Question

A3. (a) The spectrum of titanium (Z=22) can be described by the L-S coupling scheme. (i) Using the Hartree model, explain why the ground-state configuration of the titanium atom has a pair of outer electrons in the 3d sub-shell. (ii) Using the L-S coupling scheme, draw a labelled energy level diagram showing the splitting of the 3d² configuration of titanium. Explain your reasoning. (b) In the L-S coupling scheme, the fine structure corrections in atomic energy spectra are described by the following expression $\Delta E_J = \frac{C}{2} \left[ J(J+1) - L(L+1) - S(S+1) \right]$ where C is a constant and the quantum numbers S, L and J represent the total spin, total orbital angular momentum, and total angular momentum of the electrons, respectively. The spectrum of titanium contains a set of energy levels at 1.0460 eV, 1.0529 eV and 1.0666 eV relative to the ground state. Determine the quantum numbers S, L and J associated with each of these levels, and calculate the value of the constant C. (c) When a sample of titanium is placed in a 1.73 T magnetic field, a certain energy level is found to split into nine sub-levels, which are equally spaced by an energy of 1.00 x 10?? eV. Deduce the most likely values of the quantum numbers S, L and J associated with this energy level. Hint: The Landé g-factor is $g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$

          A3. (a) The spectrum of titanium (Z=22) can be described by the L-S coupling scheme.
(i) Using the Hartree model, explain why the ground-state configuration of the titanium atom has a pair of outer electrons in the 3d sub-shell.
(ii) Using the L-S coupling scheme, draw a labelled energy level diagram showing the splitting of the 3d² configuration of titanium. Explain your reasoning.
(b) In the L-S coupling scheme, the fine structure corrections in atomic energy spectra are described by the following expression
$\Delta E_J = \frac{C}{2} \left[ J(J+1) - L(L+1) - S(S+1) \right]$
where C is a constant and the quantum numbers S, L and J represent the total spin, total orbital angular momentum, and total angular momentum of the electrons, respectively.
The spectrum of titanium contains a set of energy levels at 1.0460 eV, 1.0529 eV and 1.0666 eV relative to the ground state.
Determine the quantum numbers S, L and J associated with each of these levels, and calculate the value of the constant C.
(c) When a sample of titanium is placed in a 1.73 T magnetic field, a certain energy level is found to split into nine sub-levels, which are equally spaced by an energy of 1.00 x 10?? eV. Deduce the most likely values of the quantum numbers S, L and J associated with this energy level.
Hint: The Landé g-factor is $g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$

A3. (a) The spectrum of titanium (Z=22) can be described by the L-S coupling scheme.
(i) Using the Hartree model, explain why the ground-state configuration of the titanium atom has a pair of outer electrons in the 3d sub-shell.
(ii) Using the L-S coupling scheme, draw a labelled energy level diagram showing the splitting of the 3d² configuration of titanium. Explain your reasoning.
(b) In the L-S coupling scheme, the fine structure corrections in atomic energy spectra are described by the following expression
Δ EJ = (C)/(2)[ J(J+1) - L(L+1) - S(S+1) ]
where C is a constant and the quantum numbers S, L and J represent the total spin, total orbital angular momentum, and total angular momentum of the electrons, respectively.
The spectrum of titanium contains a set of energy levels at 1.0460 eV, 1.0529 eV and 1.0666 eV relative to the ground state.
Determine the quantum numbers S, L and J associated with each of these levels, and calculate the value of the constant C.
(c) When a sample of titanium is placed in a 1.73 T magnetic field, a certain energy level is found to split into nine sub-levels, which are equally spaced by an energy of 1.00 x 10?? eV. Deduce the most likely values of the quantum numbers S, L and J associated with this energy level.
Hint: The Landé g-factor is gJ = 1 + (J(J+1) + S(S+1) - L(L+1))/(2J(J+1))

Added by Justin G.

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