Abstracts

    An approach without partial wave expansion to calculate scattering of spin-0 and spin-1/2 particles in high energy regions and those governed by long range interactions
    I. Fachruddin and A. Salam
    Zeitschrift für Naturforschung A 79, 117-132 (2024), DOI: 10.1515/zna-2023-0248

    Abstract
    Scattering of spin-0 and spin-1/2 particles is formulated in momentum space based on basis states being not expanded in partial waves. No sequential calculations with increasing angular momentum are performed to reach physical convergence, which depends on the scattering energy and the interaction range. Both nonrelativistic and relativistic cases are described. We put forward for consideration the utilization of this approach. By taking some simple interaction models we show some advantages in calculations representing those of high energy scattering as well as those of scattering governed by long range interactions.

    0248  |   Back to Articles


    ...
    ...
    ...

    ...
    ...

    ...  |   Back to Articles


    Kaon-nucleon scattering in three-dimensional technique
    A. Salam and I. Fachruddin
    Proceeding of The 4th International Conference on Theoretical and Applied Physics (ICTAP) 2014, October 16-17, 2014, Denpasar, Bali, Indonesia
    AIP Conf. Proc. 1719, 030012 (2016), DOI: 10.1063/1.4943707

    Abstract
    Kaon-nucleon (KN) scattering is formulated in the three-dimensional (3D) momentum space, in which the basis state is not expanded into partial waves. Based on this basis the Lippmann-Schwinger equation for the T-matrix is evaluated. We obtain as final equation for the T-matrix elements a set of two coupled integral equations in two variables, which are the momentum’s magnitude and the scattering angle. Calculations for the differential cross section and some spin observables are shown, for which we employ a hadrons exchange model with the second order contributions only.

    030012  |   Back to Articles


    A momentum-space formulation without partial wave decomposition for scattering of two spin-half particles
    I. Fachruddin and A. Salam
    Proceeding of The 4th International Conference on Theoretical and Applied Physics (ICTAP) 2014, October 16-17, 2014, Denpasar, Bali, Indonesia
    AIP Conf. Proc. 1719, 030037 (2016), DOI: 10.1063/1.4943732

    Abstract
    A new momentum-space formulation for scattering of two spin-half particles, both either identical or unidentical, is formulated. As basis states the free linear-momentum states are not expanded into the angular-momentum states, the system’s spin states are described by the product of the spin states of the two particles, and the system’s isospin states by the total isospin states of the two particles. We evaluate the Lippmann-Schwinger equations for the T-matrix elements in these basis states. The azimuthal behavior of the potential and of the T-matrix elements leads to a set of coupled integral equations for the T-matrix elements in two variables only, which are the magnitude of the relative momentum and the scattering angle. Some symmetry relations for the potential and the T-matrix elements reduce the number of the integral equations to be solved. A set of six spin operators to express any interaction of two spin-half particles is introduced. We show the spin-averaged differential cross section as being calculated in terms of the solution of the set of the integral equations.

    030037  |   Back to Articles


    KN scattering with hadrons exchange potential in 3D technique
    A. Salam and I. Fachruddin
    Proceedings of The XV International Conference on Hadron Spectroscopy, November 4-8, 2013, Nara, Japan

    Abstract
    We have formulated the KN scattering in a three-dimensional (3D) technique. The Lippmann-Schwinger equation is evaluated in 3D basis states without partial-wave (PW) expansion. The result is a set of two coupled integral equations for T-matrix elements in two variables, namely the momentum magnitude and the scattering angle. For the KN interaction we employ an already derived hadrons exchange model with the second order contributions only. The parameters of the model will be later determined by fitting to the experimental data for the cross section and some spin observables.

    2013_196  |   Back to Articles


    NN scattering formulations without partial-wave decomposition
    I. Fachruddin and A. Salam
    Proceedings of The XV International Conference on Hadron Spectroscopy, November 4-8, 2013, Nara, Japan

    Abstract
    Recently a new three-dimensional (3D) formulation for scattering of two spin-half particles, either identical or unidentical, is presented. In the formulation the free linear-momentum states are not expanded into the angular-momentum states and the system's spin and isospin states, respectively, are described by the product of the spin and isospin states of the two particles. We apply this technique to calculate nucleon-nucleon (NN) scattering. But as NN interaction models are usually being parametrized regarding to the NN total isospin, we take the total isospin state, instead of the product of the isospin states of the two nucleons, as part of the 3D basis state. We evaluate the Lippmann-Schwinger equations for NN T-matrix elements in these 3D basis states and end up with a set of coupled integral equations for the NN T-matrix elements in two variables, namely the magnitude of the relative momentum and the scattering angle. We show as an example the spin-averaged differential cross section, which can be calculated directly from the solution of the set of the integral equations. The recently developed 3D formulation also introduces a set of six spin operators to express any interaction of two spin-half particles. We show the Bonn NN potential in terms of these six spin operators.

