= Solution
{c}
For $5s5d$, <addition of angular momentum> gives $L=2$ and $S=0,1$. The even-<parity> <atomic terms> and <atomic fine-structure levels> are
$$
\boxed{{}^1D_2,\qquad{}^3D_1,{}^3D_2,{}^3D_3.}
$$
The <electric-dipole selection rules> exclude all connections to the even-<parity> $5s^2$ and $5s6s$ <electron configurations>. To the odd-<parity> $5s5p$ <electron configuration>, all seven allowed connections in pure <LS coupling> are
$$
\boxed{\begin{aligned}
{}^1D_2&\leftrightarrow{}^1P_1,\\
{}^3D_1&\leftrightarrow{}^3P_0,{}^3P_1,{}^3P_2,\\
{}^3D_2&\leftrightarrow{}^3P_1,{}^3P_2,\\
{}^3D_3&\leftrightarrow{}^3P_2.
\end{aligned}}
$$
Here a connection is listed independently of whether the $5s5d$ <energy level> is above or below its partner.
In a weak <magnetic field>, use the <Landé g-factor>
$$
g_J=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.
$$
For the three ${}^3D_J$ <energy levels>, this gives $1/2,7/6,4/3$; for ${}^3P_1,{}^3P_2$ it gives $3/2$. The six-component <spectral line> is therefore
$$
\boxed{{}^3D_1\leftrightarrow{}^3P_1.}
$$
A component has <frequency> displacement $(\mu_BB/h)(m_D/2-3m_P/2)$. The <electric dipole transition> with $m_D=m_P=0$ has zero <matrix element>, since the relevant $J=1\leftrightarrow1$ rank-one <Clebsch-Gordan coefficients> vanish there. The remaining allowed pairs give
$$
\begin{array}{c|c|c}
\text{polarization}&m_D-m_P&h\Delta\nu/(\mu_BB)\\\hline
\pi&0&-1,+1\\
\sigma&+1&+1/2,+3/2\\
\sigma&-1&-3/2,-1/2
\end{array}
$$
Thus all six <frequency> positions $\pm1/2,\pm1,\pm3/2$ are distinct. Perpendicular to the <magnetic field>, both the $\pi$ and $\sigma$ components are visible, with mutually perpendicular linear <polarizations>. Along the <magnetic field> the $\pi$ components disappear. This is the weak-field <Zeeman effect>, before the <Paschen-Back effect> changes the coupling scheme.
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