Linear relativistic hamiltonians

A 2(2J + 1)-component relativistic Hamiltonian H that describes free particles of mass m and spin J is said to be linear if it has the form $\text{H = h}\text{x}\text{?}\text{·}\text{p}\text{+ gm}$, where $\text{x}\text{\_.}\text{= i[H,}\text{x}\text{]}{}_{\mathrm{?}}$, h is a numerical factor, and g commutes with x and p. All such Hamiltonians are found, provided that the metric is either the unit matrix or ?₃ and provided that the theory is invariant under the discrete symmetries. If the operator Γ in the generator $\text{K}\text{=}\text{1}\text{2}\text{[}\text{x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ of Lorentz boosts is required to be local, there are only two possibilities; either Γ = 0, which generalizes the Dirac spin-$\text{1}\text{2}$ theory, or $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$, which generalizes the Sakata-Taketani spin-0 and spin-1 theories. The relationship to linear manifestly covariant equations and its significance is discussed.     

On the theory of localized spin and phonon excitations in amorphous ferromagnets

Within the framework of a perturbation theory and a quasicrystalline approximation we have solved the linearized equation of motion for the circular spin component S⁺_j = S^x_j + iS^y_j in a one-dimensional amorphous ferromagnet with periodic external excitation of the spin S⁺₀ at site j = 0. It is shown that localized spin modes of the simple form $\text{«S}{}^{\mathrm{+}}{}_{\mathrm{j}}\text{a}\text{?}\text{= S}{}^{\mathrm{+}}\text{(q0) exp[iq}{}_0\text{· R}{}_{\mathrm{j}}\text{- iω(q0) t] exp (-gk?R}{}_{\mathrm{j}}\text{?)}$ with fall-off-length κ^-1 are solutions of the ensemble-averaged equation of motion. On the other hand, we have a damping of extended spin waves according to exp(-Γt). A simple relation is derived between the fall-off-length κ^-1 of localized spin modes and the damping factor Γ of extended spin waves. Analogous results hold for phonons in amorphous materials.     

Intermittent incommensurate chaos

Near the onset of intermittent chaos from quasiperiodic motion lying on an attracting 2D torus with rotation number ρ=ω₂/ω₁=(√5?1)/2, the power spectrum of the cartesian coordinate of the intersection point on the Poincaré section is studied. The Poincaré section is distorted from the ellipse near the onset of chaos. Then a sequence of spectral lines are excited at frequencies $\text{Ω}{}_{\mathrm{i}}\text{= ρ}{}^{\mathrm{i}}\text{Ω}{}_2\text{, (i=1,2,…)}$. Their intensities are found to obey the power law $\text{Ω}{}^4{}_{\mathrm{i}}\text{or}\text{Ω}{}^2{}_{\mathrm{i}}\text{for}\text{i ? 1}$ according as the Poincaré section has a sharp wrinkle or not. A similar spectrum is obtained also in the chaotic regime ε > 0. The mean value of time intervals of quasiperiodic states between two consecutive bursts and the square root of their variance are found to be inversely proportional to ε near the onset point g3 = 0.     

Poincaré gauge invariance and the dynamical role of spin in gravitational theory

We classify, according to the number of independent gauge fields, Poincaré gauge invariant theoretical frameworks of describing gravity into three categories. One of them may provide the dynamical definition of the spin tensor S and that of the energy-momentum tensor T, resulting in the response equation of matter to gravity with the gravitational field strengths, D′ and F, coupled to the former tensors

$\text{T}{}_{\mathrm{\nu }}{}^{\mathrm{\mu }}\text{;}{}_{\mathrm{\mu }}\text{=D′}{}_{\mathrm{\mu \lambda \nu }}\text{T}{}^{\mathrm{\mu \lambda }}\text{+F}{}_{\mathrm{\mu \lambda \nu \rho }}\text{S}{}^{\mathrm{\rho \mu \lambda }}$

, where the right-hand side represents spin force densities. In the absence of spin the response reduces to the conventional one of general relativity, i.e., without the spin forces. For the electromagnetic field the phase-gauge invariance requires the same conclusion as for a scalar field. For a spin $\text{1}\text{2}$ particle there is torsion, which deflects its trajectory from geodesic; an explicit expression for torsion takes a simple form of the axial vector current $\text{ψ}\text{γ}{}_5\text{γ}{}_{\mathrm{k}}\text{ψ}$.     