    2013_142  |   Back to Articles


    KN scattering in 3D formulation
    A. Salam and I. Fachruddin
    Few-Body Systems 54, 1625 (2013), DOI: 10.1007/s00601-012-0557-1

    Abstract
    KN scattering is formulated in three-dimensional (3D) momentum space. A direct product of the relative-momentum state and the spin state is used as the basis state. The spin quantization axis is chosen along the z-axis. The interaction for the KN system is assumed to take the Yukawa-type. It consists of two terms, the central and the spin-orbit one. Calculations for the cross section based on this technique are shown, as well as comparison with the standard partial-wave calculations.

    Back to Articles


    Scattering of two spin-half particles in a three-dimensional approach
    I. Fachruddin
    Few-Body Systems 54, 1621 (2013), DOI: 10.1007/s00601-012-0513-0

    Abstract
    Scattering of two spin-1/2 particles is formulated in a three-dimensional approach based on a simple three-dimensional momentum-spin basis. Both cases of identical and nonidentical particles are considered. The azimuthal behaviour of the potential and of the T-matrix elements leads to a set of integral equations for the T-matrix elements in two variables, i.e. the momentum magnitude and the scattering angle. Observables can be directly calculated from these T-matrix elements. Some symmetry relations for the T-matrix elements reduce the number of equations to be solved.

    Back to Articles


    Scattering of two spinless particles in 3D formulation with coulomb admixtures
    F. Maulida and I. Fachruddin
    Few-Body Systems 54, 217 (2013), DOI: 10.1007/s00601-012-0354-x

    Abstract
    Scattering of two spinless charge particles for simple forces including coulomb admixtures is calculated without partial wave decomposition. The coulomb interaction being taken is of the type of screened coulomb potential. For the forces range are not infinite, the standard scattering theory is applied. The differential and total cross section is calculated and coulomb effects are shown.

    1203.0358  |   Back to Articles


    Scattering of a spin-1/2 particle off a spin-0 target in a simple three-dimensional basis
    I. Fachruddin and A. Salam
    Few-Body Systems 54, 221 (2013), DOI: 10.1007/s00601-012-0353-y

    Abstract
    Scattering of a spin-1/2 particle off a spin-0 target is formulated based on a simple three-dimensional momentum-spin basis. The azimuthal behaviour of both the potential and the T-matrix elements leads to a set of integral equations for the T-matrix elements in two variables only, namely the momentum's magnitude and the scattering angle. Some symmetry relations for the potential and the T-matrix elements reduce the number of the integral equations to be solved by a factor of one half. A complete list of the spin observables in terms of the two-dimensional T-matrix elements is presented.

    1203.0352  |   Back to Articles


    Recent developments of a three-dimensional description of the NN system
    R. Skibinski, J. Golak, D. Rozpedzik, K. Topolnicki, H. Witala, W. Glöckle, A. Nogga, E. Epelbaum, H. Kamada, Ch. Elster, I. Fachruddin
    Few-Body Systems 50, 279 (2011), DOI: 10.1007/s00601-010-0204-7

    Abstract
    A recently developed three-dimensional formulation of nucleon-nucleon (NN) scattering is briefly presented. Here the NN t-matrix is represented by six spin-momentum operators accompanied by six scalar functions of momentum vectors. A numerical example for the NN scattering cross section is given.

    Back to Articles


    Two-nucleon system in three dimensions
    J. Golak, W. Glöckle, R. Skibinski, H. Witala, D. Rozpedzik, K. Topolnicki, I. Fachruddin, Ch. Elster, A. Nogga
    Phys. Rev. C81, 034006 (2010)

    Abstract
    A recently developed formulation for treating two- and three-nucleon bound states in a three-dimensional formulation based on spin-momentum operators is extended to nucleon-nucleon scattering. Here the nucleon-nucleon t-matrix is represented by six spin-momentum operators accompanied by six scalar functions of momentum vectors. We present the formulation and provide numerical examples for the deuteron and nucleon-nucleon scattering observables. A comparison to results from a standard partial wave decomposition establishes the reliability of this new formulation.

