THEORY OF RARE EARTH

SPECTROSCOPY IN GLASSY

MATRICES -- - -

2.1 Introduction

T he development and renewed interest in solid state materials followed the

discover). of laser action in rare earth compo~u~ds, characterised by minimal

concentration quenching of emission and high fluorescence efficiency. These

observations have led to a drastic change, via reduction in size of active medium

down to the so-called mini lasers that were proposed at some point as laser sources

and also in the reduction of cost of production. As described in Chapter I glasses are

found to be good host for many active ions of rare earths. Following the report in

1961 of laser act;on in glass' many ions and host glasses were investigated and the

features and merits of giass lasers relative to crystalline materials were esttblished.

The first laser glass was barium crowri glass in which :aser action wa? obtairled in

~ d ' * ion doped in i t Aher this, number of rare earth dcped glasses were developed.

dopants includcd tio Yb. Er. Tm and 'l% 2 1 4 5 l'he laser action reported in man,

glasses during the pn.1 thirtb vears, i~cluded phosphate. borate. flwnde and tluoro

~irconatr etc in 411 thew.. theoretical predichons of laser cross-sechons and

population probabilities in rare earth doped glasses are made. These are based on the

combination of the calculation of radiative and non radiative transitions of different

rare earth ions in various oxide and fluoride glasses. And also measured various

important spectroscopic parameters such as oscillator strength, radiative transition

probabilities, life times, line strength and fluorescence cross-section etc. Transition

metal ions and organic dyes are also found to be good for lasing action in glasses.

2.2. Electronic energy level structure of free ions

Rare earth materials are classified into two groups lanthanides and actinides, each

contains fourteen elements characterised by the progressive filling of 4f- or Sf- shells

of their electronic configurations. The lanthanides, which are associated with the

filling of the 4f- shell, starting with cerium and ending with lutecium, whereas

actinides are associated with the filling of 5f- shell from thorium to lawrenciun~.

Lanthanides have assumed a great role in rare earth spectroscopy because of their

spectroscopic importance.

The spectroscopy of rare earth ions shows characteristic property with 4f electronic

configuration, which is well shielded, from the surroundings. These electronic

configuration yields very particular optical properties which are absent in other metal

ions. Electronic configuration of actinides is also more or less similar to rare earth

with electronic configuration 5r. In the neutral rare earth atom there are two or three

electrons out side the core of the normal configuration, and hence in the,firsr

spectra of rare earths very complex configuration can be expected. As an example, in

praseodymium in addition to normal configuration 4j6sZ the configurations 4 f 6 ~ 6 ~ ,

4P5dbs and 4f 6s are all of spectroscopic interest.

The neutral lanthanides ptssess the cmmorl lkature of xenon structure of electron

( IS' 2s' 2p" 3s' 3# 3d"' 4s2 46' 4d" 5sL 5p0) with two or three outer electrons (6s2 or

5d 6s') l'he lanthanides ma, be tonlzed hy the successive removal of electrm. In

the first stage of it)ntzation wltn ihe sole exception of Iutecium. results from the

removal of a 6s-electron. In the second stage of ionization, the removal of further 6s-

electron occur;, and at the third stage, all the 6s- and 5d- electrons and frequently a

4f- electrons have been removed. In atomic spectroscopyb the stages of ionization are

labeled by indicating the neutral atom, which gives rise to the first spectrum. The

chemical properties of rare e h s are greaily determined by the Y electrons. Theirf-

shell behaves as an inner shell due to the contraction offeigen function, which is

reflected in their properties. The shielding offelectrons by the sZp6 closed shell and

the failure of feigen function usually prevents it from any strong interaction with

their environment. Consequently, the surroundings of the lanthanide atom or ion are

insignificant chemically, i.e., 4felecton has little tendency to participate in any

chemical bond formation. While in the case of actinides due to the spatial extension

of the 5f- eigen function, their interaction with the environment is greater than that for

the corresponding lanthanides. The increased spatial extension of the Sfeigen

function has been made evident in electron spin resonance experiments, no such

effect was observed when corresponding lanthanide ion is substituted in a CaF2

lattice. Thefleigen function contmction is manifested in the regular decrease in the

size of lanthanide and actinide with increase in atomic number. There is also a

regular decrease in the ionic radii with increasing ionization. Lowering of binding

energy in the actinides resulted in the readily removal of Sf- electrons than 4f-

electrons in lanthanides. This is reflected in the multiplicity of oxidation states of

many actinides in contrast to very strong dominance of the +3 oxidation state in

lanthanides. Oxidation states other than +3 are known for several of the lanthanides

although these are always less stable than +3 state.'

2.3 Energy level and spectroscopy of rare earth ions

The general property of rare earth IS greatly determined by their 4f; electrons. The

electrons at the inner shell are shielded by the Ss,p,d electrons. So the effect of

I~gand field becomes weak. 'The enerby levels of rare earth ions depend on, firstly,

the static electr~c lnteract~on between electrons (H,) and secondly on the interaction

between sptn and orbital (H,,) 4f electronic configuration forms the energy levels

denoted with LSJ under the effect of the above two interactions. Figure 2.1 shows the

energy level diagram of various rare earth ions '. The energy levels of rare earth ions

in glass are similar to those of free ions. Energy level of a system of state such as

atom or ion is usually calculated by determining the Hamiltonian of the system. In

many particle system like rare earths the most common method is to apply the central

field approximation9 method to calculate the Hamiltonian of the system. Hamiltonian

of optically active electron is composed of several terms. The central field

Hamiltonian (h), which represents the interaction of electron with nucleus, the

coulomb field (Y), the interaction between the electrons, the spin orbit field &)

represents the interaction between the spin and orbital motion of the electrons and the

crystal field (I&) which represents the interaction of electron with the crystal field

from the surrounding ion in the solid. Therefore the total Hamiltonian of the system

can be written as

H,/ = H" + H , + H , + H,

Since the optical transition of interest in rare earth involves the 4felectrons, the

magnitude of the different interaction terms is different in each rare earth ions. The

other three terms gve rise to a set of states labeled by total spin S, total orbital

angular momentum L and total angular momentum J. Spin-orbit interaction has

considerable effect, so the resultant total angular momentum 'J' is considered to be a

good quantum number in almost all circumstances.

