Determination of f2~ intensity parameters

JIs JO~'IB')I 2

7.2. Carnall's ~z intensity parameters

7.3.2. Determination of f2~ intensity parameters

The £2~ parameters for induced electric dipole transitions can be determined by writing down for each transition an expression of the form

XED e 2

(~--~2U(2)+~-~4U(4)_t_~.~6U(6))

[D in esuZcm2]. (202) Dexp - 2 J + 1

SPECTRAL INTENSITIES OF f-f TRANSITIONS 165 There are several numerical methods to determine the £2 x parameters; below we describe, as examples, the standard least-squares method and the chi-square method

7.3.2.1. Standard least-squares method. The standard least-squares method minimizes the absolute differences between the experimental and the calculated values. The least- squares method is simpler than the chi-square method, but has the disadvantage that a small discrepancy in a large experimental value has the same influence as a large error in a small experimental value. Therefore the f2x parameters depend largely on the relative magnitude of the oscillator (or dipole) strengths of the transitions used in the fit. However, the parameter set should be able to predict both small and large oscillator (or dipole) strengths.

We define:

Aexp ^.^_ ( 2 J + 1)Dexp ; (203)

e 2 'XED then,

Z~exp = ff22 U(2) 4- '.Q4 U(4) 4- ff26 U(6).

(204)

It is possible to write down such an equation for each spectral transition, resulting in a system of n equations for n transitions:

= T (6)

Aexp ff~2 U~ 2) + ~r~4 U~ 4) 4- "Q6 UI , Aexp = -('22 U2 (2) 4- ~'~4 U2 (4) + ~ 6 u ~ • (6),

~2 (205)

An exp = ~ 2 U (2) 4- ~r~ 4 g n (4) + ~6Un• T(6)

We want to find a good estimate for the parameter set (Q~, g2~, g2~). The set of equations (205) is of the form

Yl = alX]l + a2x12 + "'" + a k X l k ~ Y2 = alx21 + a2x22 + "'" + a k X 2 k ,

(206)

Y n = a l x n l + a2Xn2 + " ' " + a k X n k .

The Yi are the values of the observations, the xik are coefficients and the ai are unknowns (the parameters). The observational model is a general linear model, because the dependent variable Yi is described as a function of several independent variables xik.

The function is a linear function (there are no terms of a degree higher than 1). The matrix method is used to solve the problem• This method is described in every good textbook

166 C. GORLLER-WALRAND AND K. BINNEMANS

dealing with statistics. The system o f eq. (206) can be written in terms o f the response random vector, containing the response values o f the n observations:

Y = Y2 , (207)

the model parameter vector:

a --= , (208)

the vector o f errors associated with the n observations:

E = , (209)

and the n x k design matrix:

x,2 xlk

X21 X22 • . . X 2 k ]

X = . . .. . J . (210)

\ X n l X n 2 • . . X n k /

The matrix representation for any set o f observations Yl,Y2,. • • ,Yn becomes

¥ = X a + E. (211)

Let the matrix h2

= (212)

represent the matrix o f the least-squares estimates for the parameters o f the general linear model. This matrix can be calculated as follows:

= (X3:X) -1 X T Y , (213)

where X T is the transpose o f X . The matrix X is a n x k matrix, X r is a k × n matrix, and X T X is a k x k matrix, with n ~> k. The matrix solution gives the set o f parameter

SPECTRAL INTENSITIES OF f-f TRANSITIONS 167

estimates dl, a2,..., ak in the general linear model that minimizes ~ i ( Y i - ) i ) 2 for the data collected.

After the parameter estimates (~2 ~24, s"26) have been obtained, D i ED can be calculated:

DiED=e2 ({22Ui(2)+ ~'24U}4)+ ~26U~!6)).

(214)

The quantity [XED/(2J + 1)]D~D is sometimes denoted as Dcalc, and can be compared

i

directly with the experimental dipole strength D~x p of transition i. The root mean square deviation error RMS is a measure for the goodness of the fit:

I sum o f squares of deviations ]1/2

RMS = number of observations- number of parameters J (2 l 5) For the dipole strengths, the RMS is given by

RMS

= I .~i ( O ~ z-~Dcalc) J i i 2

1/2 ,

₍₂₁₆₎

where N is the number of transitions used in the fitting procedure. The number of parameters is three (~2, ~'24 and ~"26). The ratio Dcalc/Dcxp also gives useful information about the reliability of the calculations. The error on a parameter is given by the product of the square root of the respective diagonal matrix element of the matrix ( x T x ) < and the RMS for the deviations between Ai exp and A~ alc.