    1001.1264  |   Back to Articles


    3N scattering in a three-dimensional operator formulation
    W. Glöckle, I. Fachruddin, Ch. Elster, J. Golak, R. Skibinski, and H. Witala
    Eur. Phys. J. A 43, 339–350 (2010)

    Abstract
    A recently developed formulation for a direct treatment of the equations for two- and three- nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev multiple scattering series in which order by order the spin degrees can be treated analytically leading to 3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion production threshold and applications of chiral perturbation theory 3N forces, which are to be most efficiently treated directly in such three-dimensional formulation without having to expand these forces into a partial-wave basis.

    0910.1177  |   Back to Articles


    A new way to perform partial-wave decompositions of few-nucleon forces
    J. Golak, D. Rozpedzik, R. Skibinski, K. Topolnicki, H. Witala, W. Glöckle, A. Nogga, E. Epelbaum, H. Kamada, Ch. Elster, and I. Fachruddin
    Eur. Phys. J. A 43, 241–250 (2010)

    Abstract
    We formulate a general and exact method of partial-wave decomposition (PWD) of any nucleon- nucleon (NN) potential and any three-nucleon (3N) force. The approach allows one to efficiently use symbolic algebra software to generate the interaction-dependent part of the program code calculating the interaction. We demonstrate the feasibility of this approach for the one-boson exchange BonnB potential, a recent nucleon-nucleon chiral force and the chiral two-pion-exchange three-nucleon force. In all cases very good agreement between the new and the traditional PWD is found. The automated PWD offered by the new approach is of the utmost importance in view of future applications of numerous chiral N3LO contributions to the 3N force in three-nucleon calculations.

    0911.4173  |   Back to Articles


    Treatment of two nucleons in three dimensions
    I. Fachruddin, Ch. Elster, J. Golak, R. Skibinski, W. Glöckle, H. Witala
    Proceeding of The 19th International IUPAP Conference on Few-Body Problems in Physics, August 31 - September 5, 2009, Bonn, Germany
    EPJ Web Conf. 3, 05021 (2010), DOI: 10.1051/epjconf/20100305021

    Abstract
    We extend a new treatment proposed for two-nucleon (2N) and three-nucleon (3N) bound states to 2N scattering. This technique takes momentum vectors as variables, thus, avoiding partial wave decomposition, and handles spin operators analytically. We apply the general operator structure of a nucleon-nucleon (NN) potential to the NN T-matrix, which becomes a sum of six terms, each term being scalar products of spin operators and momentum vectors multiplied with scalar functions of vector momenta. Inserting this expansions of the NN force and T-matrix into the Lippmann-Schwinger equation allows to remove the spin dependence by taking traces and yields a set of six coupled equations for the scalar functions found in the expansion of the T-matrix.

    05021  |   Back to Articles


    A formulation without partial wave decomposition for scattering of spin-1/2 and spin-0 particles
    I. Abdulrahman and I. Fachruddin
    Mod. Phys. Lett. A24, 843 (2009)

    Abstract
    A new technique has been developed to calculate scattering of spin-1/2 and spin-0 particles. The so called momentum-helicity basis states are constructed from the helicity and the momentum states, which are not expanded in the angular momentum states. Thus, all angular momentum states are taken into account. Compared with the partial-wave approach this technique will then give more benefit especially in calculations for higher energies. Taking as input a simple spin-orbit potential, the Lippman-Schwinger equations for the T-matrix elements are solved and some observables are calculated.

    1003.3711  |   Back to Articles


    Scattering of spin-zero and spin-half particles in momentum- helicity basis
    I. Fachruddin and I. Abdulrahman
    Proceeding of The Second Asian Physics Symposium, November 29-30, 2007, Bandung, Indonesia

    Abstract
    Scattering of 2 particles of spin 0 and 1/2 is evaluated based on a basis constructed from the momentum and the helicity states (the momentum-helicity basis). This shortly called three-dimensional (3D) technique is a good alternative to the standard partial wave (PW) technique especially for higher energies, where PW calculations may become not feasible. Taking as input a simple spin-orbit potential model we calculate as an example the spin averaged differential cross section and polarization.

