Scattering Of Slow Neutrons Research Articles

Slow Neutron Scattering by Ferromagnetic Crystals

Consideration is made of the predictions of the Bloch spin wave model within its region of approximate validity, for the magnetic scattering of very slow neutrons by a ferromagnetic It is found that observable peaks of intensity in the angular distribution of scattered neutrons occur due to spin wave absorption Neutrons are also scattered with phonon absorption and by an analogous calculation it can be shown that similar peaks occur due to this vibrational scattering. A possible method is suggested of differentiating between the two kinds of scattering in order to provide a check of the Bloch model

Resonant Scattering of Slow Neutrons by Cadmium

Resonant Scattering of Slow Neutrons by Cadmium

The Energy Dependence of the Resonant Scattering of Slow Neutrons from Gold

The Energy Dependence of the Resonant Scattering of Slow Neutrons from Gold

On the Theory of Slow Neutron Scattering by Solid and Liquid Helium

The slow neutron scattering properties of solid and liquid helium are studied in this paper on the assumption that these condensed phases have similar crystal structures. It is shown that, according to this limiting solid model, liquid helium, in contrast with usual structures, should exhibit a smoothly increasing total scattering cross section from the lowest group of neutron energies available at present up to those higher energies at which the asymptotic, free atom cross-section value should be approached. Neutron transmission experiments at various energies should enable one to determine whether this liquid scatters slow neutrons as a solid or not, and if the two liquid modifications have similar or dissimilar slow neutron scattering properties.

Variational Principles for Scattering Processes. II. Scattering of Slow Neutrons by Para-Hydrogen

Continuing the illustrative example of the preceding paper, the scattering of slow neutrons by two protons bound to a molecule is treated by variational methods. A numerical evaluation indicates that the Fermi approximation to the para-hydrogen cross section is reliable to about 0.3 percent.

On the Theory of Slow Neutron Scattering by Liquid Helium

Using the extreme Bose-Einstein (B.E.) and Fermi-Dirac (F.D.) models of liquid ${\mathrm{He}}_{4}$ and ${\mathrm{He}}_{3}$, respectively, the theory of inelastic or incoherent slow neutron scattering by these liquids has been studied in this paper. It is, of course, fully realized that these liquids are not ideal symmetric or antisymmetric collections of atoms. Nevertheless, they might exhibit, in their behavior with respect to slow neutron scattering, such trends which could be interpreted as resulting from their respective statistical properties.In liquid ${\mathrm{He}}_{4}$, for neutrons of kinetic energy considerably larger than $\mathrm{kT}$, $k$ being Boltzmann's constant and $T$ the liquid temperature, the incoherent differential cross section per atom is always larger, or at least equal to the free atom cross section exhibited by these atoms in the classical limit of vanishing degeneracy. At constant liquid temperatures, larger than the lambda-point temperature, i.e., in the ${\mathrm{He}}_{4}$I region, the scattering has a maximum finite limit at vanishing scattering angles. This limit depends, at these small angles, on the liquid temperature only. The liquid cross section decreases with increasing scattering angle and tends asymptotically toward its free atom value. The latter is approached faster the larger the kinetic energy of the incident neutrons. At liquid temperatures near the lambda-point temperature, the vanishing small angle scattering cross section becomes very large and becomes some ${N}^{\frac{1}{2}}$ times the free atom value at the lambda-point.In the liquid ${\mathrm{He}}_{4}$II region the very small angle scattering cross section increases as ${\ensuremath{\Theta}}^{\ensuremath{-}2}$, $\ensuremath{\Theta}$ denoting the scattering angle.At constant scattering angles the scattering cross section varies as a function of the liquid temperature, in the following way. In the limit of high temperatures, or small degeneracy, the scattering cross section tends toward its asymptotic free atom value. As the liquid temperature decreases, or the degeneracy increases, the cross section increases. At the limit of the absolute zero temperature, perfect degeneracy, the distribution of the atoms being narrowed down to practically a single state, the scattering cross section is also classical, the fundamental symmetrical correlation having vanished because of the emptiness of the higher energy levels. The constant angle cross section curves thus exhibit a maximum as a function of the temperature.In liquid ${\mathrm{He}}_{3}$, treated here as a limiting ideal antisymmetric fluid, the differential cross section is always smaller, at small scattering angles, than the free atom cross section. The latter is reached asymptotically at large scattering angles from below, in contrast with the symmetric fluid. At sufficiently large scattering angles, the constant angle cross sections vary with temperature by exhibiting a minimum between the temperature of absolute zero and the limit of high temperatures or vanishing degeneracy. Again, in contrast with the symmetric fluid, the fluid cross section remains always finite and has no anomalies.The preceding results correspond to those collision processes in which the incident neutron loses momentum and energy. Since however, in all practical cases, even the total loss of neutron kinetic energy to the target liquid ${\mathrm{He}}_{4}$, over reasonable radiation times, cannot perturb the thermal equilibrium of the liquid, the inverse scattering processes have also been studied by using the principle of detailed balance. The resulting, direct plus inverse, liquid cross sections have also been evaluated both in ${\mathrm{He}}_{4}$ and ${\mathrm{He}}_{3}$. In the latter, however, the thermal equilibrium is far from being as well preserved as in liquid ${\mathrm{He}}_{4}$, because of the large thermal neutron reaction cross section, $n({\mathrm{He}}_{3},p){\mathrm{H}}_{3}$. This reaction appears to present serious difficulties for the eventual experimental investigation of the incoherent slow neutron scattering.

