Momentum densities

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Spin-drehimpuls, m. spin angular momentum, -glied, n. (Math.) spin term, -momentdichte, /. spin momentum density. 

The one-particle distribution function fp specifies both the total particle density p and momentum density pu, where u is the average fluid velocity. 

Electron, Spin and Momentum Densities and Chemical Reactivity... 

ELECTRON, SPIN AND MOMENTUM DENSITIES AND CHEMICAL REACTIVITY... 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

This book contains a selection of chapter topics based on papers given at the 12th conference of the Commission on Charge, Spin and Momentum Density of the International Union for Crystallography, held in Waskiesiu, Prince Albert National Park, SK, Canada, July 27-August 1, 1997. The choice of topics represents some of the latest advances in the field of electron, spin, and momemtum densities and the analysis of these densities with respect to their roles in determining chemical reactivity. 

Paul G. Mezey and Beverly E. Robertson (eds.), Electron, Spin and Momentum Densities and Chemical Reactivity, 27-36 0 2000 Kluwer Academic Publishers. Printed in Great Britain... 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

In the ideal case being performed at X-ray energy transfers much higher than the characteristic energies of the scattering system, the impulse approximation [14] is applicable. In this case, the dynamical structure factor is directly connected with the electron momentum density p(p) ... 

Here, using electron field operator, momentum density is expressed as... 

The momentum density is given by the momentum wave functions and occupation number densities... 

Equation (15) is solved self-consistently employing the FLAPW method. Using the solutions, wave functions and energies, momentum densities in Equation (8) are calculated. In this step, one more drastic approximation we are going to make is that the occupation number in Equation (10) is replaced by the step function... 

It has been suggested that quasi-particle wave functions do not deviate much from LDA wave functions [26], Furthermore, in the evaluation of momentum densities shown in Figure 9, the characteristics of the quasi-particle states dominantly reflect on the occupation number densities which should be evaluated by using the general quasi-particle Green s function. In GWA, however, the corresponding occupation number densities are... 

Bauer, G.E. and Schneider, J.R. (1985) Electron correlation effect in the momentum density of copper metal, Phys. Rev., B31,681-692. 

Kubo, Y., Sakurai, Y. and Shiotani, N. (1999) Effects of self-interaction correction on momentum density in copper, J. Phys. Condens. Matter, 11, 1683-1695. 

Mueller, F.M. (1977) Anisotropic momentum densities from Compton profiles silicon, Phys. Rev., BIS, 3039-3044. 

Kobayasi, T., Nara, H., Timms, D.N. and Cooper, MJ. (1995) Core-orthogonalization effects on the momentum density distribution and the Compton profile of valence electrons in semiconductors, Bull. Coll. Med. Sci. Tohoku Univ., 4, 93-104. 

Nara, H., Kobayasi, T., Takegahara, K., Cooper, M.J. and Timms, D.N. (1994) Optimal number of directions in reconstructing 3D momentum densities from Compton profiles of semiconductors, Computational Materials Sci., 2, 366-374. 

The measurement of spectral momentum densities of solids by electron momentum spectroscopy... 

Figure 1. The electron momentum density for atomic hydrogen measured by EMS for the indicated energies compared with the square of Schrodinger wave function (solid curve) [4].

Secondly, correlations in the initial state can lead to experimental orbital momentum densities significantly different from the calculated Hartree-Fock ones. Figure 3 shows such a case for the outermost orbital of water, showing how electron-electron correlations enhance the density at low momentum. Since low momentum components correspond in the main to large r components in coordinate space, the importance of correlations to the chemically interesting long range part of the wave function is evident. 

Figure 2. The binding energy spectrum for valence electrons of ethyne and the corresponding measured and calculated self-consistent-field independent particle orbital momentum densities [5]. 

The outer most levels in C60 are due to rc orbitals . These are formed by 2p electrons which have their orbitals oriented along the radius of the molecule. The different environment inside and outside the spherical molecule causes the double-peaked structure in the momentum densities. In graphite the n band is formed by 2p orbitals oriented perpendicular to the sheets of carbon atoms. Using single-crystal graphite films we have a unique opportunity to study the effects of the orientation of these 2p orbitals in detail. 

According to theory the measured intensity should be directly proportional to the momentum density. Thus not only the peak position is meaningful, but also the area under the peak can be directly interpreted as the momentum density. To what extent is this confirmed by the experiment There are two main difficulties in verifying this claim. 

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

The same normalisation of theory to the experiment is used as in the momentum density plot. Clearly the Monte Carlo simulation compares better with the experiment than the LMTO calculation by itself, but at high binding energies there is still a significant amount of intensity missing in the theory. 

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