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64. For a two-electron atom, jj coupling is more appropriate than L-S coupling provided the coupling between:

(A) $l_1$ and $l_2$ is much stronger than that between $s_1$ and $s_2$.
(B) $l_1$ and $l_2$ is much weaker than that between $s_1$ and $s_2$.
(C) $l_1$ and $s_1$ is much weaker than that between $l_2$ and $s_2$.
(D) $l_1$ and $s_1$ is much stronger than that between $l_1$ and $l_2$.

Answer :

In the context of atomic physics, particularly when discussing two-electron atoms, coupling schemes describe how the angular momenta of electrons are combined to give the total angular momentum of the system. Two common coupling schemes are L-S coupling (also known as Russell-Saunders coupling) and j-j coupling. Let's talk about when j-j coupling becomes more appropriate over L-S coupling.

L-S Coupling

  • L-S Coupling is generally used when the electrostatic interaction between the orbital angular momenta (L) of the electrons is stronger than their spin-spin interactions. This coupling scheme assumes that the total orbital angular momentum (L) and the total spin angular momentum (S) couple strongly before interacting with each other to form the total angular momentum (J).

j-j Coupling

  • j-j Coupling is more appropriate under circumstances where each electron’s total angular momentum j (which is a combination of an electron’s orbital angular momentum l and its spin s) interacts strongly with the other electron’s total angular momentum. It assumes that each electron first forms its total angular momentum j and then these are coupled together.

Choice Explanation

Given the options:

(A) [tex]l_1[/tex] and [tex]l_2[/tex] is much stronger than that between [tex]s_1[/tex] and [tex]s_2[/tex].

(B) [tex]l_1[/tex] and [tex]l_2[/tex] is much weaker than that between [tex]s_1[/tex] and [tex]s_2[/tex].

(C) [tex]l_1[/tex] and [tex]s_1[/tex] is much weaker than that between [tex]l_2[/tex] and [tex]s_2[/tex].

(D) [tex]l_1[/tex] and [tex]s_1[/tex] is much stronger than that between [tex]l_1[/tex] and [tex]l_2[/tex].

For j-j coupling to be more appropriate, the coupling between individual orbital and spin angular momenta of each electron must be stronger than the coupling between the different electrons. Therefore, the correct answer is:

(A) [tex]l_1[/tex] and [tex]l_2[/tex] is much stronger than that between [tex]s_1[/tex] and [tex]s_2[/tex].

When the interaction between each electron's own orbital angular momentum and spin is significant, as expressed in choice (A), the j-j coupling is favored as it first forms strong individual total angular momenta for each electron before coupling them together.

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