Large Distance Limit Research Articles

Temperature dependence of Van der Waals forces in classical electrodynamics with classical electromagnetic zero-point radiation

Retarded dispersion forces are calculated at finite temperature in the theory of classical electrodynamics with classical electromagnetic zero-point radiation. Expressions are given at all separations for the forces between a polarizable particle and a conducting wall and between two polarizable particles where the particles are represented by electric dipole oscillators immersed in thermal radiation. Also given are the simple expressions holding in the unretarded short-distance limit, and in the asymptotic large-distance limit. At any finite nonzero temperature $T$, the asymptotic large-distance Van der Waals force between two polarizable particles separated by a distance $R\ensuremath{\gg}\frac{\ensuremath{\hbar}c}{\mathrm{KT}}$ is given by the potential $U(R)=\ensuremath{-}\frac{3{\ensuremath{\alpha}}^{2}\mathrm{KT}}{{R}^{6}}$ rather than the Casimir-Polder form for zero temperature, $U(R)=\ensuremath{-}(\frac{23}{4\ensuremath{\pi}})\frac{{\ensuremath{\alpha}}^{2}\ensuremath{\hbar}c}{{R}^{7}}$, where $\ensuremath{\alpha}$ is the static polarizability of each particle. Also the ${R}^{\ensuremath{-}6}$ form holds at high temperatures for any fixed separation. The finite-temperature analysis presented follows directly from earlier classical work at zero temperature since within the classical theory, classical zero-point radiation and classical thermal radiation are treated on the same footing. The classical calculations are easier than those of quantum-electrodynamic perturbation theory but have been shown generally to reproduce the quantum results for dipole-oscillator systems. The forces in the high-temperature limit are shown to agree with the results of classical statistical mechanics and with the use of the Rayleigh-Jeans law for the thermal radiation spectrum. The new results for polarizable particles fit nicely with earlier work by other authors on the finite-temperature corrections to the Casimir effect, the force between uncharged conducting parallel plates. It is emphasized that the force between conducting plates may be regarded as due to the classical boundary conditions at the conductors rather than to any discrete quantum aspects; the Rayleigh-Jeans spectrum also leads to a force between conducting plates and this force is in agreement with the high-temperature limit of previous calculations including zero-point radiation.

Operator product expansions at large distances and Regge asymptotic behavior

The class of theories is considered in which the simplest high-energy scattering processes are described by the exchange of a Regge pole or cut and in which light-cone behavior is described by operator product expansions. The scattering processes are those in which an initial particle interacts with an initial cluster of other particles to produce a final particle and final cluster. The high energy limit is that in which the energy of the initial particle relative to the energy of any particle in the clusters becomes large with fixed masses, momentum transfers and cluster subenergies. It is shown that these limits can be completely characterized by an operator statement on the product of the interpolating fields of the initial and final particles, independently of the initial and final clusters. Given then that these limits correspond to large distance limits in space-time, operator product expansions are obtained which describe in an operator manner the behavior of a product A( x) B(0) of local fields in the large distance limit x 0 → ∞ with x 2 fixed. This provides an exact operator picture of a Regge pole being built out of an infinite set of elementary spin exchanges. The crucial ingredient in the derivation is the factorization property of Regge poles or Regge cut discontinuities. The leading light-cone contribution to the scattering amplitude will then have the same factorization properties. The operator expansion is derived first for single particle matrix elements, first using commutativity of the Regge and scaling limits and second using diagonalized Bethe-Salpeter equations for the matrix elements of the local fields in the light-cone expansion. It is then deduced for single-double particle matrix elements using the second method where now helicity sums play an important role. It is finally derived for arbitrary matrix elements. Non-leading Regge poles and Regge cuts can be similarly described by using also non-leading light-cone singularities.

