On the difference between pp and pp topological cross sections up to 100 GeV/c

New 100 GeV/c$\text{p}$p data are used to find moments of the difference between the $\text{p}$p and pp topological cross sections. The mean multiplicity for annihilations at 100 GeV/c is estimated to be 9.06 ± 0.56, and the value of the quantity 〈n〉/D to be 2.75 ± 0.33. It is shown that $\text{R}{}_{\mathrm{n}}\text{= {σ}{}_{\mathrm{n}}\text{(}\text{p}\text{p) ? σ}{}_{\mathrm{n}}\text{(pp)}/σ}{}_{\mathrm{n}}\text{(pp)}$ appears at 100 GeV/c to have acquired an asymptotic form, R_n = s^?αβⁿ, with α and β constant.     

Temperature variation of the magnetic cluster structures in an iron-nickel invar alloy

Small-angle scattering of long wavelength neutrons $\text{(λ = 6.42}\text{A}\text{?}\text{)}$ from an Fe₆₅Ni₃₅ single crystal has been measured with the applied magnetic field (6.2 kG) parallel and perpendicular to the scattering vector $\text{K}$ of the elastic scattering over the temperature range from 25 to 422°C (T_c = 227°C). The scattering cross sections due to the longitudinal spin fluctuation have been analyzed by means of Guinier's approximation $\text{(dσ/dω)}{}_0\text{exp}\text{(?κ}{}^2\text{R}{}_{\mathrm{<mtext>g</mtext>}}{}^2\text{3}\text{)}$, where the forward cross section (dσ/dω)₀ is proportional to n, which is the number of atoms in a paramagnetic cluster, and R_g is the radius of gyration of the cluster. The empirical relation between n and R_g is = 0.298 × R_g^2.34 to be compared with that calculated for a simple spherical cluster model n = 1.274 R_g³.     

A mathematical interpretation of Dirac's formalism part b: generalized eigenfunctions in trajectory spaces

Starting with a Hilbert space $\text{L}{}_2\text{(}\text{R}\text{,μ)}$ we introduce the dense subspace $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ where $\text{R}$ is a positive self-adjoint Hilbert–Schmidt operator on $\text{L}$₂(R,μ). For the space $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ a measure-theoretical Sobolev lemma is proved. The results for the spaces of type $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ are applied to nuclear analyticity spaces $\text{S}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}\text{=}\text{?}\text{t>0}\text{e}{}^{\mathrm{<mtext>-t</mtext><mtext>A</mtext>}}\text{(X)}$, where e^?t$\text{A}$ is a Hilbert–Schmidt operator on the Hilbert space X for each t>0. We solve the so-called generalized eigenvalue problem for a general self-adjoint operator $\text{P}$ in X.     

Photon statistics in saturated m-photon amplification and attenuation of coherent light

The change of photon statistics of a nearly coherent single-mode light beam travelling through a saturable m-photon-amplifying or absorbing medium (m = 1, 2, …) is theoretically investigated. With the help of an approximation scheme valid for $\text{n}$ ? 1, Δn² ? $\text{n}$² ($\text{n}$ mean value and Δn² variance of the photon number) an analytical expression is derived for the quantity Δn²/$\text{n}$. The results holds for arbitrary values of the initial population inversion and of the degree of saturation. It turns out that for both amplification and attenuation processes after sufficiently strong interaction Δn²/$\text{n}$ tends to an asymptotic value.     

Evidence for assignment of 14.0 MeV state in 11B from 10B(n,n)10B

The differential cross section and polarization for neutrons scattered from ¹⁰B have been measured at E_n = 2.63 MeV (E_x = 13.85 MeV). The results of this experiment and other available neutron scattering data in the range 1 < E_n < 4 MeV are interpreted through a single-level R-matrix calculation over the region 12 < E_x < 15 MeV. Based on this analysis the most probable J^π assignment for the 14.0 MeV level in ¹¹B is $\text{11}\text{2}{}^{\mathrm{+}}$. The anomaly near E_x = 13.1 MeV can only be explained in terms of two overlapping levels having assignments of ($\text{5}\text{2}$, $\text{7}\text{2}$)^? and ($\text{3}\text{2}$, $\text{5}\text{2}$, $\text{7}\text{2}$)⁺.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.     

