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.     

The asymptotics of the gap in the Mathieu equation

We provide a simple proof that the kth gap, Δ_k, for the Mathieu operator $\text{?d}{}^2\text{dx}{}^2\text{+ 2κ}\text{cos}\text{(2x)}$ is $\text{Δ}{}_{\mathrm{k}}\text{= 8(}\text{κ}\text{4}\text{)}{}^{\mathrm{k}}\text{[(k ? 1)!]}{}^{\mathrm{?2}}\text{(1 + o(k}{}^{\mathrm{?2}}\text{))}$, a result obtained (up to the value of an integral) by Harrell. The key observation is that what is involved is tunneling in momentum space.     

Measurement of the background energy content of Nd-glass picosecond pulses with saturable absorbers

The background energy content of a mode-locked Nd-glass laser is determined with photodetectors and saturable absorbers. By comparing the signal height of a fast detector with the readout of an integrating energy meter the noise energy E_u2 outside the rise-time of the fast detector is measured. The background energy E_u1 within the rise-time is analysed by transmission measurements through two subsequent absorber cells. The obtained mean background to pulse energy (and intensity) ratios of ( and indicate a high degree of mode-locking.     

Polarization spectroscopy of the E0g+-B0u+ band system of Br2

The E-B (0_g⁺-0_u⁺) band system of Br₂ has been investigated at Doppler-limited resolution using polarization labeling spectroscopy. Merged E state data for the three naturally occurring isotopes in the range v_E = 0–16, expressed in terms of the constants for ⁷⁹Br₂, are (in cm^?1) Y_0,0 = 49 777.962(54), Y_1,0 = 150.834(22), Y_2,0 = ?0.4182(28), Y_3,0 = 6.6(11) × 10^?4, Y_0,1 = 4.1876(28) × 10^?2, Y_1,1 = ?1.607(16) × 10^?4, and Y_0,2 = 1.39(39) × 10^?8. The bond distance is $\text{r}{}_{\mathrm{<mtext>e</mtext>}}\text{= 3.194}\text{A}\text{?}$, and the diabatic dissociation energy to $\text{Br}{}^{\mathrm{+}}\text{(}{}^3\text{P}{}_2\text{) +}\text{Br}{}^{\mathrm{?}}\text{(}{}^1\text{S}{}_0\text{)}\text{is 34 700 cm}{}^{\mathrm{?1}}$.     

Interference effects in 12C(p, γ)13N and direct capture to unbound states

Excitation functions of the capture reaction ¹²C(p, γ₀)¹³N have been obtained at θ_γ = 0° and 90° and E_p = 150–2500 keV. The results can be explained if a direct radiative capture process, E1(s and d → p), to the ground state in ¹³N is included in the analysis in addition to the two well-known resonances in this beam energy range $\text{[E}{}_{\mathrm{p}}\text{= 457(}\text{1}\text{2}{}^{\mathrm{+}}\text{)}$ and 1699 $\text{(}\text{3}\text{2}{}^{\mathrm{?}}\text{) keV]}$. The direct capture component is enhanced through interference effects with the two resonance amplitudes. From the observed direct capture cross section, a spectroscopic factor of C²S(l = 1) = 0.49 ± 0.15 has been deduced for the $\text{1}\text{2}{}^{\mathrm{?}}$ ground state in ¹³N. Excitation functions for the reaction ${}^{12}\text{C(p,γ}{}_1\text{p}{}^1\text{)}{}^{12}\text{C}$ have been obtained at θ_γ = 0° and 90° and E_p = 610–2700 keV. Away from the 1699 keV resonance the capture γ-ray yield is dominated by the direct capture process E1 (p → s) to the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. Above E_p = 1 MeV, the observed excitation functions are well reproduced by the direct capture theory to unbound states (bremsstrahlung theory). Below E_p = 1 MeV, i.e., E_p → 457 keV, the theory diverges in contrast to observation. This discrepancy is well known in bremsstrahlung theory as the “infrared problem”. From the observed direct capture cross sections at $\text{E}{}_{\mathrm{p}}\text{? 1 MeV}$, a spectroscopic factor of C²S(l = 0) = 1.02 ± 0.15 has been found for the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. A search for direct capture transitions to the $\text{3512 (}\text{3}\text{2}{}^{\mathrm{?}}\text{)}$and $\text{3547 (}\text{5}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound states resulted in upper limits of C²S(l = 1) ≦ 0.5 and C²S(l = 2) ? 1.0, respectively. The results are compared with available stripping data as well as shell-model calculations. The astrophysical aspect of the ¹²C(p, γ₀)¹³N reaction also is discussed.     

