Unequal-mass Particles Research Articles

A particle grid air quality modeling approach: 1. The dispersion aspect

A particle grid air quality modeling approach that can incorporate chemistry is proposed as an alternative to the conventional partial differential equation (PDE) grid air quality modeling approach. In this approach, each particle is tagged with different species masses and particles in the same grid participate in chemical reactions. The approach is flexible and removes the advection and point source problems encountered in the PDE approach. For a typical grid size of 5 km × 5 km × 50 m used in the lowest layer of an urban air quality model, use of 2000–3000 particles of unequal masses per grid cell will yield a highly accurate grid‐averaged instantaneous concentration field that undergoes eddy diffusion for a period of about 1 day. Use of an hourly averaged concentration reduces the demand of particle per cell to about 500. Increasing the grid size also reduces the demand on the number of particles per cell. For the choice of our Lagrangian integral time scales, the time step must be small (10 s) for vertical dispersion simulation but can be large (200 s) for horizontal dispersion simulation. To reduce computation time, a time‐splitting scheme is, proposed to simulate the horizontal and vertical dispersion simulations in an alternating sequence. The present study also shows that the oft‐used second‐order‐accurate finite difference scheme for solving the diffusion equation tends to overpredict the peak of a sharply peaked concentration.

Direct N-Body Calculations

The main principles for direct integration of large point-mass systems are outlined. Most particles are advanced by the Ahmad-Cohen neighbour scheme, using fourth-order force polynomials and individual time-steps. Close encounters and persistent binaries are handled by two-body regularization, whereas extreme triple and quadruple configurations are treated as unperturbed systems by special regularization techniques. As an illustration of these methods, we discuss some recent results of an isolated system containing 1000 particles of unequal mass, with special emphasis on the post-collapse phase.

Three particles on a ring

Abstract We study the ergodic properties of three classical particles of unequal mass moving on a ring and colliding like hard points. Although it is generally believed that this system is ergodic, we show that extensive numerical calculations do not support this belief. This may be connected with our discovery that for vanishing total momentum, any initial state with one particle at rest is periodic, independent of the mass values. The analogous periodicity for two unequal mass particles bouncing in a one-dimensional box can be understood on the basis of a remarkable property of a billiard in the form of a rectangular triangle.

Unequal mass spinor-spinor Bethe–Salpeter equation

Coupled radial equations are derived for the ladder approximation Bethe–Salpeter equation describing a system of two spin- (1/2) particles of unequal masses interacting to form a bound state of total mass zero. The numerical behavior of the coupling parameter λ as a function of the mass ratio is examined for known analytical equal-mass solutions. In addition a perturbation method is employed to investigate the behavior of λ for small values of the exchange mass.

A new method for separating the centre-of-mass motion of clusters in-nuclei

A method for separating the centre-of-mass motion of a two or more particle cluster is developed. This method works for an arbitrary central shell model potential. Only the internal motion wave function must be previously defined. Formulas are obtained for the case of clusters with arbitrary angular momenta of the internal motion and with constituent particles of unequal masses. The problem of separating the relative motion of a pair of particles with previously defined centre-of-mass motion is also solved. As a byproduct, is useful integral formula for generalized Talmi-Moshinsky coefficients is obtained. Some applications of the method are demonstrated in calculating the overlap interests to radioactive states, the form factors for direct x-transfer reactions and the two-parties interaction matrix elements.

Simple Formula for the General Oscillator Brackets

An explicit formula for Talmi-Moshinsky transformation brackets of unequal-mass particles is given which is the sum of simple expressions over five variables; it is especially suited for numerical calculations.

Nuclear cluster motion in a deformed harmonic oscillator potential

Abstract A general formula for the Talmi coefficients for two particles of unequal masses in an axially symmetric oscillator potential is derived. As an example, a wave function describing an alpha cluster in such a deformed potential is expanded, by means of these coefficients, in terms of single-particle oscillator wave functions.

Regge representation for forward scattering of unequal-mass particles

The Sommerfeld-Watson transformation of at-channel helicity amplitude is investigated forz1=±1 corresponding to forward scattering of unequal-mass particles in thes-channel. It is shown that the Sommerfeld-Watson transform exists at these points under similar conditions as in the case of large |zt| and that it corresponds to a finite Regge-pole contribution. This eliminates the infinities of the conventional Regge representation atzt=±1. A modified Regge representation for helicity amplitudes is given, which guarantees a continuous behaviour of the scattering amplitude for unequal-mass particles near the forward direction.

