Interpreting data from polarized proton beams

The reactions pp → pp and pp → Δ⁺⁺n with polarized beam and/or polarized target are currently under investigation at the Argonne ZGS. We discuss how to interpret various measured quantities in terms of amplitudes whose behavior is familiar (as functions of s, t). For pp total cross sections and elastic scattering, Argonne measurements will yield Im ?₂ (s,t = 0) and the rather complicated combination $\text{2|?}{}_5\text{|}{}^2\text{+}\text{Re}\text{(?}{}_1\text{?}{}_2{}^1\text{? ?}{}_3\text{?}{}_4{}^1\text{)}$, where ?_i (i = 1, … 5) are conventional s-channel helicity amplitudes. The forward direction (t = 0) is of special interest. We find that for both pp → pp and pp → Δ⁺⁺n, polarized beam — polarized target experiments plus the rather general (testable) assumption that amplitudes with the same s-channel helicity flip quantum numbers are proportional, are sufficient to fully determine all non-vanishing amplitudes at t = 0. Numerical estimates of some observables, based on calculations in a specific model, are also given.     

Dispersion sum rules for the amplitudes of radiative transitions in quarkonium

A dispersion approach is suggested, based on the asymptotic freedom in QCD and on analyticity, which leads to sum rules for the amplitudes of radiative transitions in quarkonium $\text{(}\text{c}\text{c}\text{,}\text{b}\text{b}\text{, …)}$ and allows one to estimate these amplitudes in a model-independent way. As an example such sum rules are derived for the ${}^3\text{P}{}_0\text{?}{}^3\text{S}{}_1$ and ${}^3\text{S}{}_1\text{?}{}^1\text{S}{}_0$ transitions. For charmonium they agree with the available data and with the predictions of nonrelativistic models.     

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.     

Eigenfunction expansions of operators admitting separation of an infinite number of variables

Let A_k be a self-adjoint operator in a Hilbert space H_k (k = 1, 2, …) and let $\text{L}$ be an operator of the form $\text{L}\text{= A}{}_{\mathrm{r}}\text{? 1 ? 1 ? … + 1 ? A}{}_2\text{? 1 ? 1 ? … + …}$ acting in the infinite tensor product $\text{?}\text{k=1}\text{∞}\text{H}{}_{\mathrm{k}}$. We construct the spectral theory of these operators. In particular, the expansion is generalized eigenvectors of this operator is constructed using the eigenvectors of the operators A_k.     

Nonorthogonality corrections in the method of correlated basis functions

A set of normalized linearly independent basis functions φ₁, φ₂, …, φ_j, … generates matrix representatives $\text{H}$ and $\text{N}$ of the Hamiltonian operator and the identity. An orthonormal basis $\text{φ}{}_1$, $\text{φ}{}_2$, …, $\text{φ}{}_{\mathrm{j}}$, … generated by a Löwdin transformation is characterized by the distance in Hilbert space between $\text{φ}{}_{\mathrm{j}}$ and φ_j. The choice of positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ minimizes these distances and maximizes the diagonal elements of $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Again for positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and a finite basis, 1 ? j ? p, the analysis yields a general theorem on Trace $\text{N}$${}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ (? p for all positive and negative integral values of n except n = ?1 and ? p for n = ?1).Sufficient conditions are determined which permit the application of the binomial theorem to the evaluation of the transform of $\text{H}$. Approximate formulas for the energy eigenvalues through third order in nondiagonal matrix elements are presented in a compact form containing characteristic nonorthogonality corrections depending on the exterior or interior location of the matrix element in the perturbation formulas.     