Phase transition in strongly inhomogeneous ferromagnets

A field theoretical model is proposed to describe the critical behaviour of a strongly inhomogeneous spin system with a position dependent concentration of magnetic atoms C(R) and magnetisation M(R). Assuming a finite number of $\text{n}$ Fouriermodes $\text{C}{}_{\mathrm{<mtext>Q</mtext>}}{}_{\mathrm{<mtext>v</mtext>}}\text{v}\text{= 1}$,..., $\text{n}$, to express C(R), the quenched randomness requires to interpret {$\text{Q}{}_{\mathrm{<mtext>v</mtext>}}\text{|}\text{Q}{}_{\mathrm{<mtext>v</mtext>}}\text{|}{}^2$} on a set of invariant or marginal lengths. As consequence, M(R) can be described by n Fourier-modes M_Qv, where $\text{n ?}\text{n}$. For short range spin-spin interaction, we find for strong inhomogeneity, i.e. large n, the critical exponent between those of the related homogeneous system and those of the spherical model.     

The change of conductivity of erbium films caused by the sorption of hydrogen and oxygen

Films of erbium were evaporated onto the inside wall of a pyrex glass reaction vessel at 1O^?9 torr. Their mean thickness (200–600 Å) was deduced from their mass and geometrical area. Estimates of their surface areas were made from the physical adsorption of krypton at 78 °K (BET method) giving a mean specific surface area of 71 m^?2 g^?1. The number of surface sites was calculated from a relationship given by Brennan et al.¹⁾. The sorption of hydrogen and oxygen was studied separately, by measuring the changes in the electrical resistance of the films as a function of the amount of pure gas admitted to the reaction vessel in measured doses ～ 10¹⁸ atoms per dose. The experiments were repeated at 295 °K, 200 °K, 130–140 °K for both gases (measurements at 78 °K were not reproducible, probably because of a magnetic phase change). Hydrogen at 295 °K [as reported²⁾] caused an initial increase ΔR in the original resistance R to a maximum $\text{ΔR}\text{R}$ ～ 20 %, which was followed by a decrease to $\text{ΔR}\text{R}\text{= 0}$ and then to $\text{ΔR}\text{R}\text{< 0.}$ A similar pattern of resistance changes was observed at 200°K and 130–140°K with smaller changes of $\text{ΔR}\text{R}$, i.e. 16 % and 8 % respectively, but no significant differance in surface (θ) and bulk atomic ratios at which the maxima occurred, e.g. $\text{ΔR}\text{R}$ was a maximum for 1.6?θ?1.9.     

Bosonic construction of the Lie algebras of some non-compact groups appearing in supergravity theories and their oscillator-like unitary representations

We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups $\text{SP}\text{(2n,}\text{R}$ and $\text{SO}\text{(2n)}{}^1$ from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n?4 and stops at n = 8 leading to the Lei algebras of SU(4) × SU(1, 1), SU(1, 1), SU(5, 1), SO(12)¹ and E₇(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E₇₍₇₎ and SO(12)¹ obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)¹.     

Spatial theory for algebras of unbounded operators

In this paper, the spatial theory of ¹-automorphisms is investigated in the context of algebras of unbounded operators. In particular, it is shown that ¹-automorphisms, satisfying some order relation, of the Op¹-algebra generated by the position and momentum operators q_j, p_j(j=1,...., n) on the Schwartz space $\text{I}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ are unitarily implemented.     

Polarization transfer in the 2H(d,n)3He reaction at θ = 0°

The vector polarization transfer coefficient K_y^y′ and the tensor analyzing power A_zz have been measured for the ²H($\text{d}$$\text{n}$)³He reaction at θ = 0° over an incident deuteron energy range from 1 to 15 MeV in 0.5 MeV steps. The results agree with the previous ²H($\text{d}$$\text{n}$)³He measurements of Simmons et al. and are nearly identical to the ²H($\text{d}$$\text{n}$)³H measurements of Clegg et al. in the region of overlap. The present results provide an accurate and complete set of the observables necessary to use the ²H($\text{d}$,$\text{n}$)³He reaction as a source of polarized neutrons.     

On a condition characterizing the Lorentz and Galilei groups

We prove by topological and Lie group theoretical methods that, in the class of the subgroups of $\text{GL}\text{(n,}\text{R}\text{)(n ≥ 3)}$, the n-dimensional Lorentz and Galilei groups are characterized by a condition which is the n-dimensional generalization of an “isotropy of space axiom” previously introduced. The inclusion of the n = 3 case implies a complete solution for the hitherto unsolved problem of determining the subgroups of GL(3, R) compatible with the three-dimensional isotropy condition.     

Quantum tunneling nucleation for superparamagnetic spin cluster reversals

Usual theories for time-dependent effects in superparamagnetism (over-turning of large spin clusters) assume a flip rate ^∝ exp (-U/kT). Instead, following Lifshitz and Kagan, we calculate the flip rate due to quantum-mechanical zero-point motion. The resulting “zero temperature” flip rate is R₀ exp (-cn) for a cluster with n spins. R₀ and c depend on domain wall energy and mass; typically, $\text{R}{}_0\text{= 10}{}^{13}\text{n}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{sec}{}^{-1}$ and c = 1.     

