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CHAPTER 1
GENERAL INTRODUCTION AND REVIEW OF LITERATURE
ABSTRACT
General introduction and review of literature deals with the characteristics of
lanthanide ions, free-ion and crystal-field levels, glassy materials, glass transition
temperatures and structure of glass and crystalline materials. The advantages of
amorphous materials over crystalline materials have been discussed. To analyze the
absorption and emission spectra theoretical models have been outlined. Energy level
scheme through free-ion Hamiltonian model, intensities of spectral lines and radiative
properties of excited states using Judd-Ofelt analysis have been presented. Using
Inokuti-Hirayama and Yokota-Tanimoto models the multipolar interactions, energy
transfer processes have been discussed.
1.1 Objectives and scope of research
Spectroscopy is a branch of physics that deals with the study of interaction of
electromagnetic radiation with matter. The tremendous advancement of Science and
Technology is giving raise to the continuous appearance of new spectroscopic
techniques. Spectroscopy finds applications in the fields of physics, chemistry,
medicine, forensic, industry, agriculture, defense, telecommunications, biology,
geology, astronomy etc [1-7].
Different spectroscopic techniques are rooted in a basic phenomenon: “the
absorption, reflection, emission or scattering of radiation by matter in a selective
range of frequencies under certain conditions”.
Optical spectroscopy is an excellent tool to study the electronic structure of
absorbing or emitting centers like molecules, atoms, ions, defects etc., their lattice
locations and their environments. In other words optical spectroscopy allows us to
“look inside” of matter by analyzing the interacted electromagnetic radiation. For the
last several decades, tremendous breakthrough has been achieved in the development
of trivalent lanthanide (Ln3+
) doped materials for potential applications in photonics
[8-12]. The chemical composition of glass network former and modifier results a wide
range of variation in the optical properties of Ln3+
ions in glass environment. The
study of the optical properties of the Ln3+
ions in glass materials provides fundamental
data that includes transition positions and cross-sections, transition probabilities,
radiative life times, branching ratios, line widths etc., for the excited states. This data
is essential to estimate/design optical devices such as lasers, up converters, fiber
amplifiers, color displays, light emitting diodes (LEDs) and so on. In this direction, in
order to identify new optical devices, devices for specific utility or devices with
enhanced performance, active work is being carried out by selecting appropriate new
hosts doped with Ln3+
ions [13].
Among the most popular solid state media the rare earth (RE) ions doped laser
crystals, glasses and ceramic media have attracted the researchers with increasing
interest. Comparing with crystalline media, glasses are favorable hosts due to their
broad inhomogeneous band width, possibility of tuning the wave length, large doping
capability and easy to mould and shape in larger sizes as well [14]. Multi component
glasses which typically consist of network formers and modifiers provide wide range
of excellent optical properties for new applications by selecting and tailoring the
chemical composition. Borate glasses doped with various RE ions offer many
commercial and technological applications. Borate glasses are commonly used in a
wide range of photonic applications such as optical amplifiers, lasers, photosensitive
materials and Faraday rotators [15,16]. In lead containing glasses the non-linearities
are primarily from the Pb2+
ions, as they are highly polarizable due to the presence of
6s2
electrons. These glasses possess a large non-linear optical properties which make
them suitable for potential applications in non-linear optical devices such as power
limiters, ultra fast switches and broad band optical amplifiers [15,17]. The important
properties required for a laser medium are high gain, high energy storage capability
and low optical losses, which depend on stimulated emission cross- section,
fluorescence life-time and optical efficiency [15, 17-20].
Heavy metal oxide (HMO) glasses find their importance as host matrices for good
lasing candidates because of their low phonon energy, high mechanical and thermal
stability, corrosion resistance and good solubility of rare earth ions [15,21-23]. The
incorporation of HMOs such as PbO or Bi2O3 into the borate glass matrix leads to an
increase in its luminescence quantum efficiency [21]. The effect of lead-borate,
bismuth-borate and lead-bismuth-borate glasses on the optical properties of Nd3+
ion
have been reported [21,23-25]. Motivated by these works, in the present work, lead-
bismuth-aluminum-borate glasses, doped with Nd3+
and Pr3+
ions have been prepared
and the physical and optical properties were studied, especially the suitability of the
samples prepared as the core material for optical fiber and active medium for solid
lasers, with concentration variation of RE ions and host variation.
1.2 HOST MATERIAL
GLASSES: Glasses have a long and interesting history. The primitive cave
dwellers used the chipped pieces of obsidian, a natural volcanic glass, for tools and
weapons like scrapers, knives, axes and heads for spears and arrows [26]. The
techniques of production of colored glasses were also known to them and passed on to
the next generations as family secrets. With the advent of modern technology now a
days fine glasses are produced. Most of the glasses are transparent to the visible
light. This makes the glasses for use in sculptures in art museums and in chandeliers
aesthetically pleasing. Glass is one of the useful and versatile materials, which has
also been the focus point for intensive modern research particularly in the fields of
telecommunications and optical fibers.
Inorganic glasses have been used as optical materials for a long time due to their
isotropy and high transparency over a wide spectral range from ultraviolet (UV) to
infrared (IR). With the advent of modern technology a series of new glass forming
systems have been developed to meet the demands of the expanding field of optics
and optoelectronics. In oxide glass systems, besides traditional silicate, borate,
phosphate, tellurite, germanate glasses, new glass forming systems have also been
expanded to non-oxide glasses such as halide and chalcogenide glasses.
1.2.1 Definition of glass
Glasses are essentially non-crystalline solids obtained by freezing super cooled
liquids which exhibit short range order. According to the ASTM (American Society
of Testing Materials) standards, „glass is an inorganic product of fusion which has
been cooled to a rigid condition without crystallization‟ [27]. This definition would
exclude splat quenched glasses, glass made under high pressure, sol-gel and sputtered
glasses. In AD 1968 glass was redefined as “an amorphous solid which exhibits a
glass transition”. Glasses have two common characteristics namely absence of long
range periodic atomic arrangement and presence of time dependent behavior known
as glass transformation behavior over temperature range known as glass
transformation region[27,28]. Accordingly a glass can formally be defined as an
amorphous solid completely lacking long range periodic structure and exhibiting a
region transformation behavior [29].
1.2.2 Classification of glasses
1.2.2.1 Natural glasses: When molten lava reaches the surface of the earth‟s crust
and is cooled rapidly, natural glasses such as obsidians, pechsteins, pumice etc., can
be formed. Natural glasses can also be formed by the sudden increase in temperature
following strong shock waves e.g., tectites [30]. Glass formation is also rarely be
possible by biological process. The skeleton of some deep water sponges consists of
a large rod of vitreous SiO2 [31].
1.2.2.2 Artificial glasses: The formation of the artificial glasses takes place in various
diverse classes of materials but only some of them are of practical value. Artificial
glasses are classified as follows.
(i) Oxide glasses: Oxide glasses are the most important among the inorganic
glasses eg silicate (SiO2), phosphate (P2O5), borate (B2O3) and germinate
(GeO2) glasses.
Oxide glasses find photonic applications such as lasing material and core
material for optical fibers.
(ii) Halide glasses: BeF2 is a glass network former whose structure is based on
BeF4 tetrahedra. Fluorozirconate , fluoroborate and fluoro phosphate glasses
are the best candidates for high power lasers for thermonuclear fusion
applications.
(iii) Chalcogenide glasses: Chalcogenide glasses are formed when group VI
(S, Se and Te) elements are combined with group IV (Si and Ge) and group V
(P, As, Sb and Bi) elements. These glasses do not contain oxygen and so are
interesting due to their infrared optical transmission and electrical switching
properties. Vitreous Se possesses photoconductive properties and is used in
photocopiers (Xerography). Ge-As-Si glasses have opto-acoustic applications
and are used as modulators and deflectors for IR rays.
(iv) Metallic glasses: These are of two types viz, metal-metalloid alloys and
metal-metal alloys. These glasses have the properties of extremely low
magnetic losses, zero magnetostriction, high mechanical strength and
hardness, resistance to radiation and chemical corrosion. These materials are
used as cores in moving magnets, recording cartridges, amorphous heads for
audio and computer tape recording and high frequency power transformers.
1.2.3 Glass preparation methods
Glassy materials can be prepared by various techniques like melt quenching, gel
desiccation, thermal evaporation, chemical reaction, chemical vapor deposition,
electrolytic deposition, reaction amorphization, shear amorphization, sputtering,
glow-discharge decomposition, irradiation and shock-wave transformation. Among
these methods, melt quenching and gel desiccation techniques are widely used in the
preparation of glasses.
1.2.4 Properties of the glasses
The physical properties of a glass matrix may depend upon the previous history of
the specimen. The surface pre-treatment of the specimen has decisive importance.
