Lattice defects, ionic conductivity, and valence change of rare-earth impurities in alkaline-earth halides

4 LATTICE DEFECTS, IONIC CONDUCTIVITY, AND VALENCE C H A N G E OF R A R E - E A R T H IMPURITIES

IN A L K A L I N E - E A R T H HALIDES FRANCIS K. FONG

North American Aviation Science Center, Thousand Oaks, California

I. I N T R O D U C T I O N

This chapter deals with several important aspects of research in ionic alkalineearth halides, a class of ionic conductors with the general empirical formula MX:. Like the more thoroughly investigated alkali halides, the ionic alkalineearth halides are wide band-gap materials. Both are readily colored by various sources of radiation, by heating in an appropriate metallic vapor, and by electrolysis; both conduct electricity by the migration of lattice defects and both exhibit phenomena such as photoconductivity and thermoluminescence. Unlike the alkali halides, on the other hand, the ionic alkaline-earth halides as a class have only recently come to the attention of researchers. With the exception of calcium fluoride (which has been studied in its natural state since early in the present century), synthetic alkaline-earth halides such as SrC12, BaBr 2 and BaC1F have become available only very recently since the advent of divalent rare-earth-doped calcium fluoride lasers. Of particular interest, from a chemist's point of view, is the reduction of trivalent rare-earth ions in alkaline-earth halides to the divalent state--a valence state that is commonly known to be unstable in the rare-earths. The discussions on the following pages are grouped into six sections. In Section II some general properties of the ionic alkaline-earth halides are described. In Sections III and IV the theoretical concepts of lattice defects, conductivity, and diffusion are discussed and applied to the interpretation of experimental observations. The role of aliovalent impurity ions in the crystal lattice and the concept of charge-compensation in relation to'the determination of the type of disorder in the crystal lattice are described. Section V deals with 135

136

FRANCISK. FONG

the color center formation in several alkaline-earth halides, and Sections VI and VII are concerned with the incorporation of lanthanide fluorides in the alkaline-earth halides. In the two final sections, the reduction of the rare-earths to the divalent state and the thermoluminescent processes in gamma-irradiated samples, as well as the absorption and emission characteristics of the divalent rare-earth samples, are reviewed. The above-mentioned topics are closely related to one another, and crossreferences between the different sections are frequently made. It will be clear, for example, that an understanding of the thermoluminescent processes in CaF 2 (Section VII) requires a background knowledge of the valence reduction of trivalent rare-earth ions by gamma-irradiation (Section VI), and of the presence of localized energy states in the forbidden gap due to impurity ions (Section II). The topic of valence reduction of rare-earths in alkaline-earth halides (Section VI) is related to that of color center formation in the undoped crystals (Section V), which in turn is closely related to the topic of intrinsic disorder in crystal lattices (Sections III and IV). A conscious effort has been made to provide continuity in thought as one topic is led into another. As a result, this work tends to be much less encyclopedic than it is analytic. In this sense, far from having the intention of closing the subject by this review, the author wishes to focus attention upon several aspects of this field that might be further pursued with profit. Reviews on alkali halides in related subject matters exist in the literature. Among these are the works by Seitz (1946, 1954), Mott and Gurney (1940), Liddiard (1957), and Schulman and Compton (1962). Earlier investigations on the coloration and luminescence in naturally occurring fluorites have been reviewed by Przibram (1956). Frequent reference to these and other works have been made throughout the following text, and the readers are directed to these works for related bibliographies to the literature. II. THE ALKALINE-EARTH HALIDES Although the majority of the alkaline-earth halides may be considered as ionic, variations in the ionic character among them are significant and are dependent upon both the cation and the anion. The uniform dispositive state of oxidation of the alkaline-earths is predominantly ionic for the heavier members. It becomes increasingly covalent as the cation size decreases, until for beryllium it is predominantly covalent. The alkaline-earth halides thus differ from the alkali halides, which are largely ionic including the lithium halides. In the following, discussions will be limited to those alkaline-earth halides that are predominantly ionic in nature. Perhaps the most studied compound among the alkaline-earth halides is calcium fluoride. Not only are single crystals of CaF 2 easily synthesized, they are abundantly found in nature. In CaF2, each Ca 2 ÷ ion is surrounded by eight

137

Rare-earth impurities in alkaline-earth halides

F - ions at the corners of a cube, and each F - ion by four Ca 2 ÷ ions at the corners of a regular tetrahedron. The fluorite structure, (1) as illustrated in Fig. 2.1A, is also characteristic of SrCI2. The crystal structures of BaBr2 and BaC1F are shown in Fig. 2.1B, ¢. Barium bromide has the structure of lead dichloride, (2) PbC12. Each Ba 2÷ ion is surrounded by nine bromide ions, six

'

C .....

',

I-

-

-

,

"

\

"~,

,

BoZ~ Br'C)

O

above

o ) ..... O ) below

© A

or

Ca Sr

F CI

B¸

{_ - -

of popelr O plo~ ......

0

a{onl of pope~ F in plane

0 ...... 0

Bobelow

plon~ of paper

:>-

(:

p~one of paper C) .......

!

i--;

• t--I ( - - i 1 - - - -

[

CL ao F aa CL

CL ao F So Ct.

FIG. 2.1. Crystal structures of (A) CaF 2 and SrCI2, the fluorite structure, (B) BaBr 2 and (C) BaCIF. (After Wells <1)).

of which lie at the apices of a trigonal prism, and the remaining three beyond the centers of the three prism faces. Barium chloride fluoride has the same type of structure as PbFC1 and PbFBr. (1) Each layer in the BaC1F structure consists of a central sheet of coplanar F atoms with a sheet of C1 atoms on each side, and the metal atoms between the C1--F--C1 sheets. The plan of a layer is shown in Fig. 2.1c together with the sequence of atoms in a direction perpendicular to the layers. Within one of these complex layers, a Ba atom is surrounded by four F and four C1 atoms, but the various interatomic distances in these structures suggest that there are appreciable forces between the metal atoms in one layer and halogen atoms in a neighboring layer. (~) At room temperatures, the alkaline earth halides are highly insulating solids, the forbidden band-gap of CaF 2, for example, being 10.6 eV. As the temperature is raised, the electrical conductivity increases sharply. This conductivity is due to the diffusion of ions as evidenced by the fact that electrolysis of the crystal takes place (Sections III and IV), its constituent elements being liberated at the electrodes in conformity with Faraday's law (see Section IV).

138

FRANCIS K. FONG

Pure alkaline-earth halides are transparent from the far ultraviolet into the far infrared. The ultraviolet absorption is due to electronic transitions corresponding to the liberation of valence band electrons into the conduction band. The infrared absorption is attributed to the vibrations of the constituent ions. Chemical impurities, when incorporated in the alkaline-earth halides even in parts-per-million concentration, may sometimes give evidence of their presence by introducing absorption bands in the normally transparent spectral region. Also, defects and centers caused by mechanical deformation or irradiation with ionizing particles invariably give rise either to an extension towards longer wave-lengths of the tail of the first ultraviolet absorption bands, or to discrete bands in the region of normal transparency. In this respect, probably no other mineral shows a greater variety of colors than the fluorites. Not only does the color of a fluorite depend on its place of origin, but frequently various layers in one and the same crystal can be different. ~3) Upon ejection of valence electrons into the conduction band (which is equivalent to the transfer of valence electrons from the halide ions to the metallic ions) by the absorption of ionizing radiation, "holes" in the outermost full band of the crystal are formed. As a hole arises from an electron deficiency caused by the removal of an electron from a halide ion, it behaves like a positive carrier of electricity, and is to be distinguished from a lattice vacancy (caused by the absence of an ion from its lattice site) which, unfortunately, is also referred to as a hole by some writers. As a consequence of Pauli's exclusion principle, which requires that in the case of a full band for every electron moving in one direction there must be another electron moving with the same velocity in the opposite direction, no current can flow in a full band. Upon removal of an electron from the outermost p-orbital of a halide ion, however, an electron can migrate over from a neighboring halide ion without any energy change, the potential barrier between the two atoms being completely transparent to the electron. The process is propagated as the hole wanders freely through the crystal. Irradiation of the crystal with the proper ionizing energy, therefore, should induce "photoconductivity". In the event that the light adsorbed in the longer wavelength ultraviolet bands is not energetic enough to cause separation of the electron and the positive hole, however, an excited state of the crystal can be attained without the production of photoconductivity. This excited state is known as an "exciton" (first named by Frenkel), an electrically neutral entity in which the positive hole is bound to the electron by Coulomb attraction and free to wander through the crystal. The exciton states of an insulating crystal are analogous to the energy states that give rise to sharp line absorption in a vapor of free atoms. The sharp lines lead up to a series limit, beyond which there is a continuous absorption band. In this respect, the absorption spectrum of a nonmetal is in sharp contrast to that of a metal, which consists of continuous bands only.

Rare-earth impurities in alkaline-earth halides

139

The incorporation of impurities or the formation of lattice defects introduces localized energy states in the forbidden gap between the conduction and valence bands. The observed impurity absorption bands or color center bands involve electronic transitions between these states or between these states and the conduction or valence band. Absorption spectra due to such causes change as the character of the impurity ion or the defect changes, and sometimes provide a convenient means for the study of chemical reactions in the solid state (Sections VI and VII). Various types of impurities and trapped electrons may be distinguished as follows :(4) 1. A foreign negative ion replacing one of the negative ions of the perfect lattice. If the electron affinity of the foreign ion is less than that of the ions of the perfect lattice, the foreign ion should give rise to a series of lines leading up to a series limit. 2. A foreign positive ion replacing a cation of the perfect lattice. If the ionization potential of the foreign metal atom is greater than that of the constituent metal atoms, less work is required to bring an electron from an adjacent negative ion onto the foreign ion than on to an ion of the perfect crystal. A new absorption band may thus be expected to the long-wavelength side of the continuous absorption. 3. An electron trapped at an anion vacancy. This entity gives rise to what is known as the F-band absorption. 4. An electron trapped in the field of an interstitial ion. This entity should again give rise to a line spectrum leading to a series limit. \ \

\

\

\

\

\

\

\\\\\\CO~DDCT,O. B ~ o , \ \ \ \ \ '"' :';'I""'"'""'"'; ""'~" ....... " : " ' " ....... "';,: "[ EXCITON ... : :..;.L..= :..: ....,:..-..... ~:.;.}.. ;=.;.-.-;, : : . : ; : . : jBANDS ___ 3'4JI

~

- - - EXCITED

~V~LENCE B y N ~ FIG. 2.2. Schematic representation of the conduction and valence bands, excitation bands, and various defect and impurity centers. The numerals correspond to the discussion in the text.

5. A positive hole trapped at a cation vacancy. The negative ions adjacent to the vacant lattice point have one less than their full complement of electrons. Absorption of light excites an electron up from the full band into this vacant level, again giving rise to a line spectrum leading up to a continuous absorption. The above description is summarized by the schematic representation shown

140

F~ANCISK. FONO

in Fig. 2.2. Extended discussions of these various types of absorption have been made in the text by Mott and Gurney t4~ whose expositions pertain more directly to the alkali halides, although the basic concepts should apply equally well to the alkaline-earth halides.

III. LATTICE DEFECTS AND IONIC CONDUCTIVITY IN ALKALINE-EARTH HALIDES--BASIC CONCEPTS 3.1. The Thermodynamic Presence of Lattice Defects In thermodynamic equilibrium, the free energy

F=E-

TS,

where E, T, and S are the energy, temperature, and entropy respectively is at a minimum. Although energy is expended to forma defect against the cohesive forces of the crystal, the increase in entropy resulting from the defect formation causes the free energy to be a minimum for a definite concentration of defects at a given temperature. For vanishing values of temperature T, the entropy term becomes negligibly small, and an ideal crystal may approach a perfect state in Which no imperfections or lattice defects exist. As the temperature of a crystal is raised, however, the mean amplitude of the thermal vibrations of the atoms about their mean positions increases. These vibrations cause a certain departure from periodicity in the positions of the atoms in a crystal lattice. In addition to this, there exists in a normal crystal a number of vacant lattice sites and interstitials. An interstitial is an ion situated at an interlattice position between the normal lattice sites, as the point (b) in Fig. 3.1. The representation of an interstitial in CaF2 is shown in Fig. 3.2. The ideas of lattice X

X

X

X

X

X ~ X

xbX

×

a

X X

X

X

X X X X X FIG.3.1.Schematicrepresentationof a Frenkeldefect. defects involving lattice vacancies and interstitials were developed mainly by Frenkel, ts~ Wagner and Schottky, trJ and Jost. ~7) Two major types of defects may arise from lattice vacancies and interstitials, namely, Frenkel defects and Schottky defects. A pair of Frenkel defects is illustrated in Fig. 3.1. An atom of the crystal can leave its normal position (a) and travel to an interstitial position (b), such that there is no interaction between

Rare-earth impurities in alkaline-earth halides

141

the atom at (b) and the atoms round (a). The Frenkel disorder consists of such a pair of lattice vacancy and interstitial. The Schottky defect is illustrated in Fig. 3.3. An atom having left its normal lattice site can move to the surface of the crystal, adding to the outermost . , "- . . . . .

. . . . . . . . . . . . . . ~-;-.,, I.

.

.

.

"

i

! i i , i

I i i i

o CQ

(3 F

Flc. 3.2. Representation of an interstitial F - ion (shaded circle in the center) in CaF2,

layers of the normal crystal lattice. The formation of Schottky lattice defects increases the volume and decreases the density of the crystal, though not necessarily affecting the crystal lattice constant. In the following paragraphs, the theoretical calculation of the numbers of Frenkel and Schottky defects will be formulated for the alkaline-earth halides with the empirical formula MX2.

x

x

~ /

x

x

x

x

(x

x

x

x

x

FIG. 3.3. Schematic representation of a Schottky defect.

As a first approximation, it is assumed that the crystal M X 2 is stoichiometric, such that the component M exists in a number that is exactly one half that of the component X. The perfect crystal MX2 of a given number of molecules is taken as the standard state, for which the partition function at temperature T Z ( T ) has been constructed. It is further assumed that the states of any given imperfect configuration differ only by a constant energy W from the corresponding states of the perfect configuration, so that they possess the partition function Z(T) exp ( - W/kT), ~s) where k is the Boltzmanrl constant. We define N as the number of MX2 molecules, from which we have NM(= N) M 2+ ions and Nx(= 2N) X- ions. Let N ~ be the total number of all lattice sites; there will then be ~ N t lattice sites for M and ~ N ~ lattice sites for X. If we now let N v and N v be the number of M and X lattice vacancies, and N~

142

FRANCIS K . F O N G

and Nix the number of M 2 + and X- interstitials, respectively, we write ½N'=

NM + N v -- NiM,

and

(3.1) -} NI = N x + NV - N~ = 2NM + 2 N v - 2 N ~

Since Nx = 2N M = 2N, we have 1 N v - N~ = ~(N xV - N~).

