High-momentum Components Research Articles

Method for performing LCAO band structure calculations in crystalline solids: Application to rubidium

In this work we present a method for performing LCAO (linear combination of atomic orbitals) band structure calculations (tight binding) in crystalline solids. In the first part of the article we apply group theoretical methods to the establishment of a least-squares scheme for the calculation of the matrix of the crystal potential: This scheme is based on a well-defined choice of independent parameters for the Bloch vector-dependent matrix elements and on the considerations of the symmetries between these independent parameters and their Fourier coefficients. In the second part of this work we deal with the representation of the matrices of the identity operator and of the operator of the kinetic energy by linear combinations in terms of two center integrals: We express these linear combinations by a closed formula, which can be easily programmed on a computer, and we mention a method by which the two center integrals can be evaluated numerically fast and accurately. Finally we apply our theory to the derivation of numerical results: We determine the electronic states and the high-momentum components of Compton rates in the alkali metal rubidium and we compare the results obtained with those of augmented plane-wave (APW) calculations. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 212–225, 2000

Dual transformation for wave packet dynamics: Application to Coulomb systems

A dual transformation technique that can deal with awkward Coulomb potentials is developed for electronic wave packet dynamics. The technique consists of the variable transformation of the Hamiltonian and the transformation of the wave function with a normalization constraint. The time evolution is carried out by the alternating-direction implicit method. The operation of the transformed Hamiltonian on the wave function is implemented by using three- and five-point finite difference formulas. We apply it to the H atom and a realistic three-dimensional (3D) model of H2+. The cylindrical coordinates ρ and z are transformed as ρ=f(ξ) and z=g(ζ), where ξ and ζ are scaled cylindrical coordinates. Efficient time evolution schemes are provided by imposing the variable transformations on the following requirements: The transformed wave function is zero and analytic at the nuclei; the equal spacings in the scaled coordinates correspond to grid spacings in the cylindrical coordinates that are small near the nuclei (to cope with relatively high momentum components near the nuclei) and are large at larger distances thereafter. No modifications of the Coulomb potentials are introduced. We propose the form f(ξ)=ξ[ξn/(ξn+αn)]ν. The parameter α designates the ρ-range where the Coulomb potentials are steep. The n=1 and ν=12 transformation provides most accurate results when the grid spacing Δξ is sufficiently small or the number of grid points, Nξ, is large enough. For small Nξ, the n=12 and ν=1 transformation is superior to the n=1 and ν=12 one. The two transformations are also applied to the dissociation dynamics in the 3D model of H2+. For the n=12 and ν=1 transformation, the main features of the dynamics are well simulated even with moderate numbers of grid points. The validity of the two transformations is also enforced by the fact that the missing volume in phase space decreases with decreasing Δξ.

Semiclassical distorted wave model with Wigner transform of one-body density matrix

A semiclassical distorted wave (SCDW) model with realistic single particle wave functions in a finite range single particle potential is presented for multistep direct processes in nucleon inelastic scattering and charge exchange reactions to the continuum, making use of the Wigner transform of a one-body density matrix. The higher momentum components of target nucleons are properly taken into account in comparison with the previous SCDW model with a local density Fermi gas model. The new SCDW model still gives a simple closed-form expression for the cross section with no free adjustable parameter which allows a simple intuitive interpretation, just as in the previous SCDW model. This model is applied to the analyses of multistep direct processes of ${}^{90}\mathrm{Zr}{(p,p}^{\ensuremath{'}}x)$ reactions at incident energies of 80 and 160 MeV. The calculated double differential cross sections including up to three-step processes are compared with experimental data. The calculated cross sections at backward angles are larger than those given by the previous calculations, and the agreement with experimental data is much improved. Discussions on the mechanism of this improvement are given in terms of the momentum distribution of target nucleons.

Polarization correlation coefficient for the [formula omitted]( [formula omitted]) 4He reaction

A new approach to study the high momentum component of deuteron D-state wave function by measuring the polarization correlation coefficient, C // [ =1+ 1 4 A yy+A xx + 3 4 C y,y+C x,x ], for the 3 He → ( d → ,p ) 4He reaction is proposed. The first measurement of C // has been carried out at E d =270 MeV by using a newly constructed spin-exchange-type polarized 3He target. The obtained value is 0.223±0.044(statistical)±0.037(systematic). DWBA calculations are unable to reproduce the present results. Each component of C // is compared with data for the d+ p backward scattering and the deuteron inclusive breakup. A similarity in polarization observables is found among these reactions.

Dππdecay of thed*dibaryon

The $d^* \to d \pi \pi$ partial decay width has been calculated in a wave-function model for $d^*$ and $d$. It is found to be smaller than a previous estimate by a factor of 7. A previously proposed dependence on dibaryon sizes is confirmed. Additional dependence on the high-momentum components of dibaryon wave functions is found.