The thermal expansion and viscosity of glass also depend to some extent on the
history of the specimen. The importance of these factors has been emphasized by
Dale and Starwort [32]. In Ad 1945 Douglas [33] has given a valuable review of the
physical properties of glass.
The basic properties of glasses are:
(i) Glass is transparent but non-crystalline, a major paradox in the physics
of condensed matter.
(ii) Glass has very high resistance to water and atmospheric agencies.
(iii) Glass is electrically insulating at normal temperatures but becomes
conducting at very high temperatures.
(iv) Glass usually breaks in a direction at right angles to the direction of
maximum tensile stress.
(v) Glass is hard and yet brittle. When it cracks it shatters at the speed of
sound. It breaks suddenly when subjected to a stress exceeding its
elastic limit. Glass obeys Hooke‟s law accurately within the elastic
limit.
(vi) The coefficient of linear thermal expansion is almost constant, for
most types of glasses up to the temperature ranging 400-600 0C,
depending on the chemical constitution of the glass.
1.2.5 Glass network formers and modifies
Glasses can be prepared using different types of materials. The ability of a
substance to form a glass matrix does not depend upon any particular physical or
chemical property. It is now generally agreed that almost any substance, if cooled
sufficiently fast, could be obtained in the glassy state although in practice
crystallization intervenes in many substances.
B2O3, P2O5, SiO2, GeO2 all of which come from a certain area of the periodic table
readily form glasses on their own when their melts are cooled sufficiently fast and are
commonly known as „glass formers‟. These elements are sufficiently electro positive
to form ionic structures such as MgO and NaCl, and are not sufficiently
electronegative to form covalently bonded small molecular structures such as CO2.
Instead, bonding is usually a mixture of ionic and covalent and the structures are best
regarded as three-dimensional polymeric structures. AS2O3 and Sb2O3 produce glass
on their own when cooled very rapidly. TeO2, SeO2, MeO3, WO3, Bi2O3, Al2O3,
Ga2O3 and V2O5 will not form glasses on their own, but each will do so when melted
with a suitable quantity of certain other non-glass forming oxide. Hence they are
known as „conditional glass formers‟.
Some oxides like PbO, CaO, K2O, Na2O and Li2O produce drastic changes in the
properties (melting point, conductivity etc) of the glass network forming oxides when
added in small quantities. These oxides also modify the network structure of the glass
and hence they are termed as „network modifiers‟. Anions like halogens and oxygen
become non-bridging when they bridge two network former cations. Network
modifier cations, such as alkali, alkaline-earth and higher valance state ions are
accommodated randomly in the network in close proximity to non-bridging anions.
Glass formers, modifiers and intermediates are listed in Table 1.1. Average phonon
frequencies (ћω) of some of the network formers in glasses are given in Table1.2.
Table1.1: Glass formers, modifiers and intermediates
Glass former Modifier Intermediate
SiO2 Li2O Al2O3
GeO2 Na2O PbO
B2O3 K2O ZnO
P2O5 CaO CdO
TeO2 BaO TiO2
As2O3
As2O5
1.2.6 Differences between crystalline and amorphous solids
Basing on the atomic arrangement, solids may be broadly classified into two
categories i.e. (1) crystalline and (2) amorphous solids.
In crystalline solids both short-range and long-range order exists in the atomic
Table: 1.2 Average phonon frequencies (ћω) of some of the
network formers in glasses.
_______________________________________
Matrix ћω (cm-1
)
_______________________________________
Borate 1400
Phosphate 1200
Silicate 1100
Germanate 900
Tellurite 700
LaF3 (Crystal) 350
______________________________________
arrangements where as in amorphous solids only short-range order exists. Figure 1.1
shows the schematic representation of (a) ordered crystalline form and (b) random
network amorphous form, of the same composition.
The degree of disorder will be greater in an amorphous solid than its crystalline
counterpart. Entropy of amorphous solid is greater than that of crystalline solid.
Hence amorphous state is a non-equilibrium state. On cooling, from liquid phase to
the solid phase, a crystalline solid is obtained as a transformation from one
equilibrium state to another while amorphous solid is obtained as a transformation
from an equilibrium state to a non-equilibrium state. Amorphous materials which
melt over a range of temperature are isotropic where as crystalline materials which
have well defined melting point are anisotropic.
(a) (b)
Figure 1.1 Molecular arrangements in (a) crystal and (b) glass.
The following are the advantages of glass materials over crystalline materials in many
optical device applications.
(i) Flexibility of choosing glass composition over a wide range,
(ii) A disordered ion environment that can broaden fluorescence band width,
(iii) Uniform (isotropic) optical properties over a wide range of composition,
(iv) Ease of fabrication into complex shapes,
(v) Low fabrication cost and
(vi) Useful in producing large active lasers with good optical quality.
1.2.7 Glass transition
The transition from a viscous liquid to a solid glass is called “ glass transition” and
the corresponding temperature is known as the glass transition temperature, Tg. When
the glass is heated to a temperature above Tg, glass to viscous liquid transformation
takes place. Glass formation is due to the increase of viscosity depending on cooling
rate. Figure 1.2 represents the enthalpy–temperature characteristics for crystal, liquid
and glass. When a liquid solidifies into a crystalline state there is abrupt discontinuity
in the enthalpy at a well defined temperature called the „melting point‟, Tm , of the
material. In the case of glass formation the enthalpy of the liquid decreases at about
the same rate as in crystalline formation until there is a decrease in the expansion
coefficient in a range of temperature called „glass transformation range‟.
The liquid–glass cooling curve does not show any discontinuity. Slope of the
curve changes at Tg. Below this temperature, the glass structure does not relax at the
cooling rate used.
Fig.1.2: The effect of temperature on the enthalpy (or volume) of a glass forming
melt.
The expansion coefficient for the glassy state is usually about the same as that for
the crystalline state. Tg is a function of cooling rate and is not well defined. Slower
the rate of cooling, lower value of Tg. However, Tg cannot be lower than a particular
minimum temperature called the „ideal glass transition temperature‟, To. This can be
explained by considering the relative heat capacities and entropies of liquid and
crystalline phases of the same composition. The glass transition temperature can be
determined by differential thermal analysis (DTA) or differential scanning
calorimetry (DSC).
1.2.8 Structural features of glass
There is no single experimental technique which could produce a direct mapping
of the glass network structure due to the absence of long-range order. Hence the
elucidation of glass structure involves many different experimental and theoretical
methods. These methods yield different, often complementary, information which
can be combined to a structural model of a certain glass system.
For studying the glass structure, diffraction and spectroscopic techniques are used.
The short-range order, thereby the structural information of glasses can be obtained
by X-ray diffraction (XRD). In electron and neutron diffractions, the beam
encountering the atoms results in scattering and the structure of the order region will
be obtained. These methods lack resolution and provide the statistical average of the
spatial distribution of atoms in one-dimension. The work of Wong and Angel [28]
includes an extensive bibliography on spectroscopic studies of glasses.
Zachariasen [34] concluded that the atoms in a glass are linked together by the
same forces as in crystals. Basing on this he proposed a structure consisting of an
extended three-dimensional network made up of well defined small „structural units‟
which are linked together in a random way. According to the Zachariasen‟s rule for
the formation of the glass
(i) An oxygen atom should be linked to not more than two glass forming cations,
(ii) The number of oxygen atoms around a glass forming cation must be small,
(iii) The oxygen polyhedral share corners but not edges or faces,
(iv) In three-dimensional network, at least three corners in each polyhedral should
be occupied by anions, with cations at the centers.
The violation of any of these rules means the glass formation will be energetically
less favorable. Planar AO3 triangles are the structural units in both the cases. The
randomness of the glass phase is due to the variations of angles and distances mainly
between the structural units and to a minor extent within the structural units. The
work of Cooper [35] based on topological arguments supported Zachariasen‟s ideas.
Cooper combined Zachariasen‟s rules (iii) and (iv) and gave modified statement that
each oxygen polyhedron must be connected directly to at least three other oxygen
polyhedra. Hence, tetrahedra that share an edge and two opposite corners are no
longer excluded.
Zachariasen gave the name „network forming cations‟ for cations which form the
random network of glasses in association with oxygen. The term „network former‟ is
attributed to the oxides capable of forming glass. Oxygen ions act as bridges between
the structural units and so are called „bridging oxygens (BOs). The oxides which do
not participate in forming the glass network structure and present in the glass are
called „network modifiers‟. The three principal actions of network modifiers in
glasses for an A2O3 glass are as follows.
(i) Breaking of A-O-A bonds and creation of non-bridging oxygen,
(ii) Increasing the oxygen co-ordination of cation and
(iii) A combination of both.
The breaking of A-O-A bonds leads to the creation of non-bridging oxygen, shown
in Fig:1.3 .
Fig.1.3 Creation of non-bridging oxygens.