The Scattering of Slow Neutrons by Paramagnetic Crystals

The Scattering of Slow Neutrons by Paramagnetic Crystals

Erratum: The Scattering of Slow Neutrons by Paramagnetic Crystals

Erratum: The Scattering of Slow Neutrons by Paramagnetic Crystals

Scattering of Slow Neutrons by Deuterium Gas

Scattering of Slow Neutrons by Deuterium Gas

The Scattering of Slow Neutrons by Paramagnetic Crystals

The Scattering of Slow Neutrons by Paramagnetic Crystals

Spin Dependence of Slow Neutron Scattering by Deuterons

Spin Dependence of Slow Neutron Scattering by Deuterons

The Effect of Crystal Orientation on the Scattering of Slow Neutrons

Abstract : The transmission of slow neutrons through microcrystalline materials has been the subject of several investigations. However, the effect of nonrandom orientations of the microcrystals on neutron scattering has not been considered in detail. We shall develop a simplified scheme for taking account of the orientation effect when only a single symmetry axis needs to be considered. A comparison is made with the measurements of the transmission of slow neutrons through randomly oriented and extruded graphite. It is possible to obtain an estimate of the amount of crystal orientation in this instance.

The Resonance Scattering of Slow Neutrons in Crystals

The Resonance Scattering of Slow Neutrons in Crystals

Scattering of Slow Neutrons by Ortho- and Parahydrogen

Measurements have been made of the scattering cross sections of ortho- and parahydrogen for neutrons of energies in the range 10\ifmmode^\circ\else\textdegree\fi{}K to 30\ifmmode^\circ\else\textdegree\fi{}K. Using the Schwinger-Teller theory the results have been correlated with the scattering cross section of free protons for slow neutrons and with the range of a square-well $n\ensuremath{-}p$ potential. The measured cross sections show the energy dependence expected from the theory; they imply a free proton cross section of 19.7 barns and a force range of 1.5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}13}$ cm, the first value agreeing within experimental error with the accepted figure, but the second falling significantly below the result 2.8 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}13}$ cm obtained by Breit, Thaxton and Eisenbud for the $p\ensuremath{-}p$ force. The singlet state of the deuteron is again shown to be virtual, and the possibility of a neutron spin of $\frac{3}{2}$ is eliminated since the data in this case would imply a free proton cross section of only 7 barns.

The Scattering of Slow Neutrons by Bound Protons. III. Intermediate Energies

The calculation of the cross section for scattering of neutrons by elastically bound protons is extended to energies up to the first excited state. As in the preceding papers the results are expressed in terms of a correction factor to be applied to Fermi's approximation. This factor [Eq. (5)] has been evaluated for a number of energies and angles of scattering. In all cases the correction is small and of the order 0.3 percent or less.

Spin Dependence of Scattering of Slow Neutrons by Be, Al, and Bi

Some information has been obtained on the spin dependence of scattering of slow neutrons by Be, Al, and Bi by measuring the scattering cross section for filtered neutrons. The result is that in none of these three cases does the sign of the scattering length change when the spin orientation is changed. But in the case of Be and Bi the magnitude of the scattering length for one spin orientation may be up to twice as great as that for the other spin orientation, and in the case of Al the variation may be by a factor of three.