Contributions of operators of twist two and higher in large n moments of structure functions

Though high twist terms are becoming important as x→1, or equivalently, in large n moments, their detection in this regime in deep inelastic lepton scattering needs special caution. The high order terms in the twist two component are strongly dependent on n; one finds that at $\text{Q}{}^2\text{?Q}{}^2{}_7\text{?Λ}{}^2\text{a}{}_{\mathrm{k}}\text{exp}\text{[α}{}_{\mathrm{k}}\text{(}\text{log}\text{n)}{}^{\mathrm{<mtext>2?1</mtext><mtext>k</mtext>}}\text{(}\text{1+b}{}_{\mathrm{k}}\text{log}\text{n}\text{)]}$ the perturbative expansion is invalid whereas higher twist terms are important at $\text{Q}{}^2\text{?Q}{}_1{}^2\text{= Λ}{}^2\text{nC}$. Since $\text{Q}{}_7{}^2$ grows very fast with n the necessary requirement for any deep inelastic phenomenological analysis, namely $\text{Q}{}_1{}^2\text{?Q}{}_7{}^2$, cannot hold for too large moments. The scheme dependence of a_k, α_k and b_k is also discussed.     

An experimental investigation of the radiative structure decay K+ → e+υγ

The decay K⁺ → e⁺υγ has been investigated. For the structure-dependent part with positive γ-helicity (SD₊) the branching ratio $\text{Γ(}\text{SD}{}_{\mathrm{+}}\text{)}\text{Γ(}\text{K}{}_{\mathrm{\mu 2}}\text{) = (2.33 ± 0.42) × 10}{}^{\mathrm{?5}}$ is obtained from 51 ± 3 events observed in the kinematical region E_e ? 235 MeV, E_γ > 48 MeV and θ_eγ > 140°. For the corresponding part with negative γ-helicity we obtain an upper limit Γ(SD_?)/Γ(SD₊) < 11 (90% CL) from the sample of electrons with energies 220 MeV ? E_e < 230 MeV and with no γ in the backward direction. This upper limit implies that the ratio of structure-dependent axial vector amplitudes lies outside the region $\text{?1.8 < a}{}_{\mathrm{<mtext>K</mtext>}}\text{υ}{}_{\mathrm{<mtext>K</mtext>}}\text{< ?0.54}$.For the decay $\text{K}{}^{\mathrm{+}}\text{→}\text{e}{}^{\mathrm{+}}\text{νν}\text{ν}$ the limit $\text{Γ(}\text{K}{}^{\mathrm{+}}\text{→}\text{e}{}^{\mathrm{+}}\text{νν}\text{ν}\text{)/Γ(}\text{K}{}_{\mathrm{e2}}\text{) < 3.8}$ 90% confidence level) was found.     

Coriolis intensity perturbations in the SO2 molecule: Experimental determination of the relative signs of (∂μ∂Qi)

The Coriolis interactions between ν₁ and ν₃, and between ν₂ and ν₃ in SO₂ have been analyzed to obtain the signs of the products $\text{ζ}{}_{3.1}{}^{\mathrm{c}}\text{(}\text{?μ}{}^{\mathrm{a}}\text{?Q}{}_3\text{)(}\text{?μ}{}^{\mathrm{b}}\text{?Q}{}_1\text{)}$ and $\text{ζ}{}_{3.2}{}^{\mathrm{c}}\text{(}\text{?μ}{}^{\mathrm{a}}\text{?Q}{}_3\text{)(}\text{?μ}{}^{\mathrm{b}}\text{?Q}{}_2\text{)}$. It has been found that both of the signs of these products are positive. Then, relative signs of ($\text{?μ}\text{?Q}{}_1$) have been determined using the calculated values of the Coriolis zeta constants for the present definition of the normal coordinates. The obtained sign combination of ($\text{?μ}\text{?Q}{}_{\mathrm{i}}$) is ±(+?+), which agrees with the one predicted by the molecular orbital calculations. Using the sign combination (+?+), the polar tensors of S and O atoms were also calculated.     

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.     

Uniqueness of statistics derived from moments of irradiance fluctuations in atmospheric optical propagation

This paper describes a new and very simple technique to examine the uniqueness criteria of some probability density functions (PDF) of irradiance fluctuations used to model the atmospheric optical propagation effects. By plotting the normalized function $\text{(}\text{m}{}^{\mathrm{<mtext>?1</mtext><mtext>(2k)</mtext>}}{}_{\mathrm{k}}\text{m}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}{}_1\text{)}$ derived from the kth moment m_k, the rate of growth of m_k is studied. Examples to assess the applicability and utility of the technique include scintillations due to speckles and turbulence as well as experimental results for laboratory-simulated and outdoor atmospheric turbulence.     

