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.     

Low-energy πN partial waves,expansions of the πN invariant amplitudes about ν = 0, t = 0 and the value of the current algebra σ term

Using new experimental information on low-energy πN elastic scattering we recalculate both the low-energy partial waves and the expansion coefficients of $\text{C}\text{?}{}^{\mathrm{(\pm )}}\text{and}\text{B}\text{?}{}^{\mathrm{(\pm )}}$ about the point ν = 0, t = 0. We use the values to recalculate the πN σ term obtaining 66 ± 9 MeV. We comment on some other recent determinations of this quantity.     

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.     

Structure factor for a Bose fluid

The structure factor is calculated for a Bose fluid using an expression for S(k), which is approximately valid for all values of k. Marked fluctuations appear for $\text{k = 2.15}\text{A}\text{?}{}^{-1}$, near the roton dip in E_k, and at $\text{k = 3.65}\text{A}\text{?}{}^{-1}$ near the dip in the second branch of E_k. Our calculations are indirectly a theoretical justification for the existence of a second branch in E_k. The results presented here agree fairly well with the experimental values for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$ along with a dip at $\text{k = 3.67}\text{A}\text{?}{}^{-1}$ which has not been reported earlier. It also suggests the existence of two waves, phonons and rotons, for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$.     

Transport in two dimensions. I the self-diffusion coefficient

The Mori formalism is used to study generalized transport coefficients in two dimensions. All finite multilinear products of the single particle density and momentum density comprise the set of the variables in the calculation of the self-diffusion coefficient. A self-consistent equation, which is a non-linear integral equation, is obtained for the leading asymptotic behavior of the generalized self-diffusion coefficient. An asymptotic solution is presented which for small wavevector (k) and frequency (s) behaves like $\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(s + k}{}^2\text{D)}{}^{\mathrm{?1}}$, where D has the dimensions of a diffusion coefficient. The mean square and mean fourth displacements of a tagged particle are also calculated. The long time behavior of the momentum correlation function exhibits a tail of the form $\text{[t}\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(t)]}{}^{\mathrm{?1}}$ whose coefficient is dependent of the intermolecular potential.     

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.     

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 direct pair correlation function of liquid indium

The static structure factor S(k) of liquid indium has been measured accurately down to $\text{k = 0.8}\text{A}\text{?}{}^{\mathrm{?1}}$ using CuKα radiation with reflection geometry. The direct pair correlation function in k space is analyzed to demonstrate the utility of this technique in reducing errors in the resulting direct pair correlation function in configuration space.     

Electromagnetic waves near perfect conductors. II. Casimir effect

The Casimir free energy of the electromagnetic field in regions bounded by thin perfect conductors with arbitrary smooth shapes is studied. This free energy is expressed as a convergent multiple scattering expansion, in which the wave is damped between scatterings taking place on conductors. The result depends neither on the large box needed to render the spectrum discrete, nor on the shape of the high frequency cut-off corresponding to slightly imperfect conductivity; this establishes the universal character of quantum electro-magnetic forces on perfectly conducting thin objects.General expansions are derived for the Casimir free energy in the low and high temperature limits. They exhibit a dependence on the topology of the conducting surfaces. At low temperature, the Casimir entropy vanishes as T³ for simply connected conductors, and as T for multiply connected ones. At high temperature, the free energy has the semiclassical behavior ?$\text{C}$$\text{T}\text{log}\text{(}\text{T}\text{h}\text{?}\text{cQ}\text{)}$. The constant $\text{C}$ = $\text{(128φ)}{}^{\mathrm{?1}}\text{?dθ(}\text{3}\text{R}{}_1{}^2\text{+}\text{3}\text{R}{}_2{}^2\text{+}\text{3}\text{R}{}_1\text{R}{}_2\text{)?n}$ may be interpreted as the number of additional modes of finite frequency created by introducing the conducting surface. It depends on the topology of the surface through its genus n, and on its local curvature radii R₁, R₂. The average wave number Q depends on the shape of the surface and its size. The symmetry between low and high temperatures is recovered for parallel plates.While a plane foil is stable against Casimir stresses at zero temperature, such stresses would tend at finite temperature to create wrinkles of dimensions larger thn 2.9 $\text{h}\text{?}\text{c}\text{T}$. The same wrinkling effect exists for an arbitrary conductor at high enough temperature. The presence of a curved conducting surface transfers the free energy of radiation from the concave to the convex side at zero temperature, and does the converse at high enough temperatures. Wedges may be considered for slightly imperfect conductors; the electro-magnetic energy is lowered at finite temperature by creation of wedges, and also at zero temperature by creation of multiple wedges as in a honeycomb structure.The formalism is also applied to evaluate Van der Waals forces and torques at arbitrary temperature between remote conductors, and to show that the Casimir energy of a cylinder at zero temperature vanishes to lowest order in the multiple scattering expansion.Finally, the sphere is used as a test for studying convergence of the multiple scattering expansions. Using radial and transverse combinations of vector spherical harmonics yields a simple expression for the electromagnetic Green functions. This expression is recovered by summation of the multiple scattering expansion; the convergence domain, in the plane kR = x + iy, of the expansion includes a neighborhood of the origin, as well as the whole region $\text{y >0.0348 |x|}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$. The numerical convergence of the Casimir free energy is rapid, the lowest order term already yields a result with less than 7% error.     