These oxygen ions carry a partial negative charge and are connected to the glass
network at one end only. The resulting network gets loose and by decreasing the
connectivity a larger flexibility of the structure is obtained.
1.3 RARE EARTHS
1.3.1 SIGNIFICANCE OF LANTHANIDE IONS
The rare earth ions are divided into two series of elements, the lanthanides and the
actinides. The neutral forms of these elements have an electronic configuration made
up by progressively filling the 4f electronic shells for the lanthanides and the 5f
electronic shell for the actinides. The group of elements having atomic numbers (Z)
57 to 71 are usually classified as lanthanides (Ln) and very often referred as rare
earths (REs). They were originally extracted from oxides for which ancient name was
„earth‟ and which were considered to be „rare‟ hence the name „rare earths‟. The name
„lanthanides‟ has been derived from lanthanum which is the prototype of lanthanides.
A feature common to all the elements in the lanthanide series is xenon (Z=54)
based electronic configuration with two (6s2) or three (5d
16s
2) outer electrons.
Neutral lanthanides possess the electronic configuration [Xe] 4f n 5d
m 6s
2 where n is
the number of 4f electrons ranging from 2 for cerium to 14 for ytterbium with m=0,
with an exception of zero for lanthanum, 7 for gadolinium and 14 for lutetium where
m= 1. All the lanthanide elements have common „trivalent state‟ attained by losing
three electrons, one from 5d or 4f and two from 6s orbitals. Therefore the trivalent
lanthanide ions having most stable oxidation state have xenon core electronic
configuration [Xe] 4 f n
where n is an integer varying zero for La3+
to 14 for
Lu
3+.
Table 1.3 shows the electronic configurations of lanthanide ions along with their
ground states and valences.
The filling of 4f shell can be explained by the Hund‟s rule according to which the
term with highest quantum number S has the lowest energy and if there are several
terms with same S, the one with highest angular momentum quantum number L has
the lowest energy. Furthermore, because of spin–orbit coupling, the terms 2s+1
L split
into (2s+1)
LJ levels with J= L+S, L+S-1,…,|L-S|, where for less than half- filled shells,
the term with the smallest J lies lowest in energy. The 4f electrons are effectively
shielded by the 5s and 5p electrons, Fig. 1.4.
Hence they do not play any role in chemical bonding and so the
chemical properties of the lanthanides are much alike [33]. That is why the trivalent
lanthanides are sometimes referred to as „triple positively charged noble gases‟.
Among the 4 f n
configuration, 4 f 0 (empty f-shell), 4 f
7 (half-filled f-shell) and 4 f
14
(completely filled f-shell) are the most stable ones. Therefore, besides the trivalent
state some ions also occur in divalent and tetravalent states. Examples are Ce4+
(4 f 0),
Sm2+
(4 f 6), Eu
2+ (4 f
7), Tb
4+(4 f
7) and Yb
2+ (4 f
14).
Table 1.3: The electronic configurations of lanthanides along with their
ground states and valences.
Lanthanide Atomic Neutral atom Possible Trivalent atom
element number electronic valence Electronic Ground
configuration state configuration state
Lanthanum (La) 57 [Xe] 4f0
5d1 6s
2 3 [Xe] 4f
0
1S0
Cerium (Ce) 58 [Xe] 4f2 6s
2 3, 4 [Xe] 4f
1
2F5/2
Praseodymium (Pr)59 [Xe] 4f36s
2 3 [Xe] 4f
2
3H4
Neodymium (Nd)60 [Xe] 4f4 6s
2 3 [Xe] 4f
3
4I9/2
Promethium (Pm) 61 [Xe] 4f5 6s
2 3 [Xe] 4f
4
5I4
Samarium (Sm) 62 [Xe] 4f6 6s
2 2, 3 [Xe] 4f
5
6H5/2
Europium (Eu) 63 [Xe] 4f7 6s
2 2, 3 [Xe] 4f
6
7F0
Gadolinium (Gd) 64 [Xe] 4f7 5d
1 6s
2 3 [Xe] 4f
7
8S7/2
Terbium (Tb) 65 [Xe] 4f9 6s
2 3, 4 [Xe] 4f
8
7F6
Dysprosium (Dy) 66 [Xe] 4f10
6s2 3 [Xe] 4f
9
6H15/2
Holmium (Ho) 67 [Xe] 4f11
6s2 3 [Xe] 4f
10
5I8
Erbium (Er) 68 [Xe] 4f12
6s2 3 [Xe] 4f
11
4I15/2
Thulium (Tm) 69 [Xe] 4f13
6s2 3 [Xe] 4f
12
3H6
Ytterbium (Yb) 70 [Xe] 4f14
6s2 2, 3 [Xe] 4f
13
2F7/2
Lutetium (Lu) 71 [Xe] 4f14
5d1 6s
2 3 [Xe] 4f
14
1S0
Fig. 1.4. Approximate charge distributions of electrons in different orbitals for
Ln3+
ions demonstrating the shielding of unpaired 4f electrons
by outer filled 5s2 and 5p
6 shell electrons.
Lanthanide group ions differ in the number of electrons in the 4f shell. The
ground state electronic configuration is 4 f n
and the first excited state configuration is
4 f n-1
5d. The relative location and energy extent of these two configurations for the
rare earth ions (RE 3+
) are shown in Fig. 1.4. Due to the shielding of 4f orbitals by the
filled in 5s2
5p6
orbitals, the 4f electrons are only weakly perturbed by the charge of
the surround ligands. The spectra of RE compounds are sharp and similar to the
atomic spectra. Shielding effect also causes the unique optical properties of RE ions
[36]. John Hopkins group, under the direction of Dieke [37], generated the complete
set of energy level assignments for all RE3+
ions in an hydrous tri chlorides.
1.3.2 Free-ion crystal-field levels
An ion isolated from any interactions with its environment is known as free-ion
and its electronic energy levels are determined by the coulomb interaction between
each electron and the nucleus of the ion. The coulomb exchange and the spin-orbit
interactions among all the 4f electrons produce the multiple terms denoted by 2s+1
LJ
known as Russell-Saunders coupling. Since the free-ion is in a physical environment
having completely spherical symmetry, these interactions determine both the radial
extent and the shape of the orbital with both the energy and angular momentum being
quantized. The results of these considerations provide a set of electronic states with
significant amount of degeneracy.
Applying any type of external perturbations to this system that has a specific
spatial symmetry (electric field, magnetic field, uniaxial stress etc) can lift some of
the degeneracy, resulting in a splitting of the free-ion energy levels. The cumulative
effects of the lattice charges in the surrounding of the ion cause a perturbation that
splits the ionic energy levels into sub levels. By splitting, it is meant that the (2J+1)
fold degeneracy is partially removed which is the well known „Stark effect‟. The
perturbation that causes the splitting is an electrostatic field known as the „crystal–
field (CF)‟. Hence the multiplet terms denoted by 2S+1
LJ split into fine structure by
the strength of CF and the symmetry of the site accommodating the Ln3+
ion. The
thirty two crystallographic point groups can be divided into four general symmetry
classes as follows:
(i) Cubic: Oh, O, Td, Th, T
(ii) Hexagonal:D6h, D6, C6v, C6h, C6, D3h, C3h, D3d, D3, C3v, S6, C3
(iii)Tetragonal: D4h, D4, C4v, C4, D2d, S4
(iv)Lower symmetry: D2h, D2. C2v, C2h, C2, Cs, S2, C1
As the interaction between the 4f n
electrons and the crystal- field is low, only a
small splitting of the order of a few hundred cm-1
occurs as shown in Fig.1.5.
__________________________________________________________________
2S+1
L J(µ)
2S+1
LJ
2S+1
L 102 cm
-1
103 cm
-1
fn
104 cm
-1
SPIN-ORBIT CRYSTAL-
COULOMB FIELD
Fig. 1.5 . Schematic diagram of the splitting of RE3+
energy levels due to the
coulomb, spin-orbit and crystal-field interactions.
However, in reality, general ligand field theory gives a more accurate picture.
According to this theory, covalent bonding plays an additional role since the electrons
of the ion are partially shared with the surrounding ions or ligands. The shielding of
4f electrons reduces this effect. Ligand field theory is more appropriate to systems of
transition metal ions, which are not shielded from the surrounding environment.
Therefore CF theory is a more accurate approximation when applied to Ln3+
ions.
If the above suppositions are correct then the fact that transitions between 4f Stark
levels are forbidden for electric-dipole (ed) radiation by the parity selection rule,
which states that ed transition can only occur between levels of opposite parity. There
must be a mechanism that changes the parity of some of the levels. According to Van
Vleck [38] the CF, apart from splitting the energy levels by removing the (2J+1)
degeneracy could also mix higher configurations, such as 4 f n-1
5d, that make some of
the transitions allowed. It should be noted that the magnetic-dipole (md) transitions do
not violate the parity selection rule i.e. they can occur between the states of same
parity.
