PROTON-PROTON SCATTERING IN THE FIRST BORN APPROXIMATION WITH YUKAWA AND HULTHE’N POTENTIAL

ABSTRACT

This work is focused on proton-proton scattering in the first-Born approximation with the Yukawa and Hulthén potentials from which the scattering amplitudes and the differential scattering cross sections were evaluated. It also analyzed the variation of the differential scattering cross sections with the scattering angles at constant energy as well as the variation of the energy and the differential scattering cross sections of the potentials at constant scattering angles such that the exponential deformation of Hulthén potential was higher than that of Yukawa potential.

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INTRODUCTION

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1.1?? Particle Interaction

Particle physics is the branch of physics that deals with the study of the nature of particles that constitute matter (particles with mass) and radiation (massless particles). This branch usually investigates the irreducible smallest detectable particles and irreducible fundamental force fields necessary to explain them.

The current dominant theory explaining these fundamental particles and fields along with their dynamics is the standard model. Current understanding have explained the excitations of these particles; still the frame work for effective study and understanding of the interaction or scattering of waves and particles remains the scattering theory.

Most of the information about forces and interaction among atoms, neutrons and protons were provided by scattering experiments. In scattering event, a collimated beam of particles approach from infinity to a region where forces are exerted on the particles. The particles have well defined energies and momentum, having an assumed interaction region to be limited in extent and may result from the presence of target particle, an electric or magnetic field or some other sources of perturbation.

If there is a target particle, particles may undergo change in state, velocity and direction which is detected by devices that give us intensity as a function of scattering angle and possibly the energy of the? scattered particles if inelastic collisions are involved. Below, are examples that reveal the importance of scattering experiments:

??????????????????????????????? i.??????????? Scattering of nucleons of various energies from a nucleus gives an idea of the strength and range of nuclear forces and structure of the nucleus.

???????????????????????????? ii.??????????? Scattering of high energy electrons gives charge distribution in a nucleus and even in a nucleon.

?????????????????????????? iii.??????????? Scattering of neutrons from a nucleus reveals magnetic ????properties of the nucleus.

1.2. ??????Background of Study

Protons which are positively charged particles are made of quarks bounded by guns, although some scientists consider them as quantum fields. The solution to scattering is based on two approaches; the time independent (stationary) and time-dependent approaches. The former requires a partial differential equation for scattering of the setup wave and asymptotic solution is obtained. In the latter, transition from one state to another occurs with the help of time-dependent perturbation theory.

The transition state is specified by momentum (Aruldhas, 2009).The Schr?dinger equation is usually used in solving scattering problems. ?Other equivalent formulations are the Lippmann-Schwinger equation and Faddeev equations. The scattering solutions describe the long-term motion of free atoms, molecules, photons, electrons and protons.

1.3? ?Scattering

From the collision of beam of particles and target, scattering are classified as elastic or inelastic.

1.3.1. Elastic Scattering

In elastic collision, the total kinetic energy of a particle is conserved in the centre of mass frame, but its direction of propagation is modified by the potentials. This is achieved by transforming the centre of mass of the system, having its velocity changed in direction by some angle which is the only result of an elastic collision of mass system. The figure 1.1 below shows proton-proton elastic scattering.

???????????? Figure 1.1 Elastic proton-proton scattering???

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1.3.2. Inelastic Scattering

In an inelastic collision, there is a change (decrease or increase) in total kinetic energy that comes from the internal energy of the colliding partners. If the kinetic energy before collision is EI, after the collision it becomes; EF= EI + K, where K is the decrease in energy of the internal coordinates and EF is final kinetic energy. In the CM system, the final momentum must be zero, so the final velocities are in ration of the masses, and can be found from EF. In either case, the description of the collision is much simpler in the CM system and the final velocities can be determined by the conservation of energy and momentum, and the scattering angle ?.

Finally, the original velocity of the centre of mass is added to all velocities in order to find the result of the collision in the laboratory system. Figure 1.2 shows proton-proton inelastic scattering.

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Figure 1.2 Inelastic proton-proton scattering.