Inner Coulomb corrections to pion-nucleus scattering

A semiclassical model for inner Coulomb corrections to pion-nucleus scattering in the Δ-resonance region is suggested. Its main consequence is a change of the nuclear radius R, giving $\text{R(π}{}^{\mathrm{?}}\text{)}\text{R(π}{}^{\mathrm{+}}\text{)}\text{= 1+2Z}\text{αE}\text{k}{}^2\text{R}$. The Glauber model can be extended to include these effects.     

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.     

Inclusive production of K̄1(890) and K̄1(1420) in K̄−p interactions at 10 and 16 GeV/c

The inclusive production of K?¹(890) and K?¹(1420) is studied in K?^?p interactions at 10 and 16 GeV/c. At 10 GeV/c an enhancement in the ($\text{K}\text{?}{}^0\text{π}{}^{\mathrm{?}}$) mass distribution is found at 1.74 GeV, but no clear signal is seen at 16 GeV/c. The fraction of $\text{K}{}^0\text{'}\text{s}$ coming from decay of the $\text{K}{}^1\text{(890)}\text{or}\text{K}{}^1\text{(1420)}$ is large, being (50 ± 6)% and (45 ± 5)% at 10 and 16 GeV/c, respectively. The inclusive cross sections for K^1?(890) and $\text{K}{}^{10}\text{(890)}$ production are almost constant with energy from 8 to 32 GeV/c with values of 3.5 and 3.3 mb, respectively. The $\text{K}{}^1\text{(890)}$ production cross section is studied as a function of transverse and longitudinal variables and found to derive mainly from fragmentation of the incident K^? meson. The spectra of $\text{K}{}^0\text{'}\text{s}$ resulting from the decay of $\text{K}{}^1\text{(890)}$ are studied.     

On the mobility of very pure semiconductors at very low temperatures

The mobility μ of a very pure semiconductor at very low temperatures is investigated in terms of a model where electrons are scattered by charged impurities distributed uniformly in space, and the electron-electron interaction is taken into account by the Debye-Hueckel screening in the interaction potential. The equation for the current relaxation rate Γ, derived previously by the proper connected diagram expansion, incorporates the quasi-particle effect in a self-consistent manner. The solution of this equation at high carrier concentrations n yields the so-called Brooks-Herring formula. At lower concentrations, the solution deviates significantly from the latter. The solution is in general smaller than the standard expression for the rate based on the Boltzmann equation; and this is consistent with the existing conductivity data available. At the very low concentrations e.g. n = n₃ = 10¹³cm^?3 or lower for Ge, the mobility calculated is inversely proportional to the square-root of the impurity concentration n_s, and has a $\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$-dependence (T: temperature).

$\text{μ = 0.3597\&z.xl;h}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{k(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^1{}^{\mathrm{<mtext>?3</mtext><mtext>4</mtext>}}$

, where k is the dielectric constant. The conductivity data directly comparable with this formula are not available at present. However, the quasi-particle effect which led to this peculiar concentration-dependence should also show itself in the cyclotron resonance width; there, experiment and theory both show the ${}_{\mathrm{n}}{}_{\mathrm{s}}$-dependence for very pure semiconductors.     

The bound states of weakly coupled long-range one-dimensional quantum hamiltonians

We study the small λ behavior of the ground state energy, E(λ), of the Hamiltonian $\text{?(}\text{d}{}^2\text{dx}{}^2\text{) + λV(x)}$. In particular, if V(x) ～ ?ax^?2 at infinity and if 69-1, we prove that $\text{(?E(λ))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= ?[}\text{1}\text{2}\text{λ + aλ}{}^2\text{lnλ] ∫ dxV(x) + O(λ}{}^2\text{)}$.     

