Department of Physics

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Scatterings initiated by two coherent beams
Li, M. C. (American Physical Society, 1974-04)
A theory, which exhibits the characteristic features of scatterings initiated by two coherent beams, is discussed in the present paper. We show that the imaginary part of the scattering amplitude is an experimentally measurable quantity. To know the imaginary part of the amplitude is quite important for certain aspects of quantum mechanics, for example, the dispersion relation for a fixed momentum transfer. This relation has not yet been experimentally verified.
Spheroidal analysis of Coulomb scattering
Li, M. C. (AIP Publishing, 1974-05)
In studying the spheroidal problem of a charged particle scattered by two charged centers, we have had to deal with a differential equation. Its solution was complicated. In this paper, we study a very similar differential equation, which appears in the Coulomb problem. The solution for this equation, however, can be put in a simple form. For completeness the spheroidal analysis of the Coulomb scattering amplitude is also discussed.
Dynamic nuclear polarization in samarium-doped lanthanum magnesium nitrate
Byvik, C. E.; Wollan, D. S. (American Physical Society, 1974-08)
We report theoretical and experimental investigations of the dynamic nuclear polarization (DNP) and nuclear spin-lattice relaxation of protons in single crystals of lanthanum magnesium nitrate (LMN) weakly doped (≅0.1-1.1-mole%) with samarium ions (LMN:Sm) at 9.1 GHz and liquid-helium temperatures. Theoretical expressions for DNP and nuclear relaxation are given for a homogeneously broadened ESR line in the high-temperature limit. DNP effects due to the electron dipole-dipole reservoir (EDDR) are predicted for the case of well-resolved satellites. Our DNP experiments on LMN:Sm are in general agreement with the conventional theory, but no evidence for direct EDDR effects on DNP with well-resolved satellites is seen, despite suggestive nuclear-relaxation data. Various explanations for this disagreement are proposed. We suggest that these EDDR effects may be seen in crystals of ≅3-5-mole% LMN:Sm or in equivalent systems.
Scattering of two coherent beams by multiple centers
Li, M. C. (American Physical Society, 1974-09)
This is a continuation in the theoretical study of scattering initiated by two coherent beams. Here the scattering by multiple independent centers is discussed. The presence of these centers involves some problems, which are not present in scattering with a single beam. I show how these problems may be overcome, thus allowing the phase of the scattering amplitude to be determined experimentally.
On the spectrum of linear transport operator
Larsen, E. W.; Zweifel, Paul F. (AIP Publishing, 1974-11)
In this paper, spectral properties of the time_independent linear transport operator A are studied. This operator is defined in its natural Banach space L 1(D _ V), where D is the bounded space domain and V is the velocity domain. The collision operator K accounts for elastic and inelastic slowing down, fission, and low energy elastic and inelastic scattering. The various cross sections in K and the total cross section are piecewise continuous functions of position and speed. The two cases _0>0 and _0=0 are treated, where _0 is the minimum neutron speed. For _0=0, it is shown that _(A) consists of a full half_plane plus, in an adjoining strip, point eigenvalues and curves. For _0>0, _(A) consists just of point eigenvalues and curves in a certain half_space. In both cases, the curves are due to purely elastic ``Bragg'' scattering and are absent if this scattering does not occur. Finally the spectral differences between the two cases _0>0 and _0=0 are discussed briefly, and it is proved that A is the infinitesimal generator of a strongly continuous semigroup of operators.
Generalized optical theorem in coherent scattering
Li, M. C. (American Physical Society, 1975-03)
It is shown that the generalized optical theorem in quantum mechanics is a useful theorem in the scattering of two coherent beams.
Extension of Case formulas to L_p. Application to half and full space problems
Larsen, E. W.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1975-05)
The singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Hölder‐continuous functions to the function spaces X p = {f‖μf (μ) ‐ L p }, where the expansions are now to be interpreted in the X p norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with X p sources and incident distributions.
Crystal scattering of two coherent neutron beams
Li, M. C. (American Physical Society, 1975-10)
The crystal scattering of two coherent neutron beams is discussed. It is found that the conventional pair correlation function used in single-beam scattering is insufficient to describe the scattering of two coherent neutron beams. A generalized pair correlation function with unequal momentum transfer is introduced. It is found that the energy exchange between vibrational quanta of the crystal and incident neutrons leads to a modification of the scattering amplitude. Through the scattering of two coherent neutron beams one is able to measure directly the phase of the scattering amplitude.
