Date post: | 09-Apr-2017 |
Category: |
Technology |
Upload: | brian-walsh |
View: | 30,181 times |
Download: | 24 times |
National Aeronautics andSpace Administration
Erice, Italy (June 2005)
Judd-Ofelt Theory: Principlesand Practices
Brian M. WalshNASA Langley Research Center
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Part I: PrinciplesWhat is the Judd-Ofelt Theory?
Based on static, free-ion and single configuration approximations:• static model - Central ion is affected by the surrounding
host ions via a ‘static’ electric field.• free ion model - Host environment treated as a perturbation
on the free ion Hamiltonian. • single configuration model - Interaction of electrons between
configurations are neglected.
The Judd-Ofelt theory describes the intensities of 4f electrons insolids and solutions. The remarkable success of this theoryprovides a sobering testament to simple approximations.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
0
50
100
150
200
62 72 82 92 02
Num
ber
of
cit
ati
ons
Year
Distribution of Citations by YearB.R. Judd, Phys. Rev. 127, 750 (1962).G.S. Ofelt, J. Chem. Phys. 37, 511 (1962).
~ 2000 citations (1962-2004)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Referenced in 169 Journal TitlesTop 20 Titles # of citationsPHYSICAL REVIEW B 127 JOURNAL OF NON-CRYSTALLINE SOLIDS 108 JOURNAL OF APPLIED PHYSICS 90 JOURNAL OF CHEMICAL PHYSICS 83 JOURNAL OF ALLOYS AND COMPOUNDS 81 JOURNAL OF LUMINESCENCE 77 JOURNAL OF PHYSICS-CONDENSED MATTER 58 MOLECULAR PHYSICS 57 CHEMICAL PHYSICS LETTERS 48 OPTICAL MATERIALS 43 JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B 38 JOURNAL OF PHYSICS AND CHEMISTRY OF SOLIDS 35 PHYSICA STATUS SOLIDI A-APPLIED RESEARCH 33 OPTIKA I SPEKTROSKOPIYA 30 IEEE JOURNAL OF QUANTUM ELECTRONICS 27 PHYSICS AND CHEMISTRY OF GLASSES 27 OPTICS COMMUNICATIONS 26 SPECTROCHIMICA ACTA PART A 26 INORGANIC CHEMISTRY 24 JOURNAL OF THE AMERICAN CERAMIC SOCIETY 19
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Prelude
“Lanthanum has only one oxidation state, the +3 state. Withfew exceptions, this tells the whole boring story about theother 14 lanthanides.”
G.C. Pimentel & R.D. Sprately,"Understanding Chemistry",Holden-Day, 1971, p. 862
http://www.chem.ox.ac.uk/icl/heyes/LanthAct/I1.html ( some amusing mnemonics for the Lanthanides and Actinides)
So much for ‘Understanding Chemistry’…Let’s do some physics!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Ions in SolidsSolids• insulators (not semiconductors)• bandgaps are > 5ev (VUV photon)• produce a crystal field
Ions• replace host ions substitutionally• transition metal and lanthanide series• unfilled electronic shells• Stark splitting from crystal field• optical transitions occur within bandgap
Examples• Nd:Y3Al5O12 - Er:fiber - Cr:Al2O3 (Ruby)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Atomic Structure of Laser Ions
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Hc >> Hso (LS-coupling)Hc << Hso (jj-coupling)Hc ≈ Hso (Intermediate coupling)
Stark LevelsVo = crystal field
(Electric field of host)
!1!2!3!4
4f10
4f 95d
5S
5F
5I
5I45I5
5I8
5I65I7
ConfigurationsHo = central field
(Electrons in field
of the nucleus)
Terms 2S+1LHc = Coulomb field
(Mutual repulsion
of electrons)Levels 2S+1L
J
Hso = spin orbit
(Coupling between
spin and orbital
angularmomentum)
Ho >Hc , Hso >Vo
Atomic Interactions
Hund’s Rules*: F. Hund, Z. Phys. 33, 345 (1925) 1.) Lowest state has maximum 2S+1 2.) Of these, that with largest L will be lowest 3.) Shells < 1/2 full (smallest J is lowest), Shells > 1/2 full (largest J is lowest)
*These rules apply only to the ground state, not to excited states.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Transitions and Selection Rules• Not all transitions between atomic states that are energetically feasible are "allowed”.• Forbidden transitions are “forbidden*” to first order, which means they may occur in practice, but with low probabilities.• Selection rules for transitions depend on type of transition
– Electric dipole (E1)– Electric quadrupole (E2)– Magnetic dipole (M1)
• Wavefunctions must have correct parity (Laporte’s rule)• Symmetry plays a role in selection rules
– Vibronics, crystal field, other perturbing effects.