7.3.2.2. Chi-square method. If one wants to weigh each value of the dipole strength by its own uncertainty, the chi-square method has to be chosen as the minimizing quantity (see, e.g., Caird et al. 1981, Seeber et al. 1995, Goldner and Auzel 1996). This method minimizes the relative differences between the experimental and the calculated values rather than the absolute differences. The uncertainty in the measured intensity of each transition is often difficult to estimate. Goldner and Auzel (1996) take a constant fraction of the experimental oscillator strength for the uncertainty. The RMS (eq. 216) of this method is independent of the number or the magnitude of the included transitions. The chi-square fitting method is analogous to the matrix calculation presented for the standard least-squares method, except that the design matrix X is now given by

X =

Xll x12

01 o-1 x21 x22 o-2 o2

: ~ "..

Xn 1 Xn2

o-~ on

Xlk

al x2~

Xnk

(217)

168 C. G O R L L E R - W A L R A N D AND K. BINNEMANS

and the vector Y with the observations is now given by

(218)

The fit is applied to the quantities D~xp/Gi, where o i is the uncertainty in the experimental dipole strength D~x p. The error of a parameter is given by the square root of the respective diagonal matrix element of the matrix (XTX) -1.

7.3.3. Additivity of intensity parameters

The additivity of g23. in eq. (202) has two major advantages. First of all, in the presence of several non-equivalent sites in the host matrix one obtains an average value for each of the g23. parameters. The g23. parametrization can therefore be used also for highly disordered systems, like glasses. Only when site selective spectroscopy is used are different ~23. values found for each site (Rodriguez et al. 1995). Secondly, in the case of overlapping bands, the matrix elements of each of the transitions contributing to the overlapping band can be summed. The complex band is integrated as a whole. For instance, if transition A and transition B overlap, one finds

DED(A + B ) = e 2 ~ ~3. [UA(3.) + U~(3.)] . (219)

3.=2,4,6

7.3.4. Mixed ED-MD transitions

If the transition has partial MD character, the MD contribution DMI) has to be calculated

0 .

first. The £23. parameters are not fitted against Dexp, but against Dex p.

De°xp = Dexp 2 J + 1 XMDDMD. 1 (220)

It is assumed that the experimental and calculated magnetic dipole strengths are equal.

7.4. Successes and failures of the Judd-Ofelt theory

The £23. parametrization describes the intensity between J-levels. Each J-level is considered as 2J + 1-fold degenerate. This implies also that each of the M-levels of the ground state has an equal thermal population. If the crystal-field splitting of the ground state is not too large (<500 cm-1), the approximation is a good one at room temperature.

SPECTRAL INTENSITIES OF f-f TRANSITIONS 169 Even for larger splittings, the assumption is fairly good, since the crystal-field levels are often a linear combination of different M-levels (M-levels within one #-matrix). However, the Judd~)felt theory with g2x parameters cannot be used for low-temperature absorption spectra, especially not at 4.2 K, a temperature at which (in most cases) only the lowest crystal-field level of the ground state is populated. For europium, low-temperature spectra can be treated by the Judd-Ofelt theory, since the 7F0 ground state is non-degenerate.

At 77 K and lower, only the ground state is populated, while at higher temperatures the 7F1 level has also a non-negligible thermal population (at room temperature ~35% of the total population). One can deal with this difference in thermal population by using the

XA(T)

fractional thermal population factor (as described in sect. 3.8).

An extensive list of g2z intensity parameter sets for different (lanthanide ion)-(host matrix) systems is given in tables 10-21. Although we do not claim completeness, we have tried to compose a list which covers a broad range of different hosts. The data are covering lanthanide ions in solutions, glasses, polycrystalline samples and single crystals.

The parameter sets have been classified within one lanthanide ion according to increasing values of g22.

Over the years, the Judd-Ofelt theory has been proved to be quite successful for the intensity analysis of the trivalent lanthanide ions. A lot of pioneering work concerning the determination of experimental intensities of the f - f transitions over the lanthanide series and the systematic intensity parametrization has been done by Carnall and coworkers (Carnall et al. 1965, t968a, see also Carnall 1979). They studied the spectra of the trivalent lanthanide ions in aqueous solution (diluted HC104). In order to extend the range of measurements to the near-infrared, spectra were also recorded in diluted DCIO4.

The intensity of the transitions between J-multiplets in the spectrum can be rationalized in terms of only three parameters s'2x. One can make predictions about the intensity of transitions which cannot be observed experimentally (e.g. infrared 4f-4f transitions in aqueous solution). The parameter sets are a useful tool to compare the f - f intensity properties of different lanthanide systems. They may be used to derive relationships between spectral and structural properties for different kinds of lanthanide complexes.