Cancellation of the Singularities of the Backward Unequal-Mass Regge-Pole Terms

In terms of the contour integrals in the s plane, the Kim-Resnikoff mechanism with moving poles is studied to what extent it can be identified with that of Freedman and Wang. The method of the contour integrals is also suggested to be appropriate in under­ standing the multiple-pole theory recently given by Nakanishi. 0° Recently, Kim and Resnikoff 1 ),*) developed a method of reqUlrmg the Mandelstam analyticity for the Regge-pole term for unequal-mass particles in the' backward region (s = 0), and obtained, without introducing the daughter trajec­ tories,2)a modified Regge-pole term which is truly analytic at s = 0 and behaves asymptotically like ua(O). Furthermore, according to their result, the next largest term of the modified amplitude is proportional to which also shows up in the Freedman-Wang formulation (see Eq. (7) in refer­ ence 3)). In the latter formulation, this term results form the existence of the polar singularity of order two (<<double pole») at s = 0 in the Khuri plane, as is clear from the recent work by Nakanishi. 4 ) In the Kim-Resnikoff formulation, on the other hand, the presence of this term is connected with the implicit in­ troduction of a fixed Regge pole which probably serves as a substitute for daughter poles. There is, however, a 'criticism 5 ) on the existence of fixed Regge poles since it may contradict the miitarity condition. In this paper, we try to modify the mechanism essentially used , by Kim and Resnikoff in a plausible form which gives moving poles, and we intend to bring about a simpler understanding of the connection between the Kim-Resnikoff ,manipulation and the mechanism based on daughter poles. To this end, we will start by representing the Regge-pole term by a contour integral in the s plane, t6 the effect that the cancellation is regarded as the removal of the pinch of the integration path. We will intensionally show *) There appeared quite recently a work by Caser and Basdeyant,la) in which the technique

Study of the Kinematic Constraints and Factorization in the Regge-Pole Model for Photoproduction Processes

By enforcing factorization together with the (kinematic) analyticity requirements on the scattering amplitudes, the pole spectrum and residue-function behavior in the Regge-pole model for several photo-production processes is analyzed. It is shown that the simultaneous imposition of these requirements in all the reactions related by factorization is sufficient to limit the number of possible solutions to a few general types. In most cases, these solutions correspond exactly to those obtained from the so-called group-theoretical or $O(4)$ schemes. In a few cases, anomalous solutions which do not have any $O(4)$ counterpart are also found. There is no basic distinction between equal- and unequal-mass particles scattering in this approach.

Analyticity and the Daughter Structure of Conspiring Regge-Pole Families

By studying reactions involving unequal-mass particles with spin, we show that much of the structure of possible families of conspiring Regge poles follows simply from imposing the $t=0$ analyticity constraints. These requirements imply the necessity for both daughter and conspirator contributions, where in many cases the daughter poles are themselves conspirators. We discuss the pattern of ($t=0$) singularity cancellation by daughter trajectories in both the single-parity families and double-parity families for general mass and spin processes and demonstrate that the two have rather different structure. As an application, the reactions $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\mathrm{VV}$, $\ensuremath{\pi}N\ensuremath{\rightarrow}\mathrm{VN}$, and $\mathrm{NN}\ensuremath{\rightarrow}\mathrm{NN}$ are examined in detail from the point of view of analyticity constraints. We show that in all cases factorization of the (first) daughter residues is consistent with the required analyticity properties of the amplitudes, and that in certain (nonevasive) cases the factorization of the daughter residues is directly implied by factorization for the leading pole when the conspiracy constraints are obeyed. We conclude that our results, based on analyticity (and factorization), complement the group-theoretic $O(4)$ classification; the symmetry and analyticity methods give similar information when they overlap, but supplement each other in certain cases when one method is not readily applicable.

Singularities of a Bethe-Salpeter Amplitude

The Bethe-Salpeter equation for the scattering of two particles of unequal mass is reduced to separable form by the use of the bipolar transformation. Exact solutions are obtained for Wick's kernel, so that the $S$ matrix and the Regge trajectories are explicitly determined. The daughter trajectories are identified for this kernel and are found to be related in the expected way with one another. It is shown that they make no contribution to the scattering amplitude on the mass shell. Trajectories are found corresponding to the abnormal solutions of Wick's equation, but these do not appear in the normal scattering amplitude.

Relativistic Model of A Daughter Regge Trajectory

A model based on ladder diagrams in a $\ensuremath{\lambda}{\ensuremath{\phi}}^{3}$ theory is used to study the first daughter trajectory in the scattering of unequal-mass particles. It is found that the daughter pole moves towards the physical region less rapidly than the leading pole, and develops a smaller imaginary part at the two-particle threshold. The motion of the daughter pole is determined primarily by three-particle scattering processes. As a consequence, a model which fails to treat three-particle scattering accurately cannot give a detailed picture of the motion.

Regge Poles in the Scattering of Particles of Unequal Mass. Remark on a Paper of Freedman and Wang

An interesting analogy is presented between (i) the properties of the shadow trajectories with singular reduced residues which appear in Regge-pole models for the scattering of particles of unequal mass (Freedman-Wang), and (ii) the well-known infinite-momentum limits of Regge trajectories in potential scattering. A simple model for estimating the effects of those trajectories is presented.