* This nomenclature is historically embedded, although not entirely accurate.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Electric dipole operator (E1)(odd operator)
Magnetic dipole operator (M1)(even operator)
Quadrupole operator (E2)(even operator)
Electric Dipole Magnetic dipole Electric quadrupole
S L J (No 0 ↔ 0) Parity
ΔS = 0 ΔL= 0, ±1 ΔJ = 0, ±1 opposite ΔS = 0 ΔL= 0 ΔJ = 0, ±1 same ΔS = 0 ΔL = 0, ±1, ±2 ΔJ = 0, ±1, ±2 opposite
Multipole Selection Rules
!"=
i
ireP!!
! +"=
i
iisI
mc
eM
!!"!2
2
( ) i
i
i rrkQ!!!!
! "#$=2
1
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
A Brief History of ParityOtto Laporte (1902-1971) empirically discovered the law of parityconservation in physics. He divided states of the iron spectrum into twoclasses, even and odd, and found that no radiative transitions occurredbetween like states: O.Laporte, Z. Physik 23 135 (1924).
Eugene Paul Wigner (1902-1995) explicitly formulated the law of parityconservation and showed that Laporte’s rule is a consequence of the invariance of systems under spatial reflection. E. P. Wigner, “Gruppentheorieund Ihre Anwendung auf die Quantenmechanik der Atomspektren”. Braunschweig:F. Viewegund Sohn, 1931. English translation by J. J. Griffin. New York: Academic Press, 1959.
Wavefunctions are classified as even (+1 parity) or odd (-1 parity). By convention,the parity of a photon is given by the radiation field involved: ED ( -1), MD (+1).For mathematical reasons, the parity of any system is the product of parities of theindividual components. If the initial and final wavefunction have same parity (±1):
ED: ±1 = (-1)(±1) Parity is NOT conserved. Transition is forbidden!MD: ±1 = (+1)(±1) Parity IS conserved. Transition is allowed!
Laporte Rule: States with even parity can be connected by ED transitions only withstates of odd parity, and odd states only with even ones.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Parity Selection RulesED allowedMD forbiddenEQ forbiddens → pp → dd → ff → gs → fp → g
ED forbiddenMD allowedEQ allowed s → s d → d p → p f → f g → g
evenodd
oddeven
evenodd
oddeven
evenodd
- odd # electrons - even # electrons
g4
f3
d2
p1
s0
OrbitalAngular momentum
!= "1( )
!i
i
# !
!
!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Historical Perspective I• J.H. Van Vleck - J. Phys. Chem. 41, 67-80 (1937) (The Puzzle of Rare-Earth Spectra in Solids)
– Why are spectral lines in rare earths observable?– Electric dipole(E1), magnetic dipole(M1), quadrupole(E2)?– Concludes a combination is possible.– Suggests that crystal field makes mixed parity states (E1).