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.     

Holtzmark electric field distribution in a two-dimensional electron fluid

The Holtzmark distribution for the thermal electric field at a neutral point in a two-dimensional and uncorrelated electron fluid is given by $\text{β(1 + β}{}^2\text{)}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$, with $\text{β =}\text{E}\text{E}{}_0$, and E₀ = πnZ_pe in terms of the density n and perturber Z_pe.     

An effective procedure to determine corrugation functions from atomic beam-diffraction intensities

A computational method is described, which, starting from given difraction intensities, approaches effectively the best-fit corrugation function $\text{ζ(}\text{R}\text{)}$. Because of the approximations involved, the procedure works well for smooth corrugations with amplitudes not exceeding ～10% of the lattice constant. The method rests on two crucial observations: (i) With the full knowledge of the scattering amplitudes $\text{A}{}_{\mathrm{G}}\text{= ¦A}{}_{\mathrm{G}}\text{¦}\text{exp}\text{(}\text{i}\text{?}{}_{\mathrm{G}}\text{)}$ (absolute values plus phases), the corrugation function can be calculated to a high degree of accuracy from $\text{ζ(}\text{R}\text{) = (2}\text{i}\text{k}{}_{\mathrm{i}}\text{)}{}^{\mathrm{?1}}$ In $\text{¦?ΣA}{}_{\mathrm{G}}\text{exp}\text{(}\text{i}\text{G·R}\text{)¦}$ which is derived easily from the hard corrugated wall scattering (HCWS) equation by approximating $\text{k}{}_{\mathrm{G}}$ by $\text{?}\text{k}{}_{\mathrm{i}}$ ($\text{k}{}_{\mathrm{i}}$ and $\text{k}{}_{\mathrm{G}}$ being the wavevectors of the incoming and diffracted beams, respectively), (ii) With only the $\text{¦A}{}_{\mathrm{G}}\text{¦'s}$ (or intensities) known, approximate solutions of the HCWS equation can be obtained with a rough estimate of the relative phases of only a few intense diffraction beams; the estimate is readily performed by investigating systematically a coarse mesh of phases. In this way, approximate corrugations are found with which a full set of phases can be generated, which allows the calculation of an improved ζ(R); this step is repeated in a loop, until optimum agreement between calculated and given intensities is obtained. The effectiveness of the procedure is demonstrated for three one-dimensional model corrugations described by several Fourier coefficients. The method is finally applied to the case of H₂ diffraction from the quasi-one-dimensional adsorbate corrugation Ni(110) + H(1 × 2).     

Catalytic decomposition of DCOOD studied by reactive scattering of a modulated DCOOD nozzle beam by a polycrystalline platinum surface in ultra high vacuum

The catalytic decomposition of formic acid by a polycrystalline platinum surface was studied by use of modulated molecular beam techniques with mass spectrometric phasesensitive detection. Kinetic information about elementary surface reaction steps was obtained. The formation of CO₂ was found to be a monomolecular, whereas that of D₂ was a bimolecular process. The resulting reaction mechanism may be described as follows:

The rate constants in dependence from the surface temperature t₀ are $\text{η = 7.1 × 10}{}^3\text{exp}\text{(?}\text{9.9}\text{RT}{}_0\text{kcal/mole}\text{),}$ The sticking probability η is provided by the temperature dependence of the intensity of the nonreactive scattered formic acid molecules; the rate constants k_d1 and k_d2 are derived from the measured phase shift between reactive and nonreactive scattered particles. From the phase angle ?, the average surface residence time τ of the intermediates is computed: 3.7 ? τ_DCOO ? 0.41 msec (418 ? T₀ ? 505 K), 31.8 ? τ_D ? 11.6 msec (418 ? T₀ ? 460 K). The difference between τ_D and τ_DCOO is because of the different molecularity of desorption.     

Remeasurement of the low-energy cross section for the 15N(p, α0)12C reaction

The cross section for the ${}^{15}\text{N(p}\text{, α}{}_0\text{)}{}^{12}\text{C}$ reaction has been measured at θ_lab = 135° over the proton energy range 93 ≦ E_p ≦ 418 keV. The results are in good agreement with the less precise but much earlier measurements of Schardt, Fowler and Lauritsen (1952). An analysis of the present data in terms of a two-level calculation including the 338 keV (1^?) and 1028 keV (1^?) resonances determines a zero-energy intercept for the astrophysical S-factor of S(0) = 78 ± 6 MeV · b.     