(3.2)

We further define N i = aN l as the number of interstitial sites which, we now assume, are available for both M 2+ and X- ions. Let E v be the energy required to remove an M 2 + ion from its lattice site to a new surface lattice site, leaving an unoccupied lattice site or vacancy. Let E~ be the energy required to take an M 2+ ion from.a surface lattice site to an interstitial position. Let Exv and E L be similarly defined for a X- ion. Then, if the various imperfections do not interfere with one another, the extra energy W of any disordered configuration is W

v

v

i

i

= NME M + NME m +

N xvE xv + N xi E xi

(3.3)

The total number of distinct configurations belonging to this example is (½Nt)! (]NI)! ONV, N~, N V,N~ = N V ! (½N t v I v, 2 1 -- N M ) . N X . ( ~ - N -- N v ) ! (3.4) (tiNt)! X NiM ! N~ ! (aN t - N ~ - Nix)! where the first factor in (3.4) is the number of ways N v lattice vacancies can be chosen from a total of N~ (= ½N t) M lattice sites; the second factor is the number of ways Nxv lattice vacancies can be chosen from a total ofN~x (= -~Nl) X lattice sites; and the third factor is the number of ways N~ and N~ interstitials can be arranged on the available aN t interstitial sites. Since any degree of imperfection may be present, the complete partition function for the crystal with all its imperfections will be Z ( T ) F(T), where F(T) = ZN~ ' Nh, N~,,N~, f2 Nf,, N~,,N[, N~ exp (-- W / k T)

(3.5)

in which t2N~,N', NxLN; and W are defined in (3.3) and (3.4). The summation (3.5) is restricted by the conditions (3.1). Since it is not possible to sum the series for F(T) in (3.5) in finite terms, we turn to the approximation of determining the maximum term in the series for F(T) and replacing log F(T) by the logarithm of this maximum term. The maximum term is determined by satisfying d log (term) - c3log (term) d N v + 0 log (term) dN~ + COlog (term) d N v aN v aNh ~ x COlog(re + coN'rm)kd N ~ = 0,

(3.6)

Rare-earth impurities in alkaline-earth halides

143

which gives in our case log

NV

~

d

M+

+ (log eN~ -- N~M--

log

31, - , , x

NV

kT/

dN v

,~M TE' dN~

(3.7)

+(log ~N' - N~MN~x : Nk

E~XT) d N xi = O .

IfEV/kT >>1, EV/kT >>1, Eh/kT ~> 1 and E~/kT >> 1, which is generally tile case in practice, a N t or N t ~> N v, Nxv, Nh, N~ such that NM ~ ½N ~ and Nx =,., 2Nt 3 , eqn. (3.7) can be reduced to the simple a p p r o x i m a t e form, remembering Nx = 2NM = 2N, •

N IOg~MM

EV

dNV+

kT

l°gNxV

kT (3.8)

+

\log

N~

yM d N ~ +

log

kT

~x

Nix

kT

dN~=0,

which is restricted by the condition [from (3.2)] df=dN

v-dN~-

1 xv - ~dN 1 xi = 0 ~dN

(3.9)

At the m a x i m u m condition, therefore, df = 0 and fldf = 0, where fl is a constant. F r o m this it follows that d log (term) + fldf = 0, which yields the values N v , Nxv , N~M,and N~ for the m a x i m u m term in F(T)" V*

NM -- e ~ exp ( - - E V / k r ) , N

EV/kT),

(3.11 )

(-E~/kT),

(3.12)

Ux/kT),

(3.13)

NV* - (e p)- ~ exp ( -

2N

N~* _ (cO)_ ~ exp 3~N

Nix* - (e~) ~ exp ( 3~N

By substituting the values of NMv*,. .. . of. N.v, eqn. (3.2), we arrive at the equation for e ~ : e ~ exp ( - E V / k r )

- 3e (e ~)- ~ exp ( - E ~ / k r ) -

(3.10)

from eqns. (3.10) to (3.13) into

= (e~) - ~ exp ( - E ~ / k r ) ~ (e~) ~ exp ( -

E~/k r).

(3.14)

Considerable labor is involved in the solution of eqn. (3.14), and we shall satisfy ourselves with only two simple limiting cases of interest that exist. If effectively

144

FRANCIS K . FONG

all the imperfections consist of Frenkel defects for which all the lattice vacancies are matched in equal numbers by interstitials, the restricting condition (3.2) becomes N ~ * - N~* = ~r~rv* 2~"x _ N~) = 0

(3.15)

and eqn. (3.14) may accordingly be rewritten as e a exp ( - E~kT) = 3~ (ea) - 1 exp ( - E~/kT), (ea) -½ exp ( - E~/kT) = 3ct (ea)~ exp ( - E~/k T),

(3.16)

from which we obtain the solution for e a : e a = (3ct)½exp [(E v - E~t)/kT] =

(~)-1 exp [ - ( E v - E~x)/kT].

(3.17)

The equilibrium concentrations N v . . . . for N v . . . . may now be written : "

N-'-~M= N--~M= (3a)½N exp [ - ( E v + E~)/2kT],

(3.18)

N v = N~ = (6ot)½Nexp [ - ( E v + E~x)/2kT].

(3.19)

The energy (E v + E~) in eqn. (3.!8) is that required to take an ion of M out of the lattice and place it in a n interstitial position, i.e., the energy required to form a Frenkel defect involving M. The energy (E v + E~) in eqn. (3.19) m a y b e similarly described for X: it is the energy required to form an anti-Frenkel defect involving X. It is apparent from (3.18) and (3.19) that the n u m b e r of defects increases rapidly as the temperature is raised, as is to be expected. If all the imperfections are of the S c h o t t k y - t y p e for which N ~ = N~x" = 0, the restricting condition (3.2) becomes NV" = ~*xt ~rv',

(3.20)

and eqn. (3.14) may accordingly b e rewritten as e p exp ( - E ~ k T) = (e~) - ½exp ( - EV/k T),

(3.21)

from which we obtain e a = exp [~(E v - EV)/kT]. The equilibrium concentrations of the Schottky defects, ~

(3.22) and ~xx, are

N e x p [--(~EMt v + {EVx)/kT],

(3.23)

N~ = 2 N e x p [_(~EM~v + }E~)/kT].

(3.24)

Nv

The basic assumption of the above derivations, that N >> N~, N~, Nxr, or ~xx, is valid so long as the various energy terms such as (E~ + E~) are large compared with kT. Still a n o t h e r limiting case exists in which the imperfections consist of stoichiometric numbers of interstitial cations and anions, i.e. N ~ = ~£~ ~ ~vaX, . k n o w n as anti-Shottky defects. Occurrence of such defects has not been reported, however.

Rare-earth impurities in alkaline-earth halides

145

Lattice imperfections, whether vacancies or interstitials, possess mobility in the parent crystal. At appropriately high temperatures, they can move under the influence of a concentration gradient (in the case of diffusion) or an applied electric field (in the case of electrolytic conduction). These two types of ionic mobilities will be discussed in the following sections. 3.2. The Diffusion of Lattice Defects An interstitial or a lattice vacancy at rest is bound to its equilibrium position and is unable to diffuse through the crystal by jumping randomly from one site to an adjacent one. However, a certain fraction of the lattice defects will acquire, by thermal agitation, sufficient energy in excess of the barrier between equivalent sites. Effective mobility of the ions is proportional to the number of such energetic ions. In anticipation of a more precise theory, it is easy to see that the fraction of defects sufficiently energetic to migrate will be proportional to exp ( - U/kT), where U is a measure of the energy barrier between two equivalent sites known as the activation energy. Since the number of defects is proportional to exp ( - W/kT), where W represents one of the combinations found in eqns. (3.18), (3.19), (3.23), and (3.24) for the two special cases, the mobility p of either the interstitials or the vacancies can be expected to take the form # = #o exp [ - ( U + W)/kT],

(3.25)

where #o is a constant. If a concentration gradient exists between two parallel planes in the crystal a distance a apart, such that the number of, say, interstitials on one plane is n per cm 3, and on the other (n + a dn/dx), the number contained in a volume between these planes of unit cross section is intermediate between an and a(n + a dn/dx). The number of interstitials crossing from one plane to the other per second as a result of diffusion will be, if a is the distance between two adjacent interstitial positions, pa z dn/dx, where p = v exp ( - U/kT) is the probability per unit time position, v is the frequency with and U, as defined above, is the stitial positions. The value of thus given by

(3.26)

that an interstitial will move to a neighboring which it cab vibrate in its equilibrium position, energy barrier between two neighboring interthe diffusion coefficient of the interstitials is

D = pa 2.

(3.27)

The motion of interstitials or lattice vacancies causes the diffusion of that corresponding constituent of the crystal which forms them, giving rise to what is known as the self-diffusion of the crystal. The contribution made by the interstitials or the vacancies to the self-diffusion coefficient of the crystal is clearly proportional to the number of the interstitials or vacancies present in L

146

FRANOS K. FONt

the crystal. Thus, in the Frenkel case, remembering (3.18) and (3.19) for interstitials Ni u -~ a ,2 = (3~)~a~ vM i exp { -[~(EM 1 v + E~t) + U ~ ] / k T } , (3.28) Dh' = ~-t-'r~ and ~ X _ , a ~2 = (6~)½a~ Vx ' e x p { - [~(Ex ' v + E~,) + U~]/kT}, O~• = -~"Px

(3.29)

where the super and subscripts i, M, and X denote the interstitial, and the constituents M and X, respectively. Analogous equations can be written for D v and Dxv due to M and X lattice vacancies

,'2 D v = -N~ ~ puav,u = (3e)½a2v.Mvv e x p { - [ ~ ( E v + E•) + UV]/kT},

(3.30)

and

Nxv _Va2 Dv = ~ - t : x v,x = (6~)~a2,x vv e x p { - [ ~ ( E v + E l ) + UV]/kT},

(3.31)

where av is the distance separating two adjacent lattice vacancies. Note that whereas ai, s~ = ai, x = ai, av, M and av, x are not necessarily equal. Similarly, the contributions to the self-diffusion coefficient due to Schottky M and X lattice vacancies may be written, remembering (3.23) and (3.24): D v = av,M2vVexp {_ [(~E M1v + .~Ex)2 v + VV]/kT},

(3.32)

v + U~,]/kT}. D v = 2 a 2 x v [ exp { - [(½E~ + ~Ex)

(3.33)

In general, the self-diffusion coefficients take the form D = Do exp ( - ( / k r ) ,

(3.34)

where D o and ( are constants characteristic of each of the various defects described above. The deviations outlined above are valid strictly for higher temperatures at which the defect ions are mobile and their concentrations can approach the equilibrium values given by eqns. (3.18), (3.19), (3.23), and (3.24). At lower temperatures, the lattice defects are actually quite immobile due to the exponential term containing the activation energy U. It is clear that this is the case when the factor p = v exp ( - U / k T ) becomes small compared to 1 sec-1. An important consequence of this is that unless the crystal is cooled infinitely slowly from its melt, the concentrations of the various defects will always be greater than their equilibrium values at lower temperatures. 3.3. Ionic Conductivity due to Lattice Defects In the absence of an applied field, the lattice defects self-diffuse due to a concentration gradient. When a field is applied,they drift with a mobility that is typical for each given class of defects. The drift of lattice defects under an applied

Rare-earth impurities in alkaline-earth halides

147

field must, of course, also be subject to an activation energy U similar to that in the case of self-diffusion, and is important only at sufficiently high temperatures. Since lattice imperfections are intrinsically present at such temperatures according to the statistical thermodynamic arguments outlined in section (3.1), even the perfectly pure crystal must exhibit ionic conductivity. As before, let the probability of a defect ion having enough excess energy to " j u m p " to a neighboring vacant site be p = v exp ( - U/kT) in the absence of a field. The probability of the forward jump should be the same as that of the backward jump, as shown in Fig. 3.4A. In the presence of an applied field F, the forward and backward jumps become non-equivalent. In the direction

o --q

~=;=~,-u/kr A

-~=re- (U- ~ ZeFa)/kT "~-=re-(U+ ~-ZeFo)/kT B

FIG. 3.4. Schematic representation of the potential energy of a defect, which spends most of its time at the bottom of a potential trough. For the defect to make a jump to an adjacent equilibrium position, it must acquire energy equal to or greater than (A) U, in the absence of an externallyapplied field, or (B) (U +_½ZeFa)in the presence of an applied field F. of the field the probability per unit time that an ion makes a j u m p now becomes v exp [ - ( U - ½ZeFa)], whereas in the opposite direction it is v exp [ - ( U + ½ZeFa)],where Z is the number of charge on the ion, e the electronic charge, and a the distance between two equilibrium positions as before. The effect of the field upon the height of the activation energy is shown in Fig. 3.4B. The mean velocity of the defect ion is thus given by = v. exp(-

[/ | fZeFa'~ ] U/kWt Lexp 2 f- / - exp

= 2vaexp(-U/k

( ZeFo~]~/J

T rZeFa 1 (ZeFa] 3 )~2---~+~.T\2kW /

(ZeFa]' ...] +l\2kr / +

(3.35)

148

FRANCIS K. FONG

For (ZeFa/2kT) ,~ 1 which is normally the case in practice,* this reduces to v-

ZeFa2v k~exp(-

U/kT),

(3.36)

from which we obtain the expression for the mobility #"

# = v/F -

ZeaZv k T e x p ( - U/kT),

(3.37)

nZe2 a2v ~ exp ( - U/kT),

(3.38)

and that for the conductivity a :

O- -- he# -

where n is the number of defects per cubic centimeter. Writing the conductivities O-v,and O-v due to M and X interstitials, and M and X lattice vacancies in the Frenkel case is now only a matter of substituting Nh . . . . in eqns. (3.18) and (3.19) for n in eqn. (3.38). Thus i O-X, i O'M,

-----r-

= ¢3

t

NZMee4vh

kT

e x p { - [ ~ ( E v + EL) + Uh]/kT},

O-ix = (6e)½N Zxe2 a~v~ kW exp { - [ x exv +

+ U~x]/kr},

O-v = (3a)½-NZs~e2a~,Mvv exp kT { - [~(Ev + E~u) + U~]/kT},

(3.39) (3.40) (3.41)

and

O-v =

(6a)½N Zxe2 a2"x V~ exp kT {-[~(E~ + E~) + UV]/kT},

(3.42)

where N is the number of MX 2 molecules per cm 3, Z M = 2 and Z x = 1. Similarly, conductivities due to Schottky defects may be written by substituting N v and N~ in eqns. (3.23) and (3.24) for n in eqn. (3.38). It is clear that the electrical conductivity O-T of an ionic crystal should be the sum of all the conductivities (dh due to the various species h of lattice defects, Y',hnheph.) Since only a limited range of temperatures is available for the measurement of the characteristic conductivity, however, the whole of the conductivity may be due to the drift of one or two types of defects that possess the greatest mobilities. This will be the case, especially if the energy terms in the exponential part of the conductivity functions differ significantly * This is p e r m i s s i b l e since p u t t i n g e = electronic charge, Z = 1 for B, F = 300 v o l t s / c m = 1 e.s.u., a = 3 x 1 0 - a cm, a n d T = 6 0 0 ° K (at w h i c h t e m p e r a t u r e ionic m o b i l i t y b e c o m e s appreciable), we find

ZeFa 2kT

1 x 4.8 x 10 - 1 ° x 3 x 10 - a x I 2 x 1.4 x 10 -16 x 600

-~ 10 . 4 ,~ 1.