For Pr 3+, the Judd-Ofelt theory does not seem to work well (Goldner and Auzel 1996). Difficulties are experienced if one tries to fit both the 3F3,3F 4 ~-- 384 and the 3p2,1,0 , lI 6 +--- 3H 4 transition groups with the same set of Dx intensity parameters (Carnall et al. 1968a). Instead of determining a parameter set with the inclusion of all transitions, the 3F3,3F 4 ~-- 3H 4 transitions can be excluded. Of course, agreement will then be worse between the experimental and calculated dipole strengths for the latter transitions. Other authors prefer to exclude the 3P2 +-- 3H 4 transition (e.g., Eyal et al. 1985, Quimby and Miniscalco 1994, Seeber et al. 1995). It is also found that the Q6 parameter is appreciably larger than the value extrapolated from the other lanthanide ions in the same matrix (Peacock 1975). The problems with Pr 3+ are due to the fact that assumptions in the theory are not valid in the proper case of Pr 3+ (Carnall et al. 1968a, Peacock 1975). First of all, it has been assumed that the configurations which are mixed into the 4f N configuration are degenerate. Only the barycenters of these excited configurations are considered. The 4f 15d I configuration of Pr 3÷ starts about 45 000 cm -1 above the ground state, while those

170

_~.~

-,7

N N

C. GORLLER-WALRAND AND K. BINNEMANS

. ~ ~ ~ - ~ ~ - ~ ~

• - , ~ + ,~ . : ~ - ~ - =

'= ~ i , ' : ~ ~ ' I ! I ~ ~ = "~ " .

"x. 5 c~

~-. +1 -H

. ~ ~

~ o ¢ 0 o

I I 1 [ 1 1 1 I I

,..¢ -¢

+ ÷ ~ +

S P E C T R A L I N T E N S I T I E S O F f - f T R A N S I T I O N S 171

o ~

,.o ¢

o o

7 o o ?; o

o ~ '-~ b~ t'q

- ° -

. ~ --&

n, ._~ r..)

z z ~ z z -

-6

..= O ~

172 C. GORLLER-WALRAND AND K. BINNEMANS

,.o

a', a~ ,. a ;Z ~

~.. c ~ ~ . o t-- ~ - . .

7 7 7

o ~ ~ ~ ~ _ ~ _ ~

-~ ~ " ~ .~ ~5 ~ ? < ~ c ~ - ~ o - ~ "

~ ~ o ~ ~ ' ~ o o o

m 0 ~ o 7

~ ~ o o ~

÷ ~ Z O ~ c ; ~ °

+ + + + + + + + ± + + + +

SPECTRAL INTENSITIES OF f - f TRANSITIONS 173

o~

= o f =o~ ° ~

, ~ - ~ ~ ~ ~ ~ , m _ ~

"6"

.=- .=. W; ~ = = " = = " " ~ = = = = .m. . ~ ~

° T " T T T U ~ ~ " " + + + + o ~

174 C. GORLLER-WALRAND AND K. BINNEMANS

E

r,i

~ N N N

~ , ~

m m

N N

~2

"~oo

z z

i¢ 3 t¢~

SPECTRAL INTENSITIES OF f - f TRANSITIONS 175

~2

..=

-d

e )

o o

o ~

O O O ~

O O ~ ~ O

7. a~

~2

t t 3

~2 2:

~ - Z ~

tal)

~ -< "< L)

176 C. G O R L L E R - W A L R A N D A N D K. B I N N E M A N S

m ~ ' ~ m m m ~ m

m ~ m

.~.~ ~

- v 8 a e {

== =~ m

~ d d + d d d d d d d d d + a m ~ ~ m m m m ~ m ~

~ ~ ~ ~ ~ . . . . . . . . . .

f

" z ~.= ~ < f ...z

< ~ > .~ z T < ~ =

~--eq

S P E C T R A L I N T E N S I T I E S O F f - f T R A N S I T I O N S 177

e % ~ g 8

eq ~ "2. eq. ~

~ - ~ - ~ . ~ . a R ~ : = m

e,') eq.. d ¢-q.. e,q ..-2. e,,]. ~ ~ .

eq.. ~ ,.q % m. m. ,.q. ~, ~. ~.

z 7

a v v

~,~

~ ~ +

m tao I

7 ~ _ez c 7 ~

~ z

m e,i N b,l Z

z z ~

o c5

0 I

i ec

Im Ca

t78 C. GORLLER-WALRAND AND K. BINNEMANS

~ g ~ g g g ~

~ d ~ Q Q ~ Q Q

~ i ~ g ' ~

m m m m ~ m ~ m m N m ~ ~

' ! "q. m

0 ~ ~ ' 4 4

. . . . ~ ~ ~ ~

Z D.