The large momentum transfer reaction 12C( p,2 p + n) as a new method for measuring short range NN correlations in nuclei

The reaction 12C( p,2 p + n) was measured for momentum transfers of 4.8 and 6.2 (GeV/c) 2 at beam momenta of 5.9 and 7.5 GeV/c. We measured the quasi-elastic reaction ( p,2 p) at θ cm ⋍ 90°, in a kinematically complete measurement. The neutron momentum was measured in triple coincidence with the two emerging high momentum protons. We present the correlation between the momenta of the struck target proton and the neutron. The events are associated with the high momentum components of the nuclear wave function. We present sparse data which, combined with a quasi elastic description of the ( p,2 p) reaction and kinematical arguments, point to a novel way for isolating two-nucleon short range correlations.

Structure of 3He in backward elastic p- 3He scattering

Backward elastic p- 3He-scattering at incident proton kinetic energies T p > 1 GeV is investigated in the framework of the np-pair transfer mechanism and triangular diagram of one-pion exchange with a subprocess pd → 3 Heπ 0 using a realistic three-body wave function of the 3He nucleus. It is found that the np-pair transfer mechanism dominates owing to a rich high momentum component of the 3He wave function. We show that the experimental cross section of this process is defined mainly by the values of the Faddeev component of the 3He wave function, ϕ 23( q 23, p 1), at high relative momenta q 23 > 0.6 GeV/ c of the NN-pair in the 1 S 0 state and at low spectator momenta p 1 ≤ 0.1 GeV/ c. The spin-spin correlation parameter is calculated in the framework of the dominating mechanism for the case of a polarized target and beam. Rescatterings in the initial and final states are taken into account. Comparison with the pd → dp process is performed.

Backward elastic pd scattering at intermediate energies

The theoretical approaches to the description of backward elastic pd scattering at energies 0.5–2 GeV are reviewed. In addition to the mechanisms of neutron exchange and single pN scattering, the role of N * -isobar exchange is studied using the six-quark model for the dNN * vertex. Relativistic effects are taken into account within the Hamiltonian dynamics of systems with a fixed number of particles. The important role of Glauber rescatterings in the initial and final states in the neutron-transfer mechanism is displayed. Special attention is paid to the mechanism of Δ-isobar excitation, which, despite the large momentum transfer, is rather weakly sensitive to the high-momentum components of the deuteron wave function and dominates at incident protonenergies T p =0.5–1 GeV . The reaction pp→pnπ + is studied in the Δ-resonance region in order to determine the parameters of the amplitudes for NN↔NΔ transitions needed to calculate the Δ-isobar contribution to the process pd→dp. It is shown that the amplitude of the Δ-resonance mechanism for the process pd→dp is very sensitive to the parameters of the πNN, πNΔ, ρNN, and ρNΔ vertex form factors. The role of more exotic mechanisms is briefly discussed.

Collisional Transitory Enhancement of the High Momentum Components of a Quantum Wave Packet

A classically forbidden, transitory enhancement of the high positive momentum components of a quantum wave packet during its collision with a potential barrier is described. A quantity is defined to measure its importance; the universality of the effect is justified, and its main features are studied with the aid of an analytically solvable model. Its relation to other ``anomalously high velocities'' is also examined.

Backward elastic p[sup 3]He scattering and structure of the [sup 3]He nucleus

Mechanisms of backward elastic p 3 He scattering at energies of 0.5–2 GeV are studied using the 3-and 4-dimensional nonrelativistic diagrammatic technique. In the first step the structure of the 3 He nucleus is described using the approximation of the two-body d+p configuration, and then the three-body 5-channel wave function from solving the Faddeev equations is used. It is shown that the inclusion of the three-particle properties of the 3 He structure is crucial. Relativistic effects are taken into account using the relativistic quantum mechanics of systems with fixed numbers of particles, and are found to be unimportant for the dominant mechanism. In addition to mechanisms containing information about the structure of the 3 He nucleus at small relative internucleon separations, special attention is paid to the mechanism of sequential np -pair transfer. This mechanism in fact does not require high-momentum components of the nuclear wave functions in spite of the large momentum transfer, and it dominates under certain kinematical conditions. It is shown that when rescattering is taken into account, this mechanism allows the successful description of the available experimental data on the energy and angular dependence of the differential cross section at ∼1–2 GeV without the use of free parameters. The clearly manifested effects due to the Â{(NN)( 1 S 0 )+N} component of the wave function of the 3 He nucleus and Glauber rescatterings in the initial and final states are displayed. The results of a low-energy analysis using the unitarized K -matrix approach are also presented.

