrm{R}}\text{= 0.72}\text{meV}$.     

Reexamination of the I2 spectrum near the B(3Π0u+) state dissociation limit

The disagreement of Danyluk and King's (Chem. Phys.25, 343 (1977)) rotational constants for levels lying near the dissociation limit of B-state I₂ with the mechanical behavior predicted by near-dissociation theory is investigated. The discrepancies are shown to be much too large to be explained by either the neglect of centrifugal distortion effects in the original analysis or by rotational or spin-rotation coupling to a nearby repulsive 1_u state. These differences are therefore attributed to experimental error, a conclusion which is confirmed by more recent experimental results. A reanalysis of the best available data for levels near the dissociation limit of B-state I₂ then yields improved values for the B-state dissociation limit $\text{D}$ = 20 043.16 (±0.02) cm^?1 of the vibrational index at dissociation v_$\text{D}$ = 87.32 (±0.04) and of the long-range potential constant $\text{C}{}_5\text{= 2.88 (±0.03) × 10}{}^5\text{cm}{}^{\mathrm{?1}}\text{A}\text{?}{}^5$. This in turn implies a slightly improved ground-state dissociation energy of $\text{D}$₀ = 12 440.18 (±0.02) cm^?1.     

Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ and modified Ursell correlation functions $\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ of a classical system of particles with the two-body potential $\text{q(}\text{r}\text{) + γ}{}^{\mathrm{\nu }}\text{K(γ}\text{r}\text{)}$ are considered. The limiting values of the functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ),}\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ and $\text{γ(}{}^{\mathrm{1?k}}\text{)ν}\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γ^νK(γr). The direct correlation function c? (r, γ) is also considered and it is shown that for $\text{S}\text{≠ 0}$, $\text{lim}{}_{\mathrm{\gamma \to 0}}\text{γ}{}^{\mathrm{?\nu }}\text{c}\text{?}\text{(}\text{S}\text{γ}\text{, γ) = ?βK (}\text{S}\text{)}$, for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.     

Matthiessen's rule and its variation for cyclotron resonance width in two and three dimensions

Based on the proper connected diagram expansion, we calculated cyclotron resonance widths Γ_n associated with neighboring Landau states (n, n +1) for free electrons in interaction with more than one kind of impurities. In 3D usual Matthiessen's rule Γ_n=Γ⁽¹⁾_n+Γ⁽²⁾_n+…where Γ⁽ⁱ⁾_n represent widths calculated separately for each kind, is obtained. In 2D a new rule: is obtained.     

Microwave spectrum and structure of cyanamide: Semirigid bender treatment

An extensive study of the microwave spectrum of cyanamide has been undertaken, the analysis being based in part on semirigidbender calculations by the methods of Bunker and Szalay. Inversion lines of NH₂CN, $\text{K}{}_{\mathrm{?1}}\text{= 2}{}^{\mathrm{a}}\text{Q}$ branches and a number of vibrational satellites of the J = 2?1 transition were observed. A two-vibrational-state Hamiltonian was used to fit simultaneously the 0⁺ and 0^? microwave data and yielded rotational constants X, Y, Z, D_J, D_JK, d₁, H_JK as well as the inversion splitting and the μ_yz-connecting matrix element. Vibrational satellite data of seven isotopic species and infrared frequencies of NH₂CN were included in the semirigid bender calculations: The NCN spine is nonlinear by ca. 5° in the equilibrium structure of the molecule. Also, $\text{r}{}_{\mathrm{<mtext>NH</mtext>}}\text{A}\text{?}\text{= 0.9994 + 0.0144?}{}^2$; <HNH/2 = 60.39° ? 0.1134?²; $\text{r}{}_{\mathrm{<mtext>NC</mtext>}}\text{A}\text{?}\text{= 1.3301 + 0.0327?}{}^2$ (? is the inversion angle in rad); $\text{r}{}_{\mathrm{<mtext>CN</mtext>}}\text{= 1.1645}\text{A}\text{?}$ fixed. The inclusion of the NC bond flexing was necessary in order to reproduce the observed vibrational satellite patterns of NH₂CN, NHDCN, and ND₂CN. The barrier to inversion of the amino group is 510 ± 6 cm^?1 with minima at ±45.0 ±0.2°. The inversion dipole moment is 0.91 ± 0.02 Debye.     

Microwave spectrum and hydrogen bonding in 1-aziridineethanol

From its microwave spectrum, 1-aziridineethanol,

$\text{CH}{}_2\text{CH}{}_2\text{NCH}{}_2\text{CH}{}_2\text{OH}$

is found to have a gauche OCCN configuration, maintained by an OH?N-type hydrogen bond. The normal species rotational constants (MHz), A = 8528.87(25), B = 2069.74(2), and C = 2020.41(2), are consistent with a small (～6°) distortion from the “staggered” configuration about the central CN bond. Assuming a “normal” hydroxyl group, the hydroxy d₁ data suggest a decrease in the O?N distance upon bridge deuteration of ～0.003 Å. The dipole moment is 2.77(5) D, with 2.34(3), 1.45(6), and 0.3(3) D “a,” “b,” and “c” components, respectively.     

Some effects of spin-orbit interaction on rotational levels and rotational line intensities in vibrationally unexcited 2A, 2E, and 2F electronic states of open-shell XY4 molecules

Rotational energy levels in vibronic ground states of ²A, ²E, and ²F electronic states of open-shell XY₄ molecules, as well as rotational line intensities for allowed transitions between such states, are discussed, including the effects of spin-orbit interaction and tetrahedral splittings. Jahn-Teller effects are assumed to be small, and are only taken into account implicitly, through their contributions to various parameters in the effective Hamiltonian. Qualitative information is obtained by considering several limiting-case coupling schemes among the electron spin angular momentum S, the electron orbital angular momentum L, and the pure rotational angular momentum R. These limiting cases are similar in spirit to Hund's coupling cases in diatomic molecules, but differ sufficiently from the latter to make detailed correspondences unhelpful. Quantitative information on rotational energy levels and line intensities is obtained numerically by diagonalizing a Hamiltonian matrix set up in a basis set characterized by uncoupled moleculefixed projections of S, L, and the total angular momentum J, and symmetrized so that all basis set functions belong to a definite species in the subgroup D_2d of the true point group T_d. Hamiltonian matrix elements are determined by ladder operator techniques. Three sample calculated spectra, corresponding to $\text{p(}{}^2\text{F}{}_2\text{)-s(}{}^2\text{A}{}_1\text{)}$, $\text{d(}{}^2\text{E)-p(}{}^2\text{F}{}_2\text{)}$, and $\text{d(}{}^2\text{F}{}_2\text{)-p(}{}^2\text{F}{}_2\text{)}$ are presented. As one might expect, when the spin-orbit constant A is set equal to zero, then both qualitative and quantitative aspects of the rotational-electronic problem in open-shell XY₄ molecules can be mapped easily onto discussions of the rotation-vibration problem from the CH₄ literature.     

Concepts of convergence for a quantum law of large numbers

We introduce the notion of uniform convergence of an operator sequence relative to a state operator, and we compare it with the notion of almost everywhere convergence, introduced by Gudder. We show that a sequence (M_n) of mean-value operators

$\text{M}{}_{\mathrm{n}}\text{=}\text{1}\text{n}\text{∑n}\text{i=1}\text{1?…?A}{}_{\mathrm{i}}\text{?1?…,}$

defined on the infinite tensor product $\text{?}\text{i}\text{H}{}_{\mathrm{i}}$, converges relative uniformly to $\text{x}\text{1}$ if and only if it converges almost everywhere to $\text{x}\text{1}$.