7 ._=

s

v ~

~ +

~.z. z ..

Z <:: ~ "

z ~7

z ~

z T

I ~ ~ f

'7 '7~ d

p-.

N ~ d l ~

SPECTRAL INTENSITIES OF f - f TRANSITIONS 179

c ~

~ m ~t

¢ q ¢ q

F - c ~

- - [ ~ ~

~

• a', Ch ,-~

~ - - ~ ~ ~ ~ ~ ~ ~ ~ ~

Z Z ~ Z Z

O 0 ~ H O ~

m. m.

.z.~

~ z ~

z ~ ~

0 .-~

z e

~ 7

o ~

~ z

t ~ t ~

Z ~

~ C

o ~

z ~

~ 0

~ _ ~ ,

~cO 0 Z

o ~

Z c~ I o ~

I 0

t--

~ 9

180

o e>

C. GORLLER-WALRAND AND K. BINNEMANS

~ o ~ .... ~

~z

-& z (o

~ ~ -~ ~

~ ~ ~ o ~ ,

~7 ~

~ ~°°° ~

~ z ~ ~ z

SPECTRAL INTENSITIES OF f - f TRANSITIONS 181

~ m ~ . C

N N N N N ~ ~ ~ ~ N ~ ~ N g N

o o o o o = ~ g ~

• ~

G0 GI9

c7 ,-E -E

z ~

~ - ~ o ~ ~

~ ~ ~ ~

^{.-~ ..~}

( ~ o ~ z

o Z N Z T ~

c~ c ~ z 6

• ~ ,,d~ ~

~ Z ~ oo ~- r-- ~p

182 C. G O R L L E R - W A L R A N D A N D K. B I N N E M A N S

~ o o o o o o o o o o o o

N )

q 3

i t 3 +

,-&

Z ,Q.

, ~ + + Z 0",

r ~

"-& e-: o ~

r ] ] t¢3 0"3 ee~ O

SPECTRAL INTENSITIES OF f - f TRANSITIONS 183

,z::,

Z Z ~ ~ . ~ ~

- - T o T o ~ Z

~ ~

• q

^• ^{e ~ q e} ^{e ~} ^{e e} ^{~ e} ^{e ~}

o ~ o o o o ~ o o o ~ o

0 _ ~ z

~ m, o ~

o 7 , ~ 6 ' T q o ? ~

'T o~ d d o, f f o

O~ ~ 0 = 0

~,~ ~ + +

o ~ ~

d o ~ o = ~

7 ~ " * T C ~ ~

~+ ~ + W z 7 o m

o ~

o. [--..

7 o

o ~ =

¢7 o~ c<

184 C. GORLLER-WALRAND AND K. BINNEMANS

,2-,

g

o-, ~"

4 d d d ~ d d d d d d d ~ d d ~ d d ~ d d ~ d d ~ d

o ~ o ~

S P E C T R A L INTENSITIES OF f - f T R A N S I T I O N S 185

~D d:5

m m

O 0 ~ 0 ~

q q o ~ q ~ o o m

d d ~5

t¢~ v D ~ co m

o ~ o ~ ~.=...= z ~

~ g oo z z m ~o ~,

c7-~

9 c5~

o ~ o ? o ~

2 .~ .-= ~ z

~- c5

186 C. GORLLER-WALRAND AND K. BINNEMANS

o ,..o

~ ~ ' ~

~ ~ - -

0 0 0

6 6 6 ~ ~ = ~ ~ ~

^{o o} ^{,~ ~ 0}

~

o ~ o . o _ = < ~ o #

'~ ~ "¢ ~ C~ = " ~ - m ~ ~' >

+ + + + 4 ~ ~ ~ " ~

Z Z Z Z Z Z Z . ~ Z m ~ Z ~ Z Z Z Z Z Z Z Z Z Z Z

S P E C T R A L INTENSITIES OF f - f T R A N S I T I O N S 187

..=

'-d

¢)

._=

,.q

"7~

¢x,

188 C. GORLLER-WALRAND AND K. BINNEMANS

"-d

¢)

e¢3

¢ ¢

¢)

e n b4 b~ b a

SPECTRAL INTENSITIES OF f-f TRANSITIONS 189

"4 ,.=

¢x0 ,-¢1

~z

190 C. GORLLER-WALRAND AND K. BINNEMANS

~ ~ ~ = ~

~° o " o ~ c~

6 £ ~ o ~ 6 6 o o ~ ~ . - ~

Dalam dokumen Handbook on the Physics and (Halaman 173-200)