R a r e - e a r t h i m p u r i t i e s in a l k a l i n e - e a r t h halides

149

from one another. If only one type of defect contributes significantly to the total observed conductivity, the activation energy U for this type of defect must be much smaller than the activation energies for the others. If the activation energies for two types of defects differ by an amount no greater than kT, both types will make nearly equal contributions to the conductivity. If over the small available range of temperature the observed values of log a T are plotted against I/T, a straight line should be obtained irrespective of whether one type or more than one type of defect in the crystal is contributing to the total observed conductivity, t4) This is illustrated by the example in which the i (3.39) and a v (3.41) are nearly equal and two activation energies in, say, aM may be written respectively, as Uh = Uo + ~,

and

u~

=

Uo -

(3.43)

z.

The total conductivity ar may be written, employing eqns. (3.39), (3.41), and (3.43) as ar --

( 3ct)~N Z u e 2 exp {_[~(E M , v + E~) + U o ] / k T } [ a ~ v ~ e x p ( - x / k T ) kT

2 v + aV,MVM Setting aZViM ~ aV,MVM2 v =

(a2V)M,

exp (X/kr)].

(3.44)

eqn. (3.44) takes the form

ar = 2% exp { _ [~(E M 1v + E~) + Uo]/kT } cosh z/kT.

(3.45)

If in the small temperature range around T, the value ) d k T ,,~ 1, a r = 3a o exp { - [½(Ev + E~) + Uo]/kT}.

(3.46)

It has been established (9) that in potassium chloride both cations and anions contribute significantly to the total observed conductivity at high temperatures. It seems that at these temperatures the Schottky mechanism predominates, for which the activation energies U + and U- do not differ very much. (4) In silver chloride, on the other hand, the electrical conductivity is due largely to the drift ofAg + ions. (~°' TM The relative mobility of the ions M and X is determined by the transference number. The transference number of ion M tM is the fraction of the total current carried by that ion, and may be defined by the equations : ktM /tM + PX

tM - - _ _ ,

tM +

tx =

1.

(3.47)

A description of the classical method of measuring transference numbers for ionic crystals first employed by Tubandt et al. ~1°) may be found in the review by Liddiard. ~m For ionic crystals, the metal released at the cathode when a current is passed

150

FRANCIS K. FONG

is often deposited in the form of dendrite within the crystals. This occurs especially when too high a voltage is applied at lower temperatures. When precautions are taken to remove this difficulty, consistent values of conductivity are obtained at high temperatures. For some substances, however, the values obtained at lower temperatures may vary from one specimen to another by factors greater than 102. (4) When log a is plotted against l/T, nearly all ionic crystals exhibit a break in the slope of the plot. The lower temperature range log tr vs. 1/T plot depends on the lack of purity, and is known, somewhat unfortunately, as the "structure sensitive" range. Figure 3.5 shows the log TEMPERATURE *C 700 10-211 \

500 I

400 I

500 I

225 I

t52 I

10-3

10 -4

§ 10"5

CoF 2

10-6

1.0

12

1.4 dT

1.6 18 iOa(*~ ~)

2.0

2.2

FIG. 3.5. The electrolyticconductivityof CaF 2, SrF2, and SrC12.(AfterTaylor and

Barsis.(TM)

tr vs. 1/T plots obtained for "pure" CaF2, (13) SrF2, and SrC12.t14) In each case, except for the minor third break in the SrC12 plot, the curve may be represented by the formula (in cm- 1 _ o h m - 1) ¢rMx2 = ~1 exp ( - ~ / T ) + (g exp ( - ~ / T ) ; ~ ¢ >> c~,~ > ~ ;

(3.48)

Rare-earth impuritiesin alkaline-earthhalides

151

where M and ~ refer to the higher temperature region, and ~ and ~ correspond to the lower temperature region. While ~¢ and ~ are temperature-independent constants characteristic of the substance, ~ and ~ may vary from sample to sample. For this reason the higher temperature conductivity is usually known as the "intrinsic" or "characteristic" conductivity of the crystal. Although it was postulated decades ago that the "structure sensitive" or "extrinsic" conductivity was due to a small number of mobile ions moving over the surfaces of internal cracks ~15) (hence the term "structure-sensitive"), an alternative explanation is found(16) in Koch and Wagner's postulate that the low-temperature conductivity appears to be due to a trace of impurity in the crystal, acting in such a way as to create the presence of a number of permanent vacancies (or interstitials). If the concentration of these is c, and that of the intrinsic thermal vacancies (or interstitials) is n, the conductivity due to all the defects will be given by

aT/e

=

(n -3v c) riO exp ( - U/kT) = no~to exp [-(½W + U)/kT] + Cpo exp ( - U/kT),

(3.49)

where W is the energy of formation of the intrinsic defect. This has the same form as the empirical equation (3.48), and offers a means of separating the activation energy U from W. It may be noted that whereas the earlier postulate by Smekal t15) requires the additional term to arise from a small number of ions which could move along cracks or surfaces of misfit in a mosaic, the Koch-Wagner postulate t16) requires that the term arises because U is necessarily smaller than (½W + U). The role of divalent (Cd, Ca, Sr, Ba, Mg) halides in alkali and silver halides has been investigated by WagnerJ TM MaurerJ TM Haven,(19) Kelting,t2°~ Tetlow,(21) and others, and there remains little doubt that the presence of these divalent ions has a definite effect upon the "structure sensitive" conductivity of the host materials. The effect of impurity ions upon the conductivity of alkaline-earth halides will be discussed in detail in Section 4.2. 3.4. The Relationship between Diffusion and Electrical Conductivity The interstitial (or lattice vacancy) is free to diffuse in the crystal with a diffusion coefficient D = NDs/n where Ds is the "self-diffusion" coefficient discussed in Section 3.2, n the concentration of the defect per cubic centimeter, and N the number of AB 2 molecules per cubic centimeter. A comparison of eqns. (3.27) and (3.37) shows that D and the defect mobility/~ obey the NernstEinstein equation

I~ _ Ze D kT

(3.50)

which can also be derived as follows. Suppose the particles (of charge Z) are

152

FRANCIS K. FONG

in equilibrium in a field F, of which the potential is tp. Consider the case in which F varies only in the x direction such that F = - ~q~l~x. The number of particles n(x) per unit volume is given by the Boltzmann's law, n(x) = n o exp ( - Zeq~/k T).

(3.51)

Since in thermodynamic equilibrium no current is flowing, IznZeF -- - Z e D 0n(x)

(3.52)

fix

which on integration gives n(x) = n o exp (-ktqg/D),

(3.53)

we obtain by comparison with eqn. (3.52) Ze D - kT

which is identical with eqn. (3.50). Equation (3.50) gives the relation between/~ and D of a given type of defect diffusing through the crystal, an analogous relation between the contribution of one constituent of a crystal to the total observed electrical conductivity and the self-diffusion coefficient of that constituent, both of which depend on the number of the corresponding defects. From eqns. (3.28) and (3.39), for example, we see that ~/Ds = Ne2/kT,

(3.54)

from which we obtain the approximate relationship tr -~ 3 x 104D~,

(3.55)

letting T ,-~ 600°K, and N ,~ 1 0 2 2 . It is obvious that for eqns. (3.50) and (3.54) to be valid, #, tr, and D or Ds must pertain to the same mechanism, for only then will the exponential term occurring in the expressions for/~ and D or and Ds be the same. Once assured of this condition, one can readily estimate the value of, say a, provided that D~ is known. Further discussions on the significance of these equations will be made in Section 4.1. 3.5. T h e Pre-exponential Terms in Expressions for the Mobility, the Diffusion Coefficient, and the Equilibrium Defect Concentration

In the preceding sections we see that the equilibrium defect concentration, the mobility, and the diffusion coefficient of a lattice defect may be written respectively as

Rare-earth impurities in alkaline-earth halides

153

N h = N O exp ( - Wh/kT),

(3.56)

/~h = /~o exp ( -- Uh/kT),

(3.57)

exp ( - Uh/kT),

(3.58)

and

D h -- D O

where Wh is the energy of formation of the lattice defect h, U h the activation energy for the jump of the defect from one equilibrium site to another, and the pre-exponential terms N o, Po, and Do have been defined in eqns. (3.18) to (3.24), eqn. (3.37) and eqn. (3.27) respectively. In the case of the diffusion coefficient according to (3.27), Dh

=

a2ph = a2vh exp (-- U h / k Z )

(3.59)

or

(3.60)

D O = a2vh

The frequency factor may be estimated by the assumption that as an ion moves from one equilibrium position to an adjacent one, its potential energy varies in a simple sinusoidal manner. Thus, if x is the coordinate along the path of the moving ion, the potential V(x) may be written as V(x) = ½Uh(1 - cos 2rcx/ah)

(3.61)

where Uh is the height of the potential barrier and assumed to be equal to the energy of activation, and ah is a lattice constant. The frequency factor v is then given by v = (Uh/2ma2) ½ ~ 1013 sec- 1.

(3.62)

Since ah is of the order of 10 -8 cm, we have, according to eqn. (3.60), D O ~ 10- 3 cm 2 sec- 1.

(3.63)

In practice, however, Do may vary anywhere in the range from 0.001 to 100 cm 2 sec-1. Evidently, there exists a missing factor which should account for this discrepancy. One approach to the problem is treating the jump of a defect ion from one equilibrium site t o another as a simple rate process* as described by Eyring. (22' 2a) This theory gives for the probability of jump per second. kT Z ~

P-

h Ze

(3.64)

where Z * is the partition function of the system with the ion.in the "activated state" midway between the two adjacent equilibrium sites at the height of the potential barrier. In this activated state, the ion is confined to move in a plane which is normal to and bisects the line joining the two equilibrium positions. The partition function Ze is that of the system with the ion in an equilibrium • The quantitative formulation of the activated complex theory, first employed for chemical reactions, has been applied to a wide variety of other rate processes such as viscous flow, dielectric relaxation, internal friction in polymers as well as diffusion.

154

FRANCIS K. FONG

position without, any constrairlt placed upon the ion, which will therefore have one degree of freedom more than when it is in the activated state. Equation (3.64) can be rewritten as

kT

P-hZv

Zt

k T KS

(3.65)

where Z~ is the equilibrium partition function of the system with the ion confined to move in a plane normal to the line joining the two equilibrium positions and passing through an equilibrium site. If the activated state is in equilibrium with the equilibrium state, we have the equilibrium constant K s = Z°/Z~. The partition function Zv is that for a single linear oscillator in the direction of the jump. Recalling from elementary statistical mechanics, t24~ exp ( - hv/2kT) Zv = 1 - exp ( - h v / k T ) ' where v is the frequency of vibration of the ion in an equilibrium site, we have

hv Z~- 1 = 2 sin h(hv/2k T) ,~ -~-~, hv >>k T.

(3.66)

K * -- exp ( - AF*/kT),

(3.67)

Since where AF* is the free energy change in the transfer of one ion from a constrained two-dimensional vibration about an equilibrium position to a constrained two-dimensional vibration about a divide in a plane normal to the line joining the two neighboring equilibrium positions, we obtain upon combining eqns. (3.66) and (3.67) with eqn. (3.65), p = v exp ( - A F / k T ) = v exp (AS/k) exp ( - A E / k T ) ,

(3.68)

where AE and AS have obvious interpretations. It remains for us to show that AE can be identified with the empirically determined activation energy U. The experimental activation energy U for the diffusion coefficient D, for example, is defined as U = - kd(ln D)/d(1/T).

(3.69)

Substituting now eqn. (3.68) in (3.69), setting

D = a2v exp (AS~k) exp ( - A E / k T ) , one finds U = T 2 dAS

dAE + AE - T-d-~-- = AE,

(3.70) (3.71)

since T(dAS/dT) = dAE/dT. It is to be noted that this identhy is valid whether or not U is a function of temperature. This was first shown by Wert and Zenert25, 26) and Haven and Van Santen. t27~

Rare-earth impurities in alkaline-earthhalides

155

The coefficient Do in eqn. (3.58) should therefore, according to the above treatment, contain the factor exp(AS/k) due to the entropy of activation, which may similarly be introduced into the expressions for N h and/~h in eqns. (3.56) and (3.57). A positive activation entropy AS means that the entropy of the system in the activated state is ~greater than that of the system at rest. Although one can argue that an ion in the activated state is more loosely bound to the lattice than it is in its equilibrium position, and that, therefore, a corresponding increase in the entropy of activation could account for the large value observed for Do, one must not be too enthusiastic about such an interpretation. It is, first of all, difficult to conceive of the attainment of a metastable equilibrium state during, the jump of a defect ion from one site to another especially if the jump is a rapid one. Then, also, the assumption of an equilibrium between the activated system and the system at rest (which is essential in the treatment outlined above) may have doubtful validity. These objections have considerably less substance in the case of chemical reactions in which reaction intermediaries do exist, and the entropy of activation is more than a mere empirical factor. A different and, perhaps, more realistic approach to account for the largeness of the pre-exponential terms such as Do has been made by Mott and Gurney. ~4) In the derivations given in the previous sections, two assumptions were tacitly made : 1. The crystal is held at constant volume, so that the energies of activation are independent of temperature. 2. The vibrational frequencies of the solids are unaffected by the presence of vacancies and interstitials. The first assumption need not be made if we write the activation energy U as dUo U = U o + VoflT d V '

(3.72)

where Vo is the volume for which U is a minimum, Uo the activation energy at the absolute zero of'temperature, and fl the coefficient of thermal expansion. Hence, we obtain for the diffusion coefficient

D = va2C exp ( - Uo/kT),

(3.73)

where C is given by C = exp

k

(3.74)

dV/. i

The removal of both assumptions gives the following expression for Nh:

N'--~h= VNoB exp (-- Wo/k T),

(3.75)

where No represents the pre-exponential term before the refinement [No = (3~)~N in, for example, eqn. (3.18)],

156

FRANCIS K. FONG

and

y = (v/v') ~, B=exp

(

k

(3.76) dV]"

(3.77)

In eqn. (3.76), we suppose that each atom has ~ neighbors, and that the vibrational frequencies of the ~ neighbors of, say, a Schottky vacancy are v as before in the two directions perpendicular to the line joining the neighbor to the vacancy, but some smaller value v' parallel to this line. In eqn. (3.77), B has the same order of magnitude as C in eqn. (3.74). Similarly, we can derive refined expressions for the self-diffusion coefficient Ds, the mobility #, and the conductivity tr. In the expression for the self-diffusion coefficient, for example, all three correction factors, namely ?BC, are involved, since Ds consists of a product of the diffusion coefficient D [eqn. (3.74)] and the concentration N h [eqn. (3.75)]. The product vBC may be of the order t+~ 10 to 104, thus accounting for the large observed pre-exponential terms in the expressions for the quantities discussed above. IV. E X P E R I M E N T S C O N C E R N I N G L A T T I C E D E F E C T S IN CaF2 AND SrCI2

The knowledge gained by the many investigations of alkali halide lattice defects lends itself to the study of lattice defects in the alkaline-earth halides. Of the alkaline-earth halides, only CaF2, ~13~ SrF2, t2m SrC12,t29~ and BaBr2 t3°~ have been investigated; and of these, only CaF2 in any detail. It is therefore logical for us to place emphasis upon discussions concerning CaF2 and other compounds with the fluorite structure. As indicated in Section III, lattice defects in CaF2 may occur in four limiting cases of interest: (1) The Frenkel disorder type involving equal concentrations of interstitial positive ions and positive ion vacancies, (2) the anti-Frenkel disorder type involving equal concentrations of negative ion interstitials and vacancies, (3) the Schottky disorder type consisting of positive and negative ion vacancies in the same ratio as the ions occur in the crystal, and (4) the anti-Schottky disorder type consisting of positive and negative ion interstitials in the same ratio as the ions occur in the crystal. If the energies of formation of the various types of defects are similar in magnitude, more than one type of disorder may occur. In this case, however, the term ea in eqn. (3.14) cannot be easily solved, and the concentrations of the various defects cannot be readily calculated.

