Orbital angular momentum penetration

Lifetimes. A very definite relationship between the lifetime against a-par-ticle emission and the velocity of the a-particle has been known for many years in the form of the lifetime-range relationship the Geiger-Nuttall law. This is, in fact, no more than an expression of the velocity dependence of the barrier penetration for a given value of the orbital angular momentum. The original... 

Therefore, in the Schrodinger equation (O Eq. (2.65)) the VdouL. = 2Ze /r term must be replaced hj V = Vcoul. + VdENTRiE- According to the calculations of Winslow and Simpson (1954) the effect of the orbital angular momentum (i), taken away by the a particle, on the penetration factor is as follows (O Table 2.8) ... 

Penetration factor for different values of the orbital angular momentum (/) taken away by the a... 

Graphs of the value of r R (r) as a function of rfor the orbitals in the first three principal shells. Note that the smaller the orbital angular momentum quantum number, the more closely an electron approaches the nucleus. Thus, s orbital electrons penetrate more, and are less shielded from the nucleus, than electrons in other orbitals with the same value of n. 

Another measure of the size of an orbital is the most probable distance of the electron from the nucleus in that orbital. Figure 5.4c shows that the most probable location of the electron is progressively farther from the nucleus in ns orbitals for larger n. Nonetheless, there is a finite probability for finding the electron at the nucleus in both 2s and 3s orbitals. This happens because electrons in s orbitals have no angular momentum ( = 0), and thus can approach the nucleus along the radial direction. The ability of electrons in s orbitals to penetrate close to the nucleus has important consequences in the structure of many-electron atoms and molecules (see later). 

From Figure 5.6, the R21 wave function has no radial nodes, and the R31 function has one radial node the R41 (not shown) function has two radial nodes. The R e wave functions have n- -l radial nodes. Because the angular part of the np wave function always has a nodal plane, the total wave function has n - I nodes (n - 1 radial and 1 angular), which is the same number as an s orbital with the same principal quantum number. The R2i(r) function (that is. Rip) in Table 5.2 contains the factor a, which is proportional to r (u = Zrjao) and causes it to vanish at the nucleus. This is true of all the radial wave functions except the ns functions, and it means that the probability is zero for the electron to be at the nucleus for all wave functions with > 0 (p, d, f,...). Physically, electrons with angular momentum are moving around the nucleus, not toward it, and cannot penetrate toward the nucleus. 

The manner in which the electron density varies as we move from the nucleus outward depends on the type of orbital. The density near the nucleus is greater for the 2x electron than for the 2p electron. In other words, a 2x electron spends more time near the nucleus than a 2p electron does (on the average). For this reason, the 2x orbital is said to be more penetrating than the 2p orbital, and it is less shielded by the lx electrons. In fact, for the same principal quantum number n, the penetrating power decreases as the angular momentum quantum number increases, or... 

By the requirement of orthogonality, which affects mainly valence orbitals with low angular momentum quantum number (in a classical picture the corresponding orbits penetrate into the core region). 

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