On the criterion of the crystal-liquid phase transition

A localization criterion is proposed for the crystal-liquid phase transition (PT). According to this criterion, the PT begins when the E _d/k _b T ratio reaches a boundary value E _d(s)/k _b T _m such that a solid phase is present above it and a liquid phase is present below it in a phase diagram. Here, E _d is the energy of atom delocalization, k _b is the Boltzmann constant, T is the temperature, and E _d(s) is the delocalization energy for a solid phase at melting point T _m. This criterion is shown to generalize the Lindemann criterion of melting to the case of crystallization and the Löven criterion of crystallization to the case of melting. This localization criterion is found to be applicable for both normally melting substances and substances that melt with a decrease in the specific volume upon the transition into a liquid phase. The relation of the localization criterion to the vacancy and diffusional criteria of the crystal-liquid PT has been studied. The inequality T _N < T _m, where T _N is the temperature of the onset of crystallization, is explained using the localization criterion. The calculated values of the T _N /T _m ratio coincide well with the experimental estimates. The maximum value of T _N /T _m is likely to be most probable in crystals with a bcc structure and a small value of the Grüneisen parameter. The T _N /T _m ratio is analyzed at the points in the PT where no change in the specific volume occurs and an entropy jump is nonzero.     

Effects of the velocity dependence of the collision frequency on the Dicke line narrowing

The effect of the velocity (v) dependence of the transport collision frequency ν_trv on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance (M) and buffer (M _b) particles, β = M _b/M: it is at a minimum for β ? 1 and reaches a maximum for β ? 3. A power-law particle interaction potential, U(r) ∝ r ^?n , is used as an example to show that, compared to ν_trv (v) = const (n = 4), the line narrows if ν_trv (v) decreases with increasing v (n < 4) and broadens if ν _trv (v) increases with v(n > 4). At β ? 3, the line width can increase [compared to ν_trv (v) = const] by 5 and 12% for the potentials with n = 6 and n ? 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I(Ω) has been found to be weakly sensitive to ν_trv (v) at some detuning Ω_c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.     

Factorization and the couplings of the vacuum trajectory

The decoupling theorems associated with an isolated factorizable pomeron pole of unit intercept are re-examined. It is found that the coupling of three such poles, Γ(t, t, 0), need not vanish, precisely at the point t = 0. This is demonstrated by summing only over states in the appropriate unitarity sum, and sum rule, which are consistent with the M², s/M² → ∞ limit. The triple-Regge region then makes a constant contribution to σ_total, insteadsb of the ln lns result obtained if the isolated pole is assumed to couple also to states such that s/M² = constant. The physical implications regarding factorization and the pole-cut relationship are discussed. The relationship between higher order optical theorems (Mueller discontinuities) and particular terms in the unitarity sum for the two → two absorptive part A₂₂ is exploited. Consistent contributions to the triple-Regge region contribute constant vertex corrections to pure pole behaviour in A₂₂. There is no cut contribution and the magnitude of the vertex corrections reflects the relative amount of diffractive production. The analysis is extended to multiple fireball production where pure multipole structures emerge. The series naturally terminates if the diffractive component is sufficiently small. The implications for the behaviour of the total cross section at machine energies are discussed.     

Diagonal structure of the S-matrix in the lee model

The diagonal structure of the S-matrix in the Lee model is studied. The N2θ- and N3θ-type eigenvalues are shown to factor exactly into the products of the corresponding Nθ eigenvalues, and it follows that these eigenvalues are free of production thresholds. The (implicit) eigenvalue equation for the V2θ-type eigenvalue is given, and this eigenvalue is shown to factor asymptotically into the product of the corresponding Vθ eigenvalues. These properties are conjectured to hold for Njθ- and Vjθ-type eigenvalues. It is conjectured that the two-body type eigenphase operator gives the dominant contribution to the S-matrix at high energy.     

