Energy and momentum transfer

Here we shall define the energy, E, and momentum, k, of a neutron and address their manipulation. We recall the expression for the de Broglie energy, E, and wavelength. A, of any quantum particle 

Although the energy is strictly vhc, to be consistent with the neutron scattering literature we use a as the symbol for energy with the spectroscopic wavenumber (cm ) unit (see box). 

Energy. Although the SI unit of energy is the joule, in neutron scattering it is conventional to express neutron energy in meV or THz units. In molecular spectroscopy the wavenumber, cm, is the common energy unit and this is the unit we use. We have 1 meV = 0.2418 THz = 1.602 x 10 J = 8.066 cm  

Momentum. Although the SI unit of length is the metre and the units of momentum are kg m s, it is conventional in neutron scattering to express length in Angstrom units lA = 10 m. Consequently the conventional unit for the neutron wavelength is A and for the neutron wavevector A.  

The energy and momentum of the neutron are related through the neutron velocity. We evaluate the energy and momentum of a neutron, of mass 1.67 xlO kg, travelling with a speed, v = 1000 m s.  

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We first examine the reiationship between particie dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sampie and the incident radiation are measured. Linear response theory aiiows dynamic structure factors to be written in terms of equiiibrium flucmations of the sampie. For neutron scattering from a system of identicai particies, this is [i,5,6]... 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

At present the density effect has been quite thoroughly studied both theoretically and experimentally. There are different ways of obtaining the calculation formulas for Se. In particular, we can make allowance for the effect the surrounding medium has on the electromagnetic field of a particle by making the substitution c2—c2/e(a>) in the formula for the relativistic differential cross section of energy and momentum transfer... 

Despite the variation in scattering results for the aforementioned prototypical systems, theoretical models have been successful in predicting the degree of energy and momentum transfer in gas/surface collisions. This will be discussed in the following section. 

Under the prevalent experimental conditions of NCS, i.e.. high energy and momentum transfers, time correlations in the motion of a scatterer can be ne-... 

They are needed to describe the rate of mass, energy, and momentum transfer between a system and its surroundings. These equations are developed in courses on transport phenomena. 

In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Fmthermore, speeial attention has been paid to presenting the basic equations for the kinetie treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. 

The term 3(— l/e q, co)) is referred to as the dielectric loss function. Structures in this function can be correlated to bulk plasmon excitations. In the vicinity of a surface the differential cross section for inelastic scattering has to be modified to describe the excitation of surface plasmons. The surface energy loss function is proportional to 3(—l/e(, cu) + 1). In general, the dielectric function is not known with respect to energy and momentum transfer. Theoretical approaches to determine the cross section therefore have to rely on model dielectric functions. Experimentally, cross sections are determined by either optical absorption experiments or analysis of reflection energy loss spectra [107,108] (see Section 4.3). 

Fig. 1 The energy and momentum transfer regions accessible to different types of neutron speetrometers time of flight (TOF), backscattering (BS). and neutron spin echo (NSE).

We end this section with some comments about the energy and momentum transfer dependence expected for the cross-sections. 

We have seen in earlier chapters that there seems to be a close parallelism between the sets of leptons and the sets of quarks, at least in so far as the unified weak and electromagnetic interaction is concerned. The leptons are essentially point-like in their behaviour, and it is not inconceivable that the quarks too enjoy this property. In that case we might expect the hadrons to behave, in certain situations, in a less complicated fashion than usual. If we think of the hadrons as complicated atoms or molecules of quarks, then at high energies and momentum transfers, where we are probing the inner structure, we may discover a relatively simple situation, with the behaviour controlled by almost free, point-like constituents. The idea that hadrons possess a granular structure and that the granules behave as hard point-like, almost free (but nevertheless confined) objects, is the basis of Feynman s (1969) parton model. 

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