The effect of other interactions such as those with nucleus or crystal fields is to lift

the (a+ 1 ) degeneracy of the level and the small splitting produced may be treated as,

perturbation. The rare earth ion in a crystalline salt is situated in a potential V(r) in

the crystalline field, which arises from charge Ze on neighboring ions at positive Ri

with a potential term of V ( r ) = x e ~ , ( r - 41. This potential lifts the (2J+I)

degeneracy "'. From susceptibility measurements it is clear that these splittings due to

V(r) are much smaller than the multtplet separation. So that mixing of different

multiple1 can usually be neglected. In the absence of any magnetic field the levels of

Figure 2.1 Experimental spectra of trivalent rare earth ions in LaF, crystal (Dieke chart)

an ion with odd number of 4f electrons can at most be split into levels which are

doubly degenerate. This is applied to rare earths with half-integral 'J' ground states

e.g. ce3+. The environment of an ion with a degenerate ground state spontaneously

distorts to a lower symmetry so as to remove the degeneracy. It implies that ions with

an even number of 4f- electrons always have singlet ground state e.g. P?'. This rule

doesn't apply to excited state.

Riseberg and Weber " describe the splitting of energy levels of rare earth in

schematic fashion as shown in Figure 2.2. Typical separation due to coulomb

interaction is 2104 cm-I and it is =10'cm-' for spin-orbit interaction, which splits the

'L' terms into T states. The crystal field interaction breaks the spherical symmetry of

the Hamiltonian and remove (2J+1) degeneracy of the levels. The so-called Stark

splitting is generally a few hundred cm-' in magnitude. Radiative transition can occur

from any Stark component of one {L S J) multiplet. As an example of the energy

level splitting in rare earth, a schematic diagram of the splitting in 4f electronic

configuration is given in Figure 2.3. Because of the extreme complexity of the

Coulomb 2S+IL

Figure 2.2. Schematic diagram of the splitting of energy levels of rare earth ions

Elgure 2 3 Energy level splitting of 4j6 configuration

splitting, only a part of the diagram is shown here. In crystal spectra, Zeeman and

polarisation studies have made it possible to characterize the J manifolds to which

many of the lower lying crystal levels belong. At higher levels this becomes more and

more difficult ".

The basic energy levels of free ions are those of the configurations, which are

2(2/,+1) 2(212+ 1 ) . . . .. 2(21,,+ 1 ) degenerate, and have separation -5x 1 o4 cmei . The

detailed energy structure arises from the splitting of these levels by interaction

between electrons, here between 4fslectrons involved and also between electrons and

nucleus. In the case of rare earth three main interactions are dominant in order of

magnitude.

( I ) Coulomb repulsion between electrons

(ii) Spin orbit interaction

( I ; ; ) Nuclear hypertine and quadrapole interaction

2.3.1 Free ion in magnetic field

Ignoring hyperfine structure, the basic energy levels of a rare earth ion are those of

multiplets (E,). The application of an external magnetic field (Hz) lifts the (2J+1)

degeneracy of these levels. Due to the unfilled 4fi shell an ion has a permanent

magnetic moment p = -pIj(L+2S) which interacts with the external magnetic field.

The Zeeman splitting is much smaller than the multiplet energy separation except for

triply ionized samarium and doubly ionized europium.

2.3.2 Coulomb interaction

The coulomb interaction is the largest among three. It splits each configuration

energy into term level, which are specified by 'L' the total angular momentum

quantum number and total spin S. i'he ground state can be assigned according to

Hunds rule Each energy term and the corresponding ground level are dep~cted in the

Dieke chart in Figure 2 1 l'helr enerby separation can be calculated by the evaluation

of Slater radial integral ~ ~ ( 4 f . 4 ~ " wh~ch will be described later on For rare earth

ions the energy separation between configuration is much larger than the coulomb

interaction so their mixing probability is small.

2.3.3 Spin-orbit interaction

The coupling of the indvidual orbital angular momenta ii &id si hetween electrons is

a relativistic effect. As mentioned above due to the strong coulomb interaction in rare

earth ions, the individual li and si always couple to give L and S. The effect of the

spn orbit interaction is then to couple L and S according to Russel-Saunders scheme

L+S = J and partially lift the degeneracy of the term. The resulting (2J+l) degeaeracy

is the starting point of most theoretical considerations of the rare earths For

identifying the lowest energy levels, we have applied Land'e rule which gives a

reasonable approximation for rare earths. For ions with their .If shells less than half

full then lowest enera level has the smallest J value i.e. b-SI, while for the shell

which is more than half full it has the largest possible J ie. IL+SI

Departure from the Land'e rule has been discussed by ~ u d d l ~ when considering the

interaction between the magnetic moment of 4felectrons are concerned, but they are

very small. Deviation are also due to the partial breaking of RS coupling, which

occurs if the spin-orbit coupling is sufficiently strong to mix other terms into the

ground state. This is called intermediate coupling. But these corrections are

neglected when considering the lowest multiplet levels.

2.3.4 Hyperfine structure and quadrupole interaction

If the nucleus has spin 1 then associated with it i s a magnetic moment p, g,pN 1,

where gl is !he splitting factor and p+, :he nuclear inagneton. Due to their orbital and

spin angular momenta, 4f electron produces an effective magnetic field I(, at the

nucleus whlch interacis with p; and the resulting Hamiltonian is

This effect gives rise to hyperfine splitting of the multiplet level. Because p~ is only

111 836 p", the interaction is much smaller than those between the electrons and their

magnitude is only about -10~~-10~~cm~ ' . If the nucleus has a quadrapole moment Q

there is also an electrostatic interaction between it and 4f electrons of the form

where Q is the quadrapole moment which depends on nuclear charge density.

Apart from all these interactions some relativistic effects are also to be considered

when discussing the electronic structure of rare earth ions. The relativistic effect

become appreciable when the potential energy of an electron is not negligble

compared with its rest mass. For hydrogen like atom for which V= 2e2/r, it is clear

that such effects are most important for heavy atoms Z255 and for electrons of such

atoms with the smallest quantum numbers. Since rare earths are in the second half of

the periodic table and since the spin orbit coupling is a relativistic effect, and it is

large for the rare earths". These relativistic solutions should be employed for

calculation involving rare earths, however, these are only known numerically and

hence their use is not expected to lead to a better understanding of important physical

factors which determine the property of ions.

