Chapter 8, Problem 8D.6E

(a)

Interpretation Introduction

Interpretation:

The energy levels of ${}^{\text{1}}\text{S}$ term has to be identified.

Concept-Introduction:

Total angular momentum quantum number J:

It is the vector sum of the orbital and spin angular momenta. The possible values of J are found as,

    $\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}$

Explanation of Solution

The total angular momentum quantum number, J, is calculated as,

    $\begin{array}{l}\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}\\ \\ \text{L}\text{=}\text{0}\text{(derived}\text{from}\text{symbol}\text{S)}\\ \\ \text{S}\text{=}\text{0}\text{(Obtained}\text{from}\text{2S+1}\text{=}\text{1)}\\ \\ \text{J}\text{=}\text{0}\end{array}$

Therefore, the energy level is ${}^{\text{1}}{\text{S}}_0$

(b)

Interpretation Introduction

Interpretation:

The energy levels of ${}^{\text{3}}\text{F}$ term has to be identified.

Concept-Introduction:

Total angular momentum quantum number J:

It is the vector sum of the orbital and spin angular momenta. The possible values of J are found as,

    $\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}$

Explanation of Solution

The total angular momentum quantum number, J, is calculated as,

    $\begin{array}{l}\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}\\ \\ \text{L}\text{=}\text{3}\text{(derived}\text{from}\text{symbol}\text{F)}\\ \\ \text{S}\text{=}\text{1}\text{(Obtained}\text{from}\text{2S+1}\text{=}\text{3)}\\ \\ \text{J}\text{=}3+1\text{,}3+1-1\text{,}3-1\\ \\ \text{J}\text{=}\text{4,}\text{3,}\text{2}\end{array}$

Therefore, the energy levels are ${}^{\text{3}}{\text{F}}_{\text{4}},{}^{\text{3}}{\text{F}}_3,{}^{\text{3}}{\text{F}}_{\text{2}}$

(c)

Interpretation Introduction

Interpretation:

The energy levels of ${}^5\text{S}$ term has to be identified.

Concept-Introduction:

Total angular momentum quantum number J:

It is the vector sum of the orbital and spin angular momenta. The possible values of J are found as,

    $\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}$

Explanation of Solution

The total angular momentum quantum number, J, is calculated as,

    $\begin{array}{l}\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}\\ \\ \text{L}\text{=}\text{0}\text{(derived}\text{from}\text{symbol}\text{S)}\\ \\ \text{S}\text{=}2\text{(Obtained}\text{from}\text{2S+1}\text{=}5\text{)}\\ \\ \text{J}\text{=}\text{0}\text{+}\text{2}\text{=}\text{2}\end{array}$

Therefore, the energy level is ${}^5{\text{S}}_2$

(d)

Interpretation Introduction

Interpretation:

The energy levels of ${}^{\text{5}}\text{P}$ term has to be identified.

Concept-Introduction:

Total angular momentum quantum number J:

It is the vector sum of the orbital and spin angular momenta. The possible values of J are found as,

    $\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}$

Explanation of Solution

The total angular momentum quantum number, J, is calculated as,

    $\begin{array}{l}\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}\\ \\ \text{L}\text{=}\text{1}\text{(derived}\text{from}\text{symbol}\text{P)}\\ \\ \text{S}\text{=}\text{2}\text{(Obtained}\text{from}\text{2S+1}\text{=}\text{5)}\\ \\ \text{J}\text{=}\text{1}\text{+}\text{2,}\text{1}\text{+}\text{2-1,}\text{|1 - 2|}\\ \\ \text{J}\text{=}\text{3,}\text{2,}1\end{array}$

Therefore, the energy levels are ${}^{\text{5}}{\text{P}}_{\text{3}}\text{,}{}^{\text{5}}{\text{P}}_{\text{2}}\text{,}{}^{\text{5}}{\text{P}}_{\text{1}}$