Phonon Contribution to the Free Energy of Interacting Adatom Pairs

The free energy $\ensuremath{\Delta}F$ due to the interaction through the crystal phonon field of two identical atoms adsorbed on a (100) surface of a simple cubic monatomic lattice is studied within the framework of the Montroll-Potts model. The zero-point energy of interaction, $\ensuremath{\Delta}{E}_{0}$, is found analytically in the limit of large separation distances, and numerical results are presented for small separation distances. The leading temperature-dependent terms in the expansion of $\ensuremath{\Delta}F$ are given in closed form in both the low- and high-temperature limits. The results are given as a function of the mass and separation distance of the adsorbed atoms and as a function of the force constant binding the adsorbed atoms to the surface. For all values of the model parameters, it is found that the free energy decreases with the separation distance. Thus, two adatoms on a surface are attracted toward each other.

Stellar winds and breezes

Steady stellar winds are generally divided into two classes: (i) the winds proper, for which the energy flux per unit solid angle, E ∞ , is non-zero, and (ii) the breezes, for which E oo = 0. The breezes may be distinguished from one another by the value of the ratio, g , of kinetic to thermal energy of the particles in the limit of large distance, r , from the stellar centre, or more precisely by g=lim mv 2 r->∞ — kT ' where v ( r ) is breeze velocity, T ( r ) is temperature, m is mean particle mass, and k is the Boltzmann constant. Solutions have previously been obtained for values of g in the range 0 < g < 1, in which the breezes are subsonic everywhere with respect to the isothermal speed of sound. It is demonstrated here that two distinct solutions exist as g -> 5/3, namely (in an obvious notation) the g = 5/3 — and the g = 5/3 + possibilities. It is shown that, if g > 5/3 ( g < 5/3) the solutions are everywhere supersonic (subsonic) with respect to the adiabatic speed of sound. If 1 < g < 5/3, they possess a critical point, at which the isothermal speed of sound and the flow speed coincide. The winds are examined in the limit E ∞ -> 0, and the relation with the breezes is studied. In particular, it is shown that, for r < 0 ( E ∞ -2/5 ), the winds satisfy the stellar breeze equations to leading order, and possess a critical point at r = 0 (1). For r > 0 ( E ∞ -2/5 ), the solutions do not obey the breeze equations. They ultimately follow the Durney asymptotic law [T = 0 ( r -4/3 ), for r -> ∞] for the winds. This demonstration of how the winds merge continuously into the breezes as E ∞ -> 0 is new. The question of how the particle density ( N 0 ) and temperature ( T 0 ) at the base of the stellar corona determine the type of solution realized outside the star is examined. Even when the flow speed, v 0 , at the base of the corona is subsonic, non-uniqueness can occur. In one domain of the ( N 0 , T 0 ) plane, two distinct types of breeze are possible; in another these, together with a wind (E∞ =f= 0), are permissible. Elsewhere (large N 0 , moderate T 0 ) only a unique breeze exists or (small N 0 and/or T 0 ) a unique wind. In some domains (large T 0 ) no steady solution exists, unless the requirement that the corona is subsonic is relaxed. In this case, however, the problems of non-uniqueness are severely aggravated.

Low energy atomic collisions

The Schrödinger equation for the system H+-H developed in a previous paper is considered using new expansion functions for electronic states obtained from H2 + molecular ion electronic eigenfunctions by a unitary transformation. These new functions have the advantage of remaining orthonormal at all internuclear separations and asymptotically becoming symmetrized atomic hydrogen states. Although they are eigenfunctions of the H2 + hamiltonian only in the limit of large internuclear distance, the effect of the H2 + hamiltonian on these functions is readily found. Due to coupling which remains non-zero in the limit of large interproton distance, each independent formal solution of the H+-H equations involves more than one expansion state in this limit. These solutions may be expressed asymptotically as column vectors multiplied by incoming or outgoing spherical waves. The formal theory of scattering as developed by Gell-Mann and Goldberger has been utilized along with the projection formalism of Feshbach to obtain the correct asymptotic form of the scattering wave function. The procedure employed involves formulating the problem in terms of two-potential scattering and requires application of renormalization techniques for treating level shifts produced by the infinite-ranged coupling. This asymptotic form may be used in imposing scattering boundary conditions on numerical solutions of coupled equations for H+-H scattering. Finally, it is shown that one cannot interpret coefficients of all outgoing spherical waves as scattering amplitudes. In addition, new interference phenomena are found to result from the presence of the infinite-ranged coupling. The present formalism is shown to reduce to the usual perturbed stationary-states method in the approximation that the infinite-ranged coupling is neglected.