From combinational theory, the number of possible states for any given atom can
be calculated from [39, 40]
Ck
f =Ck14 =
𝟏𝟒!
𝐧! (𝟏𝟒−𝐧) …(1.1)
where n is the number of electrons present in a particular valence state of the specific
Ln3+
ion. Table 1.4 gives details of number of degeneracy states which arise from the
4f n
configuration.
The ionic radii of the Ln3+
ions are shown in Fig. 1.6 [41]. The large radii
mean that the charge to radius ratio (ionic potential) is relatively low which results in
a very low polarizing ability. This reflects the predominant ionic character of the
metal-ligand bonds and the co-ordination number of the Ln complexes.
Table 1.4: Degeneracy of 4 f n
configurations (The numbers shown in parentheses for 2S+1
𝑳 terms indicate the number of times that
particular term repeats.)
Configurations Ln3+ Terms(2S+1 L) Number of Number of Maximum number of
Terms J levels 2S+1LJ Crystal- Field Levels
f1, f13 Ce3+, Yb3+ 2F 1 2 14
f2, f12 Pr3+, Tm3+ 1SDGI 3PFH 7 13 91
f3,f11 Nd3+,Er3+ 2PD(2)F(2)G(2)H(2)IKL 4SDFGI 17 41 364/2
f4,f10 Pm3+,Ho3+ 1S(2)D(4)FG(4)H(2)I(3)K(2)LN 5SDFGI 3P(3)D(2)F(4)G(3)H(4)I(2)K(2)LM 47 107 1001
f5, f9 Sm3+, Dy3+ 2P(4)D(5)F(7)G(6)H(7)I(5)K(5)L(3)M(2)NO6PFH 4SP(2)D(3)F(4)G(4)H(3)I(3)K(2)LM 73 198 2002/2
f6,f8 Eu3+, Tb3+ 1S(4)PD(6)F(4)G(8)H(4)I(7)K(3)L(4)M(2)N(2)Q 3P(6)DF(9)G(7)H(9)I(6)K(6)L(3)M(3)NO 119 295 3003
5SPD(3)F(2)G(3)H(2)I(2)KL 7F
f7 Gd3+ 2S(2)P(5)D(7)F(10)G(10)H(9)I(9)K(7)L(5)M(4)N(2)OQ 4S(2)PD(2)F(6)G(5)H(7)I(5)K(5)L(3)MN 119 327 432/2
6PDFGHI 8S
Fig. 1.7 shows the Dieke diagram which gives the energy level structure of the multiplets of
the Ln3+
ions [37]. Because of the shielding effect of the outer shell electrons, these energy levels
change only slightly from host to host.
Fig.1.6. Ionic radius of trivalent lanthanides ions
1.3.3 Characteristics of the lanthanide ions
The Lanthanides exhibit a number of features in their chemistry which differentiate them
from the d-block metals. The reactivity of the elements is greater than that of the transition
metals, similar to the Group II metals.
The following are the characteristics of lanthanides.
1. Lanthanides are relatively soft metals. Their hardness slightly increases with atomic
number,
2. They have wide range of co-ordination numbers (generally 6-12, but numbers of 2,3 or 4
are also known),
3. Co-ordination geometries are determined by ligand steric factors rather than CF effects,
4. Unless complex agents are present, insoluble hydroxides precipitate at neutral pH,
5. They do not form multiple bonds like Ln = O or Ln ≡ N, known for many transition
metals and certain actinides,
6. They possess high melting and boiling points and they act as strong reducing agents,
7. Except La3+
and Lu3+
, lanthanide compounds are strongly paramagnetic,
8. The f→f transitions have small homogeneous line widths,
9. Due to well shielding of 5s2
and 5p6
orbitals, their spectroscopic and magnetic properties
are almost uninfluenced by the ligand field,
10. They have small CF splitting and very sharp electronic spectra in comparison with the d-
block metals,
11. There are many possible three and four level lasing schemes as the electronic states of the
ground 4 f n configuration provide rich optical energy level structure and
12. There are several excited states suitable for optical pumping. These excited states decay
nonradiatively to meta stable states having high radiative quantum efficiencies.
1.3.4 Color of the rare earth ions
Due to the internal transitions of 4f electrons occurring in the visible region of the spectrum,
the RE3+
ions have their characteristic colors. Main- Smith [42] tried to correlate the color
sequence in rare earth series with the 4f electronic configuration of the RE3+
ions. Table1.5 gives
similarity between the ions having 4 f n
and 4f 14-n
configurations. However, the non tri- positive
ions show wide divergence in color compared to the iso -electronic tri-positive ones. Thus the
colors of the non tri-positive rare earth ions are : Ce4+
(4f 0) - orange, Sm
2+ (4 f
6)- reddish brown,
Eu2+
(4 f 7)- straw yellow and Yb
2+ (4 f
14)- green.
Table: 1.5 Color sequence of the RE3+
ions
4fn
color 4f14-n
La (4f0) → Colorless ← Lu(4f
14)
Ce(4f1) → Colorless ← Yb(4f
13)
Pr (4f2) → Green ← Tm (4f
12)
Nd (4f3) → Pink ← Er (4f
11)
Pm (4f4) → Orange ← Ho (4f
10)
Sm (4f5) → Yellow ← Dy (4f
9)
Eu (4f6) → Pale Pink ← Tb (4f
8)
Gd(4f7) → Colorless
__________________________________________
1.3.5 General properties of rare earth ions
The following are the general properties of the rare earth ions.
(i) Rare earths are silvery-white metals which tarnish when exposed to air,
(ii) They possess high melting and boiling points,
(iii) They can burn easily in air,
(iv) They are relatively soft,
(iv) Their hardness slightly increases with increase in atomic number,
(v) Their compounds are generally ionic,
(vi) They are strong reducing agents and
(vii) They react with water to liberate hydrogen (H2) slowly in cold and quickly upon
heating.
1.3.6 Optical properties of rare earth ions
RE3+
ions are favorable candidates for luminescent device fabrication due to the following
optical properties.
(i) Luminescence of RE3+
ion spreads in various spectral ranges,
(ii) They have long emission lifetimes,
(iii) They have small homogeneous line widths,
(iv) They possess high refraction with relatively low dispersion and
(v) There are several excited states suitable for optical pumping.
1.3.7 Applications of rare earth ions in glasses
The following are some of the scientific and technological applications of RE3+
doped glasses.
(i) Communication fibers and glass lasers,
(ii) LED and color television phosphors,
(iii) Optical glasses, filters and lenses,
(iv) Light sensitive and photo chromic glasses,
(v) Coloring and discoloring agents,
(vi) Glass polishing agents,
(vii) pH Electrodes and
(viii) X-ray and γ-ray absorbing glasses.
1.4 SPECTROSCOPIC INVESTIGATIONS OF RARE EARTH IONS.
Optical absorption and fluorescence spectroscopy are the important techniques in the study of
Ln3+
doped systems because they allow the determination of natural frequencies of Ln3+
ions.
The 4f →4f transitions are very sharp due to very effective shielding of the 4f electrons by the
filledin 5s and 5p shells having higher energies than the 4f shell [41, 43, 44].
1.4.1 Absorption spectrum
The 4f electronic orbitals in Ln3+
ions are incompletely filled. So the ions absorb
electromagnetic radiation in the spectral regions of ultraviolet, visible and the near-infrared [41].
The intra–4f n
transitions, the inter- 4f n→
4f
n-1 5d
1 transitions or charge transfer transitions occur
in these regions. Also it is possible to correlate the positions of these 4f→5d transitions with the
standard (III-II) and (IV-III) reduction potentials for the lanthanides [45,46]. An easily oxidized
ligand is bound to Ln3+
ion, which can be reduced to the divalent state or when the ligand is
bound to one of the tetravalent ions then charge transfer bands result [47]. The standard
lanthanide (III-II) and (IV-III) reduction potentials have also been correlated with the energy of
the first charge transfer band [45,46]. The electrostatic interaction yields 2s+1
L terms with
separation of the order of 104
cm-1
. The spin-orbit interaction then splits these terms into J states
with typical splitting of 103 cm
-1. Finally the „J’ degeneracy of the free-ion state is partially or
fully removed by the crystalline electric field. The magnitude of the crystal field splitting
usually extends over several hundred cm-1
. In glasses the amount of splitting is of the order of
magnitude of the inhomogeneous broadening as a result of multiplicity of RE3+
ion sites.
The intra-4f n transitions are the most useful transitions in the spectra of the Ln complexes
[48]. These transitions are formally Laporte-forbidden and as a result tend to be very weak. In
addition to this the transitions that do not occur within the ground multiplet may be spin-
forbidden. Because of the shielding of the 4f electrons, the transitions that are observed are very
sharp and line like. These spectra are quite different from those of the d-transition elements.