4.1. Anti-Frenkel Defects and the Transference Numbers Perhaps the most convenient method of investigating lattice vacancies is to study them in crystals doped with ions of valencies __+1 different from that of the corresponding host iolas. If the disorder in CaF2 is of the anti-Frenkel type, the solution of substitutional aliovalent cations gives rise to the following

Rare-earth impurities in alkaline-earth halides

157

manners of charge compensation : KI

14.1)

M 1+ + Fv ~ M I + : F v and K2

M 3+ + F7 ~ M 3 + : F / -

(4.2)

where M 1+ and M 3 + represent the aliovalent impurity cations which may be compensated respectively by F - vacancies, Fv, and F - interstitials, FT. The subscripts V and i denote vacancy and interstitial, respectively. Under thermal equilibrium, the impurity-defect pairs may either be dissociated [left-hand side of (4.1) and (4.2)] or be associated as complexes [right-hand side of (4.1) and (4.2)] according to the equilibrium constants K1 and K2. On the other hand, if the disorder is of the Schottky-type, the aliovalent cations will most probably be compensated as follows: K3

M 1+ + Fv

and

~ M 1+ :Fv, K4 2M 3+ + Ca 2+ ~ M 3+ :Ca 2+ + M 3+

(4.3) (4.4)

K~

(M3+)2 :Ca2v+ In (4.4), we note that two M a+ ions are compensated by one Ca 2÷ lattice vacancy, the two equilibrium constants K 4 and K 5 represent first and second association constants of the impurity M 3 + ion with the Ca 2 ÷ vacancy. Similar equations may be written for the cases of Frenkel and anti-Schottky defects. In the Frenkel case, we have K6

K7

2M ~+ + Ca 2+ ~ M 1+ :Ca 2+ + M 1 + ~ (M~+)z:Ca 2+, and

Ka

K9

2M 3+ q- Ca 2+ ~ M 3+ :Ca 2+ + M 3+ ~- (Ma+)2 :Ca 2+. Finally, we have in the

(4.5) (4.6)

anti-Schottky case, KIo

2 M 1+ + Ca~ + ~ M I + + M 1+ :Ca~+ r " (M1+)2 :Ca 2+ ,

and

(4.7)

KI2

M 3+ + F~-

~ M 3+ :F~-

(4.8)

Zintl and Udgard ~ax) concluded from density and X-ray lattice constant measurements that the solution of YF 3 in CaF2 introduces F - interstitials. The effect upon the lattice parameter caused by the incorporation of trivalent impurities in compounds with fluorite lattices has been investigated by Croatto and Bruno t2a) and Meyer. t32) It is apparent that if the trivalent ions in the fluorite lattice are charge-compensated by anion interstitials, the density of the crystal should increase with increasing concentrations of the impurity.

158

FRANCIS K. FONG

Croatto and Bruno, ~2s) in their studies on the conductivity of SrF2 (an anionic conductor) containing up to 10~ LaFa, measured the lattice constants and densities which agreed well with calculated values of densities obtained from the lattice parameter and the assumption of an excess number of F - interstitials equal to the number of La 3+ ions. The good agreement between calculated and observed values justifies the assumed compensation mechanism. Ure found that the conductivity increases appreciably upon the addition of YF 3 in CaF 2 and that the transference number [-eqn. (3.47)] of F - in the YFa-doped crystal is close to unity. More recently, the Zintl-Udgard conclusion was re-established by Short and Roy ¢33~ from a comparison of measured pycnometric densities with calculated X-ray densities over the range 0-40 mol. ~ YF3. These experimental observations indicate strongly that the y3 + cations are, to a large extent at least, charge-compensated by F - interstitials. This implies that the major lattice defects can either be of the anti-Frenkel type [eqns. (4.1) and (4.2)] or of the anti-Schottky type [-eqns. (4.7) and (4.8)]. To distinguish between the two cases, the compensation behavior of monovalent cation-doped crystal must be determined. From Ure's experiments, it was observed that conductivity increases upon solution of NaF in CaF2. Since F has a transference number close to unity in these crystals, evidence points to the compensation behavior as represented by (4.1) and we conclude that the predominant defect in CaF 2 crystal is of the anti-Frenkel type. In the above discussion, the determination of the transference number of F - in CaF2 is crucial for the conclusion that (4.1) and (4.2) rePresent the chargecompensation behavior in mixed CaF2 crystals. Two techniques were employed by Ure "3) for the determination of the transference number. In the first approach, several transference number measurements were made on NaF-doped CaF 2 at 640°C, using the method of Tubandt, Reinhold, and Liebold. t9) A stack of pellets consisting of three BaCI 2 pellets, three CaF 2 pellets doped with NaF, and three additional BaCI 2 pellets was placed between two electrodes. After heating to temperature, a measured amount of charge was passed through the stack of pellets. Since it is known that only the C1- ions are mobile in BaC12,t3°~ the fraction of the current carried by the F - ion in CaF2 can be calculated from the change in weight of the pellets. From these measurements, the transference number of F - in CaF2 was found to be 0.98 which differs from unity by about the usual experimental error in this type of measurement. This result agrees with results obtained from similar experiments made on other substances with the fluorite structure, such as SrF 2 and BaF2, ~3°) in which the anions also have a transfer number of close to unity. The validity of Ure's second approach in obtaining the transference numbers requires some consideration. The Nernst-Einstein equation (Section 3.4)

tr/D~ = NZe2/kT

(4.9)

is clearly of great use for the confirmation of conduction mechanism and the

Rare-earth impuritiesin alkaline-earthhalides

159

type of lattice disorder, especially since both quantities a and Ds can be obtained experimentally. The diffusion constant is most commonly measured by following the diffusion of a radioactive isotope into the crystal, as noted in Section 3.4, eqn. (4.9) unless ionic conduction and self-diffusion occur by identical mechanisms. The use of the Nernst-Einstein equation by Ure to determine the transference numbers of Ca z + and F - , appears unjustified at first sight, since no precise knowledge of the mechanisms of ionic conduction or diffusion is at hand. Two obvious possibilities exist whereby processes contribute to the diffusion without contributing to an electric current. In the first place, vacancy-pairs or mutually compensating impurity-defect pairs (e.g. Na + :Fv in CaF2) can contribute to diffusion without contributing to ionic conductivity by virtue of their charge-neutrality. In the second place, adjacent lattice ions may exchange places with one another without the creation of lattice defects. Such ionic motion involving adjacent lattice ions, if it occurs, should be detected by the migration of radioactive isotopes into the crystal. Clearly, in this case, the self-diffusion coefficient Ds as obtained by the radioactive isotope technique consists of two parts Ds -- (l/N) (nla2vl exp ( - U1/kT) + n2aZv2exp ( - U2/kT),

(4.10)

where the subscript 1 corresponds to the jump of lattice defects, and the subscript 2 corresponds to the interchange mechanism described above. If U x < U2, mechanism 1 predominates, and diffusion takes place by the jump defects. If UL "~ U2, both mechanisms will contribute to the measured diffusion of the radioactive material through the crystal. If U1 > U2, the interchange process will contribute predominantly to the diffusion. In the latter case, the mobility deduced from the diffusion will disagree with that deduced from electrical conductivity which should be the smaller value of the two--for when a pair of ions of the same charge exchange places, no contribution to an electrical current has been made. Obviously, then, Ure's procedure of determining the transference number of Ca 2 + in CaF 2 by the employment of eqn. (4.9) cannot lead to a precise result, although his conclusion, tCa ~ l, is quite valid. 4.2. Conductivity of Impure Crystals Several cases of interest concern us in a discussion on conductivity of impure crystals. In Section 4.1 we mentioned the various possibilities of chargecompensation assuming that the lattice defects are of a particular type [see eqns. (4.1) to (4.8)]. It suffices here to discuss the case of impurities in host crystals, the intrinsic defects of which are of the anti-Frenkel type. Crystals with other types of defects can be treated in a very similar manner. Let ni and nv be the molar fractions of the two complementary defects (Finterstititials and F - vacancies in the case of CaFz). These quantities can be related in a solubility formula

160

FRANCISK. FONG (4.11)

nin v = K o I - n 2,

where no is a constant, and K o is the equilibrium constant in the process Ko

F7 + Fv ~ perfect crystal.

(4.12)

It has been established from experimental observations t2s,31,33) that the addition of trivalent impurity ions adds to the number of F~-, such that (4.13)

ni = nv + y,

where y is equal to the mole fraction of the trivalent impurity, assuming that one F - interstitial is added to the system with the addition of each trivalent ion. Equation (4.11) now becomes ni(n~ - y) = n~,

or

(4.14) n,=~

1+

1+~-}].

This equation clearly expresses the transition from the intrinsic region of conductivity where no >> y to the impurity controlled region of extrinsic conductivity where no '~ y. Since the observed electrical conductivity is given by the sum of the contributions from F - interstitials and vacancies, (4.15)

a T = a i "~ a V = N F ( n d l i q'- n v # v ) ,

where NF is the number of F - ions per unit volume, and/~i and/~v are the mobilities of the complementary defects. Combining eqns. (4.13), (4.14), and (4.15), we obtain

{JE(' / I ~ n°

er=NFno(#i+#v)

+ 1

Y (--

2no'(

(4,16)

where ( = I~v/l~i. The quantity NFno(Pi + #v) is the intrinsic conductivity observed in the high temperature range for a pure crystal. The contribution due to the impurity content at lower temperatures may be expressed as the ratio trT/aO, where trot= ~o

trr

= /[(~noY

Nrno(l~i + #v)

1] +

y (-1 2n o ( + 1

(4.17)

In the limit of large impurity content, y >> no, ar %

--

=

y no(1 + ()

When y ~ no, the initial gradient of the isotherm is

(4.18)

Rare-earth impurities in alkaline-earth halides

d(ar/ao) _ (1 - ~), d(y/no) (y/.o)--*o 2(1 + ~)

161 (4.19)

which is positive for ~ < 1 and negative for ( > 1. When y >> no, the isotherm becomes a straight line with the slope d(aT/aO)

d(,/no) [(,i.o)__, =½[l~]+ ~ l

(4.20)

If ~ > 1, we note that a minimum occurs in the ar vs. y curve at n o ( ~ - 1) (Y)min --

(4.21)

4~

and (O'T/O'0)mi n --

(4.22)

24~

(1 + ~)

If a monovalent cationic impurity is added to the crystal as well as a trivalent impurity, eqn. (4.13) must be rewritten in order to maintain charge-neutrality: ni = nv + (y - x) = nv + A,

(4.23)

where y has the significance as above, x is equal to the mole fraction of the monovalent impurity assuming that one permanent F - vacancy is generated for each addition of a monovalent cation, and A is the difference in concentration of the trivalent and monovalent impurities. In the above derivation, eqns. (4.14) to (4.21), the quantity y is replaced by A. Equation (4.17) now becomes

Ir¢

I

= ~/k\~no/

+ 1

A

~-1

2no" ~ + 1'

(4.24)

the quantity (ar/ao) is at a minimum when (A)min = no(~ - l)/x/(, and approaches the ordinate (at A = 0) with a slope of (1 - ~)/2(1 + (). At high values of A, the isotherm becomes a straight line of slope 1

a

1-(

It should be possible, from the ratio of the slopes at high values of A and the position of the minimum, to establish the disorder (since no minimum exists in the case of Schottky defects"4)), the concentration of intrinsic defects n o, and the mobility ratio ~. By the incorporation of various quantities of K + and La 3+ ions in CaF 2, Taylor and Barsis (14) were able to determine a well-defined minimum in the (aT/aO) vs. A plot (Fig. 4.1), thus confirming the anti-Frenkel model for the defects present in CaF 2. In a similar series of experiments for SrC12, however, the minimum in ar/ao was not observed. One probable cause of the failure to observe such a minimum could be the presence of impurity ions. Oxygen ions M

162

FRANCIS

K.

FONG

situated on anion sites, for example, should either introduce additional vacancies or act as compensation for the trivalent ions, or both. The above analysis has overlooked a particular feature of the concept of charge compensation and the law of mass action. The equilibrium constants K 1 and K 2 in (4.1) and (4.2) governing the association of the impurity ion and its complementary defect ion have been neglected. Furthermore, in the case of 90 80 70 60


50

b~'4o 3o 2o

0 700oc a 680oc

~3

#

Io

-I 0

- 8

-6

-4

-2 /~Y=

0

i 2

[K" ] - [ Y ÷ ' ' o r

i 4 La " * ÷ ]

i 6

i 8

i I0

FIG. 4.1. Electrolyticconductivityin CaF2 doubly-dopedwith K + and y3+ ions. (AfterTaylor and Barsis/TM

or

La3÷

the double-doping of La s+ and K + ions, the trivalent and monovalent ions may compensate each other without introducing corresponding concentrations of permanent defects. Obviously, then, in real situations the quantities c and A cannot simply be the concentration or the difference in concentration of the impurity ions. They must also be functions of the association constants as well, since the associated defects or impurity ions are neutral in charge and do not contribute to conductivity. For a more precise description, the approach first formulated by Stasiw and Tetlow °4) and later adopted by Etzel and Maurer "s) will now be introduced. The reaction between, say, the trivalent ions and the F - intefstitials is represented by eqn. (4.2), where K2 -

nc ni(y _ he),

(4.25)

nc being the mole fraction of the complex M 3+ :Ff-, ni that of the unassociated F - interstitials, and y that of all the M 3+ ions present.

Rare-earth impurities in alkaline-earth halides

163

By combining the charge-neutrality equation y - n¢ = n i - nv, (4.26) where n v is the mole fraction of the F - vacancies present, with eqns. (4.11) and (4.26), we obtain y = no

E

1)+

H(Z 2-

(

Z-~

,

(4.27)

where H is defined by the two equilibrium constants, H = K 2 / K ~ = K z n o,

and

(4.28)

Z = ni/n o

The dimensionless quantity Z appears in the expression gT ~--- [ Z ~- ~ Z - 1 ] / ( 1 go

-4- ~),

(4.29)

where ~ = # v / P i as before. If P v ~ Pi,* Z = ni/no = aT~gO,

(4.30)

where the subscript 0 denotes properties of the pure crystal. At large values of Z, we obtain from eqns. (4.27) and (4.30) y = Lg 2 + Fat,

(4.31)

L = n o H / g g,

(4.32)

F = n o / a o = M / P N v e p i.