Quark masses

In quark gluon theory with very small bare masses, -$\text{ψ}$$\text{M}$ψ, spontaneous breakdown of chiral symmetry generates sizable masses M_u, M_d, M_s, … We find ($\text{M}{}_{\mathrm{<mtext>u</mtext>}}\text{+ M}{}_{\mathrm{<mtext>d</mtext>}}\text{) /2 ≈ m}\text{p}\text{/ √6 ≈ 312}\text{MeV}\text{,}\text{and}\text{M}{}_{\mathrm{<mtext>s</mtext>}}\text{≈ 432}\text{MeV}$. Scalar densities have well determined non-zero vaccum expectations $\text{〈0|u}{}^{\mathrm{a}}\text{|0〉 ≡ 〈0|ψ(x) (λ}{}^{\mathrm{a}}\text{/2)ψ(x)/0〉 ≈ ?}{}_{\mathrm{\pi }}{}^2\text{M}{}^{\mathrm{a}}\text{,}\text{i.e}\text{〈0? u}{}^{\mathrm{o}}\text{/vb0〉 ≈ 8 × 10}{}^{\mathrm{?3}}\text{(}\text{GeV}\text{)}{}^3$ at an SU(3) breaking of the vacuum c′ ≡ 〈0|u⁸|〉/〈0|u^o|0〉 ≈ ? 16%     

Band contour analysis of compounds with pure isotopes: The E fundamentals of HC35Cl3

The vibration-rotation transitions for v = 1 ← 0 of NO (${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$) have been studied by using the technique of laser magnetic resonance spectroscopy. Five magnetic resonance lines are observed with three CO laser lines in the range from 1859 to 1886 cm^?1. From these, three zero-field transition frequencies, v = 1 ← 0; $\text{R(}\text{3}\text{2}\text{)}$, $\text{P(}\text{7}\text{2}\text{)}$, and $\text{P(}\text{9}\text{2}\text{)}$ are obtained with an accuracy of ±0.0007 cm^?1. The molecular constants which have been determined by borrowing centrifugal constants from a previous infrared work are $\text{B}{}_{02}{}^1\text{= 1.72004 ± 0.00006}\text{cm}{}^{\mathrm{?1}}$, $\text{B}{}_{12}{}^1\text{= 1.70212 ± 0.00010}\text{cm}{}^{\mathrm{?1}}$, and $\text{G(v = 1) ? G(v = 0) (}\text{for}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{) = 1875.8470 ± 0.0007}\text{cm}{}^{\mathrm{?1}}$.     

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 $\text{x}\text{1}$.     

Some concrete realizations of bratteli-diagrams

In this paper we give a concrete realization for some examples of A.F.-algebras, such that the associated Bratteli-diagrams are those of . We also give a realization for any quotient of 2 ideals of the current algebra and obtain direct sums of C¹-algebras of the given type.     

Combination tones in the response of single degree of freedom systems with quadratic and cubic non-linearities

The method of multiple scales is used to analyze the response of a single-degree-of-freedom system to either the combination resonance of the additive type $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ or the combination resonance of the difference type $\text{Ω}{}_2\text{? Ω}{}_1\text{≈ ω}{}_0$, where $\text{Ω}{}_1$ and $\text{Ω}{}_2$ are the frequencies of the excitation and ω₀ is the linear undamped natural frequency of the system. To the second approximation, the combination resonance of the additive type has three effects on the steady state response. First, it produces terms having the frequencies $\text{Ω}{}_1\text{, Ω}{}_2$ and $\text{Ω}{}_2\text{+ Ω}{}_1$ at first order and terms having the frequencies 0, $\text{2Ω}{}_1\text{, 2Ω}{}_2\text{, Ω}{}_2\text{? Ω}{}_1$, $\text{2(Ω}{}_2\text{+ Ω}{}_1\text{), Ω}{}_2\text{+ 2Ω}{}_1$ and $\text{2Ω}{}_2\text{+ Ω}{}_1$ at second order. Second, it produces a shift in the natural frequency of the system. Third, it produces a virtual primary-resonant excitation having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ that makes the component having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1$ be of first rather than second order. Similar effects are produced by a combination resonance of the difference type or a superharmonic resonance of order two.     