    0712.1101  |   Back to Articles


    Three-nucleon scattering at intermediate energies
    I. Fachruddin, Ch. Elster, W. Glöckle
    Proceeding of The Third Asia-Pacific Conference on Few-Body Problem in Physics, July 26-30, 2005, Nakhon Ratchasima, Thailand

    Abstract
    By means of a technique, which does not employ partial wave (PW) decompositions, the nucleon-deuteron break-up process is evaluated in the Faddeev scheme, where only the leading order term of the amplitude is considered. This technique is then applied to calculate the semi-exclusive proton-deuteron break-up reaction d(p,n)pp for proton laboratory energies Elab of a few hundred MeV. A comparison with PW calculations is performed at 197 MeV projectile energy. At the same energy rescattering processes, which are not included in the 3D calculations yet, are shown to be still important in the full Faddeev PW calculations, especially for the cross section and the analyzing power Ay. Next, kinematical relativistic effects are investigated for projectile energies up to about 500 MeV. At the higher energies, those relativistic effects start not to be negligible, especially in the peak of the cross section.

    0708.1535  |   Back to Articles


    Operator form of 3H (3He) and its spin structure
    I. Fachruddin, W. Glöckle, Ch. Elster, A. Nogga
    Phys. Rev. C69, 064002 (2004)

    Abstract
    An operator form of the 3N bound state is proposed. It consists of eight operators formed out of scalar products in relative momentum and spin vectors, which are applied on a pure 3N spin 1/2 state. Each of the operators is associated with a scalar function depending only on the magnitudes of the two relative momenta and the angle between them. The connection between the standard partial wave decomposition of the 3N bound state and the operator form is established, and the decomposition of these scalar function in terms of partial wave components and analytically known auxiliary functions is given. That newly established operator form of the 3N bound state exhibits the dominant angular and spin dependence analytically. The scalar functions are tabulated and can be downloaded. As an application the spin dependent nucleon momentum distribution in a polarized 3N bound state is calculated to illustrate the use of the new form of the 3N bound state.

    nucl-th/0402015  |   Back to Articles


    The Nd break-up process in leading order in a three-dimensional approach
    I. Fachruddin, Ch. Elster and W. Glöckle
    Phys. Rev. C68, 054003 (2003)

    Abstract
    A three-dimensional approach based on momentum vectors as variables for solving the three nucleon Faddeev equation in first order is presented. The nucleon-deuteron break-up amplitude is evaluated in leading order in the NN T-matrix, which is also generated directly in three dimensions avoiding a summation of partial wave contributions. A comparison of semi-exclusive observables in the d(p,n)pp reaction calculated in this scheme with those generated by a traditional partial wave expansion shows perfect agreement at lower energies. At about 200 MeV nucleon laboratory energies deviations in the peak of the cross section appear, which may indicate that special care is required in a partial wave approach for energies at and higher than 200 MeV. The role of higher order rescattering processes beyond the leading order in the NN T-matrix is investigated with the result, that at 200 MeV rescattering still provides important contributions to the cross section and certain spin observables. The influence of a relativistic treatment of the kinematics is investigated. It is found that relativistic effects become important at projectile energies higher than 200 MeV.

    nucl-th/0307011  |   Back to Articles


    The proton-deuteron break-up process in a three-dimensional approach
    I. Fachruddin, Ch. Elster and W. Glöckle
    Mod. Phys. Lett. A18, 452 (2003)

    Abstract
    The pd break-up amplitude in the Faddeev scheme is calculated by employing a three-dimensional method without partial wave decomposition (PWD). In a first step and in view of higher energies only the leading term is evaluated and this for the process d(p,n)pp. A comparison with the results based on PWD reveals discrepancies in the cross section around 200 MeV. This indicates the onset of a limitation of the partial wave scheme. Also, around 200 MeV relativistic effects are clearly visible and the use of relativistic kinematics shifts the cross section peak to where the experimental peak is located. The theoretical peak height, however, is wrong and calls first of all for the inclusion of rescattering terms, which are shown to be important in a nonrelativistic full Faddeev calculation in PWD.

    nucl-th/0211069  |   Back to Articles


    New forms of deuteron equations and wave function representations
    I. Fachruddin, Ch. Elster and W. Glöckle
    Phys. Rev. C63, 054003 (2001)

    Abstract
    A recently developed helicity basis for nucleon-nucleon (NN) scattering is applied to the deuteron bound state. Here the total spin of the deuteron is treated in such a helicity representation. For the bound state, two sets of two coupled eigenvalue equations are developed, where the amplitudes depend on two and one variable, respectively. Numerical illustrations based on the realistic Bonn-B NN potential are given. In addition, an `operator form' of the deuteron wave function is presented, and several momentum dependent spin densities are derived and shown, in which the angular dependence is given analytically.