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1.4.?? Proton-Proton Scattering

The proton-proton scattering cross section is an effective area that quantifies the intrinsic likelihood of a scattering event when an incident beam strikes a target proton (Fernow, 1989). For a given event, the cross section (σ) is given by; σ = ?where σ is the cross section of the event (m2), μ is the attenuation coefficient due to the occurrence of the event (m-1), n is the number density of the target particles (m-3). The differential cross section ?is a function that quantifies the intrinsic rate at which the scattered projectiles can be detected at a given angle. It is equal to the absolute square of the scattering amplitude. There are two important techniques for calculating the scattering amplitude which are:

??????????????????????????????????? i.??????????? Partial wave analysis

????????????????????????????????? ii.??????????? Born approximation

For the scope of this work we will consider only the Born-approximation.

1.5.?????? Born?? Approximation

The Born approximation consists of taking the incident field at each point in the scatterer. It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small compared to the incident field in the scatterer [Sakurav, 1994 and Roger, 2002].

The characteristics of Born approximation include:

??????????????????? i.??????????? Scattering amplitude

???????????????? ii.??????????? Differential cross section

?????????????? iii.??????????? Total cross section

1.6.?? Hulthén And Yukawa Potentials

The Hulthén potential is considered as a modification of the Coulomb potential. It is well known that the general theory of scattering is not immediately applicable to the case of Coulomb potential because it decreases too slowly as the distance increases [Mott and Massey, 1949]. The Hulthén potential is a short-range potential and it has been widely used in several branches of physics such as nuclear particle physics, atomic physics, molecular physics and chemical physics [Bechler and Buhring, 1988].

In addition, the bound state energy spectrum and the corresponding wave functions for the Hulthén potential have been investigated by variety of techniques [Qiang and Gao, 2007].The generalized Hulthén potential is near to the standard Hulthén potential which can be solved. The form of the generalized Hulthén potential is stated below:

V(r) =

On the other hand, the Yukawa potential is a potential that rises from a massive scalar field. It describes force between two nuclear particle systems; a proton and a neutron of equal mass m held by an attractive short ranged force [Brian and Graham, 2008].The Yukawa potential is given by:

V(r) =

Where the screening parameter and Z is is the atomic number. This potential can be used to compute bound state energy and normalization of neutral atoms [Hamzavi and Mohavahedi, 2012].

1.7.? Significance of Study

The aim of this work is as follows:

??????????????????? i.??????????? To solve first Born approximation using Yukawa and Hulthén potentials for their scattering amplitudes respectively.

???????????????? ii.??????????? To evaluate the differential cross sections using the Yukawa and Hulthén potentials.

?????????????? iii.??????????? To check the variation of the differential cross sections with the scattering angles and the relationship between the differential cross section and the energy (E).

?????????????? iv.??????????? To compare both potentials based on their graph plot patterns.

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2.1. Proton –Proton Interaction

The physical nature of proton-proton interactions varies with energy as a result of the decrease of the coupling constant with energy. This fact could lie behind the inability to find a unique theory to describe particle creation mechanism in (P~P) interactions over the whole available energy range. Such creation goes through phonological models in low energy region to perturbative quantum chromo-dynamics [Daniel and Adler 2003. Regge, 1959. Gribov, 1967].

The experimentalist measures the reaction cross-sections and differential cross-sections. It is the work of the theorist to deduce what is interacting and what forces are involved. Collision of small particles at high energies are governed by quantum mechanics, where particles are waves, and systems of particles exist in certain allowed energy states, and collisions take place in certain allowed angular momentum states. A quantum mechanical analysis of the scattering process yields a set of coupled partial differential equations for the partial waves. This is too difficult to solve exactly; so theorist use mathematical tricks to device approximate solutions. These calculations are contained in Foldy, Tobocman and Thaler’s publications (Foldy, 1952). Foldy’s contribution to scattering theory uses rules in devising interactions which proceed by the exchange of virtual particle. Particularly the theory set limit on the values of the isotropic spin of the exchanged “entity.” The exact solutions of the non-relativistic equations with the central potential play an important role in quantum mechanics. The study of exponential-type potential has attracted the attention of physicists.

