Isotope Shift in Magnesium

A theoretical investigation is made to see whether nuclear motion alone can account for observed fine structure in magnesium, especially for the lines $3^{1}S_{0}\ensuremath{-}3^{1}P_{1}$, $3^{1}S_{0}\ensuremath{-}3^{3}P_{1}$, $3^{1}P_{1}\ensuremath{-}3^{1}D_{2}$, $3^{3}P\ensuremath{-}3^{3}D$, and the limit of the series $3^{1}P_{1}\ensuremath{-}n^{1}D_{2}$. Both by calculation with explicit radial functions and by the choice of theoretical parameters to fit observed shifts, it is shown that the values of the core-valence parameters occurring in the theory are not at all negligible, so that a two-electron model would not be permissible. Calculations with explicit radial functions give correct signs for the specific shifts of all these lines except for the limiting shift of $3^{1}P_{1}\ensuremath{-}n^{1}D_{2}$. In the case of the line $3^{1}P_{1}\ensuremath{-}3^{1}D_{2}$ the correct sign is obtained only when one takes into account the perturbation of $(3s) (3d) ^{1}D_{2}$ by ${(3p)}^{2}^{1}D_{2}$; however, greater accuracy in the solution of the perturbation problem could not improve the agreement very much as long as the present radial functions are used. The total shift of any member of $3^{3}P\ensuremath{-}3^{3}D$ comes out much smaller than any of the other total shifts, in agreement with Meissner's observation of the sharpness of its members. A test of the theory independent of the use of explicit radial functions is carried out by adjusting the theoretical parameters so as to give the observed shifts of the five lines mentioned. According to the theory, all these parameters are essentially positive; the necessary values of the parameters do indeed all turn out to be positive, provided that the value of Jackson and Kuhn is accepted in preference to that of Fisher for the shift of the resonance line. The shift of the line $3^{1}P_{1}\ensuremath{-}5^{1}S_{0}$ and the sharpness of the lines $3^{3}P\ensuremath{-}4^{3}S$ afford independent evidence that Fisher's value is incorrect, or else that nuclear motion is not the sole cause of the shift in the resonance line $3^{1}S_{0}\ensuremath{-}3^{1}P_{1}$. There also arises an inconsistency unless the perturbation of $(3s) (3d) ^{1}D_{2}$ by ${(3p)}^{2} ^{1}D_{2}$ is taken into account. The necessary values of the parameters are shown by the estimate of corresponding $f$ values to be reasonable both as to their absolute and as to their relative values. With certain acceptable assumptions, it is shown that if the shifts in Mg are due to nuclear motion alone, then three relations may be deduced connecting the specific shifts. These are: (1) the specific shifts of the resonance line $3^{1}S_{0}\ensuremath{-}3^{1}P_{1}$ and of any member of $3^{3}P\ensuremath{-}{(3p)}^{2} ^{3}P$ must be equal; (2) (for any members) the sum of the specific shifts of the lines $3^{3}D\ensuremath{-}(3p) (3d) ^{3}D$ and $3^{3}P\ensuremath{-}3^{3}D$ must equal the sum of the specific shifts of the lines ${(3p)}^{2} ^{3}P\ensuremath{-}(3p) (3d) ^{3}D$ and $3^{1}S_{0}\ensuremath{-}3^{1}P_{1}$; (3) a complicated quadratic relation involving the specific shifts of the five lines $3^{1}S_{0}\ensuremath{-}3^{1}P_{1}$, $3^{1}S_{0}\ensuremath{-}3^{3}P_{1}$, $3^{1}P_{1}\ensuremath{-}3^{1}D_{2}$, $3^{3}P\ensuremath{-}3^{3}D$, and ${(3p)}^{2} ^{3}P\ensuremath{-}(3p) (3d) ^{3}D$, as well as the interaction parameter $\ensuremath{\alpha}$ for $3^{1}D_{2}$. With the use of known shifts, including the Jackson-Kuhn value for the resonance line, these relations lead to the prediction (for any member) of total shifts of 0.035 ${\mathrm{cm}}^{\ensuremath{-}1}$ for $\ensuremath{\lambda}2778 \mathrm{to} 2784=3^{3}P\ensuremath{-}{(3p)}^{2} ^{3}P$, of 0.027 ${\mathrm{cm}}^{\ensuremath{-}1}$ for $\ensuremath{\lambda}3890 \mathrm{to} 3900={(3p)}^{2} ^{3}P\ensuremath{-}(3p) (3d) ^{3}D$, and of 0.061 ${\mathrm{cm}}^{\ensuremath{-}1}$ for $\ensuremath{\lambda}2810 \mathrm{to} 2812=3^{3}D\ensuremath{-}(3p) (3d) ^{3}D$. It is concluded that the quantum-mechanical theory of nuclear motion, without the assumption of non-Coulomb nuclear fields or of nuclear spins, gives a good qualitative explanation of fine structure in magnesium. It appears that a measurement of the shifts of lines $\ensuremath{\lambda}2778$ to 2784, $\ensuremath{\lambda}3890$ to 3900, and $\ensuremath{\lambda}2810$ to 2812 would furnish the best quantitative test.