according to vector atom model find the value of total quantum number j for the f-electron
Answers
We need to be able to identify the electronic states that result from a given electron configuration and determine their relative energies. An electronic state of an atom is characterized by a specific energy, wavefunction (including spin), electron configuration, total angular momentum, and the way the orbital and spin angular momenta of the different electrons are coupled together. There are two descriptions for the coupling of angular momentum. One is called j-j coupling, and the other is called L-S coupling. The j-j coupling scheme is used for heavy elements (z > 40) and the L-S coupling scheme is used for the lighter elements. Only L-S coupling is discussed below.
L-S Coupling of Angular Momenta
L-S coupling also is called R-S or Russell-Saunders coupling. In L-S coupling, the orbital and spin angular momenta of all the electrons are combined separately
L=∑ili(8.9.1)
S=∑isi(8.9.2)
The total angular momentum vector then is the sum of the total orbital angular momentum vector and the total spin angular momentum vector.
J=L+S(8.9.3)
The total angular momentum quantum number parameterizes the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). Due to the spin-orbit interaction in the atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin.
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Figure 8.9.1 : "Vector cones" of total angular momentum J (purple), orbital L (blue), and spin S (green). The cones arise due to quantum uncertainty between measuring angular momentum components (see vector model of the atom). (Public Domain; Maschen).
However the total angular momentum J does commute with the Hamiltonian and so is a constant of motion (does not change in time). The relevant definitions of the angular momenta are:
Orbital Angular Momentum
|L⃗ |=ℏℓ(ℓ+1)−−−−−−√(8.9.4)
with its projection on the z-axis
Lz=mℓℏ(8.9.5)
Spin Angular Momentum
|S⃗ |=ℏs(s+1)−−−−−−−√(8.9.6)
with its projection on the z-axis
Sz=msℏ(8.9.7)
Total Angular Momentum
|J⃗ |=ℏj(j+1)−−−−−−√(8.9.8)
with its projection on the z-axis
Jz=mjℏ(8.9.9)
where
l is the azimuthal quantum number of a single electron,
s is the spin quantum number intrinsic to the electron,
j is the total angular momentum quantum number of the electron,
The quantum numbers take the values:
mℓ∈{−ℓ,−(ℓ−1)⋯ℓ−1,ℓ},ℓ∈{0,1⋯n−1}ms∈{−s,−(s−1)⋯s−1,s},mj∈{−j,−(j−1)⋯j−1,j},mj=mℓ+ms,j=|ℓ+s|(8.9.10)(8.9.11)(8.9.12)(8.9.13)
and the magnitudes are:
|J|=ℏj(j+1)−−−−−−√|J1|=ℏj1(j1+1)−−−−−−−−√|J2|=ℏj2(j2+1)−−−−−−−−√(8.9.14)(8.9.15)(8.9.16)
in which
j∈{|j1−j2|,|j1−j2|−1⋯j1+j2−1,j1+j2}(8.9.17)
This process may be repeated for a third electron, then the fourth etc. until the total angular momentum has been found.