Hyperfine structure of CH3OH

Hyperfine structure of the (0, 0, 1) - (1, 0, 1) transition of methanol has been investigated by beam absorption and of the (J, 1, 3?) → (J, 1, 3+) transitions for J = 2, 3, and 6 by beam-maser spectroscopy. The best-fit results for the spin-rotation and spin-spin coupling constants $\text{C}{}_{\mathrm{JK\tau \pm }}{}^{\mathrm{(i)}}$ and $\text{D}{}_{\mathrm{JK\tau \pm }}{}^{\mathrm{(i)}}$, respectively, are in kHz¹: $\text{C}{}_{101}{}^{\mathrm{(1)}}\text{= 2.4(10)}$, $\text{C}{}_{101}{}^{\mathrm{(2)}}\text{= ?0.6(10)}$, $\text{D}{}_{101}{}^{\mathrm{(1)}}\text{= ?13.8(9)}$, $\text{D}{}_{101}{}^{\mathrm{(2)}}\text{= 7.0(9)}$, $\text{C}{}_{\mathrm{213?}}{}^{\mathrm{(1)}}\text{= ?5.0(10)}$, $\text{C}{}_{\mathrm{213?}}{}^{\mathrm{(2)}}\text{= ?5.5(10)}$ and ($\text{C}{}_{\mathrm{J13?}}{}^{\mathrm{(2)}}\text{- C}{}_{\mathrm{J13+}}{}^{\mathrm{(2)}}\text{) = 0.98(9)}$.     

Oscillator strengths in the Cameron system of carbon monoxide

Relative oscillator strengths in the Cameron system of CO(a³Π ← X¹Σ) have been observed in absorption for six bands (υ′ = 0–5, υ″ = 0) with the result, normalized to the absolute (0, 0) band measurement of Hasson and Nicholls, $\text{?}{}_{00}\text{= (1.62±0.07) × 10}{}^{\mathrm{?7}}$, $\text{?}{}_{10}\text{= (1.96±0.09) × 10}{}^{\mathrm{?7}}$, $\text{?}{}_{20}\text{= (1.41±0.04) × 10}{}^{\mathrm{?7}}$, $\text{?}{}_{\mathrm{3\; 0}}\text{= (0.72±0.03) × 10}{}^{\mathrm{?7}}$, $\text{?}{}_{40}\text{= (0.31±0.02) × 10}{}^{\mathrm{?7}}$, $\text{?}{}_{50}\text{= (0.14±0.01) × 10}{}^{\mathrm{?7}}$. The density of CO was modulated with a motor-driven vacuum valve and synchronous fluctuations (?1 per cent) in the transmitted intensity detected with a lock-in amplifier. Peak pressure in the 21 cm absorption cell was approximately 10 torr. A curve of growth analysis was used to correct saturation effects by less than 3 per cent.     

Neutral mesons with heavy quarks and mixing angles in the six-quark model

In the framework of the six-quark model, based on the gauge group SU(2)_L × U(1) we have obtained expressions for the amplitudes of $\text{K}{}^0\text{?}\text{K}{}^0\text{,}\text{D}{}^0\text{?}\text{D}{}^0\text{,}\text{B}{}^0\text{?}\text{B}\text{B}{}_{\mathrm{S}}{}^0\text{?}\text{B}{}_{\mathrm{S}}{}^0\text{,}\text{T}{}^0\text{?}\text{T}{}^0\text{and}\text{T}{}_{\mathrm{c}}{}^0\text{?}\text{T}{}_{\mathrm{C}}{}^0$ transitions at final 4-momenta of external lines Feynman box diagrams. We have estimated the correctness of using the approximation of extremely small 4-,omenta of valence quarks, composing K⁰-like mesons. The constraints for the parameters of the unitary matrix of weak charged currents have been found at arbitrary value of the t-quark mass. Estimates of numerical values for the mixing parameters and for the parameters of CP violation for neutral systems similar to $\text{K}{}^0\text{?}\text{K}{}^0$ are given.     

The evolution of a cluster of particles and the invariants of a time-dependent oscillator

A simple physical realization is found for the time-dependent harmonic oscillator equation $\text{q}\text{?}\text{+ φ(t)q = 0}$, for its auxiliary equation $\text{g}\text{?}\text{9 + φ(t)? = h}{}^2\text{?}{}^{\mathrm{?3}}$, and for the Ermakov invariant. Through generalization of the model, a set of invariants is obtained for a system of equations of the form .     