This can be explained by examining the magnitude of the perturbations acting on the two types
of electrons [49].
In d-transition metal complexes
Inter electronic repulsions ≈ crystal–field >>spin-orbit coupling > thermal energy
In f-transition metal complexes
Inter electronic repulsions >> spin – orbit coupling > crystal–field ≈ thermal energy
This order means that the CF in the lanthanides is acting to remove some of the degeneracy
contained in the individual values of the J quantum number. This additional splitting is generally
in the order of 200 wave numbers whereas in the d-transition elements it is in the order of 10-30
thousand wave numbers.
In comparison of the spectrum of a complex Ln3+
ion with that of aquo ion three effects are
observed: (1) there are small changes usually toward longer wave lengths,(2) the bands undergo
additional (or at least different) splitting and (3) there is a significant change in the molar
absorptivity of the individual bands. These are due to the changes in the strength and symmetry
of the CF produced by the ligands. The shifts in the barycenters of the peaks in the spectra of Ln
complexes relative to the aquo–ion are caused by what has been termed as the „Nephelauxetic
effect‟ by Jorgensens [50]. These are related to the decrease in the inter-electronic repulsion
parameters in the complex. Numerous attempts have been made to relate this effect to weak
covalence effects.
Optical properties of RE3+
ions in a solid matrix are affected by changes in the environment of
the Ln3+
ion and its interaction with ligands. According to the Judd-Ofelt (JO) theory [51,52]
intensities of a set of absorption lines for a particular Ln3+
ion in any matrix is characterized by
three intensity parameters Ωλ (λ=2, 4 and 6) which depend on the symmetry of CF at the RE3+
site and the strength of covalence of the RE ion-ligand bond. Therefore it is of interest to study
the variation of these intensity parameters with the host glass composition. From these
parameters, several important optical properties such as radiative transition probabilities,
radiative life times (τR) of the excited states and branching ratios can be estimated [53, 54].
1.4.2 Fluorescence spectrum
The fluorescence spectrum can be analyzed by essentially the same procedure as for the
absorption spectrum except that the nature of the emission process will generally yield additional
information concerning the ground multiplet of the ion. When photons of electromagnetic
radiation are used for exciting the molecule, atom or ion then the emission of electromagnetic
radiation takes place while the molecule, atom or ion return to its normal level from excited level
and may result in fluorescence. Fluorescence involves optical transition between electronic states
which is the characteristic of the radiating substance. Depending upon the type of fluorescing
ion and its environment the radiation life time of the excited electronic states varies from 10-10
to
10-1
s. In the case of RE3+
ions the most striking feature of the fluorescent emission is that it
occurs in the spectral region where the crystal or glass is non-absorbing.
RE3+
ions act as very good fluorescing centers in a matrix. The fluorescence of an active
RE3+
ion is influenced by the asymmetry of the surrounding binding forces. The parameters
position (λp) intensity full width at half maximum of the emission band (∆λp) and the life time
(τR) of the fluorescing state are affected by the structure of the solid matrix. The stimulated
emission cross-section, σ(λp), one of the important parameters which determine laser
characteristics of a given transition of RE3+
ion, can be evaluated from the emission studies. The
experimental branching ratios, (βR), equal to the relative intensities of the emission bands of the
transitions with same ground level, can also be evaluated from the emission studies. All these
parameters are used to compare the theoretical branching ratios predicted from the JO theory
[51, 52]. For a given RE3+
ion, all these parameters can be varied over a wide range by changing
the composition of the material.
1.5 THEORITICAL MODELS
When a 4f ion is embedded in a solid matrix, the effect of ligand environment is minimum on
the 4f shell due to its effective shielding by the closed 5s and 5p shells. This weak perturbation
is responsible for the rich electronic spectra which provide detailed fingerprint information about
the surrounding arrangement of atoms and their interactions with the 4f electrons. Hence, it is
the prime task to evaluate the electronic energy level structure of RE3+
ions in any given solid
matrix to understand their fluorescence properties.
1.5.1 ELECTRONIC ENERGY LEVEL ANALYSIS- Hamiltonian model
The Eigen states of a system are described in terms of the wave functions of the different
electron orbitals. The energy levels of the electronic states can be found by calculating the
matrix elements of Hamiltonian between the eigen states of the system. The three types of
physical interactions described by the Hamiltonian have different magnitudes. The energy levels
of the electronic states can be found in successive steps using the techniques of perturbation
theory. The wave functions for the electronic states of the ion can be approximated by linear
combinations of products of single-electron wave functions. The physical condition is enforced
mathematically by constructing a wave function that is anti symmetric with respect to an
interchange of the electrons in two orbitals since the electrons obey Fermi-Dirac statistics and
thus must obey Pauli exclusion principle. If two of the spin-orbitals are identical the wave
function vanishes due to anti symmetry.
1.5.1.1 Free-ion Hamiltonian
In any host matrix, each interaction experienced by the Ln3+
ion can be described with the aid
of effective operators and their effect can be parameterized by using phenomenological models
[44,55,56]. The effective Hamiltonian for free-ion (HFI) is given by
Ĥ= EAVG + ∑k F
k fk + 𝜉4f Aso+ 𝛼 L(L+1)+ 𝛽 G(G2)+ 𝛾 G(R7)+ ∑iT
i ti+ ∑kP
k pk+ ∑jM
jmj
…(1.2
)
where k=2,4,6; i=2,3,4,6,7,8 and j=0,2,4. The free-ion Hamiltonian includes two body
electrostatic interactions (Fk), spin-orbit interaction (𝜉4f), two body configuration interactions
(𝛼, 𝛽, 𝛾 ), three body configuration interactions (Ti), electrostatically correlated spin-orbit
interaction (Pk) and spin-other orbit interaction(M
j). The energy of the entire configuration is
shifted by the parameter EAVG [57]. The Slater integral, Fk (k=2,4 and 6), describes the
coulombic interaction between the 4f electrons. Standard least-square methods can be used in a
semi empirical fitting approach to the 4f n
electronic energy levels structure. The quality of the fit
is estimated by rms (root-mean-square) deviation given by [58]
σrms= (𝑬𝒊
𝒆𝒙𝒑−𝑬𝒊
𝒄𝒂𝒍)𝟐𝒏𝒊=𝟏
𝒏
…(1.3)
where 𝑬𝒊𝒆𝒙𝒑
and 𝑬𝒊𝒄𝒂𝒍 are experimental and calculated energies for level i respectively and n is
the total number of energy levels considered.
1.5.1.2 Crystal-field Hamiltonian
The Ln3+
ion present in a solid matrix experiences an inhomogeneous electrostatic field
produced by the surrounding charges. The crystal field Hamiltonian, ĤCF, is expressed in
Wybourne‟s notation [55] as
ĤCF= 𝑘 𝐵𝑞𝑘𝐶𝑞
𝑘𝑞 …(1.4)
where 𝑩𝒒𝒌 =(-1)
q (𝑩𝒒
𝒌 − 𝒊𝑺𝒒𝒌 ) are the coefficients representing the functions of the radial
distances which can be varied in order to match experimental and calculated CF levels. The 𝑪𝒒(𝒌)
are the tensor operators of rank „k‟ closely related to the spherical harmonics that can be obtained
exactly [41, 55]. By the symmetry selection rules for the point symmetry at the Ln3+
ion site, the
number of HCF parameters is greatly reduced. Any surrounding that breaks the spherical
symmetry of the free ion can lead to shift and splitting of the energy levels. Hence these
conditions apply to glass materials also. The Hamiltonian for a multi electron atom is given by
Ĥ = ĤFI + ĤCF …(1.5)
where ĤFI represents the isotropic parts of Ĥ and ĤCF represents non-spherically symmetric
components of the crystal field with even parity.
1.5.2. Intraconfigurational f - f transitions
Generally transitions in the absorption spectra of Ln3+
ions are of the forced electric-dipole
type. The transitions gather intensity by mixing in states having opposite parity although
formally Laporte-forbidden. In a few cases, particularly in Eu3+
, magnetic-dipole transitions with
selection rule |∆J|= 0, ±1 but not 0↔0 have been observed. The md character has been confirmed
from the polarization properties. The spin selection rule is relaxed by spin-orbit coupling and so
the transitions that are md in origin are generally at least an order of magnitude weaker than
those that are ed in origin [59].
1.5.2.1. Induced electric–dipole transitions
An ed transition is the consequence of the interaction of the optically active ion with the
electric field vector through an ed. Such a transition has odd parity. This type of
intraconfigurational ed transitions are forbidden by the Laporte selection rule. Non-
centrosymmetrical interactions allow the mixing of electronic states of opposite parity giving
rise to weaker transitions and are called as induced ed transitions which are much weaker than
the ordinary ed transitions. The selection rules for the induced ed transitions are ∆l = ± 1, ∆τ =
0, ∆S=0, |∆L|≤ 6, |∆J|≤6, |∆J|=2,4,6 if J=0 or J’ =0.