Lower bounds for sonic booms in the midfield.

This paper presents new lower bounds for sonic booms based on minimizing either the overpressure or the shock strength of the positive part of the boom wave. The results are not restricted to the limit of large distance from the aircraft. They reduce to the Jones lower bound infinitely far from the aircraft in a uniform atmosphere but they give reductions from the Jones lower bound of the order of 50% for planned supersonic transport conditions. This is because the asymptotic results of Jones are approached very slowly (as r~1/4) in a uniform atmosphere and are never reached in the real atmosphere. The direct application of these results depends upon additionally keeping the tail shock intensity less than or equal to that of the front shock.

On a class of completely transparent nonlocal two-body potentials

We show here the existence of a large class of nonlocal two-body potentials which are completely transparent at all energies. These potentials are of the general form given by (3.10),i.e. they are super-positions of a local central potential and a sum of separable potentials. Starting from an arbitrary local potential, we show how to choose the nonlocal separable parts in such a way that the effects of the local potential are cancelled out at all energies and for all angular momenta. The total Hamiltonian therefore becomes equivalent to the free Hamiltonian as long as theS-matrix on the energy shell is concerned. At finite distances, the wave-function is of course different from the free wavefunction, but the difference goes to zero in the limit of large distances. It is perhaps worth mentioning that the transparent potentials we obtain are not pathologica; they have acceptable singularities at the origin and vanish fast at infinity. In the example we study in Sect.4, we have indeed potentials which are less singular thanr−1 at the origin, and decrease exponentially whenr→∞.

Second-Harmonic Generation of Light by Focused Laser Beams

An experimental and theoretical study is reported on second-harmonic generation (SHG) by focused laser beams in nonlinear crystals which are large compared with the extent of the focus. The gas-laser beam is in the lowest order (Gaussian) mode and very close to index-matching conditions in the crystal. The intensity pattern of the SHG observed photographically has roughly the shape of a half moon with a very sharp edge. The distribution on the bright side of the edge has been measured using a traveling slit and photomultiplier detector. Fine structure has been observed consisting of a series of fringes extending from the edge into the bright region. The nature of the fine structure depends upon where the pattern is observed. The pattern at the surface of the crystal has fringes at the positions one would expect for diffraction of a second-harmonic beam by a straight-edged screen placed at the focus. The pattern at great distance from the crystal has uniformly spaced fringes; in this case fringes have also been seen on the dim side of the edge. It has been shown that the position of the edge relative to the axis of the laser beam is sensitive to extremely fine adjustments of the crystal orientation. The power has also been measured as a function of crystal orientation, and the position of the edge has been determined for the orientation giving maximum power. The theoretical treatment of the problem, which occupies most of the paper, is based upon an exact formula for the second-harmonic field for the case in which the laser beam is in a Guassian mode. In a far-field approximation this formula is decomposed into three types of terms having different dependence upon distance. The leading term in the limit of large distance has a sharp edge and describes the gross behavior on the bright side of the edge. The other terms are associated with fine structure of the Fresnel and Fraunhofer types arising from the focus and the crystal surfaces, respectively. The theory contains a parameter to specify the phase-matching conditions which can be simply related to changes in the crystal orientation. The power is obtained as a function of this parameter, and the optimum matching condition corresponding to maximum power is thereby determined. A detailed analysis of the experimental data is presented which shows the theory to be in quantitative as well as qualitative agreement with experiment.