Breakdown of renormalizability in the Kondo problem in the high-temperature expansion of the free energy

It is shown that there are two energy scales in the Kondo problem: T _k and T ₀, one of which (T _k) is exponentially small in the coupling constant g. The second scale T ₀is proportional to the squared coupling constant. Perturbation theory is valid only in the region T? T ₀. The point T ₀ is apparently the crossover from weak to strong coupling. The first indications of the breakdown of the hypothesis of only one energy scale in the Kondo problem appear in fourth order of perturbation theory.     

Complex extension of the representation of the symplectic group associated with the canonical commutation relations

The aim of the investigation is to extend the representation of the real symplectic group associated with the canonical commutation relations into the complex symplectic group. It is shown that an extension exists to a semigroup S such that Sp(2n, R) ? S ? Sp(2n, C). The construction of the extension is achieved by extensive use of Bargmann's reproducing kernel space. We are able to give a simple geometric model for the semigroup S: it is exactly the semigroup of all complex symplectic transformations which increase the U(n, n) “length”.     

On the relationship between the Fisher-Frieden-Soffer arrow of time,and the behaviour of the Boltzmann and Kullback entropies

We provide a new derivation of the H-theorem satisfied by Fisher's information measure (I) for the diffusion equation and show that the ensuing monotonically decreasing behaviour of I yields, in turn, an H-theorem involving the second time derivative of the standard Boltzmann-Gibbs-Shannon entropy S. An upper bound for S(t) is obtained from this latter relation. Additionally, it is demonstrated that, under certain conditions, the behaviour of I may be related to that of the relative information measure of Kullback. This latter result is generalized to the case of the Fokker-Planck equation.     

On the relationship between the ionization potential and the work function: Metals

A relationship is analyzed between the ionization energy of atoms I and the work function φ of a metal composed of these atoms. The transition energy ∑=I?φ is assumed to be the sum of kinetic K and coulombic C contributions. Contribution K is calculated in the framework of a model of uniform gas of quasi-free electrons, and C is determined from experimental values of Σ. Calculations performed for a wide range of metals have shown that the dimensionless coefficients determining the Coulombic contribution C differ insignificantly between various groups of metals. The relationships obtained have been used for determining the work function of binary alloys.     

On the factorization of the pomeranchuk singularity in the Mueller-Regge diagrams

It is shown that the j plane singularity (Pomeron) which leads to a rising total cross-section like (lns)^?, ? > 0. cannot be factorizable in the Mueller diagrams for inclusive production of particles with large rapidity differences. The relevance of this result for the theories of a renormalized Pomeron is discussed.     

Boltzmann'sH theorem and the Loschmidt and the Zermelo paradoxes

The Umkehreinwand of Loschmidt and the Wiederkehreinwand of Zermelo have been reexamined. The former paradox depends on the augument that for a dynamical system, upon the reversal of the velocities of all the molecules, theH function retraces its sequence of values so thatdH/dt will change its sign. The latter paradox depends on the argument that theH function returns infinitely close to its value after a Poincare' quasi-period and therefore cannot be decreasing all the time. While the main contention of the two paradoxes is correct, that theH theorem is inconsistent with classical dynamical laws, the arguments there can be considerably simplified and the “paradoxes” answered more directly. If the distribution functionf(q _K ,p _K ,t) is governed by an equation which is time-reversal invariant (such as the Liouville equation for a closed dynamical system), then it can be shown immediately thatdH/dt=0,H=cons. In this case, both paradoxes disappear, but together with them, thedH/dt<0 part of theH theorem also has disappeared, i.e., there is no second law of thermodynamics. Iff(q _K ,p _K ,t) is governed by an equation which is not time-reversal invariant (such as the Boltzmann equation, or the Master Equation for Markovian processes), then (1) there is no argument forf andH(t) to retrace their sequence of values upon the reversal of all the velocities of the system, (2) there is no quasiperiod in whichf andH(t) return to their earlier values. In this case, both paradoxes disappear also, but then one must go beyond classical dynamics in order to maintain theH theorem.     