An algebraic approach to (3, 3)⊕(3, 3)⊕(8, 8) chiral-symmetry breaking

We examine the Hamiltonian

$\text{H = H}{}_0\text{+ε}{}_0\text{u}{}_0\text{+ε}{}_8\text{u}{}_8\text{+A S}{}_{\mathrm{\alpha \alpha }}\text{+B d}{}_{\mathrm{8\alpha \beta }}\text{S}{}_{\mathrm{\alpha \beta }}$

, where u_o, u₈ belong to $\text{(3,}\text{3}\text{)+(}\text{3}\text{,3)}$ and S_αα, S_αβ belong to (8,8), following a purely algebraic approach due to Michel and Radicati and obtain B/A = 2√3 if ε₈/ε₀ = ?√2.     

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.     

Spin alternation in one-dimensional exchange-coupled systems: Magnetic behaviour of the (12−1)N chain

We report on the magnetic behaviour of exchange-coupled closed chains (rings) of increasing length made of two alternating spin sublattices $\text{S}{}_{\mathrm{<mtext>a</mtext>}}\text{=}\text{1}\text{2}\text{, S}{}_{\mathrm{<mtext>b</mtext>}}\text{= 1}$. Taking into account the geometrical and spin space symmetries of such syste the problem is shown to be tractable up to 10 spins for distinct values of the g_a/g_b ratio between Landé factors; the results are discussed for an antiferromagnetic coupling. The limiting behaviour for the infinite chain is determined by extrapolation of the data obtained for finite rings. At very low temperatures, the analogy with the regular $\text{S =}\text{1}\text{2}$ ferromagnetic chain is underlined.     

The uniform and the strong topology on realizations of the algebra of polynomials

It is shown that under quite general assumptions on the operators A₁,…,A_n (unbounded, symmetric) and on the domain $\text{D}$ on the realization $\text{P}\text{(A}{}_1\text{,…,A}{}_{\mathrm{n}}\text{)}$ of the algebra of polynomials $\text{P}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$, the strongest locally convex topology τ_st coincides with the uniform topology τ_$\text{D}$ as well as with the strong operator topology τ_s. In the case n = 2 some conditions are given, under which these general assumptions are fulfilled.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

A further contribution to the theory of the transient hot-wire technique for thermal conductivity measurements

We study the density of states of a one-dimensional tightbinding electron model with random hopping elements. The Hamiltonian is $\text{H = -∑}{}_{\mathrm{i}}\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i+1}}\text{+a}{}^{\mathrm{+}}{}_{\mathrm{i+1}}\text{a}{}_{\mathrm{i}}\text{)}$, where the $\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}$'s are independent identically distributed random variables. It is proved that the single particle density of states D(E) diverges near E = 0 as $\text{1}\text{|(E}\text{log}{}^3\text{|E|)|}$.     

The microwave spectrum of 2,4-dioxabicyclo[3.1.0]hexan-3-one

The R band (26.5–40 GHz) microwave spectrum of 2,4-dioxabicyclo[3.1.0]hexan-3-one is reported. Rotational constants for the ground vibrational state of the common ¹²C₄¹H₄¹⁶O₃ and ¹³C₁, ¹³C₆ isotopically substituted species (the latter observed in natural abundance) have been evaluated. In addition rotational constants of the V_B = 1 to V_B = 5 quanta associated with the bending vibration of the five membered ring have been determined. A partial r_s structure has been calculated:

$\text{r(}\text{C}{}_1\text{?}\text{C}{}_5\text{) = 1.497± 0.016}\text{A}\text{?}\text{, r(}\text{C}{}_1\text{?}\text{C}{}_6\text{) = r(}\text{C}{}_6\text{?}\text{C}{}_5\text{) = 1.522 ± 0.015}\text{A}\text{?}$

,

$\text{∠}\text{C}{}_6\text{C}{}_1\text{C}{}_5\text{= ∠}\text{C}{}_1\text{C}{}_5\text{C}{}_6\text{= 60°32′ ± 1°36′, ∠}\text{C}{}_1\text{C}{}_6\text{C}{}_5\text{= 58°′ ± 1°47′}$

. With certain assumed molecular information a least squares fit yields the following parameters:

$\text{β = 68.5 ± 0.02°, r(}\text{C}{}_1\text{O}{}_2\text{= 1.408 ± 0.004}\text{A}\text{?}$

,

$\text{∠}\text{C}{}_5\text{C}{}_1\text{O}{}_2\text{= 105.8 ± 0.02°, ∠}\text{C}{}_1\text{O}{}_2\text{C}{}_3\text{= 108.10 ± 0.03°}$

,

$\text{∠}\text{O}{}_2\text{C}{}_3\text{O}{}_4\text{= 112.8 ± 0.02°}$

.     

The spectrum of HeNe+

Discharges through mixtures of helium and neon show two band groups near 4250 and 4100 Å as first observed by Druyvesteyn. These bands, assigned to the HeNe⁺ ion by Tanaka, Yoshino, and Freeman, have been studied under high resolution and have been fairly completely analyzed. The upper state of the transition is a very weakly bound state resulting from $\text{He}{}^{\mathrm{+}}\text{(}{}^2\text{S) +}\text{Ne}\text{(}{}^1\text{S}{}_0\text{)}$. There are two lower states resulting from the two components of $\text{Ne}{}^{\mathrm{+}}\text{(}{}^2\text{P) +}\text{He}\text{(}{}^1\text{S}{}_0\text{)}$. The upper of these two (${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$) is also very weakly bound while the lower of the two, the ²Σ⁺ ground state, has a dissociation energy of 0.69 eV and an r_e value of 1.30 Å. All bands in both band groups show four branches designated R_ff, Q_ef, Q_fe, and P_ee. From their analysis the rotational constants in the various vibrational levels of the three electronic states have been determined. While no spin splitting in the B²Σ⁺ state has been found the ground state X²Σ shows a very large spin splitting and the $\text{A}{}_2{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state a very large $\text{Ω-type}$ doubling. The vibrational numberings in all these states were established by the study of the spectrum of ³HeNe⁺. At the same time the hyperfine structure observed in all lines of ³HeNe⁺ confirmed the nature of the upper state B²Σ⁺ as resulting from He⁺ + Ne, i.e., by charge exchange from the ground state. The ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ component of the ²Π state has not been observed, presumably because of low intensity.     

ESR aspects of the Mn2+−Cu2+ coupled pair

Recent ESR experimental data on Mn²⁺?Cu²⁺ pairs, obtained by doping with manganese the dimeric Cu (pyridine N-oxide) Cl₂·.H₂O complex, are interpreted by means of a strong isotropic exchange model. Theg values of the pair and the hyperfine constants at the manganese and copper nuclei, for the observed effective spin S = 2, are given by the relations

$\text{g =}\text{7}\text{6}\text{g}{}_{\mathrm{Mn}}\text{?}\text{1}\text{6}\text{g}{}_{\mathrm{Cu}}$

,

$\text{A}\text{?}{}_{\mathrm{Mn}}\text{=}\text{7}\text{6}\text{A}{}_{\mathrm{Mn}}\text{,}\text{A}\text{?}{}_{\mathrm{Cu}}\text{= ?}\text{1}\text{6}\text{A}{}_{\mathrm{Cu}}$

, where g_Mn, g_Cu,A_Mn and A_Cu are the characteristic single-ion values for the quoted constants. The antiferromagnetic coupling between the Mn²⁺ spin $\text{(}\text{5}\text{2}\text{)}$ and the Cu²⁺ spin $\text{(}\text{1}\text{2}\text{)}$ is responsible for the observed g values, all measurably lower than the free electron value. Also to be noted are the enhancement of the hyperfine constant at the manganese nucleus, and the curious change of sign for that at the copper nucleus.     