2 4 Empirical analysis of free ion spectra

In the above sections we have discussed various causes of splitting of energy levels

from which various transitions between these levels can occur. Once the spectrum

has been produced and the wavelength has been measured the next step is the

analysis of energy levels. This means to identify the two energy levels between which

transition occurs and find its various properties. For the interpretation of the energy

levels, probably the greatest aid is the comparison with the theoretically calculated

energy level scheme However, there are other ways by which the interpretation can

be obtained For that it is very important to know the total angular momentum J of

the electronlr state Direct information on this can be obtained if there is any

resolvable hypertine structure. This is usually available for nuclei of odd mass

number. If there 1s only one such isotope, as for praseodymium and holmium, the

data obtained from the spectrum of the neutral element can be used. In the case of

mixture of isotopes for natural elements, artificially separated isotopes must be used.

Zeeman effect may also be of great aid for the classification, but so far this has little

use for these spectra, because in most cases the analysis of the lowest configuration

can be carried out satisfactorily without them. The analysis of the Zeeman pattern in

a very crowded spectrum demands much additional work.

As the spectra of rare earths are extremely complicated and very rich in line, a

reasonably accurate analysis will necessarily take many years, but it will yield

accurate energy levels, among them those of the 4r configuration. In the meantime

much information can be obtained from calculated levels. These calculations are

extremely valuable, as without them an interpretation of the different levels would be

virtually impossible. As more levels become known, the precision of the calculation

also improves.

2.5 Calculation of energy levels of rare eardh ions in glassy mattices

The energy levels of the triply ion~sed rare earth ions may be obtained with the

highest accuracy from the analysis of free ion spectra. In this section we shall be

primarily concerned with the energy level structure of the principal electronic

configuration encountered in the spectra of rare earth ions. Spectra of the ionised rare

earths normally involve substantially simple configuration. This can be noticed when

the atom has been ionized sufficiently to leave a normal configuration with not more

than one non-equ~valent electron outside t h e y core as in the case in all the doubly

and triply ionised rare earths. 'Thus in pr3+, when the normal configuration is 4fl the

configuration 4Pbs. 4Pbp and 4r5d are of spectroscopic importance 16.

Attempts were made to calculate the energy levels theoretically after Dieke created

the chart of energy levels. But the results were not even close to the experimental

levels measured even though the various approximations to crystal field were applied.

Much time has been spent to reach a compromise between these. The object of the

theory is to calculate the energy levels and the wavefunction of atoms even if such

complexity on the rare earth ions occurs. But in order to overcome the complications

we have avoided' some interaction factors. This is zero order approximation, which

considers the outer electrons in the central field produced by the nuclei and the 54

electrons in the complicated xenon like structure. Here we have disregarded

electrostatic interaction between outer electrons and their influence of spin.

2.5.1 Matrix diagonalization procedure

Using this method one can calculate the energy eigen values of the various

interactions by making energy matrices of electronic ineraction in the 4r

configuration and then diagonalize the resultant matrix.

Consider a quantum mechanical system of particles, which is under the influence of

some external field, the Schrodinger equation corresponding to the system is

H v = ~ : ' y l (2.4)

H is the Hamiltonian of the system of particles with wave function y~ and E is the

energy eigen value. For N electron system the Hamiltonian can be written as "

with Ze the nuclear charge of the atom. If the mass of the nucleus is assumed to be

infinite the first term in the above equation represents the sum of the kinetic energies

of all the electrons, the second tenn is the potential energy of all the electrons and the

last term is the repulsive coulomb potential energy of the interactions between pairs

of electrons. For solving thls type of problem with more than one electron we have

used the most common approximation method,'n."' the central field approximation.

In this approx~mation each electron IS assumed to move independently in the field of

nucleus and a central field made up of the spherically averaged potential fields of

each other electron. Hence each electron may be said to move in a spherically

symmetrical potential -U(rJe). The Hamiltonian Rr for the central field then

becomes

The difference H-K, is the perturbation potential. The Schrodingeh equation for the

central field is

Calculation of the energy levels of an atom or ion normally proceeds by first figuring

the matrix elements of the electrostatic perturbation potential V defined as

v=b-"2 -- U(ri)l + C- e2 i-l 5 i<J 51

The first tern is purely radial and contributes energy shifts that are same for all the

levels belonging to a given configuration without affecting the energy level structure

of the configuration. The repulsive coulomb interaction of electron will be different

for different states of the same configuration. The summation in the above equations

is over the coordinate of all the electrons.

Calculation of matrix element of the electrostatic interaction between each pair of

electrons in Legendre polynomials of the cosines of the angle o, between the vectors

from the nucleus to the two electrons as follows "'

where r, 1s the distance from the nucleus to the farther electron, r, indicates the

distance from the nucleus to the nearer electron. By spherical harmonic addition

theorcm we can wrlte

4rr 1; (cos 0, ) = -- ~ y - 4 ( 0 , ? 4 , ) > ~ - 4 ( 0 , , 4 , )

2 k + l

41r where c"', are defined by (-)I ' y4

2k+1

Thus the Equation 2.9 becomes

and the corresponding matrix element for calculating the energy level structure of a

configuration, produced by the repulsive coulomb interaction is given by

where r is another quantum number because in some configuration there will be more

than one configuration having the same quantum number. In order to distinguish we

have introduced a new quantum number r. Since the electrostatic Hamiltonian

commutes with the angular momentum operator corresponding to L', s', J,M the

matrix element will be diagonal in L and S although not in T and independent of J and

M. Hence the Equation 2.13 can be written as

Antisymmetrised eigen function for a pair of electrons can be written as ( &I., nt,lt,.

SLI. The matrix element of electrostatic interaction between the two configuration can

be written as

where (qL, ndb; SL) is the corresponding antisymmetrised eigen function for a p i r of

another electron.