The Scattering of Slow Neutrons by Bound Protons. II. Harmonic Binding—Neutrons of Zero Energy

The cross section for scattering of very slow neutrons by elastically bound protons is calculated by means of the method described in the preceding paper. The results are expressed in terms of a correction factor to be applied to Fermi's factor 4. According to Eq. (5) of the present paper this correction factor is $1+\frac{16aJ}{{\ensuremath{\pi}}^{\frac{1}{2}}}$ with $J$ defined by Eq. (6) which is suitable for numerical work. The results of numerical calculations for the effects of anisotropy in molecules with axial symmetry are presented in Fig. 1 which has to be used with Eqs. (6.2), (6.3) of the text. For scattering by the ${\mathrm{H}}_{2}$O molecule the correction to be applied to Fermi's result is only \ensuremath{\sim}.3 percent of the final value. Fermi's method is very accurate in this case.

The Scattering of Slow Neutrons by Bound Protons. I. Methods of Calculation

The mathematical problem of the scattering of slow neutrons by chemically bound protons is stated in terms of a boundary condition and its equivalent integral equation. By means of the latter it is possible to calculate corrections to the results of Fermi which can be considered as the first approximation to the more accurate equations. The method is applied to free protons, and it is shown how the well-known results for this case are obtained from the more general equations. This special case allows also an estimate (${10}^{\ensuremath{-}3}$) of the order of the corrections in the case of chemically bound protons in agreement with general estimates in connection with Eq. (8.4) of Section 7 of the present paper and the application to harmonic oscillator binding made in the succeeding paper jointly with P. R. Zilsel. The corrections discussed are greater than the inaccuracies of the boundary condition formulation which are of the order ${10}^{\ensuremath{-}5}$.

The Scattering of Slow Neutrons by Ortho- and Paradeuterium

Information relating to the spin dependence of the neutron-deuteron interaction can be obtained from slow neutron scattering experiments in ortho- and paradeuterium. Theoretical formulae have been derived for the cross sections of the various transitions among the molecular rotational levels, which involve the scattering amplitudes ${a}_{\frac{3}{2}}$ and ${a}_{\frac{1}{2}}$ for the two spin states of the neutron-deuteron system. In particular, numerical results are given for the first few transitions originating from the ground levels of the ortho- and para-systems with neutron energies not exceeding 0.05 ev. The influence of the thermal motion of the molecule is described, and explicit formulae are given for the important transitions occurring at small neutron energies, on the assumption that the ${D}_{2}$ is in gaseous form at low temperature. The ratio of the ortho- and para-cross sections, under these conditions, is examined in its dependence upon the ratio of the scattering amplitudes. If the scattering amplitudes are of the same sign, the cross-section ratio is never greater than 1.31 and attains this magnitude only for small values of ${a}_{\frac{3}{2}}$ relative to ${a}_{\frac{1}{2}}$. If, however, the amplitudes are of opposite sign, this ratio can be as large as 1.75 and always exceeds 1.11. This experiment measures only the magnitudes of the amplitude combinations ($2{a}_{\frac{3}{2}}+{a}_{\frac{1}{2}}$) and (${a}_{\frac{3}{2}}\ensuremath{-}{a}_{\frac{1}{2}}$), but not their signs, and thus leaves a fourfold ambiguity in interpretation. The possibility is discussed of determining the sign of ($2{a}_{\frac{3}{2}}+{a}_{\frac{1}{2}}$) by scattering experiments in HD. It is pointed out that the sign of (${a}_{\frac{3}{2}}\ensuremath{-}{a}_{\frac{1}{2}}$) cannot be fixed by any experiment in which the deuteron spin is unoriented in space. An alternative experimental method, involving the depolarization of neutrons, is mentioned.

Inelastic Scattering of Slow Neutrons

The total polycrystalline elastic and inelastic scattering cross sections are computed by means of the Born approximation with use of the Fermi ($\ensuremath{\delta}$ function) interaction between slow neutrons and bound nuclei. Isotopic disorder and magnetic interaction are neglected. Numerical calculation for iron scattering "300-degree" neutrons yields an inelastic cross section which rises from 0.006 at $T=0$\ifmmode^\circ\else\textdegree\fi{}K to 0.192 for a scatterer temperature of 1000\ifmmode^\circ\else\textdegree\fi{}K, in units of the "free" nuclear elastic scattering cross section. The total (elastic plus inelastic) cross section remains constant for scatterer temperature up to 250\ifmmode^\circ\else\textdegree\fi{}K, but falls off from 0.97 at $T=0$\ifmmode^\circ\else\textdegree\fi{}K to 0.90 at $T=1000$\ifmmode^\circ\else\textdegree\fi{}K. High and low temperature approximate results are also computed. Comparison of the results for inelastic scattering with the theory of the diffuse scattering of x-rays as developed by Zachariasen is successful in the limiting cases in which the two theories overlap.