1.5.2.2 Magnetic – dipole transitions
A md transition is caused by interaction of the spectrally active ion with the magnetic field
component of the light through an md. The intensity of the induced md transitions are weaker
than ed transitions. An md transition has even parity and allows transitions between the states of
equal parity (intraconfigurational transitions). Selection rules are given by ∆τ= ∆S =∆L= 0, ∆J=
0, ±1 but 0 ↔ 0 is forbidden.
1.5.2.3 Electric-quadrupole transitions
An electric-quadrupole transition arises from the displacement of charge with quadrupolar
nature. An electric-quadrupole has even parity. Electric–quadrupole transitions are much weaker
than induced ed and md transitions. So far, no experimental evidence exists for the occurrence of
quadrupole transitions in Ln3+
spectra. However, hypersensitive transitions are considered
as pseudo-quadrupole transitions because these transitions obey the selection rules of quadrupole
transitions (|∆S|=0, |∆L|= < 2 and |∆J|= < 2).
1.5.2.4 Hypersensitive transitions
The intensities of a few of the induced ed transitions in RE3+
ions are very sensitive to the
environment. These are called „hypersensitive transitions‟ and obey the selection rules of
quadrupole transitions, |∆S|=0, |∆L| < 2 and |∆J| < 2. In almost all RE3+
ions hypersensitive
transitions have been observed and are given in Table 1.6. The possible mechanism for the
occurrence of hypersensitivity was given by Jorgensen and Judd [60]. They argued that the
inhomogenety in the dielectric medium surrounding the RE3+
ion could enhance the intensity of
the hypersensitive transitions.
1.5.3 Intensity analysis of optical spectra
1.5.3.1 properties of spectral lines – Oscillator strengths
The properties of radiative transitions are manifested in absorption and emission
spectroscopy. The strength of a spectral line is characterized by a dimensionless parameter
called „oscillator strength‟ or „f number‟. The concept of oscillator strength was first introduced
by Ladenburg [61]. The ratio of the actual intensity to the intensity radiated by an electron
oscillating harmonically in three dimensions gives the oscillator strength of a transition. The
oscillator strength of an absorption transition (ƒexp) is directly proportional to the area under the
absorption curve and is given by [62, 63]
exp= 𝟐.𝟑𝟎𝟑𝒎𝒄𝟐
𝑵𝝅𝒆𝟐 𝜺(𝝊)𝒅𝝊 = 𝟒. 𝟑𝟏𝟖 ⤬ 𝟏𝟎−𝟗 𝜺(𝝊)𝒅𝝊 …(1.6)
where 𝑚 and 𝑒 are the mass and charge of an electron, c is the velocity of light, N is the
Avogadro‟s number, 𝜀(𝜐) is the molar absorptivity of a band at a wave number 𝜐 (cm-1
). The
integral value in the above equation corresponds to the area under the absorption curve. The
value of oscillator strength is obtained in cgs units.
The molar absorption coefficient 𝜺(𝝊) at a given energy 𝝊 (cm-1
) is obtained from Beer-
Lambert‟s law:
𝜺(𝝊)= (1/C𝒍) log (I0/I) …(1.7)
where C is the concentration of Ln3+
ions in mol/lit, 𝑙 is the optical path in the absorbing medium
and log (I0/I) is the absorptivity or optical density (OD) . The order of oscillator strengths of
magnetic and induced ed transitions is 10-6
.
The shape of the spectral line is also important in addition to the strength of a transition. The
sum of the initial and final energy levels of transitions determines the shape of the spectral line.
Lorentzian, Gaussian and Voigt are the three types of line shapes. The physical process that has
the same probability of occurrence for all atoms of the system produces a Lorentz line shape and
is known as homogeneous broadening. Lorentzian broadening is also known as lifetime
broadening because of the shortened lifetimes of the energy levels involved in the transition.
This type is associated with the Heisenberg uncertainty relationship relating time and energy.
This contribution is referred to as the natural line width for a transition. The process that has a
random distribution of occurrence for each atom produces a Gaussian line shape and is known as
inhomogeneous broadening. The line shape is called as Voigt profile when both the broadening
processes are present.
In determining laser characteristics the difference between Lorentzian and Gaussian line
shapes are important. In Lorentzian shapes all of the ions participate in laser emission at a
specific frequency and so single longitudinal mode operation can be obtained. In Gaussian line
shapes, several subsets of ions may laze simultaneously and so multimode operation will take
place. „Spectral hole burning‟ is exhibited by inhomogeneous broadened lines and „spatial hole
burning‟ is exhibited by homogeneouly broadened lines. The broadening line magnitude
generally depends on concentration of ions and temperature.
1.5.3.2 Mixed electric- and magnetic- dipole line strengths
The majority of 4f n
intraconfigurational transitions are induced ed type. There are certain
transitions which are neither pure electric-dipole nor pure magnetic-dipole. They contain major
„ed‟ contribution and partial „md‟ contribution [53]. The „ed‟ and „md‟ oscillator strengths are to
be calculated separately. The line strength of the electric-dipole transition can be obtained from
the expression [63-65]
𝑆𝑒𝑑 (𝛹𝐽, 𝛹′𝐽′) = 𝑒2 𝛺𝜆𝜆=2,4,6 𝛹𝐽 𝑈𝜆 𝛹′ 𝐽′ 2 …(1.8)
where Judd-Ofelt parameters 𝜴𝝀 ( 𝛌 = 𝟐, 𝟒 𝐚𝐧𝐝 𝟔) represent the square of the charge
displacement due to induced ed transition and are host dependent. The host independent doubly
reduced matrix elements 𝑼(𝝀) 𝟐 are evaluated in the intermediate coupling approximation for
the transition 𝜳𝑱 ⟶ 𝜳′𝑱′ [66].
𝑆𝑚𝑑 𝛹𝐽, 𝛹′ 𝐽′ =𝑒2ℎ2
16𝜋2𝑚2𝑐2 𝛹𝐽 (𝐿 + 2𝑆) 𝛹′𝐽′ 2 …(1.9)
The non-zero matrix elements will be those of the diagonal in the quantum numbers 𝛼, 𝑆 and 𝐿.
The selection rule ∆J= 0, 1 gives three different cases for the magnetic dipole elements. (i)
𝑱′ = 𝑱
𝜳𝑱 (𝑳 + 𝟐𝑺) 𝜳′𝑱′ = 𝒈[ 𝑱 𝑱 + 𝟏 𝟐𝑱 + 𝟏 ] 𝟏/𝟐 …(1.10)
where the Lande‟s factor 𝑔 is given by
𝒈 = 𝟏 +𝑱 𝑱+𝟏 +𝑺 𝑺+𝟏 −𝑳(𝑳+𝟏)
𝟐𝑱(𝑱+𝟏) …(1.11)
The effective magnetic momentum of an atom or an electron in which the orbital (𝐿) and
spin(𝑆) angular momenta combine to give total angular momentum( 𝐽) is given by the Lende‟s
factor.
(ii) 𝑱′ = 𝑱 − 𝟏
𝛹𝐽 (𝐿 + 2𝑆) 𝛹′𝐽′ = 1
4𝐽 𝑆 + 𝐿 + 𝐽 + 1 𝑆 + 𝐿 + 𝐽 − 1 𝐽 + 𝑆 − 𝐿 (𝐽 + 𝐿 − 𝑆)
1/2
…(1.12)
(iii) 𝑱′ = 𝑱 + 𝟏
𝜳𝑱 (𝑳 + 𝟐𝑺) 𝜳′𝑱′ = 𝟏
𝟒 𝑱+𝟏 𝑺 + 𝑳 + 𝑱 + 𝟐 𝑺 + 𝑱 + 𝟏 − 𝑳 𝑳 + 𝑱 + 𝟏 − 𝑺 (𝑺 + 𝑳 − 𝑱)
𝟏/𝟐
…(1.13)
Before computation of the magnetic dipole contribution the matrix elements must be transformed
into the intermediate coupling scheme.
