Cross Section Of Slow Neutrons Research Articles

Absolute Spin Assignments of Dy161 and Dy163 Neutron Resonances and the Hyperfine Coupling Constants in Dy163

Abstract A method and equipment are described for measuring the spin dependence of slow neutron cross sections. The neutron beam from the High Flux Beam Reactor (HFBR) is polarized by Bragg reflection from a magnetized 92% Co-8% Fe single crystal which process also serves to select the neutron energy. Nuclei in the target are statically polarized in a field of ∼ 35 kOe at temperatures ranging from 0.05 to 0.10 K. Large nuclear polarizations are obtained in polycrystalline ferromagnetic dysprosium metal because of the large magnetic hyperfine interaction. The total angular momenta, /, of the compound states corresponding to the first several resonances in Dy161 and Dy163 were determined by measuring the transmission of the target with the neutron beam alternately polarized parallel and antiparallel to the nuclear polarization. Quantitative results at two resonances of opposite spin permit absolute /-value assignments (i. e., the assignments are independent of any knowledge of the absolute signs of the nuclear magnetic moments). Results were as follows: Isotope 163 161 161 161 161 161 161 161 161 163 161 161 E0 (eV) 1.71 2.72 3.69 4.35 7.75 10.40 10.87 12.65 14.3 16.25 16.7 18.5 J 2 3 2 2 3 2 3 2 (?) 2 3 (?) 2 The hyperfine interaction constants for Dy163, obtained by measuring the nuclear polarization as a function of temperature, were A/k = 0.100 ± 0.005 K, and P/k = 0.008 ± 0.001 K where A and P are, respectively, the magnetic and the electric hyperfine interaction constants.

Total scattering cross sections of slow neutrons and the structure of liquid zinc

Total scattering cross sections of slow neutrons and the structure of liquid zinc

Phonon-Type Expansion for Slow Neutron Scattering

A phonon-type expansion of the differential scattering cross section of slow neutrons is carried out in an entirely general way. The formula thus obtained can be applied to arbitrary target systems, in particular, to non-Gaussian systems such as anharmonic crystals and molecules with an arbitrary structure. A well-defined quasiclassical approximation is presented along the line of reasoning of our previous paper. When the present formalism is applied to a system of molecules, it results in an expansion in terms of the quanta of the rotational motion of the molecule. The effectiveness of our new formula is demonstrated in numerical calculations for HCl gas.

Short-Range Order Effects in Paramagnetic Neutron Scattering

To study the effects of short-range order on paramagnetic lineshapes, we have computed temperature-dependent frequency moments for an exchange-coupled paramagnet with uniaxial, single-ion anisotropy, i.e., H=− ∑ f1,f2I(f1,f2)Sf1·Sf2−D ∑ f (Sfz)2. We have used phenomenological procedures to construct the frequency and wave-vector-dependent Van Hove correlation function. The results are discussed with reference to the inelastic scattering cross section of slow neutrons from such paramagnetic systems.

Time expansion of correlation functions and the theory of slow neutron scattering

The velocity correlation function for a classical system of particles interacting with a two-particle potential is expanded in powers of t. The coefficients in this expansion are averages of derivatives of the interaction potential and can be expressed in terms of instantaneous two, three, etc., particle distribution functions. For the special case of a harmonic crystal these expressions reduce to the respective moments of the frequency distribution function. For liquid argon the range of values of t, for which the velocity correlation is well represented by the first new terms of its time expansion, is determined from a comparison with the computer results obtained by one of us (A. R.). Similarly, the range of values of the momentum transfer is investigated, for which this time expansion leads to a useful representation of the incoherent scattering cross section for slow neutrons. For coherent scattering analogous formulae are given, but no numerical comparison with the computer results has been made. In addition, it is shown that the non-Gaussian behavior of G s ( r, t) obtained from the computer results has considerable influence on the shape of the incoherently scattered slow neutron spectrum. It is pointed out that slow neutron scattering with large values of momentum transfer can be used as an experimental technique for the determination of the mean square acceleration in any system.

