Averaged momentum distribution

Figure 2. Spherically averaged momentum distribution (in nip)) of water protons at several temperatures. Black solid and dashed lines refer to measurements in the supercooled metastable phase, while dash-dotted line is the result of a measurement at 298K. Note the appearance, in the supercooled phase data, of a peak or shoulder at high p, around p = n A , indicating proton coherent delocalization over two sites of the potential felt by protons. Experimental uncer tainties are less than 1%.

Figure 3. Top panel spherically averaged momentum distribution (4 r n(p)) of deuterons at T = 292.15K (solid line) and T = 276.15 K (dashed line). Bottom panel spherically averaged momentum distribution 4jr n p)) of deuterons at T = 276.15K (dashed line) compared to that of protons at T = 269.15K (black line), according to the shift of 7K due to the temperature difference between the density maxima of the two liquids.

Figure 4. Spherically averaged momentum distribution of water protons (4 r n(p)) at constant temperature T = 268K and several pressure values in the range 0.1-400 MPa. Note the absence of the high-momentum peak for supercooled water under pressure. The inset shows the high-momentum region of the distribution to evidence the absence of correlation between the applied pressure and the intensity of the tail of the proton momentum distribution.

The denominator in Eq. (13) can be interpreted as an average value over the momentum distribution from the initial wavepacket, that is,... 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

At this point, one may wonder why there is an interest in the atomic momentum densities and their nature and what sort of information does one derive from them. In a system in which all orientations are equally probable, the full three-dimensional (3D) momentum density is not experimentally measurable, but its spherical average is. The moments of the atomic momentum density distributions are of experimental significance. The moments and the spherically averaged momentum densities are defined in the equations below. 

Further support to the model should be found by examining the values of F], as deduced from AC or DB measurements since p-Ps annihilates in an intrinsic mode, this parameter should reflect directly the average Ps kinetic momentum. However, the data on F, are usually poorly defined [61, 64], so that the correlation with y is more conveniently sought using experimental values of r3, provided that the momentum distribution of the valence electrons participating in the o-Ps pick-off annihilation is reasonably solvent independent. Such a correlation has been effectively found for a variety of solvents at various temperatures [61], leading to ... 

Here, (...) = J2pn ri. .. ri) is the appropriate combinedquantal and thermodynamic average (over the classical probabilities pn) related with the condensed matter system. M and n(p) are the mass and momentum distribution of the scattering nucleus, respectively, and ior = q2 /2 M is the recoil energy. For convenience, h = 1. Eq. (2) is of central importance in most NCS experiments, since it relates the SCS directly to the momentum distribution. Furthermore, n(p) is related to the nuclear wave function by Fourier transform and therefore, to the spatial localization of the nucleus. It takes into account the fact that, if the scattering nucleus has a momentum distribution in its ground state, the 5-function centered at uor will be Doppler broadened. 

As might be expected, the model leads to a great simplification over the calculations required for molecules with a continuous potential energy function, as it enables the analysis to be confined to binary collisions and permits the definition of a collision frequency. Because there is no molecular interaction between collisions, the velocity distributions of two colliding molecules may be assumed to be re-established by the time a second collision occurs between them. Thus a Maxwellian distribution around the local mass velocity may be postulated for the calculation of the mean frequency of collision and the average momentum and energy transported per collision in the nonuniform state of the liquid. 

The symbol ) in Eq. (3.6) denotes initial condition averaging using the optimized LHO approximation to the phase-space centroid density. It should be noted that the momentum centroid Pq is always decoupled from the position coordinates and is described by the classical Boltzmann momentum distribution. This result is both interesting and significant for the role of the centroid variable in dynamics [4,5,8]. 

For p d l the quantity exp(ip d) would be equal to unity and one would recover the standard cross section. The average < exp(ip-d) > depends on the initial momentum p of the proton and its orientation relative to H-H vector d. For p perpendicular to d, exp(ip-d)j = 1, but for p 11 d, the < exp(ip-d) >n terms are strongly reduced if there is a large zero-point contribution to the momentum distribution n(p). The oscillations in exp(ip-d) are effectively averaged out by the large intrinsic zero-point momentum spread in of the hydrogen isotopes, which typically amounts to A = 4 for protons (see Fig. 22.6). 

GAUSsian distributions with 1/e values of about 8, 18, and 18 Mev respectively for Li, C, and 0. Due to approximations in the theory, the relative differences are more significant than the absolute values. Barton and Smith have compared the widths from He and Li, and find that the mean momentum of nucleons in He is higher than that in Li. It thus seems definite that the average momentum of the nucleons in Li is less than that of other light nuclei measured to date. 

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