Regge poles and the scattering of unequal mass particles

It is suggested that a Regge trajectory α(t) generally will also have angular momentum components α λ ( t) = α( t) − λ( λ = 1, 2,…) associated with it for all t which become redundant whenever α(t) makes a particle.

Regge Poles and Unequal-Mass Scattering Processes

It is not clear from the Regge representation that the asympotic form ${s}^{\ensuremath{\alpha}(u)}$ holds in the backward scattering of unequal-mass particles, because the cosine of the $u$-channel scattering angle remains small as $s$ increases. In this paper we use a representation for the scattering amplitude first suggested by Khuri to show that the form ${s}^{\ensuremath{\alpha}(u)}$ is valid throughout the backward region. However, in order to ensure the analyticity of the amplitude defined by the Khuri representation at $u=0$, it is necessary that Regge trajectories occur in families whose zero-energy intercepts are spaced by integers. Denoting the leading or parent trajectory by ${\ensuremath{\alpha}}_{0}(u)$, we find that daughter trajectories ${\ensuremath{\alpha}}_{k}(u)$ must exist, of signature ${(\ensuremath{-}1)}^{k}$ relative to the parent, satisfying ${\ensuremath{\alpha}}_{k}(0)={\ensuremath{\alpha}}_{0}(0)\ensuremath{-}k$. We then study Bethe-Salpeter models and find that this daughter-trajectory hypothesis is satisfied for any Bethe-Salpeter amplitude which Reggeizes in the first place. This fact follows elegantly from the four-dimensional symmetry of Bethe-Salpeter equations at zero total energy. Some phenomenological implications of the daughter-trajectory hypothesis are discussed. We have also characterized the behavior of partial-wave amplitudes in unequal-mass scattering at $u=0$ and find the hitherto unsuspected result $a(u,l)\ensuremath{\sim}{u}^{\ensuremath{-}\ensuremath{\alpha}(0)}$, where $\ensuremath{\alpha}(u)$ is the leading $u$-channel Regge trajectory.

Construction of the equivalent potential in born approximation for scattering by particles of unequal mass

In this work the construction of the configuration space equivalent potential in Born approximation is considered for scattering by particles of unequal mass. The potential agrees in essence with the one obtained using the Chew-Frautschi prescription inS-matrix theory,i.e., the range of the exchange force arising from the exchange of a particle of massMp isR0=[Mp2−(MA2−MB2)2/s]−1/2 whereMA,MB are the masses of the particles involved in the scattering ands=(c.m. energy of (A, B))2. This clarifies the confusion in the recent literature regarding this point and sets the stage for determining the importance of the energy dependence of the range parameter in studying the dynamics of strongly interacting particles. The examples considered are K* (890) exchange in πK scattering andOpen image in new window exchange in π scattering. It is shown that the energy dependence of the range parameter is not important in the former and may be crucial in the latter.

Analyticity Constraints on Unequal-Mass Regge Formulas

A Regge-pole formula is derived for the elastic scattering of two unequal-mass particles that combines desirable $l$-plane analytic properties (i.e., a simple pole at $l=\ensuremath{\alpha}$ in the right-half $l$ plane) and Mandelstam analyticity. It is verified that such a formula possesses the standard asymptotic Regge behavior ${u}^{\ensuremath{\alpha}(s)}$ even in regions where the cosine of the scattering angle of the relevant crossed reaction may be bounded. The simultaneous requirements of $l$-plane and Mandelstam analyticity enforce important constraints, and the consistency of these constraints is studied. These considerations lead to the appearance of a "background" term proportional asymptotically to ${u}^{\ensuremath{\alpha}(0)\ensuremath{-}1}$ which has no analog in the equal-mass problem. We also conclude that a necessary condition for consistency is $\ensuremath{\alpha}(\ensuremath{\infty})&lt;0$.

Consistency Questions Raised by Simultaneous Mandelstam and Angular-Momentum analyticity

As the result of a study to determine a Reggepole formula with Mandelstam analyticity for the elastic scattering of two unequal-mass particles, we were led to raise the following question: What are the constraints, if any, that follow from assuming that scattering amplitudes satisfy both the Mandelstam representation and the condition of meromorphism in the right-half angular momentum plane? To put it another way, are Mandelstam and l-plane analyticity necessarily consistent in every case? We find that if there is to be consistency, one can conclude directly that the high-energy limit of the Regge-pole position, α(∞), is necessarily negative.

Talmi Transformation for Unequal-Mass Particles and Related Formulas

Three-dimensional polynomials which occur as coefficients of the exponential in the wavefunctions of the harmonic oscillator are used in nuclear physics and kinetic theory of gases. A generating function for these polynomials is used to simplify the calculation of several integrals. These include the integrals involving products of two and three polynomials and the coefficients of the Talmi transformation. Explicit formula in terms of recoupling coefficients of angular momentum theory are obtained.