By using Equation 2.12, it is now possible to write the above equation as

(2.17)

where fK and g, separate the angular part of the matrix element of Equation 2.16 and

R K ~ represent Slater radial integral, which arises from the radial part of the one-

electron eigen function"

the Slater ~ntegral R' are defined bv"'

when &la= n& and nhlb=n.,b the right hand side of the above equation becomes

= ~ ~ ( r ~ l . , nbld (2.21)

where F ~ S are known as direct integrals. In a similar fashion we can write the

exchange Slater integral as

F% are necessarily positive and decreasing f i c t ions of K. To avoid large

denominator appearing in the matrix element calculation we can redefine the radial

F~ and G~ integral in terms of the reduced radial integral FK and GK which are related

by the expression 20

I:, 1.' - , i;, , - -odC,, =

I), r),

with K= 2.4.6 and D2 = 225, D4 ' 1089, D6 = 7361.64

when n.1, = %I, = nl, the antisymmetric eigen function can be written as

(nl,nl;SI, 4 = ( ( ~ I ) ~ ; S I , I The electrostatic interaction between the configuration n12 and (&I,, ndb) will be

given by

n . j n , , n , d ; . - ~ . ~ , ( I . L , ~ . I ~ , ~ ~ ( ~ / ~ : ~ * , . ~ J ~ ) A (2 .24)

If all the electrons are equivalent the matrix element is of the form

(n/J2; hsi,l#n )2;Lyi , :~ = x h ( / , f ) ~ ' ( n , ;n[ K

This is the general formula for the evaluation of electros

element within and between all possible two electron configuration.

In many of the systems that we shall encounter there will be more than two equivalent

electrons. In atomic systems there are only two types of electron operators that need

to be considered; those operators at a time is called F operators and those that

operates on the cordinate of a pair of electrons at a time is called G operators. In

practice it is not necessary to calculate these G operator mamx elements because it is

always possible to use the properties of the *.cle coefficients together with

appropriate recoupling of the angular momenta to reduce their calculation to those of

two electron system.2'

The matrix elements of electrostatic interaction withinfN configurations are normally

written as a linear combination of Slater radial integrals

where K is even and the f,'s are the cc~fficients of the linear combination and

represent the angular part of interaction. By taking the linear combination of the

operators, it IS possible to construct a new operators which have simple

transformation properties with respect to the symmetry group. Then the matrix

element may be written as

where e ~ , s are the angrlar part of the new operator which are related to the fKqs of

Equat~on 2 26 by the expressions

where as E ~ S are the linear combination of the FK1s

By solving the above equation we obtain the value of the FK integral as

The coefficient +(k:--O) for the /'4-N configuration are the same as those for the

congugate configuration. The coefficients of I? depend only on N, and hence the

contribution E? to the energy matrix has the effect of shifting the centre of gravity of

the entire configuration without conhibuting to the structure of the configurations.

'The effect of the elahostatlc Interaction between palls of electoms is to remove the

degeneracy of the hfferent LS tenns of the configurat~ons Dlagonalisauon of the

electrostat~c energy mamces leads to a set of energy levels, each be~ng characterid

by the quantum numbers L and S. For light elements a pure electrostatic calculation

of the energy levels may suffice for making assigment of quantum numbers to the

observed energy level schemes For heavy elements such as the lanthanides, and even

more for the actinides, this treatement is entirely inadequate. To calculate the energy

levels of a rare earth atom or ion with any reliability we must include magnetic

interactions also. These will include spiwrbit, spin-spin, spin-other orbit and other

similar interactions. Spin-orbit interaction is rather predominant among. To include

that we must add the tern to the perturbation Hamiltonian.

N

H,-, = 15(1; )($,.Ii) (2.31) # = I

where ri is the radial cordinate. SI the spin , I,, the orbital angular momentum of the

iLh electron.

In order to obtaln the energy matrix corresponding to the spin-orbit interaction we

have to first obtain the matrix elements. The basic state of eigen function in the

configuration is given by l 6

I f W U ~ S I J I C ~ ) (2 .32)

where W = (al mZ 0,) and U = (uluZ) represent irreducible representation of the group

R and its sub group G2, SLJ represent the angular momenta in LS coupling scheme

and aM represents another set of quantum number.

Thus the spin orbit interaction matrix element can be written asZZ

,v (~Nw(IL~'I,MIC ~(~xs,.I,)(~Nw'II'~'s'I.'M') ,-I (2.33)

v*I*J+a{."l } x ( I * w ~ i a s l , l l v ~ ~ ~ ~ l f ~ w . r ~ a a a y I 1 ) > z r ~ = 6(MM' )96(./.1' X I )' I! IJ

where 6(MM') and NJJ') represent the Dirac delta function and "( I" represent the 6-5

symbols. Also Enl represents the radial integral pert of the spin orbit matrix element

and IS given by"'

This integral is known as the spin-orbit radial integral. Here % =&) is the

normalised wave function of the trivalent rare earth ions 23

The values of this 4fradial wave function can be obtained by employing the Hartee-

Fock method 24 and it is found to be given by

where CI and ZI are constants .

The matrix element on the right hand side of the Equation 2.33, ie. I I v ' " ' ~ can be

obtained from the table of Neilson and ~osteZ6 for all the f configuration.

Knowing the value of all the functions on the right hand side of the Equation 2.33, all

the matrix element of the spin-orbit interaction can be evaluated.

2.5.2. Taylor series method

The energy level calculation by matrix diagonalisation method described above is

tiuitful and relaible. But it has got some complexity in its various parts of calculation.

In order to avoid this, a rather simple and accurate method for the calculation of

energy levels has usually been adopted. In this method the energy of any level in the

rare earth spectra was taken as a function of various types of intemction such as

electrostatic interaction, spin-orbit interaction etc. This energy is also a function of

zero order energy of the free ion level. Then this function was expanded using Taylor

series method in order to find out various unknown parameters involved in it. The

characteristic fature of rare earth spectra is that they always appear as group of lines.

The separation between different group of lines is of nearly a thousand in order of

wavelength Fach group contains large number of lines separated by a few hundreds

of wave numbers When these ions are embedded in different host environment and

subjected to vanous crystal field the spectral lines also got some variations in their

positions. But the centre of the group will not change with respect to the medium and

this corresponds to the free ion energy level. In almost all cases the centre of the

group will be the same for all matrices where the crystal field interaction is weak.