Multigroup neutron transport. I. Full range
Bowden, Robert L.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1976-01)
A functional analytic approach to the N_group, isotropic scattering, particle transport problem is presented. A full_range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half_range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one_group case.
Multigroup neutron transport. II. Half range
Bowden, Robert L.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1976-01)
This paper accompanies a preceding one in which a functional analytic method was used to obtain the full_range expansion in multigoup neutron transport. In the present paper the analysis is extended to obtain the half_range expansion. The method used is an extension of the work of Larsen and Habetler for the one_group case. The results are given in terms of certain matrices which are solutions of coupled integral equations and which factor the dispersion matrix.
Water-pollution monitoring using matched spatial filters
Almeida, S. P.; Eu, J. K. T. (Optical Society of America, 1976-02)
Presented are some results obtained from the application of matched spatial filtering techniques to the identification of biological specimens called diatoms (i.e., water algae). A prototype semiautomatic optical processor has been developed that utilizes the Vander Lugt type complex spatial filters. We describe the optical filter averaging technique employed and discuss our semi in situ developing method for the filters. The advantage of this method with regards to the critical position requirements for time sharing optical filters is also presented. The filter holder is mounted on X, Y stages and precision positioned under the control of a PDP-11-40 computer.
Dynamic nuclear polarization with an inhomogeneously broadened ESR line. II. Experiment
Wollan, D. S. (American Physical Society, 1976-05)
Results of proton dynamic nuclear polarization (DNP), nuclear spin-lattice relaxation, and ESR experiments in six YES: Er168 (Er168-doped yttrium ethyl sulphate) single crystals at 4.15 K and 9.1 GHz are reported here. This material approximates an inhomogeneously broadened (IHB) ESR line for certain crystal orientations relative to the applied magnetic field H⃗ 0. DNP peak enhancements Epeakss≈120 are obtained for crystals 1-3 having Er168 dilutions of 0.2%-0.01% (mole%). The steady-state enhancement Ess vs H0 curves are proportional to the ESR absorption derivative at low powers (≅ 0.3 mW), but change shape in the wings of the ESR line at high microwave powers (≅ 300 mW). Essentially idential Ess data are obtained for all three crystals. Extensive DNP pump time τDNP and ESR data for crystals 2-4, as well as limited data for the other crystals, are also given. There is no Ess or τDNP evidence in these experiments (i) for the traditional strict-shell-of-influence model of IHB SE (solid effect) DNP, (ii) for non-negligible CE (cross effect) DNP processes, or (iii) for decreases in Ess due to moderate spectral diffusion in an IHB ESR line, as proposed by Buishvili et al. The DNP and ESR data of crystals 1-3 have been fit by the least-squares technique to a muffin-tin model of IHB SE DNP which incorporates rapid nuclear spin diffusion. Moderate agreement with theory is obtained, especially if the data are interpreted with a three-spectral-region model for wide IHB ESR lines. These data have also been analyzed in terms of EDDR (electron dipole-dipole reservoir) DNP theory for an IHB ESR line with fast spectral diffusion throughout the line. There is qualitative and some quantitative agreement between this theory and the data. It is thus difficult to discriminate EDDR DNP theory from IHB SE DNP with the three-spectral-region model in these experiments. The only clearcut evidence for the latter is an order-of-magnitude theoretical calculation of the electron magnetic dipole-dipole contribution to the ESR intrinsic spin-packet width, which gives non-negligible spectral diffusion in the centers of the crystal 1 and 2 ESR lines, and negligible spectral diffusion in the wings of crystals 3 and (perhaps) 2. Conclusive evidence would require further experiments at Er168 dilutions lower than 0.01% and higher than 0.2%. Experiments are proposed for YES: Er168 to test further the three-spectral-region model, and to find high DNP by IHB SE with possible applications to polarized nuclear targets in other, more hydrogenous materials.
The relationship between normalization coefficient and dispersion function for the multigroup transport equation
Feigenbaum, M. J. (AIP Publishing, 1976-05)
An explicit formula for the discrete Case normalization coefficient is presented in terms of functions related to the dispersion function. These functions are easily determined and provide the normalization coefficient without need of prior evaluation of the eigenvectors.