    nucl-th/0101009  |   Back to Articles


    Nucleon-nucleon scattering in a three-dimensional approach
    I. Fachruddin, Ch. Elster and W. Glöckle
    Nucl. Phys. A689, 507c (2001)

    Abstract
    The nucleon-nucleon t-matrix is calculated directly as function of two vector momenta for different realistic NN potentials. The angular and momentum dependence of the full amplitude is studied and NN observables are calculated.

    nucl-th/0104027  |   Back to Articles


    Nucleon-nucleon scattering in a three dimensional approach
    I. Fachruddin, Ch. Elster and W. Glöckle
    Phys. Rev. C62, 044002 (2000)

    Abstract
    The nucleon-nucleon (NN) t-matrix is calculated directly as function of two vector momenta for different realistic NN potentials. To facilitate this a formalism is developed for solving the two-nucleon Lippmann-Schwinger equation in momentum space without employing a partial wave decomposition. The total spin is treated in a helicity representation. Two different realistic NN interactions, one defined in momentum space and one in coordinate space, are presented in a form suited for this formulation. The angular and momentum dependence of the full amplitude is studied and displayed. A partial wave decomposition of the full amplitude it carried out to compare the presented results with the well known phase shifts provided by those interactions.

    preprint (pdf)  |   Back to Articles


    Helicity formalism for NN scattering without partial wave decomposition
    I. Fachruddin and W. Glöckle
    Few-Body Systems Suppl. 12, 1-4 (2000)

    Abstract
    At intermediate energies it appears advantageous to work without partial wave decomposition and use directly the momentum vectors. We derived a finite set of coupled Lippmann-Schwinger equations using helicities. They can be used for any type of realistic NN forces, which are given in operator form. As an example we have chosen the Bonn OBEPR NN potential.

    nucl-th/9910045  |   Back to Articles


    A three-dimensional momentum space formulation for the NN system and the Nd break-up process
    I. Fachruddin
    PhD thesis (2003), supervisor: Prof. Dr. W. Glöckle, Ruhr-Universität-Bochum, Bochum, Germany

    Abstract
    A technique to perform few-nucleon calculations in momentum space without employing partial wave (PW) decompositions is developed. It is shortly called the 3D technique, and is intended to be a viable alternative to the successful PW technique, especially for higher energies.

    The development begins with the nucleon-nucleon (NN) system, where antisymmetric momentum-helicity basis states are defined, which are constructed using momentum vector states and helicity states of NN total spin. Appropiate for the momentum-helicity basis a set of six independent operators is defined to express any NN potential, which is given in operator form. Representative potentials are the modern AV18 and Bonn-B potentials, which are used in this work. The 3D technique is applied to both NN scattering and the deuteron. Comparison with PW calculations is performed to test the formulation and numerical realization of the 3D technique. In addition for the deuteron, an "operator form" of the deuteron wave function is presented, allowing for investigating some momentum dependent spin densities with analytic angular behavior.

    The development and application of the 3D technique is extended to the Nd break-up process, in which the Faddeev's scheme is used. In this work only the leading term of the full Nd break-up amplitude is considered to describe the process at higher energies, beyond ~ 200 MeV projectile laboratory energy. For simplicity the deuteron state is kept being expanded in its partial wave components, s and d waves. The leading term of the amplitude in the 3N basis states is derived in a so called physical representation, where spins and isospins of the individual nucleons are taken, and the spins are quantized along an arbitrary but fixed z axis. For the NN system a connection between the physical representation and that in the momentum-helicity basis has been obtained. This leads then to an expression for the leading term of the full Nd break-up amplitude in terms of the NN T-matrix element in the momentum-helicity basis, with which 3N observables are calculated. These are the differential cross section, the neutron polarization, the proton analyzing power and the polarization transfer coefficients. The formulation is applied to the (p,n) charge exchange reaction in the semi-exclusive pd break-up process d(p,n)pp for energies up to ~ 500 MeV. For energies below ~ 200 MeV comparison with PW calculations is performed, as well as with the ones taking the full Nd break-up amplitude to see rescattering effects. Finally relativistic kinematics is introduced into the formulation, allowing to observe some relativistic effects. Comparisons are also performed with experimental data.

    Results in this work show that the 3D technique has proven to be a good alternative to the PW decomposition and appears to be necessary at higher energies, where the PW technique is no longer feasible. In contrast to the PW decomposition the 3D technique requires much less algebraic work. For lower energies, where the PW calculations are still reliable, the 3D calculations show perfect agreement with the PW calculations.

    full text  |   Back to Articles



     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

Free Web Hosting