• L.J.F. Broer, et al., - Physica XI, 231- 250 (1945) (On the Intensities and the multipole character in the spectra of the rare earth ions)
– Considers all mechanisms.– Concludes quadrupole radiation is too weak.– Considers magnetic dipole , but as a special case only.– ED transitions dominate as suggested by Van Vleck!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
• G. Racah - Phys. Rev. 76, 1352 (1949) (Theory of Complex Spectra IV)
– Applies group theory to problems of complex spectra– Creates the tools required to make detailed spectroscopic calculations involving states of the 4f shell.– Revolutionizes the entire subject of rare earth spectroscopy.
• Subsequent developments– Racah’s methods applied to crystal field theory.– Ideas of Racah applied to transition metal ions (Griffiths).– Practical calculations assisted by computer generated tables of angular momentum coupling coefficients.
• By 1962 the stage was set for the next major development: The Judd-Ofelt theory of the intensities of RE transitions.
Historical Perspective II
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
“I suggest that the coincidence of discovery was indicativethat the time was right for the solution of the problem.”
Brian G. Wybourne“The fascination of rare earths - then, now and in the future”Journal of Alloys and Compounds 380, 96-100 (2004)
The Stage is Set
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd and Ofelt Publish (1962)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
!b= "
b+
"# V "b
Eb$ E##
% "#
!a= "
a+
"aV "#
Ea$ E##
% "# }
φa and φb have the same parity (4f N states)φβ has opposite parity (4f N-15d states)V is the crystal field (treated as a perturbation) is the electric dipole operator
!P
Mixed ParityStates
States of an Ion in the CrystalThe Crystal field, V, is considered as a first order perturbationthat ‘admixes’ in higher energy opposite parity configurations:
!"#
"$%
"&
"'(
)+
)=
* *
**
*
** ++++++++,,
EE
VP
EE
PVP
b
ba
a
ba
ba
!!!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
V = Akqrik
kq
!i
! Ykq (" i ,# i )
Racah defined irreducible tensors, C(k), which transform asspherical harmonics, having the components:
Cq
(k )=
4!2k +1
"#$
%&'1/2
Ykq
The position vector r is a tensor of rank 1, defined as r = rC (1)
Dipole Operator
Dq
(1)= !e ri
i
" Cq
(1)#$ %&i
Crystal Field
Tensor Form Dp
(t )= Atp
tp
! ri
t
i
! Cp
(t )"# $%i
Note: t is odd since only odd order terms contribute to parity mixing.Even order terms are responsible for energy level splitting.
Standard Form
Tensor Forms of Operators
!"=
i
ireP!!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
1.) The states of φβ are completely degenerate in J.
2.) The energy denominators are equal ( Ea-Eβ = Eb-Eβ )
Assume an average energy for the excited configurationabove the 4fN, that is, the 4f N-15d.
Assume that the difference of average energies, ΔE(4f-5d),is the same as the difference between the average energyof the 4f N-15d and the energy of the initial and final statesof the 4fN
These assumptions are only moderately met, but offer agreat simplification. Otherwise, the many fold sum ofperturbation expansions is not suitable for numericalapplications.
J-O Theory Assumptions 1&2
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
4f N and 4f N-15d configurations
0
5
10
15
20
25
30
35E
nerg
y(×1
04 cm
-1)
Ce58
Pr59
Nd60
Eu63
Sm62
Pm61
Gd64
Tb65
Ho67
Dy66
Yb70
Tm69
Er68
Atomic Number & Symbol
4fn configuration
4fn-15d1 configuration
Lanthanides in YLF:
K. Ogasawara et al., J. Solid State Chem.vol. 178, 412 (2005
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
!""
# !" = 1
II.) Closure can be used ( the excited configuration forms a complete orthonormal set of basis functions)
I.) Energy denominators can be removed from the summations
III.) Angular parts of the electric dipole operator and crystal field
Cq
(1)= ! C
(1)!! Uq
(1)
Cp
(t )= ! C
(t )!! Up
(t )
Uq
(1)Up
(t )= (!1)1+ t+"+Q
"Q# (2" +1)
t 1 "! ! $!