1.5.3.3 Judd-Ofelt theory
Judd and Ofelt independently derived expressions for the oscillator strengths of induced
electric dipole transitions of ƒ𝒏 configurations [51,52]. Since their results were similar and
published simultaneously this theory is known as Judd-Ofelt theory. According to J-O theory the
intensity of the forbidden ƒ − ƒ electric dipole transitions can arise from the admixture of
configurations of opposite parity (e.g., 4ƒ𝒏−𝟏𝒏′𝒅′ and 4ƒ𝒏−𝟏𝒏′𝒈′) into the 𝟒ƒ𝒏 configuration. It
was considered that the odd part of the crystal- field potential is the perturbation for mixing
states of different parity into the 𝟒ƒ𝒏 configuration. The experimental oscillator strength is given
by [15, 39, 51, 64, 67]
ƒ𝒆𝒙𝒑 = ƒ𝒆𝒅 + ƒ𝒎𝒅 …(1.14)
The total oscillator strength of an absorption band is obtained from the expression
ƒ𝒆𝒙𝒑 𝜳𝑱, 𝜳′𝑱′ =𝟖𝝅𝟐𝒎𝝂
𝟑𝒉(𝟐𝑱+𝟏)
(𝒏𝟐+𝟐)𝟐
𝟗𝒏𝑺𝒆𝒅 𝜳𝑱, 𝜳′𝑱′ + 𝒏𝟑 𝑺𝒎𝒅(𝜳𝑱, 𝜳′𝑱′) …(1.15)
where n is the refractive index of the medium, m is the electron mass, 𝝂 is the wave number of
the transition in cm-1
, h is the Plank‟s constant, (2J+1) is the degeneracy of the ground state
2S+1LJ,
(𝒏𝟐+𝟐)𝟐
𝟗 is the Lorentz local field correction which accounts for dipole-dipole correction.
The intensities of the magnetic dipole transitions which are weak are relatively independent of
the surrounding Ln ions. Therefore the experimental oscillator strengths are almost equal to the
electric dipole oscillator strengths.
ƒ𝒆𝒙𝒑 = ƒ𝒆𝒅 …(1.16)
Hence the experimental oscillator strengths can be equated to the calculated oscillator strengths.
ƒ𝒆𝒙𝒑 𝜳𝑱, 𝜳′𝑱′ = ƒ𝒄𝒂𝒍 𝜳𝑱, 𝜳′𝑱′ = 𝟖𝝅𝟐𝒎𝝂
𝟑𝒉(𝟐𝑱+𝟏)
(𝒏𝟐+𝟐)𝟐
𝟗𝒏𝑺𝒆𝒅 𝜳𝑱, 𝜳′𝑱′ …(1.17)
The experimental oscillator strengths are evaluated from the obtained spectra and used to find the
J-O intensity parameters Ωλ (𝛌 = 𝟐, 𝟒 𝐚𝐧𝐝 𝟔) by least square fit. The quality of the fit is
determined by the rms deviations between the measured and calculated oscillator strengths.
The intensity of ƒ − ƒ transitions in rare earth complexes is ligand dependent. Hence many
authors tried to correlate the intensity parameters with the chemical nature of ion-ligand bond,
with the properties of the ligand itself or with the structure of the complex. According to Zahir,
Ω2 depends on the asymmetry of the rare earth ligand field [3,15,68,69]. Oomen and van Dogen
suggested that Ω2 depends on the short-range effects i.e., the covalency of the ligand field. It
depends on the structural changes in the vicinity of the lanthanide ion [70]. 𝛀𝟒 and Ω6 follow
the same trend and mainly depend upon long-range effects. These parameters are related to the
bulk properties of the host material and the indicators of viscosity of the rare earth doped glasses
[15, 17, 19, 20]. For the first time Kaminski et al.[71] introduced the parameter, spectroscopic
quality factor (𝝌) ,which is useful for predicting the stimulated emission in any laser active
medium and is given by
𝝌 = 𝜴𝟒
𝜴𝟔 …(1.18)
1.5.4 Radiative properties
The radiative properties of excited states of RE3+
ion are predicted by the J-O parameters
using refractive index. For the transition 𝜳𝑱 ⟶ 𝜳′𝑱′ the radiative transition probability can be
obtained from the equation [54, 63,72, 73]
𝑨𝑹 𝜳𝑱, 𝜳′𝑱′ =𝟔𝟒𝝅𝟒𝝂𝟑
𝟑𝒉(𝟐𝑱+𝟏) 𝒏(𝒏𝟐+𝟐)𝟐
𝟗𝑺𝒆𝒅 𝜳𝑱, 𝜳′𝑱′ + 𝒏𝟑𝑺𝒎𝒅(𝜳𝑱, 𝜳′𝑱′) …(1.19)
The total radiative transition probability is given by
𝑨𝑻 𝜳𝑱 = 𝑨𝑹 𝜳𝑱, 𝜳′𝑱′ 𝜳′ 𝑱′
…(1.20)
The radiative lifetime of an excited state is obtained from the expression
𝛕𝑹 𝜳𝑱 = 𝛕𝒄𝒂𝒍 𝜳𝑱 =𝟏
𝑨𝑻 𝜳𝑱 …(1.21)
Strong emission probabilities and more number of transitions from an energy level result
shorter lifetimes due to faster decay. The difference between the predicted and experimental
lifetimes is due to the non-radiative process (WNR) either by multiphonon relaxation rate (WMPR)
or energy transfer rate (WET). The quantum efficiency can be estimated from the expression
η=𝛕𝒎
𝛕𝑹=
𝑨𝑹
𝑨𝑹+𝑾𝑵𝑹 …(1.22)
The experimental branching ratios are obtained from the relative areas of the emission bands.
The branching ratios corresponding to the emission from an excited level 𝛹𝐽 to its lower level
𝜳′𝑱′ is given by [74-76]
𝛃𝑹 𝜳𝑱, 𝜳′𝑱′ =𝑨𝑹 𝜳𝑱,𝜳′ 𝑱′
𝑨𝑻 𝜳𝑱 …(1.23)
The peak stimulated emission cross-section, σp 𝜳𝑱, 𝜳′𝑱′ between the states 𝜳𝑱 𝐚𝐧𝐝 𝜳′𝑱′
can be obtained from the equation
σp 𝜳𝑱, 𝜳′𝑱′ =𝛌𝑷𝟒
𝟖𝝅𝒄𝒏𝟐∆𝝀𝒆𝒇𝒇𝑨𝑹 𝜳𝑱, 𝜳′𝑱′ …(1.24)
where λP peak wave length of the transition and ∆𝝀𝒆𝒇𝒇 is its effective line width. The large values
of stimulated emission cross-sections indicate the good lasing transitions.
1.5.5. Excited State Decay
The study of excitation and relaxation of RE ions due to intra 4f electronic transitions gives a
deeper insight into mechanisms involved in excitation. Fig.1.8 shows a typical time-resolved
intensity spectrum. The curve is described by integral solutions to appropriate rate equations
which account for the possible excitation and de-excitation mechanism. An excited RE3+
ion
may relax to the initial ground state through radiative transition, phonon emission, by
transferring its excess energy to a nearby RE3+
ion or by a combination.
The fluorescence decay curves can be fitted to a single exponential function at very low
concentrations of doped ions, where the interaction between the RE3+
ions is negligible. A
perfect single exponential decay indicates that the energy transfer between luminescent ions is
not dominant and the lifetime of the excited level can be simply determined by finding the first
e-folding times.
Fig.1.8: A typical time-resolved excitation and de-excitation curve of RE3+
ions.
The fluorescence intensity as a function of time is given by
𝑰 𝒕 = 𝑰𝒐 𝒆−𝒕/𝛕 …(1.25)
where 𝑰𝒐 is the fluorescence intensity when t=o or τ represents the lifetime of the excited state
and is reciprocal to the probability of a spontaneous emission from the excited state to the ground
state. A logarithmic plot of the intensity versus time helps to determine lifetime. It is evident
from Fig.1.8 that after time t= τ the intensity of the excited state becomes 𝑰𝒐 /e.
1.5.6. Nonradioactive relaxation
An excited RE3+
ion may relax to the lower energy state via a non- radioactive process. The
deviation from a single exponential decay is due to non-radiating energy transfer to the 4f
electron system provided only a single local environment is present. The energy transfer
between different RE3+
ions by cross-relaxation and excitation migration between the same
types of RE3+
ions are typical examples. Another example is the population of a multiplet by
another multiplet, which is energetically located above, but within the same ion.
The total decay rate is given by
𝟏/( 𝝉𝒎 ) = AR + WNR …(1.26)
where τm denotes the measured lifetime of the emitting state and (AR) and (WNR ) are the
radiative and non-radiative decay rates respectively.
The quenching of lifetime is mainly influenced by the non-radiative decay rates. There are
mainly four non-radiative decay processes contributing to the reduction of measured lifetime of
the emitting level.
WNR= WMPR+ WET+ WCQ+ WOH …(1.27)
where WMPR, WET , WCQ and WOH denote the non-radiative decay rates corresponding to the
multi-phonon relaxation (MPR), energy transfer between donor to donor or donor to acceptor,
concentration quenching (CQ)and hydroxyl (OH-) groups respectively.