On the reaction pp → ppη near the threshold

The reaction pp → ppη near the kinematical threshold is considered in the framework of the model, which includes the pp final states interaction (FSI). The Coulomb and strong effects of the pp FSI are calculated correctly in the model “Coulomb + Yamaguchi potential”. The pp FSI is shown to provide the main contribution to the energy dependence of the total cross section of the pp → ppη reaction close to threshold. The energy behavior of the total cross section is predicted in agreement with the data in the excess energy region < 15MeV. The predictions for different spectra are also performed.     

On the dynamical structure of the Dicke maser model

For suitable states of the Dicke maser model we study the time evolution of the mean photon number n(t) in the limit N → ∞. Here N is the number of maser active atoms. Our starting point is a well-tuned cavity with only one mode of the radiation field excited. Introducing new dynamical variables we are able to exploit fully the conservation laws so as to get a simple but completely rigorous solution. We find that n(t) is periodic, and given by a Weierstrassian elliptic function, provided a net polarization is present in the cavity. No approximation is involved.     

Note on the difference between the a-cyclic and c-cyclic version of the XY model

The transverse susceptibility of the c-cyclic version of the one-dimensional XY model with respect to an infinitesimal magnetic field in the x-direction is investigated in more detail. Special attention is paid to the c-cyclic version of the one-dimensional Ising model. The c-cyclic susceptibility χ_xx is evaluated explicitly. The autocorrelation function of the magnetization M_x turns out to be time dependent in the c-cyclic Ising model.     

Prediction of the one pion exchange contribution to the processπ − p→π − pω with the help of the Veneziano model

The one pion exchange contribution to the differentialπ ^? p→π ^? pω cross-section is calculated assuming that theπ→πω amplitude associated with one of the vertices is described by the Veneziano formula, and that only a trivial dependence of the latter on the mass of the exchanged pion needs to be taken into account. The remaining unknown overall constant is then fixed by considering the decayω→3π. A rough comparison with preliminary data from a 16 GeV/cπ ^? p experiment shows that the one pion exchange model could possibly describe the reactionπ ^? p→π ^? pω forπω masses above 1.4 GeV.     

On the divergence of the perturbative expansion for the anomalous magnetic moment

Following the analysis performed by Lautrup, the behaviour of the graphs for the anomalous magnetic moment is studied in the case in which two photon lines (with insertion of n ? 1 electron-positron bubbles) are exchanged. The behaviour with n is a divergence of the type αⁿn! ln n, very similar, therefore, to the case of one photon line.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

Theoretical interpretation of the measured X-ray spectrum of beryllium in the region of the Raman band

The X-ray spectrum of beryllium in the region of the Raman band displays characteristic peaks, similar to the ones observed in graphite. An ab initio unrestricted Hartree—Fock cluster calculation is performed to investigate the bonding character of the electronic structure of beryllium and to identify the “anomalous” peaks. These peaks are found to be due to the 3sp_xP_y?3sp_z splitting of the 3ps hybridized band. A similar splitting of the 2sp orbitals is responsible for the dip in the density of states curve at the Fermi level.     

On the self-consistency requirement in the low density expansion of the optical potential in nuclear matter

We critically discuss the choice of the auxiliary potential U which is introduced in the low density expansion of the mass operator M(k, w). This choice is related to the analytical properties of M(k, w) in the complex w-plane and we take due account of momentum conservation in the intermediate states appearing in the diagrams associated with M(k, w). We also provide a computation of the one-hole line, rearrangement and renormalization contributions to the optical potential, of the hole state spectral function and of the momentum distribution in nuclear matter. We use a real auxiliary potential which is self-consistent up to the order considered here, i.e. which takes into account the rearrangement and the renormalization corrections. Rearrangement is treated rigorously. The dependence of the obtained results on the choice of the nucleon-nucleon interaction, namely the Hamman-Ho Kim one in our calculation, is discussed.     

Momentum distributions in the nucleus

The nuclear form factor F(q) and one particle momentum distribution p(q) can be shown to have a power law decrease for large momenta. For the form factor F(q) we show that it is q/A that must be large for this asymptotic behavior to be important. For only q large the form factor, in a simple model, is shown to decrease exponentially in q. A similar behavior for p(q) is proposed.     

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.