Angular Momentum in Many-Electron Atoms

Find the possible J values when angular momenta with quantum numbers j =1, 

To add more than two angular momenta, we apply (11.39) repeatedly. Addition of 71 = 1 and 7 2 = 2 gives the possible quantum numbers 3,2, and 1. Addition of 73 to each of these values gives the following possibilities for the total-angular-momentum quantum number  

We have one set of states with total-angular-momentum quantum number 6, two sets of states with / = 5, three sets with 7 = 4, and so on. 

EXERCISE Find the possible J values when angular momenta with quantum numbers 7i = 1.72 = 1. and 73 = 1 are added. (Answer J = 3,2,2,1,1,1,0.) 

Total Electronic Orbital and Spin Angular Momenta 

Total Electronic Orbital and Spin Angular Momenta. The total electronic orbital angular momentum of an -electron atom is defined as the vector sum of the orbital anguleu momenta of the individual electrons  

Although the individual orbitd-angular-momentum operators L, do not commute with the atomic Hamiltonian (11.1), one can show (Bethe and Jackiw, pp. 102-103) that L does commute with the atomic Hamiltonian [provided spin-orbit interaction (Section 11.6) is neglected]. We can therefore characterize an atomic state by a quantum number L, where L(L -I- 1) is the square of the magnitude of the toted electronic orbital angular momentum. The electronic wave function il/ of an atom satisfies L tfr = L(L -I- The total-electronic-orbital-angular-momentum quantum number L of an atom is specified by a code letter, as follows  

The total orbital anguleu momentum is designated by a capital letter, while lowercase letters are used for orbited angular momenta of individual electrons. 

EXAMPLE Find the possible values of the quantum number L for states of the carbon atom that arise from the electron configuration ls 2s 2p3d. 

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Section 11.5 Angular Momentum in Many-Electron Atoms 323... 

The energy of the electron in a hydrogen atom is determined solely by its principal quantum number. In many-electron atoms, the principal quantum number and the angular momentum quantum number together determine the energy of an electron. 

ANGULAR MOMENTUM COUPLING IN MANY-ELECTRON ATOMS... 

The simple L, S model falls victim to the Z" increase of the L S coupling between electron spin (S) and the angular momentum (L) of the orbitals in many-electron atoms at about Z = 30. Heavier elements need to be described by the /, J scheme where J = L + S. 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

The elements of the theory of angular momentum and irreducible tensors presented in this chapter make a minimal set of formulas necessary when calculating the matrix elements of the operators of physical quantities for many-electron atoms and ions. They are equally suitable for both non-relativistic and relativistic approximations. More details on this issue may be found in the monographs [3, 4, 9, 11, 12, 14, 17]. 

The role of graphical methods to represent angular momentum, 3 nj- and jm-coefficients was emphasized in the Introduction as one of the most important milestones in the theory of many-electron atoms and ions. 

The Bohr model of one-electron atoms Bohr postulated quantization of the angular momentum, L = m vr = nh/lir, substituted the result in the classical equations of motion, and correctly accounted for the spectrum of all one-electron atoms. E = —Z jrd (rydbergs). The model could not, however, account for the spectra of many-electron atoms. 

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). 

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