The energy of a level, while considering the electrostatic and spin-orbit interactions

only, can be written as a function of the Slater radial integral FK and spin-orbit

interaction parameter (&)

4 = f(FK.<,,.)

where K = 2,4,6. If represents the zero order energy of the level j, then the new

energy E, can be expressed as

Using Taylor series expansion we can expand the above equation by taking only the

first order denvative of E.

where PtzFZ. PZ - F4, P1=F6, and P,=bt

Using Equation 2.36 the energy (F?) of different levels can be calculated. This

method was s u m by Wong '' in 1961. This mehtod consists of first evaluating

at{, various partial denvat~ves (-)and the delta values (API) For evaluating the

a/; various partial denvatives we have used numerical techniques2'. From Equation 2.39

we have

The number of levels observed for the rare earth ion IS represented by the parameter

j which may be high or numerous. This results in the formation of 5' number of

linear equations for the corresponding experimentally observed energy levels. The

unknown in these equations are AFz, AF4. s 6 and A&f. Since the number of

equations in this case are very much larger than the number of unknowns, the method 28 . of least square analysis IS employed to find the solution of this set of linear

equations.

The method of least square analysis consists of first finding a least quantity (6) such

that the difference between

aE. AE, and z---Lw is an extremum.

aF:

On taking the square of all such equations, and summing over 7, the above equation

becomes.

The method of least square analysis says that, the sum of these squares should be

minimum with respect to the constant Mi. By taking pmhal derivatives of Equation

2.42 with respect to the constant Pi, the above set of equations can be reduced to the

following four equations with four unknowns

where k - 1,2,3,4

Strictly speaking AP, represents the variations in the P, parameters of the free ion due

to the interaction of the surrounding matrices. Hence the values of the parameters P,

for any matrix can be obtained by adding these m e c t i o n factors to the free ion

parameters. Thus the original parameters Pi for any matrix assumes the form

Fz = + &, F4 = ~ ' 4 + M4, F6 = P6 + m6, b = ro4r + ALr

where Pz, p 4 , FI)6 and represent the constant free ion parameters

Both the above described methods of energy evaluation have their own advantages

and limitations. The 4r energy matrix diagonalisation procedure is found to be

much more efficient than the Taylor series expansion method. The accuracy of this

method results from the considerations of all the interactions that many electron

systems have. But when we want to calculate the energy values of the same rare earth

ion in different matrices, the method is found to be disadvantageous. The various

parameters of energy calculation viz. Slater integrals, spin-orbit interaction

parameters, etc. are found to be matrix dependent. Since this matrix dependence of

the various interaction parameters are not incorporated in the energy matrix

formulation, we have to reconstruct the energy matrix every time. This is an

extremely t d o u s pmcess and hence errors will always accompany the energy level

calculations. In the Taylor series expansion method, even though the matrix

dependence of the F,, F,. F, and Lf parameten is taken into account, the dependence

of other interaction parameters is not accounted. If the dependence of all these

parameters is incorporated into the calculation, then the evaluation of the partial

derivations will be extremely difficult

2.6 Optical and physical properties of laser glasses

The design and development of new host acttve medium is required to satisfy various

optical and phys~cal condrttons in order to obtain efficient opt~cal amplification. In

the last two sections we have descrikd in detail about the calculation of energy levels

of dopants like rare earth ion in various medium. Using these theories we can study

the nature of laser active ions by evaluating its energy and assignments with respect to

their positions. But all these lelvels are not capable of producing emission except a

few, thereby causing the optical amplification in the medium. Our present interest is,

by taking all these observed energy levels, to calculate the various optical properties

in glass system. In the present study we have undertaken these evaluation of optical

properties in borate and phosphate glasses.

Before going in deta11 of our study, we have listed some important and fundamental

optical constants of rare earth ions in general glass systems. They are AA

fluorescence band width FWHM at lasing wavelength, & effective fluorescence

bandwidth. peak fluorescence wavelength for lasing transition, up, stimulated

emission cross-section at kp, G, fluorescence gain of optical transition of interest, r~

fluorescence life time for dopant in sec-I, R L Q v Q,, Judd-Ofelt parameters of dopant

in a given composition [cm-'1, branching ratio etc. These parameters are of great

importance in the design and development of new laser active medium. Judd and

Ofelt put forward a theory to fit for the evaluation of these parameters and it is found

that this theory is very successful for rare earth ion in glassy matrices. By using the

1.0. theory, the data obtained from glass samples of various compositions can be used

to predict accurately the performace of a system.

2.6.1 Judd-OfeR theory

The rare earths are usually doped ~ n t o solid state host in the trivalent state. The

absorption spectra of these ions in the optical region arise from the transition within

4f configurat~on mainly Origin of these levels and its location are determined by

coulomb, spin-orbit and crystal field interactions. Fach of these interaction has

typical tenns and energy separation valurs. in rare earth, the crystal tield interaction

with 4jelectron 1s weak, owing to the shielding effect of outer 5s and 5p shell

electmns. Hence crystal field potenttal can be expanded in a spherical harmonic

series.

V, , = ~ ~ ' , ( ( ' ~ , ) i (2.45) L.Y.,

where the coefficients B ~ , are parameters that describe the strength of the crystal

field components, the cKq are tensor operator components and the summation is over

the P electron of the ion. The value of K is restricted to Ki 6 for f shell. Opcal

transition in rare earth ions has been found to be predominantly electric dipole in

origin. For transition between f electrons, no change in parity is evolved and by

Laporte's rule electrical dipole transitions are forbidden. The electric dipole

transitions are found experimentally and are indeed predominant as a consequence of

the fact that the odd harmonics of the local crystal field can admix states of opposite

parity in to 4r. This occurs if the rare earth ion is located at a site that has no

inversion symmetry ".

By using perturbation theory the consideration of the odd parity terms in the crystal 30 . field potential grves rise to first order am level eigen state for they configuration as

where (0") = I f ' [ ~ ~ I ~ ] ) . L I ~

The summation above is over all opposite parity states including 4~''n11', whre n'l' =

5d or 5g as well as 3 d Y 4 r ' . Calculation of the intensity or line strength for a

transition involves considemtion of the electric-dipole matrix element of the operator

P. They are given by

Ih where p component 1s

In &tion to the values B , ~ in VCF, which have not been calculated satisfactorily, the

energy levels Ep of the excited state configuration are difficult to obtain. It has been

shown independently by ~ u d d ~ ' and Ofelt " that it is possible to make the calculation

of electric dipole probabilities tractable by assuming certain approximations in the

Equation 2.45. The first is replacement of (:K,I@,)(@,,I~"p by a tensor operator

components U(t) where t is even. Second, the 4f and excited opposite parity

configurations are treated as degenerate with a single average energy separation.

