Judd–Ofelt theory

Judd–Ofelt theory is a theory in physical chemistry describing the intensity of electron transitions within the 4f shell of rare-earth ions in solids and solutions.^[1][2][3] It provides a mathematical framework for predicting and analyzing the spectra of rare-earth ions in solids and solutions, in particular branching ratios, radiative lifetimes, and oscillator strengths.

Theory

Judd-Ofelt theory may be used to predict and analyze the intensities of intra 4f electronic dipole transitions. Such transitions are disallowed by Electronic Dipole Transition Selection rules in free space, as the initial and final states have the same parity. However, intra 4f transitions have been observed. The transition strengths and the transition changing the orbital angular momentum quantum number are not consistent with them being Magnetic Dipole Transitions.

This apparent discrepancy is reconciled by treating the crystal field an ion in a solid experiences as a perturbation to the free space Hamiltonian. This perturbation mixes free space electronic states of opposite parity (namely the rare earth ion's ground 4fⁿ electronic configuration with the opposite parity 4f^n-15d). Thus Electronic Dipole transitions between these crystal field perturbed electronic states do not violate this parity change selection rule.

The theory quantitatively describes this mixing using three phenomenological parameters particular to the host crystal, denoted as ${\displaystyle \Omega _{\lambda }}$ (where ${\displaystyle \lambda =2,4,6}$). These parameters account for the asymmetric nature of the crystal field and enable the calculation of transition probabilities, oscillator strengths, and radiative lifetimes of excited states, which are crucial for the development of various photonic devices such as lasers and optical amplifiers.^[4]

Like Russell-Saunders Coupling (LS-Coupling), Judd-Ofelt theory can be simplified to a list of selection rules. The rules for Judd-Ofelt Induced Electric dipole transitions are listed in the following table and compared to LS-coupling Magnetic and Electric dipole transitions.^[5]

Term symbols in LS-Coupling describe the total orbital angular momentum (${\displaystyle L}$), total spin (${\displaystyle S}$), and total angular momentum (${\displaystyle J=S+L}$). ${\displaystyle \pi }$ is the parity of an electronic configuration.

Judd-Ofelt Induced	LS-Coupling
Electric dipole	Magnetic dipole	Electric dipole
${\displaystyle \Delta S=0}$
${\displaystyle |\Delta L|\leq 6}$	${\displaystyle \Delta L=0}$	${\displaystyle \Delta L=0,\pm 1}$ but ${\displaystyle L_{i}=0\not \leftrightarrow L_{f}=0}$
${\displaystyle |\Delta J|\leq 6}$ but ${\displaystyle |\Delta J|=2,4,6}$ if ${\displaystyle J_{i},J_{f}=0}$	${\displaystyle |\Delta J|\leq 1}$ but ${\displaystyle J_{i}=0\not \leftrightarrow J_{f}=0}$
No restriction	${\displaystyle \pi _{i}=\pi _{f}}$	${\displaystyle \pi _{i}=-\pi _{f}}$

Example: Europium (III)

For example let us apply these selection rules to triply ionized Europium (Eu³⁺). Eu³⁺ has an electronic configuration of [Xe]4f⁶. The ground term symbol for this configuration (according to Hund's Rules) is ⁷F₀.

Applying the above selection rules, transitions from this state to ⁵D₂, ⁵D₄, and ⁵D₆ are allowed, but not to ⁵D₁, ⁵D₀, or ⁵L₇ (violating the restrictions on ${\displaystyle J}$).^[5]

History

The theory was introduced independently in 1962 by Brian R. Judd of the University of California, Berkeley, and PhD candidate George S. Ofelt at Johns Hopkins University.^[2] Their work was published in Physical Review and the Journal of Chemical Physics, respectively.^[6][7] Judd and Ofelt did not meet until 2003 at a workshop in Lądek-Zdrój, Poland.^[1]

Judd-Ofelt Theory has become a standard tool in the field of lanthanide spectroscopy, providing insights into the optical properties of rare earth-doped materials and aiding in the design of materials for color display systems, fluorescent lamps, and lasers.^[8] Their work was cited approximately 2000 times between 1962 and 2004.^[1] Brian M. Walsh of NASA Langley places Judd and Ofelt's theory at the "forefront" of a 1960s revolution in spectroscopic research on rare-earth ions.^[2]

Author	Title	Year
Brian Judd	Optical Absorption Intensities of Rare-Earth Ions[9]	1962
George S. Olfelt	Intensities of Crystal Spectra of Rare-Earth Ions[10]	1962