1.5.6.1. MULTI-PHONON RELAXATION
The non-radiative processes competing with luminescence are energy loss due to the local
vibrations of surrounding atoms and electronic states of atoms in the vicinity, such as energy
transfer, which may be resonant or phonon assisted. The other type of energy loss is given by
the exponential energy gap law[17]
WMPR = B exp (-α ΔΕ) …(1.28)
where ∆E is the energy gap between the luminescent level 𝑱 and the closest lower level 𝑱′ and
the parameters B and α are host dependent and independent on the chosen RE ion. Several lattice
phonons are emitted in order to bridge the energy gap if the energy gap between the excited and
the next lower electronic levels is larger than the phonon energy. The most energetic vibrations
are responsible for the non-radiative decay which conserves energy in the lowest order.
The stretching vibrations of the glass network polyhedral are the most energetic vibrations.
These distinct vibrations are active in multiphonon process, rather than the less energetic
vibrations of RE ion- ligand bond. The less energetic vibrations may participate in cases when
the energy gap is not bridged totally by the high energy vibrations. It was found that the
logarithm of the multiphonon decay rate decreases linearly with the energy gap. The average
phonon frequencies ( ħ𝝎 ) of some of the network formers [17] are presented in Table 1.7.
Table: 1.7. Average phonon frequencies ( ħ𝜔 ) of some of the network formers in glasses.
______________________________
Matrix ħ𝝎 (cm-1
)
Borate 1400
Phosphate 1200
Silicate 1100
Germanate 900
Tellurite 700
LaF3 (Crystal) 350
_________________________________
1.5.6.2. Energy transfer
An excited center can also relax to the ground state by non-radiative energy transfer to a
second nearby center. Non-radiative energy transfer is often used in practical applications such
as to enhance the efficiency of phosphors and lasers. Some interaction mechanism is needed to
allow energy transfer from the excited donor D* to the acceptor A. The rate of energy transfer
from the donor centers to the acceptor centers is given by [77, 78]
WDA = 𝟐𝝅
ħ 𝝍𝑫𝝍𝐀∗ 𝑯𝒊𝒏𝒕 𝝍𝐃∗𝝍𝑨
𝟐 𝙜𝑫 (𝝂)𝙜𝑨 (𝝂) 𝒅𝝂 …(1.29)
where 𝜓D and 𝜓 D* refer the wave functions of the donor center in the ground and excited states
respectively, 𝝍A and 𝝍 A* are the wave functions of the acceptor center in the ground and
excited states respectively and Hint is the donor - accepter interaction Hamiltonian.
The overlap between the normalized donor emission line-shape function gD (𝝂) and the
normalized acceptor absorption line-shape function gA (𝝂) is represented by the integral in
Eq.1.28. This term is needed for energy conservation being a maximum when D and A are
centers with coincident energy levels, a case that is called resonant energy transfer, Fig 1.9(a).
However, when D and A are different centers, it is usual to find an energy mismatch
between the transitions of the donor and acceptor ions, Fig 1.9 (b). In this case lattice phonons
of appropriate energy ( ħ𝝎 ) assist the energy transfer process which is known as phonon-
assisted energy transfer. This electron-phonon coupling must also be taken into account together
with the interaction mechanism responsible for this energy transfer.
Fig 1.9. The energy level schemes of donor (D) and acceptor (A) centers for (a) Resonant
energy transfer and (b) phonon-assisted energy transfer.
The interaction Hamiltonian (Hint ) in Eq 1.28 possesses different types of interactions namely
multipolar (electric and/or magnetic) interactions and /or a quantum mechanical exchange
interaction. The dominant interaction strongly depends on the spacing of the donor and acceptor
ions and on the nature of their wave functions. So far, the experiments on REs have been
interpreted mostly in terms of electric multipolar interactions between the ions.
The energy transfer mechanism for electric multipolar interactions can be classified into three
types basing on the character of the involved transitions of the donor and acceptor centers.
When all the transitions in D and A are of electric dipole character then electric dipole-dipole (d-
d) interactions occur and they correspond to the long range order and the transfer probability
(a)
D* A*
D A
(b)
D* …………….. ħω
A*
D A
varies with 1/R6 where R is the spacing of D and A. The dipole-quandrupole (d-q) interaction
varies as 1/R8 while quandrupole-quandrupole (q-q) interaction varies as 1/R10 .
In the similar way as in multipolar electric interactions, energy transfer probabilities due to
multipolar magnetic interactions also behave. Hence the transfer probability for a magnetic
dipole-dipole interaction varies with 1/R6 . The higher order magnetic interactions are influenced
at short distances only. In any case, the multipolar electric interactions are stronger than
magnetic interactions. The basic theory for energy transfer proposed by FÖster [77] and Dexter
[78] was modified by Inokuti and Hirayama [79] to account for energy transfer between 4f shells
of RE3+
ions. According to them the luminescence intensity decay is obtained from
𝑰 𝒕 = 𝑰𝒐 𝒆𝒙𝒑 −𝒕
𝝉𝒐− 𝑸
𝒕
𝝉𝒐 𝟑/𝑺
…(1.30)
where t is the time after excitation, 𝝉𝒐 is the intrinsic decay time of the donors in the absence of
acceptors and Q is the energy transfer parameter given by
𝑸 = 𝟒𝝅
𝟑𝜞 𝟏 −
𝟑
𝑺 𝑵𝒂𝑹𝒐
𝟑 …(1.31)
Q depends on S and the Euler‟s function 𝜞 which is equal to 1.77 for dipole-dipole (S=6),
1.43 for dipole-quadrupole (S=8) and 1.30 for quadrupole-quadrupole (S=10) interactions. Na is
the concentration of the acceptors, which is almost equal to total concentration of lanthanide
ions. Ro is the critical transfer distance, defined as the donor-acceptor separation for which the
rate of energy transfer to the acceptors is equal to the rate of intrinsic decay of the donors. CDA is
the dipole-dipole interaction parameter which describes an elementary energy transfer of direct
donor-acceptor interaction between RE3+
ions at the distance R0 and is given by
𝑪𝑫𝑨 =𝑹𝒐
𝑺
𝝉𝒐 …(1.32)
Yokota-Tanimoto [80] proposed an alternative theoretical approach to explain the energy
transfer mechanism at higher concentrations. They have obtained a general solution for the donor
decay function (migration) which includes both diffusion within the donor system and donor-
acceptor energy transfer via dipole-dipole (S=6), dipole-quadrupole(S=8) and quadrupole-
quadrupole (S=10) coupling.
1.5.6.3. Concentration quenching
In principle, when the concentration of luminescence centers increases there will be increase
in the absorption efficiency resulting an increase in the emitted light intensity. This is true up to
critical concentration of the luminescent centers. Above this concentration, the luminescence
intensity starts decreasing. This process is known as “concentration quenching” of
luminescence.
In general, luminescence concentration quenching arises due to efficient energy transfer
among the luminescent centers. At a certain concentration, the average distance between the
luminescent centers favors energy transfer and so the quenching starts. Two mechanisms are
generally invoked to explain the luminescence concentration quenching.
The excitation energy can migrate about a large number of centers before being emitted due
to efficient energy transfer. These centers can relax to their ground state by multi-phonon
emission or by infrared emission. Thus, they act as an energy sink within the transfer chain and
so the luminescence concentration quenching takes place.
Luminescence concentration quenching also takes place when the excitation energy is lost
from the emitting state via cross relaxation mechanism, by the resonant energy transfer between
two identical adjacent centers, due to particular energy-level structure of these centers. A simple
possible energy level scheme involving cross-relaxation is shown in Fig.1.10. We suppose that
for isolated centers radiative emission E3 E0 dominates. When there are two nearby similar
centers, a resonant energy transfer mechanism may occur. In this, one of the centers (donor)
transfers part of its excitation energy (E3 - E2) to the other center (acceptor). This resonant
transfer becomes possible when the energy for the transition E3 E2 is equal to that of the
transition Eo E1. As a result of this cross-
E3
E2
E1
E0
Fig 1.10. Cross-relaxation between pairs of centers.
relaxation, the donor center will be in the excited state E2 while the acceptor center will be in the
excited state E1. Then from these states a non-radiative relaxation or emission of photons with
energy other than E3 - E0 will occur; in any case the emitted energy is less than (E3 - E0) i.e., E3
E0 emission is quenched.
1.5.6.4. EFFECT of OH group
The effect of the OH group can be estimated by the infrared transmittance spectra. When
RE3+
concentration increases the OH group contents also increase and the possibility of the
interaction between OH and RE3+
ions increases. Hence the influence of OH group increases.
The presence of OH groups decreases the radiative lifetime of RE3+
ions due to energy
transfer form RE3+
to its nearby OH groups. The energy transfer rate, WHO , is proportional to the
acceptor and donor concentrations [81]. The effect of the OH group can be minimized if the
glasses are prepared under dry O2 atmosphere [82].