Neutron momentum transfer vector

The reader will appreciate that the scattering law of a particular vibrational transition, given by Eq. (2.32), is written in terms of the scalar (or inner, or dot) product of the neutron momentum transfer vector, Q, and the atomic displacement vector, . This product is a function of the angle between the vectors and allows some interesting single crystal effects to be examined. 

An incident monochromatic beam scattered by a sample is analyzed with a detector at a general position in space (see Fig. 2). Incident and scattered neutrons are regarded as plane waves whose wavevectors are ki and kf, respectively. ( jfco = 27t/Ao and kf = 2tt/ /, where Ao and A/ are the incident and scattered wavelengths, respectively.) The momentum transfer vector is Q — ki kf. 

The previous derivation of the reflection and transmission coefficients correctly describes the intensity of reflected neutrons at any value of momentum transfer vector. However, there is a useful alternative derivation, which gives a highly analytical function describing the reflectivity. This derivation is based on the Born approximation and is often referred to as reflectivity in the kinematic limit. Suppose there are two arbitrary but different SLD profiles pi(z) and p2 z) and one wishes to determine the separate reflectivities Ri(Q,z) and -R2(Gz) for the two scattering potentials. The solution to the problem is described by combining Eqs. (3.15) and (3.17)... 

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

Here k and k are incident wave vector and scattering wave vector and g is an important vector proportional to momentum transfer vector. Nowadays neutron science is involved in many areas of science including chemistry, crystalline materials, complex fluids, disordered materials, magnetism and superconductivity, polymers and structural biology. 

However, the neutron scattering cross sections depend on the momentum-transfer vector and vibrational displacements of scattering nuclei in normal vibrations, and the importance of measuring anisotropy of cross sections was emphasimd by Summerfield (1965). Accordingly, the anisotropy of down-scattering cross sections of a stretch-oriented sample of polyethylene was measured by Myeis, Summerfield, and King (1966) with a triple-axis spectrometer. As shown in Fig. VII.1, two peaks... 

Although single crystal diffraction is the most powerful method for studying the structure of crystals, the powder diffraction method is also frequently used to reveal structural information for crystalline materials. In this case the sample is not a single crystal, but is instead a powder which contains numerous microcrystals, such that all orientations are equally likely. The observed diffraction pattern is then an average of Equation [28] over all possible relative orientations between the momentum transfer vector Q and the crystal lattice. The measured diffraction pattern does not depend on the orientation between the sample and the neutron beam, and it is usual to identify each Bragg peak in the diffraction pattern by its d-spacing, di,y, defined by... 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The momentum transfer hQ, respectively the wave vector, is given by Q= k -kf where k and kf are the wave vectors of the incoming and outgoing (scattered) neutrons. They relate to the neutron wavelength k j=2Tt/Aij. The neutron momenta a.rep ij=m Vi f=fikif. Therefore ... 

Using a monochromatic incident beam, the intensity for neutron scattering measured at each frequency (energy transfer ho = hu) depends on the orientation and magnitude of the final wave vector. In one dimension, the scattering function at momentum transfer Qx and energy transfer fuolj for a transition between states j(a )) and f(x)) can be written as ... 

Consider an incident neutron of wave vector k that is scattered by the sample into a solid angle d l. If the wave vector of the scattered neutron is kh the momentum transferred to the neutron from the sample is equal to hq given by... 

Here is the momentum transferred from the neutron and x is one of the reciprocal lattice vectors of the palladium lattice. Thus, the incoherent scattering sees all the vibration modes but the coherent scattering selects one particular phonon for a particular experimental value of Q. It is now clear that both the incoherent and coherent one-phonon scattering will depend on the shape of the optical dispersion curves and hence will be influenced by hydrogen-hydrogen interactions. Indeed, one of the first observations of inelastic scattering from a hydride [10] interpreted the shape of the optical peak in terms of a frequency distribution broadened by H-H interactions. 

In these equations, hco is the neutron final energy, Or and aj. are, respectively, the incoherent and coherent scattering lengths for the rth nucleus, [co/(q)] is the Bose population factor, K is the momentum transferred by the neutron, 2H is the Debye-Waller factor for the rth nucleus, and 2nris a reciprocal lattice vector the remaining terms have been previously defined. 

The quantity X represents the wavelength of the radiation, and 6 is the angle of scatter. The quantity K, sometimes written or Q, is variously called die wave vector or the range of momentum transfer, especially for neutron scattering. 

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