(4.33)

where and

By fitting experimental data of ar vs. y at constant temperature to eqn. (4.31), the quantities L and F are evaluated. The mobility #i can be obtained directly from eqn. (4.33) since the values of the electronic charge, e, the Avogadro's number Nv, the molecular weight, M, and the density, p, of CaF2 are accurately known. The equilibrium constant, K 2, is K 2 = L / F 2,

(4.34)

from which the heat of formation of the complexes, AH, can be obtained from the temperature dependence of K 2 by means of the van't Hoff equation, AH -

Rd(ln K2) d(1/r)

(4.44)

Since in reality ~v ~ /~i as indicated by conductivity measurements on * This condition exists in NaCl (a Schottky-disordered lattice with Na + and CI- vacancies) at low temperatures, for which ]~Clg <~ ~l'Nav"The data of Tubandt (Part III, Hdb. der Exp. Physik, Vol. XII, 1932) indicate that the contribution of the chloride ion in NaC1 to the conductivity is negligibly small below 400°C.

164

FRANCIS K. FONG

NaF-doped and YF3-doped CaF 2 samples;tls) however, a re-examination of the above approach appears advisable. Ure pointed out that if one sets n~ _ a r no t7 o

(1 + ~),

(4.45)

eqns. (4.31) to (4.44) are valid even for #v > #i. It is only when this condition fails that the more complicated eqns. (4.27) and (4.29) must be employed in order to evaluate the various experimental parameters. With this theoretical model, Ure obtained from conductivity measurements on N a F and YFs-doped CaF2 samples, by employing the method of least squares, the results : ~i = (1.34 x 107)T -1 exp [ - ( 1 9 + 4) x 103/T] cm2/sec, volt, and

no = 2.96 x 1025 exp [-(16.3 x 103)/T] cm -a.

Values of the various experimental parameters such as the activation energies, entropies, and heats of formation of impurity complexes are listed in Table 4.1. TABLE 4.1. ACTIVATION ENERGIES, ENTROPIES, AND HEATS OF FORMATION IN C a F 2. (The entropy of activation here has the significance discussed in Section 3.5.)

Process

F - vacancy mobility

F - interstitial mobility Density of anti-Frenkel defects Association of F - vacancy with Na ÷ ion Association of F - interstitial with y3+ ion

T (°C)

200 300 400 600 690-920

Activation Activation Heats of energy entropy formation (kcal/mole) (cal/mole deg) (kcal/mole) 20 17 14.5 12 38

_ + _ -I+

1.5 1.5 1 1 8

Entropy of formation

16 ___ 4 10 + 3 5 +2 2.5+1 21 -I- 7

640-920

65+5

200-600

1.7

690-920

27 + 4

33

The conditions under which eqn. (4.45) holds must n o w be examined. It follows from eqn. !4.29) that for eqn. (4.45) to be valid, ( Z - 1 _ uv no _ 0,

(4.46)

lh ni

which means that either/~v//h or no/n~ must be zero. Since/~v ~ #~, the validity of eqn. (4.45) requires that

n--2°= 0. ni

(4.47)

Rare-earth impurities in alkaline-earth halides

165

This is feasible at lower temperatures, at which the densities of the thermally generated defects are very low. Using (4.31), the results in Table 4.1 for F vacancies were derived from conductivity experiments carried out in the temperature range 200-600°C, in which the intrinsic conductivity of the pure crystal was negligibly low (Fig. 4.2). The results for F - interstitials, however, 10_2700°C

500

400

500

250

200

1 uE E

~-

_>

10-4

o

j

g

tO-5

c~ i-Ld W 10-6

io-71

I0

i

I

~

12

I

i

14

16

i

I

18

i

I

i

20

22

103/T (OK)-I

Electrolytic conductivity of NaF-doped CaF 2. (After Ure. (13))

FIG. 4.2.

TEMPERATURE, =c

E

~.'~"

\

--o--

y:o58% y= IOO %

b

uJ I0 -6 10-7

~ I 08

I

I I0

I

. I

I

12

I I4

I 16

103/ T {eK)-I

FIG. 4.3. Electrolytic conductivity of YF3-doped CaF 2. (After Ure. (13))

166

FRANCIS K. FONG

were obtained from measurements at temperatures so high (640-920°C) that intrinsic conductivity of the.pure Crystal is no longer negligible (Fig. 4.3). The condition (4.47) is no longer valid, and eqns. (4.27) and (4.29) instead of (4.31) were employed for the evaluation of experimental parameters. 900

io-3L'-.

700 "

TEMPERATURE, °C 500 400 300

' ~ ~

225

K~:, F'IN'TERSTITIAIIswITHy.+l. ~ OF-VACANCIES WITH Na +

IN,

.

:

alE 10-5

i03

_~ '0-° I /

~'~

~0-7/ .8

I I0

I 12

I I 1.4 1.6 103/T (OK)-I

I 1.8

I 20

103~ I0

FIG. 4.4. Temperature dependence of mobility and association constant K 2 of F interstitials and vacancies. (After Ure. tl aj)

950 I

TEMPERATURE,=C 800 I

650 I

E u 1019

b.

1018 =o

1017 0.8

I

I

0.9

1.0

I.I

I0 / T(=K) -I

FIG. 4.5. Temperature dependence of the density of defects in CaF 2. (After Ure. ~1a~)

The association constants, K2, for F - interstitials and vacancies, the mobilities, ~i and Pv, obtained by Ure are shown in Fig. 4.4 as a function of temperature. The temperature dependence of the density of intrinsic anti-Frenkel defects in pure C a F 2 is shown in Fig. 4.5. The role played by unintentional impurities such as O 2- ions has been

Rare-earth impuritiesin alkaline-earthhalides

167

emphasized by Ure "3~ and by Taylor and Barsis. (a4) Ure observed that when a contaminated sample is cooled or heated rapidly in the region between 350 ° and 600°C, an activation energy of 13 + 2 kcal/mole is observed. This is just the activation energy for conductivity observed for NaF-doped crystals. Since 0 2- and F - ions are of comparable size, a CaO molecule could go into solid solution introducing F - vacancies in much the same way as does NaF. Stockbarger ~aS~ found that CaF 2 reacts with water vapor at an appreciable rate at 100°C, and the rate increases rapidly with increasing temperature. For the more hygroscopic SrC12 or BaBr 2 crystals, the rate of hydrolysis must be considerably higher. The mechanism of the hydrolysis of CaF 2 has been investigated by Bontinck.t36) 4.3. Some Concluding Remarks The task of determining the type of disorder, the defect mobilities, the activation energies, the equilibrium association constants, and the heats of complex formation in CaF2 has not been an easy one. Most of the results listed in Table 4.1 were obtained by choosing parameters to give the best curve-fitting, and sometimes the values of a desired quantity (F- interstitial mobility, for example) t~3~ are very sensitive to slight changes in the data. Thus, it is not clear why AH for F - interstitials and trivalent impurity ions should be over an order of magnitude higher than that for F- vacancies and monovalent ions. The pre-exponential factor for the intrinsic defect density no in CaF2 has been found to be of the order of 1025 as compared to that of 1023 obtained by Etzel and Maurer ~ls) for NaC1. The experimental value of 2.8 + 0.2 eV (65 _+ 5 kcal/mole, Table 4.1) for the heat of formation of anti-Frenkel defects in CaF2, on the other hand, has been well born out by the theoretical calculations of Franklin, ~3~ who found the value to be 2.7 eV calculated by the method of Tharmalinghamtas) using the Born model of ionic solids, including coulombic, repulsive (Born-Mayert39~) potentials and polarization (monopole-dipole and dipole-dipole energies). V. COLOR CENTERS IN ALKALINE-EARTH HALIDES Lattice defects in ionic crystals constitute traps for electrons and holes that give rise to color center absorption bands. The study of color centers, particularly in alkali halides, has been actively pursued for nearly half a century. The importance of this study is evident: not only does it provide insight into many problems of lattice defects in simple ionic crystals, but also much light has been shed upon the more complicated, technologically important materials such as luminescent powders, semi-conductors, and, more recently, laser crystals. Much has been unveiled in the field of color centers by the work of Pohl ~4°) and others, whose results provide a fairly complete understanding of these

168

FRANCISK. FONG

phenomena in alkali halides. Definitive reviews of this subject may be found in the works of Pohl t4°) (1937), Mott and Gurneyt4) (1940), Seitz ~41) (1946 and 1954) and Schulman and Compton t42) (1962). Although alkaline-earth halides resemble closely alkali halides as ionic crystals and coloration in natural fluorites has been studied extensively since the nineteenth century,ca) progress has been made only relatively recently since synthetic crystals of these substances became available. 5.1. H i s t o r i c a l N o t e s ~3) Popularly known as the fluorites, CaF 2 was noted for its abundance of different hues and brilliant luminescent properties. In 1853 Kenngott14al remarked on the "variety of color usually regularly distributed and sharply separated in a very small space, a distribution which occurs in scarcely any other mineral". Violet-banded fluorite from Derbyshire, locally known as Blue John, has frequently been polished and used for purposes of adornment. Indeed, the very word "fluorescence" is derived from the word "fluorite", on account of the brilliant luminescence from many English fluorites. The distribution of colours in fluorites has been described and illustrated by Steinmetz,t44) who proposed that the sulfide inclusions are probably responsible for the coloration of these fluorites. The extraordinary hardness of the dark colored fluorite crystals established by Rexer 14s) lends support to the view that impurity is an important factor in the coloration. Berthelott461 proposed that natural fluorites could be colored by ionizing radiation. Henricht47) supported this view by noting the appearance of free fluorine with the crushing of the Wolsendorf "foetid fluorspar", as a sign of the reduction of F - ions to fluorine in CaF2. As early as 1830, Pearsall t48) observed that the natural color distribution in bleached fluorite crystals could be reproduced by exposure to light. Belar~49) carried out measurements of the radiation-coloration and bleaching of colorless fluorites from Cumberland, and observed that the saturation value of the coloration depended on the intensity of the radiation. He noted that the halflife of the radiation-induced color was of the order of 17 days in the dark at room temperature and that a stable final value was observed after approximately a lapse of 30 days. He found that the fluorite was bleached within 28 hr at 100° to 180°C, and that the color could also be readily bleached by exposure to light. In 1905, W6hler ~5°) demonstrated that fluorites could be colored additively by heating in the vapor of calcium. The additive coloration* of fluorite was further investigated by Haberlandt. t51) A detailed study of the coloration of the fluorites by Ca vapor and by solid-state electrolysis (850°C, 50 V) was made by Mollwo,ts2~ who observed two well-defined bands which he named the * The term "additivecoloration"impliesthat the crystalbecomesnon-stoichiometricby being heated in Ca vapor, additional Ca 2 + ions beingincorporatedinto the crystallattice.

Rare-earth impurities in alkaline-earth halides

169

7- and [~-bands. With this observation, we begin our discussions of the more recent developments in the investigation of color centers in alkaline-earth halides. 5.2. Calcium Fluoride Due perhaps to the greater accessibility of CaF 2 in the single-crystal form, the coloration of this material has been investigated more than that of any other alkaline-earth salt over the past three decades. In the pioneering work of Mollwo,(52) who studied the properties of color centers in natural fluorite crystals formed by additive and electrolytic coloration, several similarities were found between the color centers in CaFz and those in the alkali halides. The absorption bands obtained by Mollwo (additive coloration of CaFz with Ca vapor followed by rapid quenching) are shown for three different temperatures in Fig. 5.1. The bands designated as ~- and [3-bands were always found

Wavee l ngth (m~) 3I 2I k

~

-186"

i

0.

o

+00", 350 400 500 600 Energy (ev) FIG. 5.1. MoIlwo's absorption spectrum of additively colored and quenched natural CaF2 crystals at three different temperatures.

together. Whereas the peak of the [3-band varies only slightly with temperature, that of the m-band has a pronounced temperature dependence similar to that of the F-band in the alkali halides. Concentrations of color centers as high as 1022/cma have been obtained by the additive coloration technique, at which concentrations Mollwo observed that changes in density are easily measurable. The drift mobilities in an electric field of the centers associated with the Qt- and [3-bands are both of the order of 2-5 x 10 -4 cm2/volt-sec in the temperature

170

FRANCIS K. FONG

range of 650--840°C, which is of the same order of magnitude as that of the F center in the alkali halides. The 0t- and 13-bands were not observed in the work of Smakula, {53} who studied the X-radiation-induced coloration of CaF2 in high-purity synthetic crystals. Instead, he found at least four bands which are shown in Fig. 5.2A. Liity {s4) found that the Smakula bands could also be obtained by additive 400

580 JL

A

B 177"K)

550 600

670

i 77OK 7:50 3:$0:oo

k,x,~

Aq ' 3;o' ~;o' ,;o ',;o WAVELENGTH ~. imp)

Flo. 5.2. Additive coloration (at 850°C) of C a F 2 : the "Smakula spectrum" (A) was obtained when the sample was furnace-cooled; spectrum (B) was obtained when the same crystal was reheated at 850°C with Ca vapor and then quenched; (C) the "Smakula spectrum" was again observedwhen the sample from (B) was reheated and furnace-cooled. (After Fong and Yocom.<55>)

coloring, and reported the following observations. The Smakula bands were obtained regardless of the cooling rate at low temperatures (500-600°C) at which the vapor pressures of excess Ca are restricted to low values. If higher additive-coloring temperatures were employed and the crystals were allowed to cool slowly, the same Smakula-type absorption bands appeared superimposed on a broad "colloidal" type absorption. On the other hand, if the

Rare-earth impurities in alkaline-earth halides

171

crystals were quenched from the higher additive-coloring temperatures, the Mollwo spectrum, consisting of the ct- and [3-bands, was obtained. Liity concluded from these results that the Smakula-type absorption bands, produced by X-irradiation or by additive coloring with a small excess of Ca, were due to traces of unknown impurities. Only after the exhaustion of these impurities could the ct- and [3-bands, which according to Lfity were characteristic of the CaF2 structure, be formed. The conclusions of Liity were refuted by Smakula who showed that an identical four-banded absorption spectrum was produced by electron-beam or X-irradiation, by additive coloring, and by electrolytic coloring in the purest synthetic CaF2 obtainable. The Mollwo-type spectrum was not produced even on quenching from a high additive-coloring temperature. Recently, Fong and Yocom (Ss) observed that the coloration of CaF 2 is dependent upon the conditions under which the additive-coloring experiment is conducted, and the absorption spectra they obtained are shown in Fig. 5.2. The Smakulatype spectrum was obtained for the sample (heated at 850°C for 1 hr) which was slowly cooled in a furnace (Fig. 5.2n); but when a similar sample was quenched from 850°C by being plunged into a cold water bath, an appreciably different absorption spectrum (Fig. 5.2B) was observed. This spectrum, however, was easily reconverted to the four-banded spectrum (Fig. 5.2c) when the same crystal was reheated with Ca vapor at 850°C and allowed to furnace cool. This interconvertibility clearly suggests that for given temperatures, there are equilibrium configurations whereby the electrons can be trapped. The rapid quenching of the additively colored CaF 2 causes the freezing of the trapped electron in an equilibrium configuration characteristic for the temperature (850°C in this case) at which the additive experiment has been conducted. As the sample is reheated and allowed to cool slowly, equilibrium conditions are allowed to re-establish with the lowering temperature. The four-banded spectrum presumably corresponds to the stable configuration at lower temperatures. Since color centers arise from lattice defects, the existence of which is temperature dependent (Section III), one would expect different coloration behavior at different temperatures. Aside from the difference in the absorption spectrum, the increased intensity in absorption of the quenched sample is to be noted (Fig. 5.2B). This corresponds, qualitatively at least, to the higher densities of lattice defects at higher temperatures. The enhancement of CaF 2 coloration was also achieved by Schulman et al. from doping the crystal with monovalent ions. (s6) Figure 5.3 shows the results obtained with a single crystal of NaF-doped CaF 2 compared to a pure CaF 2 crystal of the same thickness X-rayed under the same conditions. The solution of NaF introduces F - vacancies in CaF2 (Section IV), which should be expected to enhance the colorability of the crystal. Adler and Kreta (57) showed that spectral changes from the Smakula spectrum were produced after irradiating a crystal which had been heated in air.