%&'
()*
t 1 "p q Q
+,-
./0UQ
(" )
can be combined into an effective tensor operator
and
Advantages of the Assumptions
The 3j symbol ( ) is related to the coupling probability for two angular momenta.The 6j symbol { } is related to the coupling probability for three angular momenta.The Wigner 3-j and 6-j symbols are related to Clebsch-Gordon coupling coefficients.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Nevertheless, combining the tensors for the electric dipoleand crystal field terms in a combined tensor operator,
fN! JM UQ
(" )fN #! #J #M = ($1)J $M
J " #J$M Q #M
%&'
()*fN! J U (" )
fN #! #J
, UQ
(! )
can be simplified further by the Wigner-Eckart Theorem:
Reduced Matrix Elements
The matrix elements on the right side have been tabulated:“Spectroscopic Coefficients of the p N, d N, and f N Configurations,”C.W. Nielson and G.F. Koster, M.I.T Press, Cambridge, MA (1963).
The 3-j and 6-j symbols have also been tabulated:“The 3-j and 6-j symbols,” M. Rotenberg, R, Bivens, N. Metropolis, J.K. Wooten Jr., Technology Press, M.I.T, Cambridge, MA (1959).
Geometry Physics(transformations) (Dynamics)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
“Full Judd-Ofelt Theory”
Y (t,!) = 2n! r "n "! "n "! r
tn!
Ea# E$n!
% ! C(1) "! "! C
(t )!1 t !
! "! !
&'(
)*+
This is the “Full Solution” of the Judd-Ofelt Theory. This formcan be used to find electric dipole matrix elements between mixedparity states for individual Stark level to Stark level transitions.
Application of “Full Judd-Ofelt Theory”:R.P. Leavitt and C.A. Morrison, “Crystal-field Analysis of triply Ionized lanthanumtrifluoride. II. Intensity Calculations.” Journal Of Chemical Physics, 73, 749-757 (1980).
Where,
( ) ( ) ( )!! ""#
$%%&
'()
(""#
$%%&
'+))=
))
tp Q
batp
QMJ
ba UMQM
JJ
pQq
ttYAeP
*
* ++**
**,, )(1
,121!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Q = 0 (π-polarized, E ⊥ c)Q = ±1 (σ-polarized, E || c)
M = -J, -(J-1), …, 0, …, (J-1), J
Oscillator strength (f-number) for electric dipole transition:
J ! "J#M Q "M
$%&
'()M "M
*J "! "J
#M "Q "M
$%&
'()=
1
2! +1+! "! +Q "Q
1 ! t
q Q p
"#$
%&'Q
(1 ! )tq Q )p
"#$
%&'=
1
2t +1* t )t * p )p
3.) Sum over Stark split J-levels (Assumes all Stark levels equally populated)
4.) Sum over dipole orientations (Assumes optically isotropic situation)
J-O Theory Assumptions 3&4
L.J.F. Broer, et al., - Physica XI, 231- 250 (1945)( ) ! """##
$
%&&'
( +
+=
2
22
2
2
3
2
123
8MJPJM
n
nn
eJ
mcf ))
*+ !
"
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
“Approximate Judd-Ofelt Theory”
Defining Ωλ as:
f =8! 2
mc
3h"(2J +1)nn2+ 2
3n
#$%
&'(
2
(2" +1)t=1,3,5
)p
)Atp
2
(2t +1)Y2(t,") *a U
(" ) *b
2
"=2,4,6)
!" = (2" +1)Atp
2
(2t +1)t=1,3,5
#p
# Y2(t,")
f =8! 2
mc
3h"(2J +1)nn2+ 2
3n
#$%
&'(
2
)" *a U(" ) *b
2
"=2,4,6+
This is the “Approximate Solution” of the Judd-Ofelt theory.It can be used to find electric dipole matrix elements betweenmixed parity states for manifold to manifold transitions.