High-Missing-Momentum Components in the4He(e,e′p)3HReaction

The cross section of the He-4(e, e'p)H-3 reaction has been measured for missing momenta 220 less than or equal to p(m) less than or equal to 690 MeV/c to study high-momentum components of the nuclear wave function and the reaction mechanism for this transition. The zero predicted in the plane-wave impulse approximation (PWIA) cross section, due to the underlying pt momentum distribution, is found to be washed out. Three types of calculations indicate that this is caused by final-state interactions and contributions from two-body currents. The calculations reproduce the high p(m) (600-690 MeV/c) data, although due to different ingredients in the models.

Yields and transverse momenta of the 6Li fragments in the emulsion at 4.5 GeV/c per nucleon

We present the results of the study of interaction of the relativistic 6Li nucleus with the momentum 4.5 GeV/c per nucleon with the photoemulsion. Yields of the 1H (3He) and 2H (4He) isotopes due to the fragmentation of 6Li are found to be almost equal. Cross sections for the charge exchange and pickup reactions are found to be σexch=9±2 mb. The distributions of the fragment transverse momenta projected onto the emulsion plane are used to obtain the nucleon Fermi momentum of 6Li, PF, this value being equal to 129±8 MeV/c. The high momentum component in the transverse momentum distributions of 3He and 4He isotopes is observed.

Polarized 3He project at RIKEN

A new approach to study the high momentum component of deuteron D-state wave function by measuring the polarization correlation coefficient, C ∥, for the 3 H e( d , p) 4He reaction has been proposed. In order to carry out the experiment, a spin exchange type polarized 3He target was constructed at RIKEN Accelerator Research Facility (RARF). The density of the target was 2.2 × 10 20 cm −3 (8.2 amagats), and the polarization was about 10%. The spin flip system worked well. The experiment has been carried out in November, 1996, at RARF, using a polarized deuteron beam of 270 MeV.

Backward elastic p 3He scattering at energies 1–2 GeV

The two-body transfer amplitude for the rearrangement process i + { jkl} → j + { ikl}is constructed on the basis of technique of four-dimensional covariant nonrelativisitic graphs. The developed formalism is applied to describing backward elastic p 3He scattering in the energy range 0.5–1.7 GeV. Numerical calculations performed using the 5-channel wave function of the 3He nucleus show that the transfer of a noninteracting np-pair dominates and explains satisfactorily the energy and angular dependence of the differential cross section at energies 0.9–1.7 GeV. A weak sensitivity to high momentum components of the 3He wave function, in spite of large momentum transfer as well as a very important role of rescatterings in the initial and final states, are established.

Electroinduced two-nucleon knockout and correlations in nuclei

A model to calculate cross sections for electroinduced two-nucleon emission from finite nuclei is presented. Short-range correlations in the wave functions and meson-exchange contributions to the photoabsorption process are implemented. Effects of the short-range correlations are studied with the aid of a perturbation expansion method with various choices of the Jastrow correlation function. The model is used to investigate the relative importance of the different reaction mechanisms contributing to the A(e,e′pn) and A(e,e′pp) process. Representative examples for the target nuclei 12C and 16O and for kinematical conditions accessible with contemporary high-duty cycle electron accelerators are presented. A procedure is outlined to calculate the two-nucleon knockout contribution to the semi-exclusive (e,e′p) cross section. Using this technique we investigate to what extent far semi-exclusive (e,e′p) reactions can be used to detect high-momentum components in the nuclear spectral function.

On the Coulomb and higher-order sum rules in the relativistic Fermi gas

Two different methods for establishing a space-like Coulomb sum rule for the relativistic Fermi gas are compared. Both of them divide the charge response by a normalizing factor such that the reduced response thus obtained fulfills the sum rule at large momentum transfer. To determine the factor, in the first approach one exploits the scaling property of the longitudinal response function, while in the second one enforces the completeness of the states in the space-like domain via the Foldy-Wouthuysen transformation. The energy-weighted and the squared-energy-weighted sum rules for the reduced responses are explored as well and the extension to momentum distributions that are more general than a step-function is also considered. The two methods yield reduced responses and Coulomb sum rules that saturate in the non-Pauli-blocked region, which can hardly be distinguished for Fermi momenta appropriate to atomic nuclei. Notably the sum rule obtained in the Foldy-Wouthuysen approach coincides with the well-known non-relativistic one. Only at quite large momentum transfers (say 1 GeV/ c) does a modest softening of the Foldy-Wouthuysen reduced response with respect to that obtained in the scaling framework show up. The two responses have the same half-width to second order in the Fermi momentum expansion. However, when distributions extending to momenta larger than that at the Fermi surface are employed, then in both methods the Coulomb sum rule saturates only if the normalizing factors are appropriately modified to account for the high momentum components of the nucleons.