Study of the photon spectrum in the decay KS0→π+π−γ

The branching ratio $\text{Λ(}\text{K}{}_{\mathrm{<mtext>S</mtext>}}{}^0\text{→π}{}^{\mathrm{+}}\text{π}{}^{\mathrm{?}}\text{γ)}\text{Λ(}\text{K}{}_{\mathrm{<mtext>S</mtext>}}{}^0\text{→π}{}^{\mathrm{+}}\text{π}{}^{\mathrm{?}}\text{)}$ has been determined to be (2.68±0.15)×10^?3 for photon energies $\text{E}{}_{\mathrm{\gamma }}{}^1$ greater than 50 MeV in the K_S⁰ rest frame. The decay K_S⁰→π⁺π^?γ is found to be dominated by the internal bremsstrahlung transition. The branching rato of a possible direct transition is found to be less than 0.06 × 10^?3 at 90% confidence level for $\text{E}{}_{\mathrm{\gamma }}{}^1\text{> 50}\text{MeV}$.     

The law of corresponding states for metals

Using the similarity of the effective potentials seen by ions in metals a reduced phonon equation of state is derived. It is shown that the melting point T_m(0) and the atomic volume Ω₀ at T = 0 K and at p = 0 are suitable macroscopic parameters for scaling ? and σ characterizing the interatomic potentials of metals having similar structures. The temperature and pressure dependence of thermodynamical quantities reduced with the above parameters are discussed and the results are compared with the experiment. It is shown that the pressure dependence of the reduced thermodynamic quantities can be described by the pressure dependence of the scaling parameters T_m(p) and Ω₀(p).The general form of the reduced equation of state (containing the electronic contributions as well) obtained gives that the reduced pressure is a universal function of the following reduced variables: the volume, temperature, de Broglie wavelength, Gibbs free energy of electrons $\text{3}\text{5}\text{zE}{}_{\mathrm{fo}}\text{?}$ (E_fo is the Fermi energy at T = 0 K) and depe of the valence z as well. It is shown that $\text{E}{}_{\mathrm{fo}}\text{?}$ is a function of $\text{Ω}{}_{\mathrm{o}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ and $\text{(}\text{E}{}_{\mathrm{fo}}\text{/?}\text{)Ω}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is approximately constant within the same sub-group of the periodic table.     

Measurement of Debye-Waller factors by Raman scattering from inter-term excitons and exciton-multiphonon states in FeF2

We have observed inter-term Raman scattering from ⁵T_2g→⁵E_g Frenkel excitons in antiferromagnetic FeF₂. It differs qualitatively from previously observed intra-term scattering in a sharply reduced zero-phonon cross section and the appearance of relatively strong exciton-phonon scattering. Since the Raman process is fully allowed, it is possible to measure excited state Debye-Waller factors, D, and we find $\text{D(Λ}{}_3{}^{\mathrm{+}}\text{+ Λ}{}_4{}^{\mathrm{+}}\text{,}{}^5\text{E}{}_{\mathrm{g}}\text{) = 0.04}$ and $\text{D(Λ}{}_1{}^{\mathrm{+}}\text{,}{}^5\text{E}{}_{\mathrm{g}}\text{) = 0.03}$.     

The giant Gamow-Teller resonance states

The mean energy of the giant Gamow-Teller resonance state (GTS) is studied, which is defined by the non-energy-weighted and the linearly energy-weighted sum of the strengths for Σ^A_i = 1^τi?σⁱ? Using Bohr and Mottelson's hamiltonian with the ξl· σ force, the difference between the mean energies of GTS and the isobaric analog state (IAS) is expressed as$\text{E}{}_{\mathrm{GTS}}\text{?E}{}_{\mathrm{IAS}}\text{,≈ 2〈π¦Σ}{}^{\mathrm{A}}{}_{\mathrm{i}}\text{=1ξ}{}_{\mathrm{i}}\text{l}{}_{\mathrm{i}}\text{· σ}{}_{\mathrm{i}}\text{¦π〉/ (3T}{}_0\text{-4(k}{}_{\mathrm{\tau }}\text{?k}{}_{\mathrm{\sigma \tau }}\text{) T}{}_0$. The observed energy systematics is well explained by k_τ?k_{στ≈ 4/A MeV}. The relationship between the mean energies and the excitation energies of the collective states in the random phase approximation for charge-exchange excitations is discussed in a simple model. From the excitation energy systematics of GTS, the values of k_στ and the Migdal parameter g′ are estimated to be about $\text{k}{}_{\mathrm{\sigma \tau }}\text{=}\text{(16–24)}\text{A}\text{MeV and}\text{g′ = 0.49–0.72}$, respectively.