Then it is possible to invoke clossure over the summation in the above equation and

we obtain

where Y(t,q,p) is a phenomenological parameter which includes crystal field

parameters, radial integrals and average energy separahon

For intermediate coupling the line strength S of an electric dipole transition is given

by

S = (YT, I ~ Y , ) ' (2.50)

ie. - -.

where ((liul()an: the doubly reduced unit tensor operators calculated in the

intermed~ate coupling approximation. Parameters Rk called Judd-Ofelt intensity

parameters contain the energy denomenato~ the odd symmetry crystal field terms,

and mdal integ~al. The value of these parameters is determined by a wmputerised

least square fining routlne by one of the Equations 2.50 and 2.51 by eleminating S.

With the determination of these parameters it is a simple task to determine the

radiative probability, branching ralios. tluorescence life time and other spectroscopic

parmeters.

2.6.2 Absorption and emission characteristics of active media

A host activated medrum doped with an impurity ion, which has its own characteristic

descrete energy level srtucture. This scheme of discrete (Stark) energy level which

caused physically various quantum processes occuring in the activated medium with

the external radiation field. Basic to this connections are the energy transitions

between different activated or doped ion levels, which determine whether the

electromagnetic energy is absorbed or emitted by the host medium. In his famous

paper in terms of thermodynamics, Einstein postulated the existance of the following

elementary processes in a quantum system with discrete state; such as spontaneous

emission induced or stimulated absorption and emission. The latter two processes are

only possible in the case of incident electromagnetic radiation since all transitions

between energy levels are random. Einstein probability functions characterise the

transitions probability per unit time between descrete state in a quantum system. For

low level systems these functions can be expressed in the following manner '* 42 = B21 (2.52)

or generally

where i b d j indicate the initial and final states with their corresponding degeneracies

are g, and g,, 'c' is the velocity of light in vacuum and n is the refractive index of the

m d u m . The factors before All represent the number of radiative oscillators

(oscillation modes) per unlt volume emitting over a single Frequency range. The

parameters A and H &fine Einstein spontaneous and induced transition probabilities.

In order to understand more clearly the main laser propemes. it is better to

characterise the ~ndwed radiative transitions by a parameter called cross-section, o,

rather than by probability. In order to pass from probability to cross-section, we

introduce the notion of radiation intensity J(v) which determines the total number of

photons passing through a square centimeters of the surface of a medium per unit

time.

B,,(I(vo) = ueJ tvu 1

where U(v,,) is the density of emisslon energy at v,,. The above equation can be

modified as

If an emission spectrum consists of a single narrow line with the form factor g(v), the

parameter a. can have various forms

Hence

For a lorentz luminescenece line of width Avh the Equation 2.57 takes the form "

C2 -- = A, do2 ue = A,, 2 2

47r2v n Av,,,,, 4nZ2n26v,,

This equat~on IS one of the various forms of the well known Fuchtbauer-Landernburg

formula. Uslng the above formula we can calculate the emission cross-section for a

particular transltlon

2.6.3 Oscillator strength and life tlrne

In continuation of the spectroscopic properties of rare earth ions in glassy matrices we

have to deal wlth some other ~mportant parameters which are of great interest in

defining the potentiality of the dopant ion and also the host medium. When a parallel

monochromat~c laser beam of frequency vz1 and density Jo incident on a two level

isotropic med~um of length I. the following expression can be deduced

where k, =DN represents the absorption co-efficient per unit mass per unit frequency

interval and D is the optical density or absorbance of the medium. By differentiating

the above equation with respect to I and considering the population of the two level

we can modify the Equation 2.60 as

Using the Equation 2.54 we can write

As the case N,>>N, is more frequent in practice, induced emission can be ignored.

Then, integrating the above equation yields

This is the integrated absorption co-efficient and is one of the basic equations of

absorption spectroscopy. Comparison of the Equations 2.57 and 2.63, shows that the

absorption coefficient is propotional to the lower level population and to the

transition cross-section

Knowing the number of pwticles at the lower level with a large energy gap AE

between the levels, N, is equal to C , the total concentration of the activator ions. We

can readily lind the transition cross-section by measuring the absorption coefficient .

'The above consideration leads to an Improtant corollary, ie. two independent methods

can be employed to determine the transition cross-section ie,.from absorption. By

wing Equat~ons 2.59 and 2.63

Experimentally the former method is usually employed when the terminal level of the

transition is concerned, because it is practically unpopulated. The co-efficient A2,

and B2, can be ~ t t e n in terms of elecbic and magnetic dipole moment 3 3 The

analogs of these quantities in classical electrodynamics are the amplitudes of time

variations in the electic and magnetic moments of the system.

Impurity ion currently used in existing laser media primarly undergoes electric dipole

transitions. For such transitions spontaneous emission probability is given by 33

where P: is the squared amplitide of a matrix element of the dipole moment P. The

theories of quantum and classical oscillators are in good agreement. In place of

transition probability a specific characteristic function can be introduced, the

oscillator strength. The term has the advantage that, it can be used to relate the

classical and quantum mechanical Weatmenis.

The relationsip between oscillator sbength and transition probability can be written as

where m and c are the electron mass and charge respectively. By substuting this

equation in 2.66 we get

In terms of integrated absorption coefficient we can modify the above equation as

From the above two equations we obtain

The above equation is known as Kravertz formula for oscillator strength. Here NI

represents the number of active ions present in the medium and is propotional to the

concentration of the ions in the medium. In practical absorption spectroscopy

concentration IS expressed in moles per l i t b t and absorption (A) in terms of molar

exctinction coefficient E(v). These two terms are related via.

where c is the concentration of ions in the medium and I is the optical path length.

By measuring the concentration in moles per litre the Equation 2-70 becomes

mL' te . j ; , (v) = --;-j&(v)a,

ni?