Use of time integration to calculate the differential scattering cross sections of slow neutrons

Use of time integration to calculate the differential scattering cross sections of slow neutrons

Damping theory and its application to neutron scattering by anharmonic crystals

The damping theory is developed in terms of projection operators, and an expression for the transition probability per unit time which is suitable to line-shape studies in general is obtained. This expression replaces the usual energy-conserving delta-function by a Lorentzian distribution whose width is equal to the difference of the widths of the initial and final states. Using this expression, a formula for the scattering cross-section of slow neutrons by an anharmonic crystal is derived. Expressions for the width and shift of lines corresponding to multiphonon events in general, and to a one-phonon event as a special case, are obtained, and investigated in the zero- and high-temperature limits. The results are compared to those obtained previously. It is found that the inverse of the phonon lifetime is expressible in terms of the difference of the widths of the crystal states, as obtained by Brout. The shift formula is shown to be in agreement at all temperatures with that obtained by Maradudin and Fein, and others. The width formula agrees with theirs only in the zero-temperature limit. In the high-temperature limit, the present formula predicts a quadratic temperature-dependence in contrast with the linear dependence predicted by theirs. This discrepancy is discussed in terms of the concept of phonon lifetime. The present theory predicts a larger width at finite temperatures, and therefore is expected to yield a better agreement with the large experimental widths observed by Brockhouseet al., than that reported by Maradudin and Fein.

Coherent Inelastic Scattering of Slow Neutrons by a Polyatomic Liquid Element

An expression for the coherent inelastic-scattering cross section of slow neutrons by a diatomic liquid element has been determined by extending Vineyard's approximation to systems with internal degrees of freedom. The range of validity of the expression obtained has been discussed, taking into account also the energy resolution of the experimental setup. The energy spectrum of neutrons scattered by a sample of liquid bromine has been measured with a triple-axis spectrometer for a fixed value of the modulus of the momentum exchange. Reasonable experimental evidence for the existence of a peak associated with the excitation of the first vibrational level in liquid bromine was obtained in the neighborhood of an energy transfer $\ensuremath{\hbar}\ensuremath{\omega}=0.04$ eV; this is approximately the value established by measurements of conventional spectroscopy in the gaseous phase.

Analysis and Interpretation of the Slow Neutron Cross Sections of the Fissionable Nuclei

Analysis and Interpretation of the Slow Neutron Cross Sections of the Fissionable Nuclei

Slow-neutron cross sections of hafnium

Slow-neutron cross sections of hafnium

Renormalisation effect in the electronic specific heat of vanadium

The lattice vibrational spectrum of vanadium calculated from the differential inelastic scattering cross section for slow neutrons is presented. The lattice specific heat and corresponding Debye temperatures are shown. The lattice specific heat above 150 deg K derived from neutron data is compared with caloric measurements, and it is found that the electronic specific heat constant found at low temperature is not valid at higher temperatures. It is proposed that this effect might be attributed to the renormalization of the free energy of electrons, which is caused by the electron-phonon interaction. To give a qualitative justification of this proposal the expression for the free energy of the electrons was evaluated. (C.E.S.)

Small-Angle Scattering of Neutrons by FeCl2 near the Néel Temperature

Small-angle scattering cross-section for slow neutrons by FeCl 2 above but near the Néel temperature is calculated with a method which is an extension of Zernike's theory of propagation of order. The result is compared with the experimental data and a reasonable agreement is found if the ratio of the two exchange coupling constants in this crystal is assumed to be about 0.08.

On the Evaluation of the Thermal Neutron Scattering Law

The magnitude of the scattering cross section for slow neutrons by atomic systems (gases, liquids, and polycrystalline solids) is governed by the correlated motions of atoms in the system. A major ...

Temperature-dependent green functions and neutron scattering in liquid helium II

The object of this paper is to calculate the partial scattering cross section of slow neutrons from liquid helium II, using temperature-dependent Green Functions. The theory of temperature-dependent Green functions is briefly reviewed. In Section II an expression is obtained for the scattering cross section as a function of momentum lost by the neutron and the temperature. In Section III, the line widths of the scattered neutrons are considered. Finally, the results are discussed and compared with experiment, where this is possible.

Total Slow Neutron Cross Section Measurements of Titanium, Zirconium, and Hafnium

Total neutron cross section measurements have been made on metallic samples of Ti, Zr, and Hf with the MTR crystal spectrometer. The Ti cross section was measured in the energy region 0.025 to 45 ev and values of 4.19 ± 0.04 b and 6.09 ± 0.13 b were obtained for the bound atom scattering and 2200 meters/sec absorption cross sections. The Zr cross section was measured in the energy region 0.02 to 78 ev and a value of 6.40 ± 0.07 b was obtained for the bound atom scattering cross section. The Hf cross section was measured from 0.04 to 3.5 ev and a value of 102 ± 3 b for the 2200 m/sec absorption cross section is reported. Energy values of 1.095 ± 0.005 ev and 2.378 ± 0.016 ev were determined for the peak cross sections of the first two Hf resonances. Values of σ0 = 6030 ± 80 b, Γ = 0.068 ± 0.001 ev and gΓn = 0.163 ± 0.003 mv were obtained for the 1.095 ev resonance.