SED
= !" #aU(" ) #
b
2
"=2,4,6
$ is called the Linestrength.
Judd-Ofeltparameters
Oscillatorstrength
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
In principle, the Judd-Ofelt parameters can be calculated “ab-initio” if the crystal structure is known, and hence, Atp:
!" = (2" +1)Atp
2
(2t +1)t=1,3,5
#p
# Y2(t,")
Y (t,!) = 2n! r "n "! "n "! r
tn!
Ea# E$n!
% ! C(1) "! "! C
(t )!1 t !
! "! !
&'(
)*+
! C(1) !! = ("1)!
! 1 !!0 0 0
#$%
&'(2! +1( )
1/22 !! +1( )
1/2
! C(t ) !! = ("1)!!
!! t !
0 0 0
#$%
&'(2 !! +1( )
1/22! +1( )
1/2
3-j and 6-j symbols can be calculated for ! !!= 3 (4f ) and = 2 (5d)Radial integrals between configurations and crystal field components, Atp, are difficult to calculate. Instead, Judd-Ofelt parameters are usually treated as phenomenologicalparameters, determined by fitting experimental linestrength data.
Judd-Ofelt Parameters
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
“The two papers of 1962 represent a paradigm that hasdominated all further work on the intensities of rare earthtransitions in solutions and solids up to the present time.”
Brian G. Wybourne“The fascination of rare earths - then, now and in the future”Journal of Alloys and Compounds 380, 96-100 (2004).
Intermission
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Part II: PracticesThe Judd-Ofelt theory, in practice, is used to determine a setof phenomenological parameters, Ωλ (λ=2,4,6), by fitting theexperimental absorption or emission measurements, in aleast squares difference sum, with the Judd-Ofelt expression.
The Judd-Ofelt parameters can then be used to calculate thetransition probabilities, AJ′J, of all excited states. From these,the radiative lifetimes, τr, and branching ratios, β, are found.
Collect MatrixElements|<U(λ)>|2
Least SquaresFitting
Judd-OfeltParameters
Ωλ
TransitionProbabilities
AJ′J
τr and β
! (")d"manifold
#
AbsorptionMeasurements
! "2( )
!#k
= 0
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
ji! 0
!i! 0
j1" j
2# j
3# j
1+ j
2
!2" !
3# j
1# !
2+ !
3
!1" !
3# j
2# !
1+ !
3
!1" !
2# j
3
1 t !
! "! !
#$%
&'(
ji! 0
mi" j
i
m1+ m
2+ m
3= 0
ji,m
i
j1# j
2" j
3" j
1+ j
2
(1, 1/2 integer)
J ! "J#M Q "M
$%&
'()
!J " J # $
j1
j2
j3
!1!2!3
!"#
$%&= 0
Unless:
j1
j2
j3
m1
m2
m3
!"#
$%&= 0
Unless:! = 2,4,6
t = 1,3,5,7 !J " 6
!L " 6
!S = 0
J = 0 : #J $ even
#J = 0 : J $ even
Electric Dipole
Magnetic dipole ΔS = 0 ΔL = 0 ΔJ = 0, ±1 sameElectric quadrupole ΔS = 0 ΔL = 0, ±1, ±2 ΔJ = 0, ±1, ±2 opposite
S L J (No 0 ↔ 0) Parity
ΔS = 0 ΔL ≤ 6 ΔJ ≤ 6 opposite ΔJ = 2,4,6 (J or J′ = 0)
Selection Rules Revisited
New Selection Rules From Judd-Ofelt Theory
! " 1+ t
!! " ! # 1
Only d or gorbitals canmix parity
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd-Ofelt Analysis I
Sm=3ch(2J +1)
8! 3e2"
n3
n2+ 2
#$%
&'(2
) (")d"manifold
* Sjm
SED
= !" #aU(" ) #
b
2
"=2,4,6
$
Components of1 x N matrix
Mij - components of N x 3 matrix for square matrix elements of U ( 2 ), U ( 4 ), U (6 )
Ωi - components of 1 x 3 matrix for Judd-Ofelt parameters Ω 2, Ω 4, Ω 6
!(0)
= M†M( )
"1