The ã3A2 ← X̃1A1 absorption spectrum of thioformaldehyde: A vibrational and rotational analysis

The results of a vibrational and rotational analysis of the banded $\text{a}\text{?}{}^3\text{A}{}_2\text{←}\text{X}\text{?}{}^1\text{A}{}_1$ transition in $\text{CH}{}_2\text{S}\text{CD}{}_2\text{S}$ are presented. Only three of the six vibrational modes are active in the spectrum with $\text{ν′}{}_2\text{=}\text{1320}\text{1012}$, $\text{ν′}{}_3\text{=}\text{859}\text{798}$, and $\text{2ν′}{}_4\text{=}\text{711}\text{516}\text{cm}{}^{\mathrm{?1}}$. The spin forbidden transition gains intensity primarily by a mixing of the ${}^1\text{A}{}_1\text{(π}{}^1\text{,π)}$ and ${}^3\text{A}{}_2\text{(π}{}^1\text{,n)}$ states. This is confirmed by a rotational analysis of the 0₀⁰ band of both isotopes. The rotational analysis shows that the coupling in the $\text{a}\text{?}{}^3\text{A}{}_2$ state is near Hund's case b and that the spin constants are nearly 10 times greater than those observed for CH₂O. A $\text{CNDO}\text{2}$ calculation shows that this difference is due to the greater spin orbit coupling of S in CH₂S and to the smaller energy differences between the $\text{B}\text{?}{}^1\text{A}{}_1\text{(π}{}^1\text{,π)}$, $\text{b}\text{?}{}^3\text{A}{}_1\text{(π}{}^1\text{,π)}$, $\text{X}\text{?}{}^1\text{A}{}_1$, and the $\text{a}\text{?}{}^3\text{A}{}_2\text{(π}{}^1\text{,n)}$ states. The r₀ structure calculated from the rotational constants is $\text{r}{}_{\mathrm{<mtext>CS</mtext>}}\text{= 1.683}\text{A}\text{?}$, $\text{r}{}_{\mathrm{<mtext>CH</mtext>}}\text{= 1.082}\text{A}\text{?}$, β_HCH = 119.6°, and α (out of plane) = 16.0°. A simultaneous fit of the vibrational levels in ν′₄ of CH₂S and CD₂S to a double minimum potential function yielded a barrier to molecular inversion of 13 cm^?1 and an equilibrium out-of-plane angle of 15°.     

Nonlinear temperature and velocity distributions in the two-dimensional Bénard problem

The nonlinear temperature and velocity profiles of a two-dimensional Boussinesq fluid heated from below are shown to be the sum or terms with periodicity $\text{Λ}\text{cr}\text{n}$, with n = 1, 2, 3; their amplitudes are a function of $\text{? =}\text{(R}{}_{\mathrm{<mtext>a</mtext>}}\text{? R}{}^{\mathrm{<mtext>cr</mtext>}}{}_{\mathrm{<mtext>a</mtext>}}\text{)}\text{R}{}^{\mathrm{<mtext>cr</mtext>}}{}_{\mathrm{<mtext>a</mtext>}}$. The results are in agreement with the observations of Bergé and Dubois.     

Rotation-vibration analysis of the Coriolis-coupled ν5g, ν6g, ν8g and ν9 bands of H2CCO

Medium resolution infrared grating spectra of gaseous ketene, H₂CCO were recorded between 1000 and 400 cm^?1, both at instrument temperature (40°C) and with cooling (?40°C). Interferometric Fourier spectra were also measured at ?70°C with resolution 0.22 cm^?1 between 450 and 330 cm^?1. The K structure of the fundamentals ν₅, ν₆, ν₈, and ν₉ was assigned. These fundamentals are coupled by a-axis Coriolis interactions. These couplings were analysed on the symmetric top basis for setting up the perturbation matrix and by utilizing the K-dependent Coriolis shifts of levels. A preliminary analysis of the Coriolis intensity anomalies was also undertaken.Band center values from combination differences are $\text{ν}{}_5{}^0\text{= 587.30 (27)}$ and $\text{ν}{}_6{}^0\text{= 528.36 (39)}\text{cm}{}^{\mathrm{?1}}$. Synthetic spectra indicate the band origins of ν₈ and ν₉ to be close to 977.8 and 439.0 cm^?1, respectively. Estimates of Coriolis coupling constants obtained from synthetic spectra are $\text{ζ}{}_{58}{}^{\mathrm{a}}\text{= + 0.33 (5)}$, $\text{ζ}{}_{68}{}^{\mathrm{a}}\text{= + 0.714 (20)}$, $\text{ζ}{}_{59}{}^{\mathrm{a}}\text{= ? 0.774 (20)}$, and $\text{ζ}{}_{69}{}^{\mathrm{a}}\text{= ? 0.30 (2)}$. Approximate ratios of unperturbed vibrational transition moments obtained from spectral simulations are $\text{M}{}_8{}^0\text{:±iM}{}_5{}^0\text{:±iM}{}_6{}^0\text{:M}{}_9{}^0\text{≈ +2:?9:+10:+0.5}$.     