Neutron Scattering from a Liquid on a Jump Diffusion Model

The incoherent inelastic scattering cross section of slow neutrons from liquids is calculated using a simple model in which the liquid is assumed to have appreciable short range order in a quasi-crystalline form. Diffusive motion takes place in large discrete jumps, between which the atoms oscillate as in a solid. The model predicts a definite, easily calculable cross section which is not dominated by diffusion effects as when continuous diffusion is assumed, but shows a characteristic variation with angle which could be looked for experimentally. The related pair correlation functions are dominated at small r and t by vibrational effects. Although simple and extreme the model explains several aspects of the observations of Brockhouse and Pope in 1959 and others. A brief discussion of the coherent scattering cross sections for the model is given although explicit formulae are not obtained.

Proton Motions in Ammonium Halides by Slow Neutron Cross-Section Measurements

Total neutron cross sections were measured at room temperature for NH/ sub 3/ gas at l15 psia, NH/sub 4/F, NH/sub 4/Cl, NH/sub 4/Br, and NH/sub 4/I, and the scattering cross sections per hydrogen atom plotted vs. neutron wavelength ( lambda n). At short lambda n. the cross sections approach the values for the free atoms, while at long lambda /sub n/n they may be represented bv sigma s = a + b lambda /sub n/. The slopes b of the ammonium halides were determined from the graph and correlated with other properties of the halides, e.g., torsional frequency and potential barrier to rotation. The order-disorder transitions between phases I, II, and III of the halides are discussed. It is concluded that neutron cross section measurements can give evidence of the dynamics and chemical binding of hydrogen in molecules and crystals. (D.L.C.)

Resonance Absorption of Nuclear Gamma Rays and the Dynamics of Atomic Motions

The theory of resonance absorption of nuclear $\ensuremath{\gamma}$ rays is generalized for an arbitrary system of interacting particles by expressing the relevant transition probability in terms of a space-time self-correlation function; and thus relating the resonance line shape to the incoherent differential scattering cross section for slow neutrons. Two limiting cases: (i) a gas and (ii) a solid have been considered. Discussion regarding the justification of the use of a classical self-diffusion function for a liquid is given and expressions for the broadening of the resonance line due to diffusive motions of the atoms of the interacting system are derived. It is suggested how M\"ossbauer-type experiment could be used to give information regarding the diffusive motions of atoms in a solid and also, under more favorable circumstances, in a liquid.

Effects of Chemical Binding on Nuclear Recoil

The recoil of a chemically bound nucleus is considered for slow neutron scattering and for the resonant absorption of neutrons or gamma rays. The Doppler-broadened resonance line shape is derived in terms of the time-dependent self-correlation function describing the motion of a nucleus due to the interatomic forces. This explicitly relates the resonance line shape to the differential scattering cross section for slow neutrons in the Fermi pseudopotential approximation. Within this formulation an expansion for large nuclear recoil is naturally suggested. For the case of a crystal, this expansion can be directly related to the expansion associated with the central limit theorem of probability theory and can therefore be proved to be asymptotic in nature. The expansion parameter is ${(\frac{{K}_{\mathrm{av}}}{R})}^{\frac{1}{2}}$, where ${K}_{\mathrm{av}}$ is the average kinetic energy of a nucleus and $R$ is the recoil energy for a free nucleus at rest. The leading term of the expansion is the weak binding limit originally obtained by Lamb. In this limit the Doppler-broadened line shape is the same as would obtain for an ideal monatomic gas of the same mass with an effective temperature ${T}^{\ensuremath{'}}=(\frac{2}{3}){K}_{\mathrm{av}}$. For noncrystalline systems, a similar expansion with the same leading term can be obtained by a rearrangement of the terms in an expansion used by Wick to study the slow neutron total cross section. The relation of the present expansion to Wick's expansion is discussed.

Scattering of Slow Neutrons by Water

The motions of hydrogen atoms in water are considered in terms of the ${\mathrm{H}}_{2}$O molecule at the basic dynamical unit. Vibrations, hindered rotations, and translations of the molecule are included. For the numerical calculations, the hindrance of the translations is neglected, and the hindered rotations are replaced by a torsional oscillation with a single energy, $h\ensuremath{\nu}=0.06$ ev. When certain approximations are made in the average over molecular orientation, this model allows for the computation of differential and total slow-neutron cross sections. The computed cross sections are in good agreement with most of the available slow-neutron scattering data. The features of the high-resolution experiments directly associated with the hindrance of the molecular translations in the liquid are not reproduced.