Dynamic nuclear polarization with an inhomogeneously broadened ESR line. I. Theory
Wollan, D. S. (American Physical Society, 1976-05)
Theoretical investigations of the dynamic nuclear polarization (DNP) and the nuclear spin-lattice relaxation time Tn in diamagnetic dielectric crystals diluted with electron paramagnetic impurities are reported here. The steady-state DNP enhancement Ess and the DNP pump time τDNP are calculated for the solid effect (SE), the cross effect (CE), and DNP by the electron dipole-dipole reservoir (EDDR), assuming (i) a predominantly inhomogeneously broadened (IHB) ESR line with uniform spin-packet widths (for SE and CE, but not EDDR DNP), (ii) rapid nuclear spin diffusion, (iii) the electron and nuclear high-temperature limit, (iv) no phonon bottleneck, and (v) isotropic electron impurities or axially symmetric electrons whose symmetry axis (c axis) is aligned parallel to the applied magnetic field H⃗ 0 DNP results for IHB SE, which include DNP leakage factors, predict Ess comparable to the ideal enhancement in favorable cases, with Ess vs H0 curves not proportional to the ESR line-shape derivative at high microwave powers, contrary to early theories of IHB SE. The effects of electron jumps between spin packets are discussed. DNP and Tn are calculated for the cross effect for both the well-resolved and unresolved limits. We get larger SE DNP and smaller CE DNP for the unresolved CE than do Hwang and Hill, and consider CE DNP leakage factors they omitted. The inequality Tn2≤τDNP≤Tn is found for the well-resolved CE, while τDNP≈Tn is estimated for the unresolved CE, different from the general SE case where τDNP≪Tn often occurs. DNP by EDDR is extended from the pure homogeneously broadened (HB) ESR limit to the IHB case with fast spectral diffusion, as suggested by Abragam and Borghini, and is further extended from electronic isotropy to axially symmetric electrons with the c axis parallel to H⃗ 0 Various limits of Ess and τDNP are compared and contrasted for SE, CE, and EDDR DNP, to enable experimenters to differentiate these three DNP mechanisms, if possible. We propose a three-spectral-region model for wide ESR lines, which may exhibit IHB behavior in the wings of the line, HB EDDR effects near the ESR absoption peak, and CE effects in between.
On the solution of nonlinear matrix integral equations in transport theory
Bowden, Robert L.; Zweifel, Paul F.; Menikoff, R. (AIP Publishing, 1976-09)
The coupled nonlinear matrix integral equations for the matrices X (z) and Y (z) which factor the dispersion matrix Λ (z) of multigroup transport theory are studied in a Banach space X. By utilizing fixed‐point theorems we are able to show that iterative solutions converge uniquely to the ’’physical solution’’ in a certain sphere of X. Both isotropic and anisotropic scattering are considered.
Direct phase determination in crystallography
Li, M. C. (American Physical Society, 1976-12)
It was shown theoretically that through the diffraction of two coherent beams, one is able to measure both the magnitude and phase of the structure factor directly. Here some experimental comments concerning the performance of the proposed experiment are presented.
Vlasov theory of plasma oscillations: linear-approximation
Arthur, Michael D.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1977)
A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function _ has first and second order zeroes. Special care is taken in the study of real zeroes of _ culminating in new results for this case. For a simple zero of _ with nonvanishing imaginary part the van Kampen-Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.
Conservative neutron-transport theory
Bowden, Robert L.; Cameron, W. L.; Zweifel, Paul F. (AIP Publishing, 1977-02)
A functional analytic development of the Case full_range and half_range expansions for the neutron transport equation for a conservative medium is presented. A technique suggested by Larsen is used to overcome the difficulties presented by the noninvertibility of the transport operator K _1 on its range. The method applied has considerable advantages over other approaches and is applicable to a class of abstract integro-differential equations.
Universality in the percolation problem—Anomalous dimensions of φ4 operators
Amit, D. J.; Wallace, D. J.; Zia, Royce K. P. (American Physical Society, 1977-05)
We consider critical systems, such as the percolation problem, whose symmetry permits an invariant interaction of third order in the fluctuating fields φ. In the renormalization-group approach one is naturally led to look for infrared-stable fixed points which yield ε expansions in 6−ε dimensions, with ε=3 as the physical value. Since the Gaussian fixed point becomes unstable to φ4 interactions for ε>2, it is important to check that the fixed point obtained in the ε expansion remains stable to such perturbations. We report the calculation to first order in ε of the corrections to scaling induced by (stability with respect to) φ4 interactions in a general class of such theories. The results indicate that φ4 interactions remain irrelevant in the percolation problem.
Doppler-free two-photon absorption of two coherent beams
Li, M. C. (American Physical Society, 1977-12)
Proposed and performed experiments on the Doppler-free two-photon absorption of two coherent laser beams are discussed. The coherent phenomenon in experiments follows directly from basic quantum mechanics. It is shown that, to excite the coherent absorption, the laser beams need not be spatially or temporally separated.

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