M†Sm
! 2= Sj
m " Mij#i
i=1
3
$%&'
()*j=1
N
$2
Sjt= Mij!i
i=1
3
"
! " 2( )!#k
= $2 M jk
j=1
N
% Sjm $ Mij#i
i=1
3
%&'(
)*+= 0 MINIMIZED
LEAST SQUARES DIFFERENCE
N = number of tramsitions
Matrix Forms
Judd-Ofelt Parameters1 x 3 Matrix
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd-Ofelt Fit (Ho:YLF)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
280 330 380 430 480 530 580 630 680
!ab
(1
0-2
0 c
m2)
Wavelength (nm)
Visible absorption spectrum of Ho:YLF
(! and " polarization)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd-Ofelt Analysis II
A !J J =64" 4
e2
3h(2 !J +1)3
#nn2+ 2
3
$%&
'()
2
SED
+ n2SMD
*
+,,
-
.//
With the Judd-Ofelt parameters, the ED transition probabilityfor any excited state transition (J´→ J) can be calculated
Radiative lifetime(natural decay time)
1
!r
= A"J J
J
#
Branching ratio(fraction of total photon flux)! "J J =
A "J J
A "J J
J
#
MD transitions are normally orders of magnitude smaller than ED transitions. SinceED transitions for ions in solids occur as a result of a perturbation, some MDtransitions will make significant contributions.
Transition probability(Einstein A coefficient)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Magnetic Dipole Contributions
fn[SL]J = C S,L( ) f nSLJ
SL
!
LS-coupledmatrix elements
G.H. ShortleyPhys. Rev. 57, 225 (1940)
Intermediate coupled wavefunctions(linear combination of LS states)
Intermediate coupled matrix elements
MD Linestrength
Magnetic dipole contributions can be easily calculated using anappropriate set of intermediate coupled wavefunctions fortransitions obeying the selection rules ( ΔS = 0, ΔL= 0, ΔJ = 0, ±1).
[ ] [ ]2
2
22
JLSfSLJSLfmc
S nn
MD!!!+"
#$
%&'
=!!"
[ ] [ ] ( ) ( )( ) ( )
( )
2/122
22
14
1112
!"
!#$
!%
!&'
(()
*
++,
-()
*+,
-
+
..++.++=///+
J
SLJJLSJLSfSLJSLf nn
!!
[ ] [ ] ( ) ( )!""
"""+="""+LSSL
nnnn JLSfSLSLJfLSCLSCJLSfSLJSLf,
2,,2!"!!
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd-Ofelt Results (Ho:YLF)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Testing the J-O theory
!em
"( ) = !ab
"( )Z!
Zu
exp EZL#hc
"$%&
'()kT
*
+,-
./
Branching ratios can be measured directly from emissionspectra. Use reciprocity of emission and absorption toindirectly “measure” the radiative lifetimes.
By comparing the measured emission cross section
! "( ) ="5
8#cn2 $r/ %( )
3I& "( )
2I! "( ) + I# "( )'( )*"d"+
with the emission cross section derived from absorption, thequantity (τr/β) can be determined and the radiative lifetimeextracted for comparison with the Judd-Ofelt theory
D.E. McCumberPhys. Rev. 136, A954 (1964).
P. MoultonJ. Opt. Soc Am. B 3, 131 (1986).
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Reciprocity of Ho:YLF (5I7 ↔ 5I8)
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Accuracy of J-O theory (Ho&Tm)Results are somewhat betterin Ho3+ than Tm3+. Overall,the accuracy of the Judd-Ofelttheory is quite good, despitethe approximations used.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Special Case I: Pr3+ ion(A failure of the standard Judd-Ofelt theory?)