Vector and tensor meson production in K−p interactions at 32 GeV/c

Inclusive production of vector and tensor mesons is studied in a K^?p experiment at 32 GeV/c in the MIRABELLE bubble chamber. The $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>0</mtext>}}\text{(890)}$, ?⁰ and ω cross sections are comparable, about 4 mb each. The $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>0</mtext>}}\text{(1420}$ and cross sections are also comparable, about 1 mb each. The $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>o</mtext><mtext>?</mtext><mtext>+</mtext>}}$(890), Φ, $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>o</mtext><mtext>?</mtext><mtext>?</mtext>}}$(1420) and f cross sections beam fragmentation; ? production is almost forward-backward symmetric in the c.m.s. The p_T production slopes of $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>o</mtext><mtext>?</mtext><mtext>?</mtext>}}\text{(890)}$ and ? are similar, the Φ slope is shallower. Vector and tensor mesons alone are responsible for ?50% (?60%) of final-state pions     

Nucleon-nucleus optical potential in a relativistic theory of nuclear matter

From a relativistic model of nuclear matter the optical potentials for nucleon scattering from ⁴⁰Ca and ⁹⁰Zr are obtained. These potentials are derived from the properties of the target nucleus and are essentially universal. This means that the integrated strength of the optical potential $\text{J}{}_{\mathrm{A}}\text{= (}\text{1}\text{A}\text{) ∫}\text{d}{}^3\text{r U}{}_{\mathrm{<mtext>op</mtext>}}\text{(r)}$ is very weakly dependent on A. The optical potential for antiparticle-nucleus scattering is also computed.     

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H₀ and H = H₀ + V(r) acting in L²(R³, dx) with V(r) a radial potential satisfying certain technical conditions, and for ? a real function on R with ?′ > 0 except on a discrete set, we prove that the Moller wave operators

exist and are independent of ?. The scattering operator

$\text{S = (Ω}{}^{\mathrm{+}}\text{)1Ω}{}^{\mathrm{?}}$

is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. $\text{V(r) =}\text{sin}\text{r}\text{r}$ and many others.     

Strong vacuum polarization and self-damping of e+e− annihilation

If the hadronic contribution to vacuum polarization, which is proportional to the ration of e⁺e^? annihilation into hadrons to that into μ⁺μ^? rises asymptotically as a power $\text{β >}\text{1}\text{2}$ of the squared four-momentum, the ratio $\text{σ}\text{(e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→hadrons)}\text{σ}$ point $\text{(}{}^{\mathrm{\sigma }}\text{point =}\text{(4πα}{}^2\text{)}\text{(3s))}$ is bound to be less than $\text{?}\text{(3}\text{tg}\text{πβ)}\text{(8α)}$ and this limit is approached sooner for higher β. Other models of vacuum polarizations are also considered together with their possible origin and implications.     

On the mobility of a very pure semiconductor including the low impurity concentration limit

The low temperature mobility μ limited by charged impurities is calculated by solving the equation for the relaxation rate previously derived. The calculated μ behaves like $\text{μ = 2.03 κ}{}^2\text{(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{?3}}\text{z}{}^{\mathrm{?2}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{?1}}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>?1</mtext><mtext>2</mtext>}}$ In $\text{[38.2κ}{}^2\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{/z}{}^2\text{e}{}^4\text{h}\text{?}\text{n}{}_{\mathrm{s}}\text{]}$ for lowest concentrations n_s<10¹¹cm^?3 for Ge and

$\text{μ = 0.360}\text{h}\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{κ(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>?3</mtext><mtext>4</mtext>}}$

for intermediate concentrations n_s ～ 10¹²?10¹⁴cm^?3.     

On the continuity of generalized eigenfunctions of the Schrödinger operator

We prove that the generalized eigenfunctions of the Schrödinger operator are continuous for potentials obeing the following assumptions: V=V⁺?V^?,V^±≥0,$\text{V}{}^{\mathrm{+}}\text{∈L}{}^{\mathrm{p}}{}_{\mathrm{<mtext>loc</mtext>}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$, $\text{V}{}^{\mathrm{?}}\text{∈L}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$,$\text{p >}\text{l}\text{2}$.