Neutron phonons

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... 

The method of neutron spectroscopy is the most efficient tool for study frequencies of the crystal lattice vibrations. This method is based on the scattering of the low-energy, so-called heat neutrons by the nuclei of solids. The wavelength of the normal vibrations and of the heat neutrons are values of the same order as the energies are. As a result of the interaction of the low-energy neutrons with solids, quanta of normal vibrations of the crystal lattice (phonons) are created or, conversely, annihilated. The collision neutron-phonon changes the state of the neutron essentially and this change can be detected experimentally. 

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

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.

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

The third term in Eq. 7, K, is the contribution to the basal plane thermal resistance due to defect scattering. Neutron irradiation causes various types of defects to be produced depending on the irradiation temperature. These defects are very effective in scattering phonons, even at flux levels which would be considered modest for most nuclear applications, and quickly dominate the other terms in Eq. 7. Several types of in-adiation-induced defects have been identified in graphite. For irradiation temperatures lower than 650°C, simple point defects in the form of vacancies or interstitials, along with small interstitial clusters, are the predominant defects. Moreover, at an irradiation temperatui-e near 150°C [17] the defect which dominates the thermal resistance is the lattice vacancy. 

Figure 3 Phonon dispersion curves obtained by inelastic neutron scattering revealing precursor behaviour prior to the 14M transformation in Ni-AI. The dip at q = 1/6 [110] (a) deepens upon cooling and (b) shifts under an external load .

The density of states (DOS) of lattice phonons has been calculated by lattice dynamical methods [111]. The vibrational DOS of orthorhombic Ss up to about 500 cm has been determined by neutron scattering [121] and calculated by MD simulations of a flexible molecule model [118,122]. 

The Debye temperatures of stages two and one were determined by inelastic neutron scattering measurements [33], The total entropy variation using equation 8 is in the order of about 2 J/(mol.K). Although smaller in value, such variation accounts for 10-15% of the total entropy and should not be neglected. We are currently carrying on calculations of the vibrational entropy from the phonon density of states in LixC6 phases. 

Figure 5. Model spectra of a naked neutron star. The emitted spectrum with electron-phonon damping accounted for and Tsurf = 106 K. Left panel uniform surface temperature right panel meridional temperature variation. The dashed line is the blackbody at Tsurf and the dash-dotted line the blackbody which best-fits the calculated spectrum in the 0.1-2 keV range. The two models shown in each panel are computed for a dipole field Bp = 5 x 1013 G (upper solid curve) and Bp = 3 x 1013 G (lower solid curve). The spectra are at the star surface and no red-shift correction has been applied. From Turolla, Zane and Drake (2004).

Chen, Y., and W. A. Sibley, 1969. A study of zero phonon lines in electron-irradiated, neutron-irradiated, and additively colored MgO, Philos. Mag., 20, 217-223. 

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

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