Pr3+ ions suffer from several problems in applying Judd-Ofelt theory1.) Large deviations between calculations and experiment observations.2.) Negative Ω2 sometimes obtained, in opposition with definition.3.) Ω2, Ω4, Ω6 highly dependent on transitions used in fit.
These inconsistencies are usually explained by the small energy gap(~ 50,000 cm -1) between the 4f N and 4f N-15d configurations in Pr3+
Solutions:1.) Modify the standard theory:
E.E. Dunina, et al., Sov. Phys. Solid State 32, 920 (1990).2.) Remove 3H4 → 3P2 from the fit, or augment fit with fluorescence β’s
R.S.Quimby, et al., J. Appl. Phys. 75, 613 (1994).3.) Use normalized least squares fitting procedure:
P. Goldner, et al., J. Appl. Phys. 79, 7972 (1996).
!"# = "# 1+ $Eij % 2E4 f( ) E5d
0 % E4 f( )&
'()
! 2= Sj
m " Mij#i
i
$%&'
()*
! i
+
,-
.
/0
j=1
N
$2
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Special Case II: Eu3+ ions(Beyond the standard Judd-Ofelt theory)
The ED transitions 7F0 ↔ 5DJodd, 7FJodd ↔ 5D0 and 7F0 ↔ 5D0 in Eu3+ are “forbidden” in standard JO-theory. They violate the selection rules:
• ΔS = 0• If J = 0 then J′ is even• If J′ = 0 then J is even• 0 ↔ 0
These transitions are primarily MD, but all three do occur as ED with low intensity in the spectra of some materials.
This implies that the standard Judd-Ofelt theory is incomplete. These ‘forbidden’ transitions provide an ideal testing ground for extensions to the standard theory.
What mechanism or mechanisms could be responsible? Are they meaningful!
G.W. Burdick, J. Chem Phys. 91 (1989).
M. Tanaka, Phys. Rev. B, 49, 16917 (1994).
T. Kushida, Phys. Rev B, 65, 195118 (2002).
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
The Atom, Issue 2 August 1962(DC Comics)
Europium’s Peculiar Properties(Adventures of The Atom)
Coincident with the publications ofJudd and Ofelt, who were both alsointerested in Europium’s peculiarproperties.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Extensions I1) J-mixing: The wavefunctions of the J ≠ 0 state are mixed into the J = 0 state by even parity terms of the crystal field. Explains the radiative transition 7F3 ↔ 5D0
in Eu3+. J.E. Lowther, J. Phys. C: Solid State Phys. 7, 4393 (1974).
2) Electron correlation: Electrostatic interaction between electrons is taken into account. Goes beyond the single configuration approximation and electron correlation within the 4f shell is incorporated by configuration interactions. Contributes to “allowing” the “forbidden” 0 ↔ 0 transitions such as 7F0 ↔ 5D0 in Eu3+. K. Jankowski, J. Phys B: At. Mol. Phys. 14, 3345 (1981).
3) Dynamic coupling: The mutual interaction of the lanthanide ion and the crystal environment are taken into account. Goes beyond the static coupling model. Explains hypersensitive transitions (transitions highly sensitive to changes in environment). M.F. Reid et al., J. Chem Phys. 79, 5735 (1983).
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
4) Wybourne-Downer mechanism: Involves spin-orbit interaction among states of the excited configurations, leading to an admixing of spin states into the 4f N configuration. This accounts for the observed spin “forbidden” transitions ΔS = 1 B.G. Wybourne, J. Chem. Phys. 48, 2596 (1968). M.C. Downer et al., J. Chem. Phys 89, 1787 (1988).
5) Relativistic contributions: Relativistic treatment of f → f transitions in crystal fields. Reformulation of crystal field and operators in relativistic terms. Importance unknown. L. Smentek, B.G. Wybourne, J. Phys. B: At. Mol. Opt, Phys. 33, 3647 (2000). L. Smentek, B.G. Wybourne, J. Phys. B: At. Mol. Opt, Phys. 34, 625 (2001).
