LEVEL SHIFTS FROM COULOMB INTERACTIONS OF ATOMS IN A GAS
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Date
1975
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Ohio State University
Abstract
We consider $N_{a}$ identical atoms in a gas of volume V; each atom has a $J^{ w}=0^{+}$ ground state and L excited $1^{-}$ levels. The 3 $N_{a} L$ excited states are labelled $|{R}_{I}\alpha\nu\rangle$ ; they have at $\underline{R}_{I}$ partner $\alpha(\alpha = x,y,z)$ of the $\nu^{th}$ $1^{-}$ level, of excitation energy $E_{\nu}$, and have at all other atomic positions unexcited atoms. The atoms are assumed not to overlap. The Hamiltonian is $H = H^{o} + H^{int}$, where $H^{o}$ is the sum of $N_{a}$ one-atom terms and $H^{int}$ is the sum of $N_{a}(N_{a}-1)/2$ Coulomb Interaction terms. Here the Hamiltonian matrix $$\langle{R}_{I}\alpha\nu|{H}| {R}_{I}^{\prime}\alpha^{\prime} \nu^{\prime}$$ is the Kronecker product $G \times \varepsilon$ of a 3 $N_{a}$ by 3 $N_{a}$ “geometrical” matrix G and an L by L “electronic” matrix $\varepsilon$. Similarity transformation of H by L $\Gamma \times I_{T}$, where $\Gamma$ is the matrix of eigenvectors of G ($\Sigma_{I}^{\prime} \alpha^{\prime} G_{I\alpha, {I^{\prime}} {\alpha^{\prime}}}\Gamma_{I^{\prime}}{\alpha^{\prime}},n=g_{n}\Gamma_{I\alpha,n}$ and $I_{L} = (\delta_{\nu\nu^{\prime}}$) is the L-dimensional unit matrix, reduces H to 3 $N_{a}$ diagonal L by L blocks with zeros elsewhere. Furthermore, the $n^{th}$ such block of H-W has matrix elements $(H-W)_{\nu\nu^{\prime}} = (E_{\nu}-W) \delta_{\nu\nu^{\prime}} + g_{n}e^{2}Z_{\nu{o}}Z_{\nu{o}^{\prime}}r_{s}^{3}$, where $-eZ_{\nu{o}}$ is the transition moment between the $v^{th}$ excited and the ground state of any atom, and $r_{s}$ (defined by $4\pi r_{s}^{3/3} = V/N_{a}$) is a scale length introduced to make G and g dimensionless, The determinant of H-W, det(H-W), is $\Pi_{n}\{(\Pi_{\nu}\lambda_{n\nu}(1+\Sigma_{\nu}1/\lambda_{n\nu})\}$, where $\lambda_\nu = (E_{\nu} - W)/(g_{n}e^{2}Z_{\nu{o}}Z_{\nu{o}}/r_{s}^{3}$). The secular equation det $(H-W) = 0$ has a simple graphical interpretation for the energy eigenvalues $W_{nv}$ and a surprisingly simple equation for the coefficients of the excited states. All the above can be carried through explicitly if the atomic positions $R_{I}$ form a lattice; the states are then the “exciton states” of solid state physics. For the argon resonance line the shift is about 1 in $10^{5}$ at 1000 torr.
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Author Institution: Oak Ridge National Laboratory, The University of Tennessee