172

FRANCIS K, FONG

The presence of Mollwo's 0t-band observed under these conditions indicates that it is connected with the incorporation of oxygen or hydroxide, which results from partial hydrolysis of the crystal during the heating (Section IV). Bontinck (58) found that neutron bombardment produced a spectrum different from the Smakula-type absorption. A band peaking at 520 mB, 0.6

CoF2 : No ~

~

pure

0.5 "~ 0 . 4 0.3

8 o.z O.l 0 2000

i

I

5000

I

I

4000

i

I

5000

I

I

6000

i

7000

Wavelength, X ~.

FIG. 5.3. Effect of Na ÷ impurity on the X-ray-induced absorption spectrum of CaF 2. (After Schulman and Compton. t42~)

apparently identical with Mollwo's 13-band, was formed. Bontinck concluded from thermal bleaching, bleaching with polarized light, and lattice parameter measurements on additiv.ely colored crystals that Mollwo's 13-band arises from a configuration identical with the F center, and that the or-band is due to a similar configuration adjacent to an F - interstitial. Bontinck's .assignment of the ct- and ~-bands, as Przibram la) pointed out, is difficult to reconcile with the work of Smakula and others, in which these bands have not been observed even at high concentrations of color centers produced by additive coloring. In the recent work of Fong and Yocorn, it was found that gamma-irradiated CaFz crystals displayed a host of color center bands, the formation of which depended upon the radiation dosage. The absorption spectra of the irradiated samples are represented in Fig. 5.4. At radiation dosage of l0 s rad, no color bands were produced (Fig. 5.4A). It appears that CaF 2 is very insensitive to radiation, massive doses being required to initiate coloration. At 5 x 105 rad, the familiar four-banded spectrum was observed (Fig. 5.4B). It is to be noted that, although identical in band positions, this spectrum is considerably less intense than that obtained by additive-coloring (Fig. 5.2A). This is consistent with the earlier observation that large doses of X-radiation were required to form comparably low color center concentrations. (42) At 106 rad, new absorption bands became observable (Fig. 5.4c). At even higher dosages (5 × 107 rad), very intense absorption bands in the infrared were developed, while the bands at wavelengths shorter than 600 m#, with the exception of the one at 534 m/~, diminished or vanished (Fig. 5.49).

Rare-earth impurities in alkaline-earthhalides

173

The band peaking at 534 m/~ observed in high-dosage gamma-irradiated CaF2 is not the same as the [3-band (525 m/~) observed by Mollwo in his additive and electrolytic experiments (52) and by Bontinck with neutron bombardment. (Se) Whereas the [3-band is largely unaffected by temperature (Fig. 5.1), the 534 nap band shows significant temperature dependence. (55) Compared to I.O

ALL SPECTRA ~ ,_?1 ~

--

~-

]

/

0.8

6, 344

o.s

554

~

0.5

558

~~

/ \ 4:51 / / ~

O

/ .

./

/¢

/

I~.

//D

/~

CA O

I

i

5 O0

i

500

i

i

I

t

i

i

700 900 WAVELENGTH ~. (m/z)

i

i

I I O0

1500

FIG. 5.4. Room temperature gamma-ray-inducedcoloration of C a F 2. (A) 10 5 rad; (B) 5 x 10 5 rad; (C) 106 rad; (D) 5 x l0 T rad. (After F0ng and Yocom. (sS))

i.o.9

. . . . . .

"

A - INITIAL SPECTRUM ( 5 x I0 rod ) B -(ROOM CONDITIONS1 3 WEEKS

.8

C - 8 HOURS OF UV IRRADIATION

o[.

(2537')

'

]

'

l

l/

:

//lISa'

/

LI

S

/f ~

~z .5 a .4

I l/

I

tu

_~ . 3

A

.I

----

0300

I

&

~

I 400

~ 500

~ 600

i t ~ ~ ~ ~ 700 800 900 IO00 I100 1200 1300 1400 WAVELENGTH X(mp.)

FIG. 5.5. Bleaching of high-dosage gamma-ray-induced absorption bands in CaF2. (After Fong and Yocom.(ss)) the spectrum at liquid nitrogen temperatures (77°K), the room temperature (300°K) spectrum exhibits an appreciable broadening of the half width and a slight shift of the peak frequency to a lower energy, (55) a phenomenon which is well known for the F-band of the irradiated alkali halides. (42) Unlike the bands

174

FRANCIS K. FONG

¢N

¢N

e~

g

0

©

t~

0

d

o

z

z

o .o t~

<

t~

Z

o 0

,.2

'~0

.

~xxxN m

~

.~

Rare-earth impuritiesin alkaline-earthhalides

175

associated with chemical impurities, the bands induced by high-dosage gammairradiation are relatively unstable. They are readily bleached by heat and light, the bands in the infrared being the most unstable (Fig. 5.5). Due to the fact that these bands are not observed at very high radiation dosages, the authors concluded that they are indicative of the severe crystal structure damage caused by the radiation. The very complicated picture of the coloration effects in CaF 2 is manifested in a large number of different absorption bands observed, which are listed in Table 5.1. Coloration experiments have also been reported for the similar compounds SrF 2 and BaF2 .t59'6°) A closer comparison of these substances in the various coloration effects described above is very much warranted, especially since techniques of crystal growth are now well-developed for all these materials. 5.3. Strontium Chloride, Barium Bromide and Barium Chlorofluoride The coloration effects in synthetic SrC12, BaBr 2, and BaC1F crystals were

o

I 3O0

5(]0 700 WAVELENGTH X (m,ul

900

FiG. 5.6. Coloration of SrCI2. (A) Gamma-rayinduced; (B) additive(750'C, furnacecooled). (AfterFong and Yocom.tSs))

reported by Fong and Yocom. t55) Color centers in SrCI 2 have been produced by gamma-irradiation at room temperature and by Sr vapor treatment at 750°C. Both BaBr 2 and BaC1F were colored additively. The resulting absorption spectra of these crystals are shown in Figs. 5.6, 5.7 and 5.8. Unfortunately,

176

FRANCIS K. FONG

m e a s u r e m e n t s were m a d e with the s a m p l e s in a pyrex glass dewar, a n d the s p e c t r a were l i m i t e d to the range b e l o w 300 m#. 580

~

_ _ \\"~

i

i

i

i

i

i

400

500

600

700

800

900

I

WAVELENGTHli (m/c)

i

I000 II00

FIG. 5.7. Absorption spectra of additivelycolored BaBr2 at two different temperatures. (After Fong and Yocom.(Ssl)

510

I "A 77"K

|

~=

755782

I |

',',ol,~:,a , /k ,,w \

44

i $

k~7,,, \

'

r~"\

i

= -/

\A "i'

W',,J

j

I'~°

it, v A

\

300"K i

500

i

~o

i

9~o

i

WAVELENGTHX (m/c)

,,~o

i

FIG. 5.8. Absorption spectra of additivelycolored BaCiF at two different temperatures. (After Fong and Yocom.(55))

177

Rare-earth impurities in alkaline-earth halides

Unlike CaF2, the spectrum observed for SrC12 by gamma-irradiation does not vary with radiation dosage. For all dosages employed, the same bell-shaped absorption band at 378 m p was obtained (Fig. 5.6). The additively colored SrCl 2 crystals displayed the same spectrum, except that indications of colloidal formation were apparent. For additively colored BaBr 2, the spectrum consists of two bands (Fig. 5.7) peaking at 580 and 723 m/~. The presence of the two bands may be related to the structure of BaBr2. Barium bromide has the crystal structure of PbC12 (Section II). There exist two different sites for the bromide ion and, consequently, centers comprising electrons trapped at the two different types of bromide ion vacancies can conceivably give rise to absorption bands peaking at two different wavelengths. The spectrum observed for barium chlorofluoride is surprising in its large number of absorption bands (Fig. 5.8). Another striking feature is its strong temperature dependence for the two temperatures, 77 ° and 300°K, employed (Fig. 5.8). The possibility of the existence of both fluoride ion and chloride ion vacancies in barium chlorofluoride and its tetragonal structure (Section II) may undoubtedly contribute to the complexity of the multi-banded spectrum, as may also the presence of small quantities of impurities. V1. RARE EARTHS IN ALKALINE-EARTH HALIDES 6.1. Natural Occurrence of Rare Earths in Fluorites The trivalent ions of the rare-earths, according to Goldschmidt, t61) have radii comparable to that of the calcium ion (1.06 A) which is an important factor in the frequent incorporation of rare-earth impurities in naturally occurring fluorites. Przibram and his school (62) examined the coloration and luminescence properties of rare-earth contaminated fluorites, t3~ and concluded that many color bands in fluorites are due to rare-earth ions that have been reduced to the divalent state by radioactive radiations. As early as 1904, TABLE 6.1. CONCENTRATION OF RARE EARTHS IN FLUORITES AND IONIC SIZES (3)

Origins of fluorite

Yb

Aare-Massiv

10-4

Weardale Shinden (Japan)

I0 3

Hundholmen

lO-S

Ionic radius of the trivalent ion (A)

Er

Y

Eu

Small trace ~10 -3

10-2

Small trace Small trace

10-1

Small trace

I 1.00

1.13

1.06 i

I

1.04

178

FRANCIS K. FONG

Humphreyst6a) proved the presence of Y and Yb spectrographically in fluorites that exhibited the yellow-green low temperature fluorescence of Yb 2+. Wild(64) established spectroscopically the presence in fluorites of Yb, Eu, Y, Er, and Mn in varying concentrations. Wild's results, together with the sizes of several trivalent rare-earth ions,t64) are listed in Table 6.1. Indeed, the affinity of CaF 2 for ya + ions is so great that it is very difficult to get rid of the last traces of these impurity ions, which have been found to be responsible t65' 66) for the Smakulatype absorption (Section V) in irradiated synthetic CaF 2 crystals of high purity. The trace quantities of trivalent impurity ions are largely charge-compensated by F - interstitials (Section IV). Alternatively, they might also be compensated by 0 2- ions which may be incorporated during crystal growth by the hydrolysis of CaF 2 (Section IV). In the following discussions, rare-earth impurities have been added intentionally so as to make a controlled study of the coloration of the rare-earth contaminated samples. 6.2. The Lanthanide Series The term "rare earths" actually embraces two long series of elements which are characterized by the progressive filling of the 4f or 5fshells of their electronic configurations. One of the series (associated with the 4f shell) is known as the lanthanides, and the other (associated with the 5fshell) is known as the actinides. Discussions in this and the following sections will be restricted to the lanthanides. Members of the lanthanide series (or the first inner transition series) are characterized by the progressive filling of the 4f shell of their electronic configurations. They occur as a group of fourteen elements which commence with the element lanthanum. In their natural state, the lanthanides possess the common feature of a xenon electronic structure (1s22s22p63s23p63dX°4s24p64dl°5s25p6) with two or three outer electrons (6s2 or 5d6s2). It has been shown that the energy and spatial extension of the 4f-eigenfunction drop abruptly at the commencement of the lathanides. In lanthanum, the 4f-eigenfunction is essentially located outside the xenon structure, whereas in neodymium the 4f-eigenfunction has contracted so that its maximum lies inside the 5s25p 6 closed shells of the xenon structure. At the commencement of the lanthanide series, a deep potential well develops near the nucleus and the 4f-electrons are drawn from the outer shells of the atom into the interior. No such effect occurs for the eigenfunctions of s-, p-, or d-electrons. With the addition of each electron to the 4f shell of the lanthanides, the nuclear charge increases correspondingly by one. Due to the imperfect shielding of 4f-electrons, the entry of electrons into the 4f-orbitals causes no significant change in the overlying 5s, 5p, 5d, and 6s orbitals. This results in a regular increase in the effective nuclear charge across the lanthanide series. The increase in effective nuclear charge is paralleled within the series by a monotonic decrease in size for ions of comparable charge,t6T) and, with a few significant exceptions (Sm, Eu, and Yb), for the neutral atoms themselves. This size decreaset6a) is known as the lanthanide contraction,

179

Rare-earth impurities in alkaline-earth halides

i

t~

O O

O O

o Z

5

)-

[]

©

Z ~a

+

©

e'~

7

~q +

+~

0

0

Z

~8

i

ae- + +

180

FRANCISK. FONG

the effects of which are significant. Such small differences in properties as exist among the lanthanides in a given state of oxidation may be ascribed to size differences resulting from this contraction. These differences form the bases for most chemical separational procedures. The lanthanides may be ionized by the successive removal of electrons. With the exceptiofi of lutecium, the first stage Of ionization results from the removal of a 6s-electron. The second stage of ionization involves the removal of the other 6s-electron. At the third stage, both the 6s-electrons and a 5d- or a 4f-electron have been removed to leave, apart from the xenon structure, a 4f" configuration where n = 1 for cerium and increases regularly to n = 14 for lutecium. The normal electronic configuration for the neutral lanthanides, their three stages of ionization, and several empirically known values of ionization potentials are given in Table 6.2. For the sake of simplicity, the closed shells of the xenon structure have been omitted in designating the electronic configurations of the rare earths. Only the f"-shell and the electrons outside the xenon structure are written. Nearly all the normal configurations given in Table 6.2 have been either established or inferred from spectroscopic or magnetic data. t69) The tripositive oxidation state dominates throughout the lanthanide series, although the less stable dipositive and quadrupositive oxidations have been known to exist in several of the lanth.anides,tT°' 71) The stable existence of the 2 + and 4 + oxidation states in some lanthanides can be ascribed to the added stability associated with an empty, half-filled, or filled 4f shell. Thus cerium and terbium occur in the 4 + state having the stable configurations 4f ° and 4f 7 respectively. Similarly, europium and ytterbium occur in the 2 + state with the stable configurations 4f 7 and 4f 14, respectively. Also, the relatively frequent occurrence of the 2 + state in samarium (4f 6) is attributed to the fact that the stable configuration 4f 7 is approached. The importance of environmental effects in obtaining the less common oxidation states will become evident in Section 6.3. 6.3. Valency of the Rare-earth Ions in Alkaline-earth Halides When YF3 is added to a CaF 2 melt from which a mixed crystal is being grown, the yttrium ions remain in the trivalent state. This is borne out by the conductivity and density measurements (Section IV) in which it is found that the incorporation of the y3+ ions introduces F - interstitials in the lattice. For species such as Eu a+ or Sm 3+ ions, however, it is conceivable that the impurity ions may be reduced to the divalent state, if the gain in ionization energy should offset the loss of electrostatic energy in the Madelung potential. The details of the system of Cd 2 +, Ca 2 +, and Sr 2 + in KC1 and NaC1 as investigated by Bassani and Fumi t72) have been reviewed by Liddiard,t12~ and our discussion on the solution of trivalent rare-earth ions (henceforth denoted as R a +) in alkaline-earth halides (MX2) will proceed in a similar fashion.

