Question

(a) Show that Hund's rules for a shell of angular momentum \(l\) containing \(n\) electrons can be summarizcd in the formulas: $$ \begin{aligned} &S=\frac{1}{2}[(2 l+1)-|2 l+1-n|] \\ &L=S|2 l+1-n| \\ &J=|2 l-n| S \end{aligned} $$ (b) Verify that that the two ways of counting the degeneracy of a given \(L S\) -multiplet give the same answer: ie, verify that $$ (2 L+1)(2 S+1)=\sum_{|L=5|}^{L+S}(2 J+1) $$ (c) Show that the total splitting of an \(L . S\) -multiplet due to the spin orbit interaction \(\lambda(L \cdot S)\) is $$ \begin{aligned} E_{J_{\max }}-E_{J_{\min }} &=\lambda S(2 L+1), & & L>S \\ &=\lambda L(2 S+1), & & S>L \end{aligned} $$ and that the splittings between successive \(J\) -multiplets within the \(L S\) -multiplet is $$ E_{J+1}-E_{\jmath}=\lambda(J+1) $$

Short answer

Hund's rules can be summarized as given by the formulas, and their derivations follow from the rules' concepts of maximum spin, maximum orbital angular momentum, and energy levels for subshells. Degeneracy is verified by equating the product of degeneracies of L and S with the sum of degeneracies of all possible J states. The total and successive splittings are calculated according to the formulas for LS multiplets, considering the spin-orbit interaction.

Step-by-step solution

Derive Hund's Rules Formulas

Recall Hund's first rule: the term with maximum spin multiplicity has the lowest energy. The maximum spin S occurs when the maximum number of electrons have their spins parallel. The maximum spin is given by half the number of unpaired electrons. For a shell of angular momentum l with n electrons, if n is less than or equal to (2l+1), all electrons can be unpaired, giving a spin S=(n/2). If n is greater than (2l+1), there will be (n-(2l+1)) pairs of electrons with opposite spins, so S=[(2l+1)-(n-(2l+1))]/2. This can be summarized as S=(1/2)[(2l+1)-|2l+1-n|].

Calculate the Total Orbital Angular Momentum L

Hund's second rule states that for a given spin multiplicity, the term with the largest value of the total orbital angular momentum L lies lowest in energy. To calculate L, we know that it is proportional to the number of holes in the subshell which is |2l+1-n|, and as L must be zero or a positive value, and the spin value is S, it's just the product of these two, giving us L=S|2l+1-n|.

Determine the Total Angular Momentum J

Finally, Hund's third rule tells us that for subshells that are less than half-filled, the term with the smallest value of J lies lowest in energy, while for those more than half-filled, the term with the largest value of J lies lowest. J is found using the absolute difference between the orbital angular momentum L and the spin S, but because we've defined L in terms of S already, J is simply |2l-n|S.

Verify the Degeneracy of LS Multiplet

To verify the degeneracy equality, we use (2L+1)(2S+1) and compare it with summing over all the possible values of J, which range from |L-S| to L+S, as \(2J+1\). This is due to the fact that J can have all integer values in that range, and each value of J corresponds to a state with a degeneracy of (2J+1). We should sum the degeneracies for all these states to verify that it gives the same total degeneracy as (2L+1)(2S+1).

Show the Total Splitting of LS Multiplet

To find the total splitting of the LS multiplet, we compute the difference between the energy of the state with maximum J, \(E_{J_{max}}\), and the energy of the state with minimum J, \(E_{J_{min}}\). The splitting depends on the relationship between L and S. If \(L > S\), then the total splitting is given by \(\lambda S(2L+1)\), and if \(S > L\), it is given by \(\lambda L(2S+1)\).

Calculate the Splitting Between Successive J-multiplets

The splitting between successive J-multiplets within the LS multiplet is given by the energy difference \(E_{J+1}-E_J\). For each increase in J by 1, the corresponding increase in energy is \(\lambda(J+1)\), where \(\lambda\) is the spin-orbit coupling constant.

Key concepts

Spin-Orbit Interaction

In quantum mechanics, the spin-orbit interaction is a relativistic effect that occurs when an electron's magnetic moment, associated with its spin angular momentum, interacts with the magnetic moment induced by the electron's movement around the nucleus—which is related to its orbital angular momentum. Imagine an electron circling a nucleus; to the electron, it seems as though the nucleus is spinning around it, generating a magnetic field. This magnetic field then affects the electron's spin.

This interaction causes energy levels to split into fine-structured sublevels within an atom. The magnitude of the energy shift due to spin-orbit interaction is characterized by the spin-orbit coupling constant, \(\lambda\), which varies depending on the atom and its electronic state. In multi-electron atoms, the effect of spin-orbit interaction is vital for determining the energy and spectral properties of the atom and plays a significant role in Hund's rules for determining the ground state electronic configuration.

Angular Momentum

Angular momentum is a fundamental concept in physics related to the rotational motion of an object. In atomic physics, we often discuss two types of angular momentum: the orbital angular momentum and the spin angular momentum. The orbital angular momentum refers to the motion of electrons around the nucleus, analogous to the Earth's orbit around the Sun. This momentum is quantized in atoms, meaning it can only take on certain discrete values, described by the quantum number \(l\).

The spin angular momentum, on the other hand, is an inherent property of particles like electrons and is related to the particle's spin. The quantum number \(s\) represents it. Both types of angular momentum can impact each other through the spin-orbit interaction mentioned earlier. Understanding angular momentum is essential not just in atomic physics but also in understanding the structure and behavior of molecules and condensed matter systems.

LS Multiplet Degeneracy

In the context of atomic physics, LS multiplet degeneracy pertains to the number of quantum states that have the same total orbital angular momentum (\(L\)) and total spin angular momentum (\(S\)). Within an energy level, electrons can occupy states that have the same energy, which we refer to as 'degenerate' states. The total number of these states is determined by the values of \(L\) and \(S\), with the degree of degeneracy given by the formula \( (2L+1)(2S+1) \).

According to Hund's rules, for any LS multiplet, the total spin \(S\) is maximized first, followed by maximizing the total orbital angular momentum \(L\) consistent with \(S\), and finally, the total angular momentum \(J\) depends on whether the shell is more or less than half-filled. The multiplet degeneracy is essential in spectroscopy as it explains the number of spectral lines that can appear due to transitions between quantum states within the multiplet.