2.2. Applications Of Hulthén And Yukawa????????? Potentials

Hulthén potential is one of the short-ranged potential, studied in both relativistic and non-relativistic quantum mechanics. Another potential of great importance is Yukawa potential. This is also short-range and has application in high energy physics. Also in atomic and molecular physics, Yukawa potential is used as a screened Coulomb potential and it is also referred to as Debye-Huckel potential in plasma physics.

The Hulthén potential is one of the most important exponential model potential in physics and it is extensively used to describe interaction systems in nuclear and particle physics. Hence it is natural that research into this aspect attracted much attention and interest within the frame of work of relativistic and non relativistic quantum mechanics (Zhu, 2006).

However, Schr?dinger, Klein-Gordon and Dirac equations with the Hulthén potential cannot be analytically solved except for (S- state), due to the centrifugal term [Hammann and Messouber, 1996]. Consequently, to obtain arbitrary (L-state) solutions, several techniques are used to solve the Schr?dinger equation with this potential to obtain analytical solution for non-zero angular momentum. These methods include numerical and Quasi-analytical methods (Patil, 2001).

The generalized form of the Hulthén potential can be reduced to the standard Hulthén potential also to the Woods-Saxon potential by transforming the corresponding parameter; the latter plays an essential role in microscopic and nuclear physics since it can be used to describe the interaction of a nucleon with a heavy nucleus.

An alternative method of solving Schr?dinger equation has been proposed, called Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reducing to a generalized equation of hyper-geometric type. This method has been used to solve the Klein-Gordon, Duffin-Kemmer-Petiau equations with exponential-like potentials such as Woods-Saxon, Hulthén and Rosen-Morse [Berkdermir and Server, 2006].

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2.3. Other Applications of Hulthén Potential

Using the Born approximation method, as modified by the coulomb potential having the wave equation for the S-states can be solved exactly in the case of the Hulthén potential (Rosenfeld, 1957). Another concern is the determination of discrete bound from zeros of the Fredholm determinant for scattering problems. This has recently been done in work on field theories (Roger, 2013).

The Hulthén potential has the general form ?where a real positive parameter. A well known application of this potential in atomic and nuclear physics is the screened coulomb potential. In this case, it is written as; V(r)

Where Z is the screening coulomb charge and ?is the screening parameter. For small values of the screening parameter, it is approximated as V(r) , showing clearly the screening effect. The exact solutions of one dimensional Schr?dinger equation for some physical potentials are very important since they provide all necessary information for quantum system under consideration.

The bound state energy spectra of these potentials have been investigated by various methods such as factorization methods, supersymmetric quantum mechanics (Vahidi, 2011), Nikiforo-Uvarov method [Ikot and Akpabio, 2011].

The non relativistic Hulthén problem has a closed from analytic solution only for S-wave (l 0). Several techniques were used to obtain approximate solutions in the case where the angular momentum is not zero. Various methods were employed in obtaining the solution of the non relativistic Hulthén problem. Super-symmetric quantum mechanics, shape invariance path integration, and dynamical group are four methods among many which were used in the search for exact solutions of the wave equation with the Hulthén potential.

The relativistic problem, on the other hand, did not receive adequate attention. Most of the limited work on this problem was in the context of the Klein-Gordon equation (Znoji, 1981).

2.4.? Deformation Hulthén And Yukawa Potentials??

The relationship between the scattering angles and the differential scattering cross section has attracted the interest of many theorists and experimentalist.

Various works have displayed the exponential decay of the modified Coulomb potential (Griffiths, 2005). A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. The decrease in the value of the deformation parameter function increases the deformation of the potential. The potential energy function is defined such that the force of the field is equal to the negative of the gradient of the potential function.

Although the Yukawa potential involves an inverse function of distance with an exponential decay, it is not precisely the form that gives rise to the exponentially attenuated inverse distance squared form that has the proper form for a particle based field.

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