Second-order contributions of diamagnetic interaction to the energy shift and splitting are studied in the basis of the angular momentum eigenfunctions for two kinds of Rydberg states in atoms: (i) hydrogen-like degenerate states and (ii) nondegenerate Rydberg states with small magnetic quantum number, m ? 3, in many-electron atoms. General formulae are presented for calculations of higher order energy corrections in degenerate states with the use of reduced Green's function. The analytical expressions are derived for the second-order radial matrix elements of operator r2 and for the second-order diamagnetic susceptibilities of hydrogen-like states with m ? n-5. The quasiclassical quantum defect method is used to calculate the irreducible components ?nl(p) of the diamagnetic susceptibility ?nlm(2) for Rydberg states of alkali atoms. The numerical results are presented for susceptibilities of degenerate hydrogen substates, and for susceptibilities of alkali atoms in Rydberg s-, p-, d-states. The asymptotic dependence of susceptibilities in alkali atoms on the effective principal quantum number ? is determined numerically and the deviation from that of the hydrogen-like states is discovered. The data is also presented for the scaling parameters cl(p), determining the irreducible parts of the diamagnetic susceptibilities in states with high ?, according to the asymptotic formula ?nl(p) = cl(p)?11.
We study the effect of diamagnetic interaction on the probability of radiative transitions of an atom from states split by a field. We write the analytic expressions for the diamagnetic corrections to the matrix elements of transitions belonging to the Lyman and Balmer series and of transitions between arbitrary nondegenerate states in hydrogen. We also discuss the perturbation theory for transitions from degenerate diamagnetic states. The theory is based on expanding in powers of the field strength the eigenfunctions and eigenvalues of the matrix of diamagnetic interaction in the subspace of states with given principal and magnetic quantum numbers. The field changes the coefficients in both the superposition and the degenerate basis. To derive the analytic expressions for the higher-order matrix elements, we use the Sturm expansion of the reduced Coulomb Green’s function. We also elaborate on the features of the frequency dependence of the corrections to the radiative matrix elements, which correlate with the structure of the diamagnetic spectrum of excited levels. Finally, we establish that the magnetic field acts selectively on the diamagnetic components of emission lines: as the field strength increases, an increase in the intensity of certain lines is accompanied by a decrease in the intensity of the other lines.
With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ ( R < R F ) this dependence transforms to the formula for electric dipole interaction energy Δ E d − d = C 3 / R 3 and for R > R F it transforms to the formula for the van der Waals interaction energy Δ E VdW = − C 6 / R 6 . The van der Waals constant C 6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J . The numerical values of the irreducible components of tensor C 6 were calculated for rubidium atoms in the same Rydberg states | nlJM 〉 with large quantum numbers n . We present the calculated resonance interaction energy of two rubidium atoms in the states |43D 5/2 M 〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P 3/2 M 〉 and of the other in the state |41F 7/2 M 〉.
1.General mathematical theory.---Modifications are given in the general derivation of the Langevin-Debye formula by means of quantum mechanics published in part I which are required by the moment now being magnetic rather than electric. The main additions are the appearance of a diamagnetic term, and allowance for the fact that the magnetic moment consists of two parts arising respectively from orbital electronic motions and from the electrons' internal spins. These parts cannot always be treated as rigidly coupled to form a "permanent" resultant moment. The Hamiltonian function used for the internal spin is that of a spherical top.2.Diamagnetism.---The writer's previous claim that Pauli's formula for the diamagnetic susceptibility can be applied to molecules as well as atoms is shown to be invalid because of the anomalous fact that the square of the angular momentum, unlike the average angular momentum, does not vanish even in
S states when there is more than one nucleus. Pauli's formula is instead an upper limit to the diamagnetism in non-monatomic molecules and is a good approximation when the Schr\"odinger wave function has nearly as much symmetry as in an atomic
S state.3.Paramagnetism of atoms.---Limiting values for the paramagnetic susceptibilities of atoms are $\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}$ and $\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}$ where $\ensuremath{\beta}$ is the Bohr magneton and $\ensuremath{\alpha}$ is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to
kT, and are valid regardless of whether the magnetic field is strong enough to change the quantization by producing a Paschen-Back effect. The formula for small multiplets yields susceptibilities slightly different from those given by Laporte and Sommerfeld's expression for this case, and is much simpler, as they overlooked the contribution of the portion of the magnetic moment which is perpendicular to the invariable axis.4.Paramagnetism of molecules.---The susceptibility is calculated on the basis of the Hund theory of molecular quantization. Formulas are given applicable to his couplings of type (a) and type (b) provided in the former the multiplet intervals are either very large or very small compared to
kT. The experimental susceptibilities for
O2 and Cl
O2 are in accord with the assumption that the normal states are respectively
3S and
2S terms. In the particular case of
S terms the numerical results are the same in the atomic and molecular formulas, but, unlike previous theories, it is not necessary to suppose the orbits are as freely oriented in molecules as in atoms. Polyatomic molecules may have lower paramagnetic susceptibilities than diatomic ones because the dissymmetry causes large fluctuations in electronic angular momentum.5.Paramagnetism of nitic oxide.---Spectroscopists have recently found that the normal states of the NO molecule are
2P terms separated by 120.9 ${\mathrm{cm}}^{\ensuremath{-}1}$. This permits an unambigous calculation of the susceptibility of NO which agrees with the experimental value within 1.5 per cent. Deviations from Curie's law are calculated which result from the doublet interval being comparable with
kT. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.
We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to
n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states.
The diamagnetic and Stark shifts of strongly perturbed Rydberg spectra such as those of the alkaline earths are expressed in terms of multichannel---quantum-defect parameters. The theory is applicable when the shifts produced by the external fields are smaller than the neighboring Rydberg separations. It is applied to evaluate the diamagnetic shifts in Ba, an atom in which Rydberg and doubly excited channels are strongly mixed. In such complex spectra, the diamagnetic effect can serve a useful purpose as a tool to disentangle the interacting series. The radial sizes and, therefore, the diamagnetic shifts are quite different for singly and doubly excited states even when they are at the same energy. In fact, for the field strengths considered ($\ensuremath{\simeq}40$ kG), the diamagnetic contribution of the doubly excited configuration is quite negligible. As a result, the observed shift is a direct measure of the amount of singly excited configuration in the state of interest. Exactly similar conclusions apply to Stark shifts. The last part of this paper considers states of very high excitation further up in the spectrum which are no longer in a perturbative regime. Spectra of atoms in a magnetic field have shown equally spaced resonances near the ionization threshold with characteristic spacings. By the consideration of the effects of electric fields which are also often simultaneously present, a criterion is established for the nature of these spacings depending on the relative strengths of the crossed electric and magnetic fields. The spacing seen in any particular experiment is shown to depend on the mass and quantum defect of the atom, the strengths of the external fields, and experimental conditions such as the temperature of the atomic vapor whose absorption spectrum is being studied.