Analysis software

Judd–Ofelt intensity parameters can be calculated from absorption spectrum of any lanthanide by the RELIC analysis software.^[3] The intensity parameters and derived quantities (oscillator strengths, radiative transition probabilities, luminescence branching ratios, excited state radiative lifetimes, and estimates of quantum efficiencies) of Eu³⁺ doped compounds, can be obtained by the JOES application software from their emission spectrum.^[11][12] Theoretical Judd-Ofelt intensity parameters for Eu³⁺ can be obtained using the LUMPAC software.^[13] Additionally, the JOYSpectra web platform provides these parameters for all Ln³⁺ ions.^[14][15]

See also

• Parity (physics)
• Bert Broer
• Otto Laporte
• Giulio Racah
• John Hasbrouck Van Vleck
• Eugene Wigner
• Brian Garner Wybourne

References

1. Walsh, Brian M. (June 2005). Judd–Ofelt Theory: Principles and Practices (slideshow). The International School of Atomic and Molecular Spectroscopy. Erice, Italy. Retrieved 18 November 2015.
2. Walsh, Brian M. "Chapter 21: Judd-Ofelt theory: Principles and practices". In Di Bartolo, B.; Forte, O. (eds.). Advances in Spectroscopy for Lasers and Sensing. Springer Netherlands. pp. 403–433. Retrieved 18 November 2015.
3. Hehlen, Markus P.; Brik, Mikhail G.; Krämer, Karl W. (April 2013). "50th anniversary of the Judd–Ofelt theory: An experimentalist's view of the formalism and its application". Journal of Luminescence. 136: 221–239. Bibcode:2013JLum..136..221H. doi:10.1016/j.jlumin.2012.10.035.
4. Burdick, Gary; Downer, M. (January 1993). "One-photon rare earth optical transitions: recent theoretical developments". Digital Commons @ Andrews University: 97–131. Retrieved 6 November 2023.
5. Binnemans, Koen (2015). "Interpretation of europium(III) spectra". Coordination Chemistry Reviews. 295: 1–45. doi:10.1016/j.ccr.2015.02.015.
6. Judd, B. R. (1962). "Optical Absorption Intensities of Rare-Earth Ions". Phys. Rev. 127 (3): 750. Bibcode:1962PhRv..127..750J. doi:10.1103/PhysRev.127.750.
7. Ofelt, G. S. (1962). "Intensities of Crystal Spectra of Rare-Earth Ions". J. Chem. Phys. 37 (3): 511. Bibcode:1962JChPh..37..511O. doi:10.1063/1.1701366.
8. "Judd-Ofelt theory: principles and practices". SearchWorks catalog at Stanford University. Retrieved 6 November 2023.
9. Judd, B. R. (1962). "OPTICAL ABSORPTION INTENSITIES OF RARE-EARTH IONS". Physical Review. doi:10.1103/PhysRev.127.750.
10. Ofelt, G. S. (1 August 1962). "Intensities of Crystal Spectra of Rare-Earth Ions". Journal of Chemical Physics. 37 (3): 511–520. Bibcode:1962JChPh..37..511O. doi:10.1063/1.1701366. ISSN 0021-9606.
11. Ćirić, Aleksandar; Stojadinović, Stevan; Sekulić, Milica; Dramićanin, Miroslav D. (January 2019). "JOES: An application software for Judd-Ofelt analysis from Eu³⁺ emission spectra". Journal of Luminescence. 205: 351–356. Bibcode:2019JLum..205..351C. doi:10.1016/j.jlumin.2018.09.048. S2CID 105828989.
12. "JOES – Judd-Ofelt from emission spectrum Software – OMAS Group".
13. Dutra, José Diogo L.; Bispo, Thiago D.; Freire, Ricardo O. (2014). "LUMPAC lanthanide luminescence software: Efficient and user friendly". Journal of Computational Chemistry. 35 (10): 772–775. doi:10.1002/jcc.23542. PMID 24532191.
14. Moura Jr., Renaldo T.; Carneiro Neto, Albano N.; Aguiar, Eduardo C.; Santos-Jr., Carlos V.; de Lima, Ewerton M.; Faustino, Wagner M.; Teotonio, Ercules E.S.; Brito, Hermi F.; Felinto, Maria C.F.C.; Ferreira, Rute A.S.; Carlos, Luís D.; Longo, Ricardo L.; Malta, Oscar L. (2021). "JOYSpectra: A web platform for luminescence of lanthanides". Opt. Mater.: X. 11: 100080. doi:10.1016/j.omx.2021.100080.
15. "JOYSpectra a web platform for luminescence of lanthanides".