Local-density-derived semiempirical nonlocal pseudopotentials for InP with applications to large quantum dots

In the same way that atomic calculations have been used previously to extract bare ionic pseudopotentials, self-consistent bulk calculations can be used to construct screened atomic pseudopotentials. We use such a method to construct screened nonlocal atomic pseudopotentials for InP. A series of bulk, local-density-approximation (LDA) calculations are performed on a few InP crystal structures, covering a range of unit-cell volumes, to produce bulk potentials {${\mathrm{V}}_{\mathrm{LDA}}$ (G)}. By solving a set of linear equations, we extract from these crystalline potentials the corresponding screened atomic 'spherical LDA' (SLDA) potentials ${\mathrm{v}}_{\mathrm{SLDA}}^{\mathrm{\ensuremath{\alpha}}g}$(|q|) for sites \ensuremath{\alpha}=In or P. In combination with the nonlocal part of the usual LDA pseudopotentials, these SLDA potentials give band structures and wave functions that are virtually indistinguishable from the self-consistent LDA results for bulk InP. In the next step, we apply linear changes to the local SLDA potentials (while keeping the nonlocal potentials at their LDA values), to fit the band structures to experiment. Interestingly, this removal of LDA eigenvalue errors requires only small and subtle changes in the potential---mostly an upshift in the region near the cation core, with nearly no change at the bond center. Furthermore, the linear changes to the SLDA potentials result mostly in an upshift of the conduction bands with little effect on the valence bands. Because only small changes in the potential suffice to fit the bands to experimental results, the wave functions remain virtually unchanged relative to those in the original LDA calculation. Hence, we obtain semiempirical pseudopotentials which can produce ab initio LDA-quality wave functions with experimentally measured band structures, effective masses, and deformation potentials. The potentials obtained here were deposited on an FTP site and can be used by interested readers. Since the resulting pseudopotentials are 'soft' (with small high-momentum components), they can be applied within a plane-wave basis in combination with a Gaussian correction to large systems for which LDA calculations are prohibitively expensive. As an illustration, we apply our InP screened atomic pseudopotentials to calculate quantum size effects on the band gaps of InP dots with sizes up to 700 atoms. Good agreement is found between the theoretical and the experimental band gaps. Fitting the calculated band gaps ${\mathrm{E}}_{\mathrm{g}}$ (in unit of eV) versus the effective dot sizes D (in unit of \AA{}) gives ${\mathrm{E}}_{\mathrm{g}}$ =1.45+37.295/${\mathrm{D}}^{1.16}$ . This prediction differs significantly from the quadratic size dependence ${\mathrm{D}}^{\mathrm{\ensuremath{-}}2.0}$ expected from simple effective-mass theory.

Weak decay form factors of heavy pseudoscalar mesons within a light-front constituent quark model

The transition form factors describing the semileptonic decays of heavy pseudoscalar mesons are investigated within a relativistic constituent quark model formulated on the light-front. For the first time the form factors are calculated in the whole accessible kinematical region, adopting meson wave functions derived from various effective q q interactions able to describe the meson mass spectra. It is shown that the decay rates of the B → Dℓ ν ℓ, D → Kℓ ν ℓ and D → πℓ ν ℓ weak decays are mainly governed by the effects of the confinement scale, whereas the B → πℓ ν ℓ decay rate is sensitive to the high-momentum components generated in the meson wave functions by the effective one-gluon-exchange interaction. Our results are consistent with available experimental data and predictions of lattice QCD calculations and QCD sum rules.

SOME CONSEQUENCES OF DRESSING NUCLEONS

A representation of nucleon spectral functions in nuclear matter is developed. Various contributions are identified including a quasiparticle term, broad background contributions below and above the Fermi energy, and a high-energy tail which has no significant momentum dependence. A convenient analytical energy dependence supplemented by a numerical dependence on momentum allows computationally tractable studies of the consequences associated with the considerable dressing of the nucleons. Applications are presented for high-momentum components as a function of energy, density of states, dressed particle-hole response, and the dressed particle-particle/hole-hole propagator.

Charge form factor of pi and K mesons.

The charge form factor of $\ensuremath{\pi}$ and $K$ mesons is evaluated adopting a relativistic constituent quark model based on the light-front formalism. The relevance of the high-momentum components of the meson wave function, for values of the momentum transfer accessible to CEBAF energies, is illustrated. The predictions for the elastic form factor of $\ensuremath{\pi}$ and $K$ mesons are compared with the results of different relativistic approaches, showing that the measurements of the pion and kaon form factors planned at CEBAF could provide information for discriminating among various model of the meson structure.