Optical absorption intensities of rare-earth ions

TLDR: In this paper, an expression for the oscillator strength of a transition between two states of the ground configuration $4{f}^{N}, on the assumption that the levels of each excited configuration of the type $4 {f} n{n}^{\ensuremath{'n}d$ or $4

Abstract: Electric dipole transitions within the $4f$ shell of a rare-earth ion are permitted if the surroundings of the ion are such that its nucleus is not situated at a center of inversion. An expression is found for the oscillator strength of a transition between two states of the ground configuration $4{f}^{N}$, on the assumption that the levels of each excited configuration of the type $4{f}^{N}{n}^{\ensuremath{'}}d$ or $4{f}^{N}{n}^{\ensuremath{'}}g$ extend over an energy range small as compared to the energy of the configuration above the ground configuration. On summing over all transitions between the components of the ground level ${\ensuremath{\psi}}_{J}$ and those of an excited level ${{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$, both of $4{f}^{N}$, the oscillator strength $P$ corresponding to the transition ${\ensuremath{\psi}}_{J}\ensuremath{\rightarrow}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$ of frequency $\ensuremath{\nu}$ is found to be given by $P=\ensuremath{\Sigma}{T}_{\ensuremath{\lambda}}\ensuremath{\nu}{({\ensuremath{\psi}}_{J}\ensuremath{\parallel}{U}^{(\ensuremath{\lambda})}\ensuremath{\parallel}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}})}^{2},$ where ${\mathrm{U}}^{(\ensuremath{\lambda})}$ is a tensor operator of rank $\ensuremath{\lambda}$, and the sum runs over the three values 2, 4, and 6 of $\ensuremath{\lambda}$. Transitions that also involve changes in the vibrational modes of the complex comprising a rare-earth ion and its surroundings, provide a contribution to $P$ of precisely similar form. It is shown that sets of parameters ${T}_{\ensuremath{\lambda}}$ can be chosen to give a good fit with the experimental data on aqueous solutions of Nd${\mathrm{Cl}}_{3}$ and Er${\mathrm{Cl}}_{3}$. A calculation on the basis of a model, in which the first hydration layer of the rare-earth ion does not possess a center of symmetry, leads to parameters ${T}_{\ensuremath{\lambda}}$ that are smaller than those observed for ${\mathrm{Nd}}^{3+}$ and ${\mathrm{Er}}^{3+}$ by factors of 2 and 8, respectively. Reasons for the discrepancies are discussed.

Figures

Table IV. Oscillator ' 3+ strengths for Er

Fig. 1. A comparison between experimental and theoretical oscillator strengths of transitions in aqueous solutions of NdC13. The lengths of the horizontal lines running

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Lawrence Berkeley National Laboratory
Recent Work
Title
OPTICAL ABSORPTION INTENSITIES OF RARE-EARTH IONS
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https://escholarship.org/uc/item/0hd516pd
Author
Judd, B.R.
Publication Date
1962
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UNIVERSITY
OF
CALIFORNIA
Lawrence
Radiation
Laboratory
Berkeley)
California
Contract
No. W-7405-eng-48
OPTICAL
ABSORPTION
INTENSITIES
OF
RARE-EARTH
IONS
B. R.
Judd
January)
1962
UCRL-10019

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.,
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iii
OPTICAL
ABSORPTION
INTENSITIES
OF
RARE-EARTH
IONS
Abstract
....
I.
Introduction
II.
Matrix
Elements
III.
Approximations
IV.
Solutions
of
Rare
Earth
Ions.
v.
Other
Contributions
to
P
VI.
Comparison
with
Experiment.
VII.
Variation
of
T
2
.....
.
Contents
VIII.
Environment
of
a
Rare-Earth
Ion
in
Solution.
IX.
Calculation
of
the
Parameters
T~
X.
Discussion
.
XI.
Conclusion
Acknowledgments
References
. . .
.
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