Reduced electron mass

In laser enrichment, one takes advantage of the fact that the atomic energy levels of different isotopes differ slightly (called the isotope shift). This is due to the differing reduced electron masses for the different isotopes and the change in the overlap between the wave functions of the inner electrons and the nucleus, due to isotopic... 

Here p, = p2 is the reduced electron mass, MN is the nuclear mass, n and r2 are relative coordinates described in the center-of-mass transformation, eqn.(8), and we have written the momentum operator p as —iV. For the values of the nuclear masses see Table(l). 

For the He atom there has been an extremely large number of calculations using all kinds of different methods. The most accurate value has, again, been found with the ECW-SOS method used for H and is a = 1.38319 au[33] (with the atomic unit based on a reduced electron mass appropriate for the He nucleus) the experimental value is 1.38320 0.00007 au[34]. Other accurate values are to be found in Refs. [27], [31], and [35]. We recall that Pauling s value, using his theoretically determined screening constant, was 1.343 au. 

The simplest example is that of tire shallow P donor in Si. Four of its five valence electrons participate in tire covalent bonding to its four Si nearest neighbours at tire substitutional site. The energy of tire fiftli electron which, at 0 K, is in an energy level just below tire minimum of tire CB, is approximated by rrt /2wCplus tire screened Coulomb attraction to tire ion, e /sr, where is tire dielectric constant or the frequency-dependent dielectric function. The Sclirodinger equation for tliis electron reduces to tliat of tlie hydrogen atom, but m replaces tlie electronic mass and screens the Coulomb attraction. 

In most applications, the reduced mass is sufficiently close in value to the electronic mass that it is customary to replace fx in the expressions for the energy levels and wave functions by m. The parameter /fjte is... 

The effect of the spin-orbit interaction term on the total energy is easily shown to be small. The angular momenta L and S are each on the order of h and the distance r is of the order of the radius ao of the first Bohr orbit. If we also neglect the small difference between the electronic mass We and the reduced mass the spin-orbit energy is of the order of... 

It is common practice to use the electronic mass, m rather than the reduced mass fi. This amounts to the assumption of an infinitely heavy nucleus, and numerically makes a difference of less than 0.0005. For heavier nuclei the difference is even less. 

To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H20)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunc-tions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. 

In El mass spectrometry, the molecular ion peak can be increased to a certain degree by application of reduced electron energy and lower ion source temperature (Chap. 5.1.5). However, there are compounds that thermally decompose prior to evaporation or where a stable molecular ion does not exist. The use of soft ionization methods is often the best way to cope with these problems. 

This equation is exactly the same as the equation seen above for the radial motion of the electron in the hydrogen-like atoms except that the reduced mass ji replaces the electron mass m and the potential V(r) is not the coulomb potential. 

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

One trivial improvement of the Dirac formula for the energy levels may easily be achieved if we take into account that, as was already discussed above, the electron motion in the Coulomb field is essentially nonreiativistic, and, hence, all contributions to the binding energy should contain as a factor the reduced mass of the electron-nucleus nonreiativistic system rather than the electron mass. Below we will consider the expression with the reduced mass factor... 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

The largest contribution to the uncertainty of the indirect mass ratio in (12.34) is supplied by the unknown theoretical contributions to hyperfine splitting. This sets a clear task for the theory to reduce the contribution of the theoretical uncertainty in the error bars in (12.34) to the level below two other contributions to the error bars. It is sufficient to this end to calculate all contributions to HFS which are larger than 10 Hz. This would lead to further reduction of the uncertainty of the indirect value of the muon-electron mass ratio. There is thus a real incentive for improvement of the theory of HFS to account for all corrections to HFS of order 10 Hz, created by the recent experimental and theoretical achievements. 

If internal nuclear motion is neglected, then the reduced mass p is approximated by the electron mass m, and a in these formulas is replaced by a0 (the Bohr radius), where... 

Another reason for the deviation of the calculated data from experimental may be the fact that the true value of the parameter ae may differ from the value calculated according to the formula ae = h(2mle) 1/2 due to the difference between the effective electron mass for tunneling in a condensed medium and its real mass (see Chap. 3, Sect. 3), as well as because in fact the electron can tunnel under a barrier whose height corresponds to the ionization potential of the reduced acceptor, 7a, rather than of the donor, 7d. 

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