The probability of spontaneous transition is associated with one of the most important

spectroscopic parameters, the life time of the excited state, which characterises the

rate of luminescence decay after excitaton has been discontinued. It is significant in

laser physics because it is used to predict whether a given luminescence medium is

suitable for various laser type or not.

Consider the case where we have only one possible channel of deexcitation from the

excited level, the life time of the excited level can be written as

But in general case we have a number of such luminescencee channels originated

from the excited slate In such case we can write it as

Besides purely radiative channel of deactivation, non radiative transition " which

tends to reduce T , ~ can also occur. I n d d taking account of non radiative bansition

characterised by the probabilities d,,, the life time can be written as

the above equation can be modified in the following fashion

Due to the presence of these nonradiative term in the system the values of is

reduced by a factor of 1

-- The intensity of lum~nescence is also

I + C ( d,,

1 C(A,,) reduced due to the non radiative process. This process is known as luminescence

quenching, and is easily characterised by another spectroscopic parameter, quantum

yield of luminescence. This is the ratio of the number of emitted photons to the

number of systems that reach the state j and is expressed by

if the non radiative term is absent in a case, the quantum yield becomes unity

2.7 Idtensity of spectral lines of trivalent rare earth ions

In trivalent lanthan~des electronic tranist~ons are attributed to the three differznt types

namely electric dipole, ma~metic dipole and electic quadrapole in character. In some

cases transition may have contribution from more than one mode of the above. In the

electric dipole transitions between levels same electronic configuration may be

attributed to non centrally symmetric interaction of the activator ion with the

surrounding crystal field In the earher work performed by Broer et.a13' revealed that

the probabilities of electric and magnetic dipole transitions greatly exceed that of

electric quadruple transitions. This was confirmed by a number of researchers

experimentally and theoreti~all~"~'~ Investigations show that in most cases the

emission properties of laser materials doped with trivalent rare earth ions are due to

electic dipole transitions. In certain cases, considerable contributions can also be

made by magnetic dipole transition.

In order to understand more about this, one has to look for the dependence of these

transition on the surrounding matrix and its properties. These transtions are functions

of dipole moments. The sum of the squared matrix elements of the electric dipole

moment operator (p) or magnetic dipole moment operator is termed as the line

strength (s) of the correspond~ng transitions.

Thus the electric dipole and magnetic dipole line strengths are given by the equations

where f3 is the Bohr magneton. In terms of the line strength, Einstein coefficients can

647T4vz3 be written as A,, = l l g,

3hc3 S,,

2.7.1 Electric dipole transitons

Electric dipole transitions between the states of the 4f electron configuration of an

isolated trivalent rare earth ion are prohibited by the parity selection rule ". This

prohibition can be more or less avoided due to the noncentrally symmetric

interaction^'^ of the ions with surroundings, which mix states of opposite parity. One

way in which this occurs is if the rare earth ion is located at a site that has no

inversion symmetry According to Riseberg and Weber " the effect can occur for the

point-pup symmetries: C,, C,, C2, Czv. C3, Cxv. CdVr Cf,, Chv, DM. D3. D3b D4, Ds,

D,, S4, T, T2, 0 The static and dynamic portion of the crystal field potential (V,,)

are examples of the noncentrally symmetric interactions. The static terms are the odd

terms of the crystal field potential and are denoted by v,~. On the other hand the

dynamic terms are the result of the noncentrally symmetric vibrations of the

surrounding field. The static terms induce pure electronic transitions while the

dynamic terms induce electron vibrational transitions. Thus the static portion is

responsible for the main contribution to the probability for radiative transitions of

trivalent rare earth ions in any medium.

If (A1 and (A') represent the two eigen states of a trivalent ion, then these states can be

represented as linear corn bin at ion^'^ of the wavefunctions of the ground state

configuration 14y ~JJJ,) with the wave functions of the excited p) configurations of

the opposite parity. This can be written as

( A ) = (4fNyrl.l , 1 - 1 1 4 1 ~ ( + = 1 6 7 ~ X B I) 0 E(~ . / '"@J, ) - E ( B )

(A') = (4 f Vvq J' J' , I - C IB )(@rdbf " V ' J ' J ~ )

B E(4fN(v ' J 'J ' , ) - E(/3)

Here 10) represents a state of opposlte parity at an energy E(P) and ~(4y ~JJJ,)

represents the ground state energy

For an electricd~pole transition between two levels A and A' with energies ~ 4 p

tqJJ,) and ~ ( 4 p tqlJ'J',) the intensity is proportional to (AIPIA') where P is the

electric-dipole moment operator. Thus, from Equation 2.79 we get

(2 80)

It 1s d~fficult to calculate the nght-hand s ~ d e of the above equatlo ,cause, for the

purpose, it IS necessary to know not only the E(P) energles and v. &ons of the

excited IP> configuration levels, but also the odd term of the crystal field potential

which is responsible for the opposite purity mixing.

A satisfactory way of simplifying the above equation was first suggested by Judd 'O

and Ofelt " in 1%2. According to their theory, it is better to replace the energy

difference terms in the denominator of Equation 2.80 by the constant AE which is

independent of W, J, and P. This IS equivalent to the assumption that elechpn

configuration splitting is negligible compared with the energy gap between the levels.

The line strength (S) of the electric dipole transition is given by

since (~1.1 A, = ~ Q , ~ ( ~ L I ~ ~ ~ I ( " ~ ~ yv./.)(1 1=2.4.6

where Q, are a set of parameters known as m d f i e d Judd-Ofelt parameters.

Thus the electric dipole linestrength (Sod) is given by

To facilitate calculations by the above equation another designation of the line

strength (Sod) is adopted and is related to the conventional one by the relation

Substituting Equation 2.82 in 2.78~1 y~elds

n(n2 + 2)' where represents the local field correction factor for the transition

9

probability in the simple tsotropic tight bindng approximation " Also from Equations 2.69 and 270

Usually the above equations are written in terms of another set of the so called Judd-

Ofelt (J-0) parameters denoted by T, and the relationship between the modified J - 0

parameters (Q) and J-0 parameters (T,) is given by

T - 8 d m n2 + 2)2

" 3h(2J+l) 9n Q A

The Equations 2.85 and 2.86 becomes

Thus, to describe the line intensities of transitions 0bSe~ed in the absorption and

luminescence of materials containing trivalent rare earth ions in the J-0

approximation, it is sufficient to have three parameters TA or RA. As a rule, these

parameters are determined fmm the analysis of the optical absorption spectra. Fmm

Eguation 2.82 it can be seen that line strength S is related to the QA parameters by a

linear relationship. Since the electric dipole moment operator p is a vector, it can

have the x, y, and z components and hence S can be represented by a vector equation

as

%- Have S is the vector whose components are the calculated line strengths; R is the

vector whose components are the intensity parameters Q ; and 'H is the matrix

elements of 11"'

If the absorption band is a superposition of several intermanifold transitions, the

matrix element can be taken to be the sum of the corresponding squared matrix

elements because of the additivity of the integrated absorption coefficient of this

band.