3Σu−-3Σu+ coupling in the O2B3Σu− predissociation

The role of $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ spin-orbit mixing in the O₂ Schumann-Runge predissociation is investigated. The $\text{2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ state is found to cross the $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}$ state near 2.0 Å with an interaction matrix element of approximately 55 cm^?1. This state contributes to the widths of the Bv ≥ 6 levels, but introduces only small level shift perturbations. When the partial widths due to the ${}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ interaction are added to the previously calculated widths due to the ⁵Π_u, ³Π_u, and ¹Π_u states, reasonable agreement is obtained with experimental measurements on O¹⁶O¹⁶ and O¹⁸O¹⁸. The possibility of non-Lorentzian line profiles and the dependence of the width on rotational quantum number is investigated. The approximation of the spin-orbit matrix element by its value at the crossing point is shown to be a good approximation for calculating the second difference perturbations.     

QED distributions for hard photon emission in e+e− → μ+μ−γ

The hard photon emission in e⁺e^? → μ⁺μ^?γ is investigated to order α³. Formulas for a number of distributions are obtained, when neglecting terms of order (m_e/?)² and (m_μ/?)². Both charge-even and charge-odd contributions are calculated. The total contribution to the charge asymmetry parameter

$\text{? =}\text{[}\text{d}\text{σ(θ)}\text{d}\text{O}{}_{\mathrm{\mu }}{}^{\mathrm{+}}\text{?}\text{d}\text{Oσ(π?θ)}\text{d}\text{O}{}_{\mathrm{\mu }}{}^{\mathrm{+}}\text{]}\text{[}\text{d}\text{σ(θ)}\text{d}\text{O}{}_{\mathrm{\mu }}{}^{\mathrm{+}}\text{+}\text{d}\text{σ(π ? σ)}\text{d}\text{O}{}_{\mathrm{\mu }}{}^{\mathrm{+}}\text{]}$

does not exceed 5% for the c.m.s. energy 2? = 3 GeV.     

The g-factor difference for the 61+ and 88+ states in 210Po

The difference in g-factors for the 6₁⁺ and 8₁⁺$\text{(πh}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}{}^2\text{)}$ states in ²¹⁰Po has been measured as $\text{(g}{}_6\text{? g}{}_8\text{)}\text{g}{}_8\text{= 2.0 ± 0.7\%}$. This result represents a small violation of additivity. A value of g₈ = 0.909 ± 0.011, independent of g₆, was also obtained.     

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.     

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.     

Spectrum and structure of the He2 molecule: Characterization of the singlet and triplet states associated with the UAO's 4s, 4dσ, 4dπ, and 4dδ

The emission spectrum of the He₂ molecule has been rephotographed in the ～4000–～5700 Å region and the $\text{4d(}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^3\text{Π}{}_{\mathrm{u}}\text{,}{}^3\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$, $\text{4d(}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^1\text{Π}{}_{\mathrm{u}}\text{,}{}^1\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$, $\text{4s}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$ and $\text{4s}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$ transitions analyzed. The 4dδj³Δ_u, 4dπj³Π_u, $\text{4dσj}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and $\text{4sh}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states have been characterized through v = 2 and the 4dδJ¹Δ_u, 4dπJ¹Π_u, $\text{4dσJ}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, and $\text{4sH}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states for v = 0. The term levels for these perturbed and l-uncoupled states have been confirmed (a) by analyses of bands with common levels from Δv = 0, ±1 sequences and (b) by analyses of the transitions between the above states from 4d and 4s and the $\text{c}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ and $\text{C}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ states associated with 3pσ. Molecular constants are reported which have been partially corrected for the effects of l-uncoupling and the homogeneous perturbations between the state pairs J, H and j, h.