Extensions II
Review ArticlesEarly development:
R.D. Peacock, Structure and Bonding, vol. 22, 83-122 (1975).Later developments
L. Smentek, Physics Reports, vol. 297, 155-237 (1998).
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
SummaryPhysical Mechanisms: (Not a complete list) Crystal field influence based on static model. Second order in the perturbation. (This is the standard Judd-Ofelt theory). Crystal field influence based on static and dynamic model. Second order. Electron correlation based on static and dynamic model. Third order. Spin-orbit interaction. Intermediate coupling and Third order effects. Relativistic effects.
Remaining Problems: Estimating the relative importance of each mechanism is considerable. (Many competing mechanisms producing various effects. Entangled situation) Ab-initio calculations still not entirely successful. - Theory of f - f transitions not yet complete. - Calculation of Radial integrals and knowledge of odd crystal field parameters. - Vibronics (Vibrational lattice-ion coupling) Multitude of mechanisms and new parameters abandons simplicity. - Simple linear parametric fitting to observed spectra is lost. - Physically meaningful descriptions can be obscured.
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
“It has been in a very real sense the first step in thejourney to an understanding of the rare earths andtheir much heavier cousins, the actinides, but like manyjourneys into the unknown, the end is not in sight.”
What’s next?
Brian G. Wybourne“The fascination of rare earths - then, now and in the future”Journal of Alloys and Compounds 380, 96-100 (2004).
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Judd and Ofelt Finally Meet40 years after publications
B.R. Judd G.S. Ofelt B.G. Wybourne
Ladek Zdroj,Poland - June 22, 2003“4th International Workshop on Spectroscopy.Structure and Synthesis of Rare Earth Systems.”
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
2007 School of Atomic andMolecular Spectroscopy
2007 will be the 45th anniversary of the simultaneous publications of Brian Judd andGeorge Ofelt. A special session is certainly worth considering in the next course.
“The fascination of the Rare Earths - 45th Anniversary of Judd-Ofelt theory”Possible invited lecturers:
Brian R. Judd - The Johns Hopkins University, Baltimore, MD 21218, USAE-mail: [email protected]
George S. Ofelt - 824 Saint Clement Road, Virginia Beach, VA 23455, USAE-mail: gsofelt@pilot,infi,net
Lydia Smentek - Vanderbilt University, Box 1547, Station B. Nashville, TN 37235, USAE-mail: [email protected]
G.W. Burdick - Andrews University, Berrien Springs, MI 49104, USAE-mail: [email protected]
Francois Auzel -UMR7574, CNRS, 92195 Meudon Cedux, FranceE-mail: [email protected]
Sverker Edvardsson, Mid Sweden University, S-851 70, Sundsvall, SwedenE-mail: [email protected]
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Rino DiBartolo-Thank you for your years of wisdom and my first lecture onJudd-Ofelt theory in your office in ‘old’ Higgins Hall. Alsofor inviting me to Erice these last 10 years.
Norm Barnes-Thank you for helping me see the laser side of life. Thediscussions we have had over the years remain with me.
Acknowledgements
National Aeronautics andSpace Administration
International School of Atomic and Molecular SpectroscopyErice, Italy (June 2005)
Brian G. Wybourne (1935-2003)Professor Brian G WybourneCommemorative Meeting: Symmetry, Spectroscopy and SchurInstitute of Physics, Nicolaus Copernicus University, Torun, PolandJune 12-14, 2005.
A commemorative meeting in honor of Professor Brian G. Wybourne will beheld in Torun, Poland from 12-th to 14-th June 2005. The aim is to celebrateBrian's academic life and his contributions to many aspects of physics andmathematics. This meeting will bring together friends, students, collaboratorsof Brian as well as people interested in the results and consequences of his research.
Dedication