Infrared diode laser spectroscopy of the CF radical

The fundamental bands of the CF radical in the $\text{X}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{X}{}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ electronic states were observed by using an infrared tunable diode laser as a source. Zeeman modulation could be used in detecting lines not only in the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ state, but also in ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, because the CF radical deviates considerably from Hund's case (a). From the least-squares analysis of the observed spectra, the following molecular constants were obtained: B_e = 1.416 704 (37) cm^?1, α_e = 0.018 419 (50) cm^?1, $\text{r}{}_{\mathrm{e}}\text{= 1.271 977 (17)}\text{A}\text{?}$, D_e = 6.68 (15) × 10^?6cm^?1, p₀ = 0.008 580 (21) cm^?1, p₁ = 0.008 52 (11) cm^?1, and $\text{ν}{}_0\text{= 1286.1281 (5)}\text{cm}{}^{\mathrm{?1}}$, with three standard errors in parentheses.     

KL0-KS0 transmission regeneration on deuterons and neutrons in a momentum range of 10–50 GeV/c

The energy dependence of the K_L⁰-K_S⁰ transmission regeneration amplitudes on deuterons and neutrons in the momentum region 10–50 GeV/c is determined. The moduli of the modified transmission amplitudes are momentum dependent. These dependences are fitted by the expression A_jp^?nj, where A_j and n_j (j = d, n) are constants:

$\text{A}{}_{\mathrm{<mtext>d</mtext>}}\text{=2.88 ±0.04}\text{mb}\text{, n}{}_{\mathrm{<mtext>d</mtext>}}\text{=0.546±0.030,}\text{for deuterons}\text{,}$

$\text{A}{}_{\mathrm{<mtext>n</mtext>}}\text{=1.97 ±0.14}\text{mb}\text{, n}{}_{\mathrm{<mtext>n</mtext>}}\text{=0.530±0.019,}\text{for neutrons}\text{,}$

The amplitude phases do not depend on the kaon momentum and are equal to $\text{?}{}_{\mathrm{<mtext>d</mtext>}}\text{= (?130.9 ± 2.7)°}$?_n = (?132.3 ± 1.7)°. The mean value of the ratio of the total cross-section differences for K⁰ and K⁰ interactions with neutrons and protons is determined. The residues of the partial ω and ? amplitudes, which contribute to the kaon-nucleon interaction amplitudes, are also obtained.     

Thermal expansion of zirconium—II

The generalised Gruneisen parameters γ′ = ?? log ω/??′ and γ″ = ?? log ω/??″ have been calculated for various normal mode frequencies using the force model for Zirconium obtained on Keating's approach. There is a general agreement between the normalised frequency distribution curve of Zirconium obtained on the present model and that of Bezdek et al. The temperature dependence of the effective Gruneisen functions $\text{γ}\text{?}{}_{\mathrm{\perp }}\text{(T)}$ and $\text{γ}\text{?}{}_{\mathrm{|}}\text{(T)}$ has been calculated using the procedure of Blackman. The high temperature limits of $\text{γ}\text{?}{}_{\mathrm{\perp }}\text{(T)}$, $\text{γ}\text{?}{}_{\mathrm{|}}\text{(T)}$ and $\text{γ}\text{?}{}_{\mathrm{v}}$ are in good agreement with those obtained by Goldak et al. from an analysis of the thermal expansion data of Zirconium.