Rare-earth impuritiesin alkaline-earthhalides

181

To a perfect lattice consisting of stoichiometric numbers of cation (M 2+) and anion (X-) sites, one molecule of RX3 is added. If the rare-earth ion enters the lattice in the trivalent state, there will be one X- interstitial formed in addition to one new cation and two new anion sites (see Section IV for chargecompensation in an anti-Frenkel disorder lattice). On the other hand, if the impurity ion enters the lattice in the divalent state,, no X- interstitial will be introduced. If we represent the first case by R3+(M, Xz.+z) + X7 and the second by R 2 +(M.X2,+ 2), where the suffices n and 2n + 2 represent the numbers of the M 2 + and X- ions, respectively, and X~- denotes the X- interstitial, the difference between these two states may be expressed by the equation : 1 R3+(MnXzn+2) + X/- + U = RZ+,M ~ nX2n+2) + ~X2(gas ).

(6.1)

We note that the reduction process !s accompanied by the evolution of X2 gas. The quantity U in eqn. (6.1) may be calculated by means of a Born-Habertype cycle : X/- -~- R3+,M t nX2n+2) ~ R2+(MnX2n+2) 4- -~2 (1)I+E~ X/- + R3+(M,X2,+0v + X-(gas) ]

(5)1-~ ½Xx + X~- + RZ+(M.XE.+Ov

X°.~ + e + R3+(M.X2.+ 1)v + X/in which the subscript V represents the presence of a X- vacancy. The five steps in the above scheme may be briefly described as follows : (1) Removal of one X- ion from the crystal lattice to set it at rest at infinity. The amount of work required is Eov. (2) Removal of one electron from the X- ion to yield a neutral X ° atom. The energy required for this process is equal to the electron affinity of the X ° atom, A. (3) Formation of ½X2 molecule from X ° yielding the heat of formation ½/4. (4) Reduction of the R 3 + ion to the divalent state, which gives an amount of energy equal to the third ionization energy of the R atom, I. It is also necessary here to take into account the loss or gain of electrostatic energy, M, in the Madalung potential due to the change in valence of R. (5) Recombination of the X- vacancy and the X- interstitial, which yields an amount of energy, 4, equivalent but opposite in sign to the energy of formation of a pair of anti-Frenkel defects, defined as (E v + E{~). Here, E v is the energy required to remove one X- ion from the crystal lattice to set it at rest at infinity, and E{~ is that required to return the X- ion at infinity to an interstitial position.

182

FRANCISK. FONt

By balancing the energy terms for the above cycle, we obtain U=E~+A-½H-(I-M)-~=A-½H-(I-M)-Eg

(6.2)

If U is negative in value, the rare-earth ion will enter the lattice in the divalent state. If U is positive, it will enter as a trivalent ion. Whereas the quantities A, H, and I are known empirically, M and Eg can be calculated from the lattice theory. Bassani and Fumi evaluated the quantity M for divalent cations in NaC1 and KC1 and found it to be of the order of - 15 eV. For alkaline-earth ions to enter the lattice of alkali halides reduced to the monovalent state, it is estimated t121 that the second ionization potentials of the alkaline-earth metals should be of the order of 23 eV or more. Since/(Ca +) = 11.87 eV and l(Sr +) = 11.03 eV, however, it is to be expected that Ca or Sr should enter the lattice of NaC1 and KC1 in the divalent state. Although no estimates of the quantities M and E/o have been reported in the literature for rare-earth contaminated alkaline-earth halides, eqn. (6.2) is useful in a qualitative discussion of several experimental observations. Under ordinary conditions of crystal growth, most rare-earth ions, with the exception of Eu and to a much lesser extent, Sm, enter alkaline-earth halides in the trivalent state. In the presence of a reducing agent, t73) however, it has been repeatedly observed that Sm 3+ and Tm 3÷ ions can be readily reduced to the divalent state. The reduction is achieved by growing the crystal under an atmosphere of H2, or alternatively, with a stoichiometric amount of the appropriate rare-earth metal or alkaline-earth metal. For example, a single SrC12 crystal nominally containing 2 mol. ~ of Tm 2+ ions was grown from a melt containing 2 mol. ~ of TmC1 a and 1 mol. ~ of Sr metal, t73) It was observed recently in the writer's laboratory that Tm a + ions were reduced to Tm 2 + ions in appreciable quantities in CaF2 when the crystal was grown from a graphite container without any other sources of reducing agents. Evidently, the reducing agents play an important role in lowering the value or, in some cases, even changing the sign of U in eqn. (6.2) which is rewritten as U=A-½H-(I-M)-E~-

W,

(6.3)

where W is the gain in energy attributed to the role played by the reducing agent. When H E is employed as a means of reduction, the quantity - W is equal to the heat of formation of HX from H 2 and X2, which, in the case of HC1, has the standard state value t74~ -22.06 kcal/mole = - 9 . 5 7 x 10-1 eV. Since the other quantities in eqn. (6.3) are of the order of several electron volts, the gain in energy W necessary for reduction to take place is comparatively small. Apparently the quantity U in electron volts in the cases of Tm, Sm and, particularly, Eu has near-zero values, and from eqn. (6.2), we obtain the approximate relationship -M+E~ =A-½H-I. (6.4)

Rare-earth impurities in alkaline-earth halides

183

Of the three quantities on the right-hand side of equation (6.4), A and H are known empirically. For the C1- ion, for example, A = 3.70 eV and H -- 2.48 eV. Unfortunately, no accurately determined values exist in the literature for the quantity 1, the third ionization potential of the rare earths. According to the atomic spectra ~75) o f L a III (La 2 +), La II (La 1+), and La I (La°), it is possible by means of extrapolation of data for the first members of the spectral series (the electron levels 6s, 7s, and 5d, 6d for La II and La I) to determine approximately the corresponding ionization potentials which are given as follows: ILa o =

5.61 +_ 0"03 eV, ILa + = 11.43 _+ 0"07 eV, and ILa2+ = 19.1 _+ 0'1 eV.

Similarly, the first and second ionization potentials of Sm, Eu, and Yb have been found :(75, 76~ Ismo = 5.6

Ism+ = 11.4

IEuo = 5.67 __+0 01 eV, IEu + = 11.24 _+ 0 02 eV, and /Vb o =

6.25

Ivb + =

12.10 eV

The close agreement of the values for the first and second ionization potentials for La, Sm, Eu, and Yb indicates that the ionization potentials of all atoms of the rare-earth elements should be close to each other3 TM These and earlier values of ionization potentials are listed in Table 6.1. Of the third ionization potentials, that (19.1 eV) of La has been determined. ~75) It is safe to assume, in view of the similarities observed among the rare earths, that the third ionization potentials of the lanthanides should be approximately equal to or greater than 19.1 eV, depending upon whether the 4f °, 4f 7, or 4f 14 configuration is approached or not. Without going into any details, one quick glance at the normal configuration of the trivalent rare-earth ions (Table 6.1) suggests that /yb2+ > /Tm2+ > /ERE+ > In02+ > /Dy2+ > /Tb2+ > /Gd2+,

(6.5)

IEu2+ > lsm2+ > Ipm2+ > Isd2+ > IPr2+ > lce2+ > ILa2+"

(6.6)

or

We may further expect that l~d~+ ~ 1La~+ = 19.1 eV, since the changes in the configurations Gd z + (4fTSdl) ~ Gd 3 +(4f 7) and La 2 +(4f°5d 1) ~ La 3 +(4f °) both involve one 5d electron and the stable configurations 4f ° and 4f 7. The above postulated order of third ionization potentials is consistent with

184

FRANCISK. FONG

the empirical observation that ease of reduction of the rare-earths from the trivalent state to the divalent state ranges in the order of Eu > Sm, Tm > Nd, Pr, Er, Ho, Dy. Since most of these may be readily reduced to the divalent state (see Section 6.4), the spread in values of the third ionization potentials is probably not too large. When the value of U in eqn. (6.2) is close to zero, the extent of the reduction depends critically upon factors influencing the various equilibria in the cycle described above. For example, by the continuous removal of C12, one of the products of the reaction, the reduction process is favored. Likewise, the exertion of an external constraint upon the system, such as the irradiation of u.v. light, may greatly facilitate the reaction in a preferred direction. These ideas are borne out by the experimental work of Pinch, t77~ who demonstrated the reduction of Tm a + during the growth of TmS+-contaminated SrC12 crystals by the irradiation of u.v. light in an evacuated reaction assembly. The influence of the environment upon the valency is illustrated in the different behavior of Eu in alkaline-earth oxides, sulfides, and halides. In the absence of reducing agents, rare earths dissolved in the alkaline-earth hosts usually exist in the trivalent state, as indicated by the optical spectra of the resulting crystal. Europium is an outstanding exception. In the oxides MgO, CaO, SrO, and BaO, the Eu ions are predominantly trivalent, but in the corresponding chalkogenides and halides such as CaF 2 and SrCl2, they exist in appreciable numbers in the divalent state. Brauer tTa~ has shown, in the ease of the ehalkogenides, that this is consistent with expectations based upon the lattice theory. 6.4. Solid State Valence Reduction of Rare Earths A. Ionizing-radiation-induced reductions. The reduction of rare-earth ions in fluorites by ionizing-radiation was investigated many years ago by Przibram and his school3 a~ The radiation-induced reduction processes resemble those of the coloration of undoped CaF2, which has been achieved by electron bombardment, tSa'59,66~ by X-irradiation, t56's9~ and by gamma-irradiationt55~ (Section V). When rare-earth ions enter the lattice of CaF 2 in the trivalent state, they are largely compensated by F - interstitials according to the equilibrium (4.2). Both the unassociated impurity ion M a+ and the eomplexed species M a+ :F 7 have been observed by paramagnetic resonance, tTa-a2~ In the case of the unassociated trivalent ion, paramagnetic resonance shows the cubic symmetry of the CaF2 lattice. After CaF2 crystals containing 0.05 % Tm were irradiated by X-rays at 77°K or room temperature, Hayes and Twidelltaa) observed an isotropic ESR spectrum with a two-line hyper-fine structure which they successfully assigned to the Tm 2 ÷ ion which resulted presumably from the trapping

Rare-earth impuritiesin alkaline-earth halides

185

of an electron on an unassociated T m 3+ ion. Divalent holmium obtained by exposure of the crystals to ionizing radiation has also been observed ta4' 85) by means of paramagnetic resonance by Sabisky ta4) and Hayes et al. taS) Przibram, t3) Butement, ts6) Wood, (sT) McClure, ~ss) and others ~s9- 91) observed that divalent rare-earth ions in ionic crystals display a group of diffuse absorption bands (Section 6.5) in the visible spectral range. By observing changes in the absorption characteristics of the crystals, Fong t92) investigated the effects of gamma-irradiation upon CaF 2 crystals doped with DyF 3 of various concentrations from 0 to 0.5 mol. %. His results may be briefly summarized as follows : (i) The coloration behavior of the impure CaF 2 crystals depends upon the DyF 3 concentration (Fig. 6.1). At low concentrations of contamination .9

,

,

,

.7

,

,

i

i

i

i

C

0

i

300

400

i

5OO

i

600 700 800 WAVELENGTH). (m/=)

i

900

i

IO00

I00

FIG. 6.]. Effect of Dy3÷ impurity concentration o n the gamma-ray-inducedabsorption spectrumof CaF2. (A)Beforeirradiation,(B)undoped CaF2 at 106rad, (C) CaF2 : 2 x 10- 3 mol. % DyF3 at 104 rad, (D) CaF2 : 5 x 10- 3 tool. % DyF3 at 104 rad. (After Fong.~92~)

(,-~2 x 10-3 mol. % DyF3), a spectrum consisting of a band peaking at 498 m/t with a "spike" on the high-frequency side (at 450 m/t) was obtained (Fig. 6.1c). At higher concentrations ( > 5 × 10-3 mol. % DyF3), the characteristic spectrum showing the diffuse 4f-5d absorption bands (Section 6.5) of Dy 2÷ ions was observed (Fig. 6.1D). It was postulated that the low Dy-concentration spectrum could be associated with the presence of 0 2- ions in the crystal lattice (Section 4.2), and the band peaking at 498 m/t with its characteristic spike at 450 m/t was accordingly named the RO band. This result is consistent with that of Sabisky's paramagnetic resonance investigations393) (2) The low Dy-concentration spectrum changes with increasing dosage of irradiation. At high dosages ( > 107 rad), the bands in the infrared spectral

186

FRANCIS K. FOr~G

region become magnified. This dosage-dependent behavior is reminiscent of that observed in the " p u r e " CaF 2 crystals (Section 5.2). (3) The concentration of Dy E+ ions attainable in a sample of a given total Dy content saturates with the radiation dosage (Fig. 6.2). This phenomenon is similar to the saturation of color center formation when " p u r e " crystals are irradiated.

~

715m F •

•

.

OW

t'- >. (nQ

<

a:

"

|

i

0

2

4

5 7 2 m F.

9

910mF

I

I

I

r

6 8 I0 12 14 16 G A M M A - RADIATION DOSAGE (x 106 RAD)

I

18

I

20

F;G. 6.2. Saturation of reduction at a given total Dy concentration with radiationdosage.The wavelengthscorrespondto major Dy2 + absorptionpeaks.(AfterFong.(92))

~t~ta6 ~

15mF •

o~5 ~8 ~,.~ta z4 ~- >

4 5 8 m~

z_t~ 3 ta > ~ .<

5 7 2 mF

< 2

d

• •

910 mF

I

TOTAL

t

I

i

I

I

.I

.2

.5

.4

.5

DYSPROSIUM CONCENTRATION IN COF2 (MOLE PERCENT)

• FIG. 6.3. Saturation of reduction at a given radiation dosag e with total Dy concentration. The wavelengths correspond to major Dy 2÷ absorption peaks. (After F o n g . (92))

(4) The concentration of Dy 2+ ions attainable saturates as the total Dy content increases (Fig. 6.3). This is consistent with the postulate that only Dy 3+ ions unassociated with F - interstitials can trap electrons to give Dy 2÷ ions. Recalling the discussions of Section IV, Dy 3÷ ions should enter the CaF 2 lattice compensated by F - interstitials according to the equilibrium

R a r e - e a r t h i m p u r i t i e s in a l k a l i n e - e a r t h halides

187

K2

Dy 3 + + Fi- ~ Dy 3+ :F~-

(6.7)

where K2, the equilibrium constant for the association of Dy 3+ and FT, can be written as (y -

n.)