Change of photon statistics due to multi-photon absorption

Simple approximative equations governing the temporal behaviour of both the mean photon number, $\text{n}$, and its mean square deviation, Δn², in the process of k-photon absorption (k = 1, 2, …) are derived and solved for initial photon distributions characterized by $\text{n}$ ? 1 and Δn² ? $\text{n}$^-2. It is readily shown that such an initial distribution, in the course of attenuation, tends to a distribution for which $\text{Δn}{}^2\text{= k (2 k - 1)}{}^{-1}\text{n}$. Hence, for k > 1, the distribution is narrower than a Poisson distribution which means that photon antibunching occurs. The feasibility of a Hanbury Brown and Twiss type experiment allowing to detect this effect utilizing two-photon absorption is discussed, and an estimation of the required order of magnitude of the two-photon absorption cross-section is presented.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

On the spaces of positive and negative frequency solutions of the Klein-Gordon equation in curved space-times

In a space-time $\text{(V}{}_{\mathrm{n}}\text{×}\text{R}\text{;g)}$ with V_n closed (n ≠ 2) satisfying certain global conditions, we can write the Klein-Gordon equation, relative to a suitable class of atlases, in the evolution form du/dt = T^-1(t)u, on Sobolev spaces K^l(V_n) = H^l(V_n) × H^l?1(V_n), where the spectrum of T^-1(t) is imaginary. Following papers by T. Kato and J. Kisyński we prove the existence of the evolution operator for this equation. The space $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(V}{}_{\mathrm{n}}\text{)}$ has a natural strongly-symplectic structure ω. We determine the explicit form of complex-structure-positive operators of this structure. We prove that any two such operators, say J₁, J₂, are symplectically equivalent, (i.e. there is a symplectic transformation S such that J₂ = SJ₁S^-1). Spaces of positive and negative frequency solutions are then unique modulo symplectic equivalence. Each operator J determines a regular kernel on space-time which satisfies the properties of the kernel postulated by A. Lichnérowich in his program of quantization of fields in curved space-times. We carry out explicit calculations in the case of Robertson-Walker space-times. If an additional condition is satisfied by the given space-time, a unique complex-structure-positive operator can be selected in a natural way. This condition is satisfied by globally stationary space-times.     

New inversion techniques for the triton photoeffect

The photoeffect cross section for a particle in a model potential has been calculated analytically along with five energy-weighted moments. Two sets of orthonormalized polynomials R_n and S_n are developed with the weighting functions $\text{exp}\text{(?x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{exp}\text{(?x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. respectively. These polynomials are used to invert the calculated moments to get the cross section corresponding to 5, 4, 3 and 2 moments.The cross sections found by S-inversion are in good agreement with the model; those using R-inversion give somewhat poorer agreement.Clare and Lally calculated triton moments and used Laguerre inversion to find the triton cross section. We use their moments with R-inversion and S-inversion. Our resulting cross section agrees reasonably well with Gorbunov's and Faul's measurements of the ³He photoeffect. We find better agreement than that found by Clare and Lally.     

On universality for area-preserving maps of the plane

We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form

$\text{Λ}{}^{\mathrm{?1}}\text{°Φ°Φ°Λ = Φ}$

and

$\text{det}\text{DΦ = 1}$

, Φ:$\text{R}$²→$\text{R}$². In particular we argue that the scaling transformation Λ:$\text{R}$²→$\text{R}$² is conjugate to the transformation Λ₀:(x, y)→(λx, μy), with λ² ≠ μ, and in fact λ² >μ. We present some numerical evidence that

$\text{δ = 8.721}$

…,

$\text{?}\text{1}\text{λ}\text{= 4.018}$

…,

$\text{1}\text{μ}\text{= 16.36}$

…, where δ is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2^k described above.     

Energies of the n2S12 and n2D32,52 states of Cs

Using the technique of Doppler-free two-photon spectroscopy the energies of the n²S_{$\text{1}\text{2}$} (12 ? n ? 35) and n²D_{$\text{3}\text{2}$,$\text{5}\text{2}$} (11 ? n ? 48) states of CsI have been measured with a thermionic detector. The absorption spectrum of molecular iodine was used as a reference giving us a total estimated accuracy of about 2 × 10^-7. Taking also into account the energies of the 7–11 ²S_{$\text{1}\text{2}$} and 5 and 6 ²D_{$\text{3}\text{2}$,$\text{5}\text{2}$} states measured by other authors and using an extended Ritz-formula we found the ionization limit to be E_i = 31406.468 ± 0.006 cm^-1.