Neutron momentum vector

But the molecular velocity, Vmoi, is zero along any direction, including that of the neutron momentum vector, when averaged over every direction that the molecular motions take in the sample. Therefore,... 

Here it is convenient to recall the expression for the de Broglie wavelength. A, of a neutron and the related neutron momentum vector, k. Readers should familiarise themselves with aspects of the manipulation of k ( 2.3). The neutron velocity is v. 

Fig. 2.2 A diagram of the momentum vectors in a scattering event shown in real space, above, and their relationship to the scattering triangle, shown below. The incident, i, and final, f, neutron momenta, k and the transferred momentum Q, are shown in the figure.

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. 

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... 

Much of the wave nature of solids is carried out in reciprocal space because the momentum vector of photons, phonons, and particles such as electrons and neutrons can be expressed directly in terms of their vector k as fc = 2tt/A. The reciprocal lattice vectors A, B, and C are constructed such that they are perpendicular to the direct lattice vectors a, b, and c and their lengths are proportional to the inverse length of the direct lattice vectors. The lattice points in reciprocal space correspond to planes in direct space and a reciprocal translational vector is defined as Gm = hA -I- fcB - - C, where hk are fhe Miller indices of these planes. It is shown that Ghu is perpendicular to the hkl) plane and the distance of this plane from the origin is given by d M = 2-n/ Ghkt -... 

Scattering experiments invoive processes in which incident particies (X-rays or neutrons) with wave vector ki (energy Ej interact with the sampie and emerge with wave vector kf (energy )), obeying the conservation iaws for momentum and energy,... 

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 ... 

As mentioned before, the scattered intensity arises from the interaction between neutrons and nuclei. It decomposes into two terms, namely the coherent and the incoherent contribution. The coherent intensity depends on the scattering vector q which is the difference between the momentum of the incident and the scattered neutrons. Its norm is q=(47t/A.) sin(0/2), 0 being the... 

Thermal neutrons have energies comparable to the excitation energies of molecular solids and because of their mass they carry momentum. This momentum or wave-vector, k, is conventionally represented through the characteristic deBroglie wavelength, X and hence, k = 2n/X. Typical neutron wavelengths match the interatomic distances in solids, ca. 2 A and, unlike the... 

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. 

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 ... 

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)... 

---