Nuclear probability distribution

Figure 2.2 Nuclear probability distributions associated with the wavepacket supported by the adiabatic channels W (R,t) formed under aco = 943.3 cm 1 = 5 x 10 W/cm laser pulse with 3 = 0 on the left panel and 8 = n/lor the right panel. The wavepackets are taken at four times within the first optical cycle (a) t = 1/4T (b) t = 1/2T (c) t = 3/4T (d) t = T.

Figure 2.2 illusfrafes the dynamics of W and of the associated nuclear probability distributions. For 5 = 0 the field is at its peak intensity at f = 0, when the initial wavepacket is prepared on the inner repulsive edge of fhe affracfive potential (close to Cg in this region). Only its tail penetrates the gap region (R 4 a.u.) which, at that time, is widely open between W+ and W. At f = T/4, the wavepacket components reach the gap region with a gap now closed due to the vanishing field amplitude, preventing thus any... 

In an electron diffraction experiment the data are obtained as an average over the vibrational motions of the molecule at a particular temperature. If multiple scattering effects are neglected (see Chapter 1, p. 23), so that only scattering terms from pairs of atoms are included, the electron diffraction intensity depends on the nuclear probability distribution Pair) for each pair of atoms i and j in the molecule. 

The emission line is centered at the mean energy Eq of the transition (Fig. 2.2). One can immediately see that I E) = 1/2 I Eq) for E = Eq E/2, which renders r the full width of the spectral line at half maximum. F is called the natural width of the nuclear excited state. The emission line is normalized so that the integral is one f l(E)dE = 1. The probability distribution for the corresponding absorption process, the absorption line, has the same shape as the emission line for reasons of time-reversal invariance. 

Fig. 1. The lowest singlet (So) and first excited singlet (Si) surfaces of two hypothetical molecules. Vibrational wavefunctions for one and three vibrational levels, respectively, are indicated. Top part of a indicates schematically the time development of the nuclear geometry probability distribution after initial excitation...

For visualization purposes we have made plots of pair distribution functions, defining the electron-nuclear radial probability distribution function D(ri) by the formula... 

Fig. 3. Z-scaled electron-nuclear distribution functions for H, He, Li, and Ne (a) radial probability distribution D(r ) Z (b) radial density /o(ri)/Z. The curves can be identified from the fact that higher maxima correspond to higher Z.

Like the Coulombic forces, the van der Waals interactions decrease less rapidly with increasing distance than the repulsive forces. They include interactions that arise from the dipole moments induced by nearby charges and permanent dipoles, as well as interactions between instantaneous dipole moments, referred to as dispersion forces (Israelachvili 1992). Instantaneous dipole moments can be thought of as arising from the motions of the electrons. Even though the electron probability distribution of a spherical atom has its center of gravity at the nuclear position, at any very short instance the electron positions will generally not be centered on the nucleus. 

Due to the finite size of the nuclear charge distribution, the relative distance between the nucleus and the electron is not constant but is subject to additional fluctuations with probability p r). Hence, the energy levels experience an additional shift... 

Exercise. The neutrons in a nuclear reactor behave as free particles until they are absorbed, scatter, or cause fission and thereby produce more neutrons. The master equation for the joint probability distribution of the occupation numbers nk of the phase space cells X is... 

We note first that the masses of the nuclei are much greater than those of the electrons, Mproton = 1836 atomic units compared to electron = 1 atomic unit. Therefore nuclear kinetic energies will be negligibly small compared to those of the electrons. Typically, the amplitude of nuclear vibration is of the order of 1 % the spread of an electron s probability distribution. In accordance with the Bom-Oppenheimer approximation, we can first consider the electronic Schrbdinger equation... 

The electron probability density along the line passing through the nuclei is graphically represented as in Fig. 4.5 for H2 and the isoprobability contours for a plane containing the intemuclear axis are similar to those of Fig. 4.7 for the same ion. If we seek a distribution similar to the radial probability distribution for atoms. Fig. 6.1 is obtained (ref. 65). It shows the circular distribution of electron density for different distances from the inter-nuclear axis. It is found that the electronic charge is concentrated in a circular doughnut around the H-H axis, with a maximum at about 37 pm from the axis and about 50-55 pm from each nucleus. 

The equilibrium bulk magnetisation comes from the net moment on summation of contributions from individual spins. A sample containing spin- /2 nuclei has two states (energy levels) with the nuclear spins distributed between these according to the Boltzmann distribution which gives the probability of occupation of the different states. 

This expression looks just like the continuous-wave differential cross-section for a fixed nuclear geometry in a single electronic state (1), weighted by the probability distribution of nuclear geometries in the populated electronic states, summed up over the duration of the incoming X-ray pulse and, finally, weighted by the... 

Total radial probability distributions for the helium, neon, and argon atoms are shown in the following graph. How can one interpret the shapes of these curves in terms of electron configurations, quantum numbers, and nuclear charges ... 

In textbooks, justifications for this choice of LCAO function, often, involve considerations of the electron probability densities for different separations of the nuclei and positions of the electron. In the region of one nucleus, especially for large separations of the nuclei, the probability distribution should correspond to that for an occupied Is atomic orbital about the nuclear position. At very large separations, we do not know whether to describe the electron in terms of one atomic orbital or the other. Thus, the most appropriate function is the form given in equation 6.2. 

Notice that the nuclear wavefunction has no parametric dependence on the electronic coordinates. Clearly the probability distribution of the nuclei must involve the electrons since it is the electrons which hold the nuclei together. This basic asymmetry in the two sets of particles shows itself in the equations which the two wavefunctions satisfy ... 

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