2.7.2 Magnetic dipole transitions

For a magnetic dipole transition the linestrength (&) is given by

where the magnetic dipole matrix element I(JIM]J1)( is given by 20

where g, = 2 gives the gyromagnetic ratio of electron spin, and P = 0 1 gives the 2-

components (cr polarised) and x, y components (rr-polarised) of L and S. The values

of the reduced matrix elements JJL+2S(I is given by the following equations.

(a) when J - J' or AJ = 0

(a SUIF+2SllaSU) = gh [J(J+I)(~J+I)]'

(b) when J' = J-I or AJ = -1

(a SLJ111,+2S[laSLJ-I) = h ([(S+L+J+ I )(S+L+ I-JXJ+S-L) x (J+L-s)]/~J}")

(c) whenJ '=J+I o r h l = + l

( ~ S W J J L + ~ ~ S J J ~ S U + ~ ) = ~ ( [ ( S + L + J + ~ X S + L + ~ - L X L + J + I - S ) . ( S + L - J ) ] / ~ ( J + ~ ) ) ~

(2.93)

Thus the magnetic dipole linestrength (Smd) is given by

Substitution of Equation 2.94 in 2.66 will give the magnetic dipole transition

probability A,,.'""' as

Thus the total transition probability (AT) and oscillator strength (fT) due to the

electric4pole and magnetic dipole contribution are given by

AdJJ') = Ajy(ed) + AJy(md) (2.97)

The total oscillator strength ( f ~ ) can be directly measured from the absorption

spectrum which in turn can be utilised in evaluating the Judd-Ofelt intensity

parameters (Q).

The parameter that charaderises the possibility of exciting stimulated emission in a

given channel is the intermanifold luminescence branching ratio

The other two important parameters viz. the stimulated emission cross-section (ue)

and the quantum yield (q) can be calculated using the Equations 2.59 and 2.77.

2.8 Selection rules for optical transitions

The optical trans~tions of rare earth ions in various host medium are due to the intra

-y transition which are predominantly of electric-dipole in nature character. For free

ion electricdipole transtions between states of the same c0ntiguratiq.n are strictly

parity forbidden and any observed spectra of crystals or solutions must concern itself

with non centrosymmetric interactions that lead to a mixing of states of oppoiste

panty. The probability of a radiative transiton between two states depends on the

value of the matrix elemnts (blqa) where D is either D. or 4A and a and b are two

states. Because D, is an odd parity operator the matrix element of D, will be zero,

when states a and b have same parity, hence electric dipole transitions are

forbidden between these states. This is Laporte selection rule. In the optical region

the maximum value of the matrix element of D. is over two order of magnitude larger

than the maximum value of the matrix element D,. Elecmc dipole processes, then,

ought to be the dominant radiative mechanism in the optical region.

In all the transitions of rare earth ions, we have observed in different host medium

there are some selection rules corresponding to every transitions. In the case of free

ion various selection rules are summarise. below

For elecctric dipole transition

Angular momentum L:AL=O,fl

Total angular momentum J: AJ = 0, f I

Spin quantum number S: AS=O

And for magnetic dipole transiotion

AL=O,+1,+2,AJ=O,and AS=O

But under the influence of any external field the above selection rules become

Forced electric dipole : (a)Al=* l,(b)AL, AJ521

(c) AS 4 for O f 9 0 transition

Forced maghetic dipole: (a)AL=O, (b)AJ=O,f l.(c)AS=O

and for quadruple A J = O , _ < ? 2

Violation of the above selection rules are due to:

1 . Selection rule AS 0 is completely valid when L+S interaction is weak. With the

coupling of l . tS become strong, it may be violated, the higher the atomic

number, the larger the multistate splitting and higher the possibility of violation

of this selection rule.

2. There may be quadrapole radiatron AL 4, + I , +2 and magnetic dipole radiation

when AL. -0, + I , fl when the electric &pole radiation is forbidden the intensity

is very low

3. The selection rules can be changed at high electirc field or magnetic field.

Further breakdown of the selection rules due to the ligand field action may also occur.

If the ligand diitribution is not completely symmebised the orbital may be partially

mixed, e.g. d and p, f and d orbitals mixed. So some of the transitions become non

forbidden. Another cause is that due to the vibration of crystal lattices the central

ions shift from their equilibrium position losing their temporary symmetry, then

producing mixed orbitals. When the littice vibration causes electron-vibration

spectra, not only new spetral lines produced but also the intensity of the spectral lines

is increased.

2.9 Hypersensitive transition

It was observed that in a few cases the Judd-Ofelt approach did not yield any

satisfactory results. In certain cases some transitions obeying the quadrupole

selection rules (AJ ~=0, k2) are usually considerably more intense than expected and

sensitive to the surroundings. These transitions are generally called hypersensitve

transitions. These hypersensitive transitions are characterised by high value of their

oscillator strengths. The large changes for oscillator strength can be expected only for

transition with AJS 2. These levels are normally excluded for the calculation of Judd-

Ofelt parameters. This limitation of J - 0 theory in accounting for the hypersensitive

transition was studied and remedied by Meson and Peacock

We have used the thwretical formulations described above for the optical

characterisatlon of rare earth doped glassy systems The work presented in the

following chapters describes the analys~s of the optical spectra and the spectroscopic

properties of neodymtum ions in borate glasses and samarium ions in phosphate

glasses. Various spectroscopic pmaeters have been evaluated from the absorption

and fluorescence data. The analysis has also led to the identification of certain

potential transition for optical amplification and the evaluation of the corresponidng

gain parameters.

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