K z -

(6.8)

, rlirl u

n. being the concentration of unassociated Dy 3+ ions, ni that of the unassociated F - interstitials, and y that of all the Dy 3+ ions present. Since at low temperatures the concentrations of intrinsic defects are very low (Section IV), ni =

nu,

or

(6.9) (y - n.) K2

--

2 t~u

From eqn. (6.9), we obtain the expression for the concentration of unassociated or r e d u c i b l e Dy 3 + ions as a function of the total Dy content y : 1

n, = 2 ~ 2 Ix/(1 +

4Kzy)

-

1].

(6.10)

For all values of n, and y, dn, - (1 + 4Kzy) -~. dy When

4 K E y ~ O,

(6.11)

the initial gradient of the n, vs. y plot is dnu = 1. dy ,*r~,~~

(6.12)

When y ~ ~ , the plot saturates with dnUdy 4r2~,~ = 0.

(6.13)

As the temperature increases, two opposing factors set in. On the one hand, K2 tends to decrease with increasing temperature, which favors the unassociated form of Dy 3 +. On the other hand, the presence of intrinsic F f defects at higher temperatures tends to push the equilibrium (6.7) to favor the associated form of Dy 3+ :F 7 according to the law of mass action. Quantitative investigations of these effects are currently underway at the writer's laboratory. (5) Of the four major 4 f - 5 d absorption bands, absorption of light corresponding in wavelengths to the sharp band peaking at 458 m# (Fig. 6.4) causes reconversion of Dy 2 + ions to the trivalent state at both room and liquid nitrogen temperatures. The optical bleaching behavior of a gamma-irradiated sample is shown in Fig. 6.4.

188

FRANCIS K. FONG

(6) The gamma-irradiated sample bleaches readily with brilliant thermoluminescence when it is heated above room temperature. The thermoluminescence is due to the 4f--* 4femission of Dy 3÷ ions (Fig. 6.5). The thermoluminescent processes will be discussed in some detail in Section VII. The observations that the Dy 2÷ concentratio~a saturates with radiationdosage [result (3) above] and total Dy content [result (4) above] were confirmed

,i8(".%~,N8) . ~J~A [ ~ I t B/

,7

(A) INITIAL SPECTRUM (RADIATION DOSAGE: 5xlOTRA~) (IB) 18 HOtJRS 3 IRRADIATION tA-~ .2. '.-~_--l ATBLEACHING ~L.I ~vM'au~:~.j" FREQUENCIES (D) 200HOURS (458mF)

.5

~.4

k'.E o

R

i

0300

400

i

i

i

i

i

700 800 900 WAVELENGTH Mm#)

500

i

1000

i

1100 1200

FIG. 6.4. Bleaching of gamma-radiation induced Dy 2 + ions. (After Fong. (92))

A-THERMOLUMINESCENCE

o_

d

B 400

450

500

550 600 650 700 WAVELENGTH k (rap.)

750

FIG. 6.5. Spectra of thermoluminescence of gamma-radiation-induced Dy 2+ and fluorescence of Dy 3+ in CaF 2 at 87°C. (After Fong. (°2))

by Sabisky (93) who monitored the reduction of Ho 3+ in CaF 2 by comparing the Ho 2÷ paramagnetic line (s4) with the resonance of Cr 3÷ in A120 3. The Cr concentration and the total Ho concentrations were obtained by chemical means. Sabisky noted that in crystals containing 2 x 10 -3 mol. ~ HoFa, no Ho 2 ÷ resonance was observed, [result (1) above]. His results on the concentration saturation effect are summarized in Fig. 6.6, which shows that in a crystal containing a total Ho concentration of 3.8 x 10- ~ mol. ~ , only 3 ~ of the Hi 3+ ions were reduced by gamma-irradiation at 5 x 106 rad. Sabisky also

189

Rare-earth impurities in alkaline-earth halides

observed that the amount of rio 3+ reducible to the divalent state in CaF 2 irradiation is not temperature dependent for temperatures from 77°K to room temperature. The reduction of rare-earth ions by ionizing-radiation results in divalent ions which are unstable with respect to light and heat. This is a necessary outcome since the crystal remains essentially stoichiometrically unchanged before and after the irradiation. Thus, for every trivalent ion reduced to the L

I

i

1

I

E u

[

I

I

0 ~t

+-I

t.d >

"6

z 2o

7\

U

x~x

Ul

(~

5 8

x

I

I

0.05 0.1

I

I

0.5

0.2 RATIO

OF

I

005

0.4

TOTAL HOLMIUM

TO C A L C I U M

L

~

I

0.1

w

-

I

0.2

0.3

0.4

(%)

FIG. 6.6. Concentration of Ho 2+ as a function of total Ho concentration at g a m m a radiation dosage 5 x 106 rad. (After Sabisky. (93))

divalent state, a corresponding hole-center must necessarily be formed elsewhere in the crystal. The hole-electron recombination can subsequently be activated by the absorption of light or heat. In the following, two techniques are described by which stable divalent rare-earth ions can be readily prepared.

B. Additivereduction.

When a crystal containing trivalent rare-earth ions is heated in the corresponding alkaline-earth metal vapor at sufficiently high temperatures (> 750°), the rare-earth ions are readily reduced to the divalent state (94) as the charge-compensating X- interstitials are removed from the crystal lattice to form additional MX 2 layers on the crystal surface. Valence reduction of this type may be described by a reaction cycle similar to that outlined in Section 6.3. The energy U required (or gained) by the additive reduction process [recall equation (6.3)] can be empirically determined from simple equilibrium considerations. The free energy of a monatomic vapor of M atoms metal containing n 1 atoms per unit volume may be written as Fv=

1

[ {27rmkT'n"3/2` /

nlkT l o g / T ]

1

+ log 1 + 1 .

nl

(6.14)

190

FRANCIS K. FONG

The change in free energy caused by the removal of one atom from the vapor is

2r~mkT 3/2 AFv= kT[log(--~--)+ l o g l 1.

(6.15)

Since for every M atom that settles on the crystal, two R 3+ ions are reduced according to the equation, M + 2R 3÷ = M 2+ + 2R 2÷,

(6.16)

we have ½n2 excess M 2÷ ions if there exist n2 R 2÷ ions in the crystal. The entropy of the system is, if there are n3 R3+ ions, S = k log [(N + in2) -k n 2 q- n3]! (N + ½n2)! n2 ! n3 !

'

(6.17)

where N is the original number of MX 2 molecules. When the number of excess M 2 + ions increases by one, the change in entropy is AS = k log [(N + ½n2) + n 2 + na]n 2 (N + ~1n 2 ) n 22 The increase in free energy in the crystal is thus A t s = U - kT log [(N + ½n2) +1 n2 +2 na]n 2 (N + ~n2)n 2

(6.18)

At equilibrium AFv + AFs = 0 and we obtain 1 22 (2nmkT~-3/2 (N + ~nE)n [(N + ½n2) + n2 + na]n 2 = n l \ h2 ,] exp ( - U / k T ) .

(6.19)

For N ~> n2, and setting n 2 + n a = n o which is constant,

n2 (N + no)(2rcmkT~ -3/2 K = nln2 N k, h 2 J exp(-U/kT),

(6.20)

where K is the equilibrium constant for the reduction process, and n~, n2 and n 3 are measurable quantities. Since the pre-exponential term is only weakly temperature dependent compared to the exponential term, we obtain the quantity U from eqn. (6.20): U = - k d(ln K)/d(1/T).

(6.21)

The value of U is now being determined at the writer's laboratory.

C. Electrolytic reduction. When a moderate potential ( ~ 6-10 V) is applied to a crystal containing trivalent rare-earths at elevated temperatures (> 600°C),

Rare-earth impurities in alkaline-earth halides

191

the rare-earths can be readily reduced to the divalent state ~95- 97~ as the chargecompensating X - interstitials are removed from the crystal lattice by the anode. A variety of conducting materials such as graphite, gold, platinum or calcium m a y be suitably employed as the electrodes. It was observed ~96) that electrolytically reduced samples contain much higher densities of R 2 ÷ ions than those reduced by ionizing radiation. Furthermore, saturation in R a ÷ concentration with increasing total R content was not evidenced in electrolytically reduced samples (Fig. 6.7). If the saturation effect ELECTROLYTICALLY INDUCED Dye÷ 4f-Sd ABSORPTION

- - - - - GAMMA INDUCED Dy~4f-Sd ABSORPTION AT 5 x 106 RAD to

÷ ÷

z

.J

==

B/1 /

6 l/~/I'1 ~" 4

4588m~

2 ~'~"

TOTAL

910 rnp.

DYSPROSIUM CONCENTRATION IN CaF 2 (MOLE PERCENT)

FIG. 6.7. Relative optical density of Dy2+ absorption bands plotted against the total Dy concentration in CaF 2 samples reduced by electrolysis and gamma-radiation. (After FongJ9~) is indeed due to reasons elaborated in Section 6.4, it becomes clear that in the electrolytic reduction process the associated R 3 + ions as well as the isolated ones are reduced. This can be readily understood in view of the law of mass action and the equilibrium condition K2

M 3+ + X/- ~ M 3+ :X/-.

(6.7)

The M3+:X7 complexes can neither contribute to electrical conductivity

192

FRANCIS K. FONG

(Section III), nor trap electrons and be reduced. As M 3+ and X7 ions are being continuously depleted by electrolysis, however, the equilibrium readjusts itself so as to cause a continuous dissociation of the M 3+ :X/-- complexes. The same phenomenon, of course, must also occur in the additive reduction process. The above interpretation is supported by spectroscopic data. (94'96) The absorption spectra (94) of the Tm(II + III)t system in CaF2 are shown in Fig. 6.8.

.....

B J ~ J i ~ A f t e r

duotlo°

irradiation

After electrolysis or additive treatment.

I I ~1 I I 1.06 1.10 1.14 1.18 1.22 1.26 Wavelength k (/~)

20DO0 ~

3

TEfi 15,000

w

.~3F3

5,000 0

3H4 FT/2

Tm3÷3H6

FIG. 6.8. Evidence for the selective reduction of Tm 3+ ions in CaF2. (After Kiss and Yocom. [94))

In the range of 1.12 to 1.25~t shown, the absorption spectrum of Tm 3+ corresponds to the aH 6 ~ 3H 5 transition (Fig. 6.80). The line at 1.116 p is the absorption from the ground 2F~ to the 2F~ state of Tm 2+ ions. The fact that the 3 H 6 ~ 3 H 5 absorption lines due to Tm 3+ ions has not diminished in intensity after photochemical reduction of Tm a÷ ions by gamma irradiation has taken place suggests that these absorption lines are due to Tm 3+ :F[complexes with the F - interstitials distorting the cubic site and removing the center of inversion symmetry, and that the transitions are therefore induced electric-dipole transitions with oscillator strength of the order of 10-5. These Tm 3+ ions are to be distinguished from the Tm 3+ ions in cubic sites (i.e. unassociated Tm 3+ ions) which give rise to a much weaker magnetic dipole absorption (with oscillator strengthf ~ 10 -7) spectrum.The disappearance of the trivalent thulium aH 6 --* 3H 5 absorption spectrum after the electrolytic reduction clearly indicates that complexed trivalent thulium ions in noncubic sites as well as unassociated thulium ions in cubic sites have been reduced by the electrolytic process. tHere, the chemist's notations (II) and (III) for the divalent and trivalent states, respectively, have been retained.

Rare-earth impurities in alkaline-earthhalides

193

6.5. Absorption and Emission Characteristics of Divalent Rare-earths The divalent state of rare-earths in crystals was first investigated nearly three decades ago by Przibram and his school, (3) who concluded that divalent europium on irradiation with ultraviolet light gives a violet fluorescence with a diffuse band with its maximum about 4000/~, and that divalent samarium has a diffuse red band peaking at 6300 A, a line spectrum with a red line at 6900 A and others in the infrared. More quantitative studies of the spectroscopic properties of am 2 +, Eu 2 +, and Yb 2 + in H 2 0 , polycrystalline BaC12 and SrCI 2 were made by Butement, t86~ who discussed the absorption and emission characteristics of the divalent rare-earths in terms of two types of transitions: (i) 4f--* 5d,

or

4f--* 6s, etc.,

and (ii) internal 4 f ~ 4f. Due to the large perturbation of the external shells by its environment, type (i) should give rise to diffuse absorption bands which should remain diffuse at low temperatures. Qualitatively, it is known that the temperature necessary to reduce the trichlorides and the rate of reaction of the dichlorides with water both increase in the order Eu, Yb, Sm. This suggests that type (i) absorption for Sm 2 +, Eu 2 +, and Yb 2 + should set in at a lower frequency than for trivalent rare-earths, and that the frequency at which absorption begins should be lowest for Sm 2 +. This was indeed observed to be the case. ~86) Type (ii) involving internal 4f transitions should give narrow line-like bands and being forbidden will be of low i n t e n s i t y f ~ 10 .5 to 10 -6 in an asymmetric field. For Yb 2+, the 4fshell is full, and Pauli's principle excludes internal transitions, Accordingly, there should be no type (ii) bands, and none have been observed. Eu 2 + and Sm 2 + are isoelectronic with Gd 3+ and Eu 3 +, respectively. Accordingly, their narrow, weak bands have been correlated to those of the corresponding isoelectronic Gd 3+ and Eu 3+ ions. The absorption spectra at room temperature of divalent rare-earths in CaF2 were measured by McClure, and are shown in Fig. 6.9, in which positions of some 4f---, 4ffluorescence lines are also shown. The absorption spectra are characterized by broad absorption bands occurring mainly in the visible region, which are to be compared with corresponding trivalent rare-earth broad-band transitions occurring at much higher frequencies. Clearly, it is necessary to see if the onset of the divalent rare-earth spectra is consistent with the postulate that they are due to type (i) transitions. To do this, McClure and Kiss (88) calculated the excitation energy of these spectra in a first approximation as the energy difference between the " H u n d rule", single-determinant states of the configuration f " - l d and f". The general formula for the excitation energy may be written, t88) AE = [g(5d) - E(4f)] + [Y,F~, - IF,Fk] + [S' - S] + [AC ° + C 4] o

(6.22)

194

FRANCIS K. FONG

where E is an orbital energy and the prime refers to the 5d4f" states, the ZF k are the appropriate combinations of Slater integrals, S' and S are the spin-orbit term depression of upper and lower states, AC ° and C 4 the difference of the zeroth crystal-field terms between upper and lower states and fourth-order cubic-crystal-field stabilization of the upper state. The first term in brackets 5000

15000

I0000

'

Ll'

8000(cm'l)

, t~r--f ,

tF'r/2

d,, ,

I

i

I

25000

30000

2F5/2

,

42001¢~1| I I I t-~l

20000

i

i

9000(cm-I) ,

I 50100(cm'l)

fl,3

Tin2+

f12

Er 2+

fll

Ho2 ÷

! 4I

~

4600 (on; I ) 1

55,oo(,©m'v! t_._t. 4,o~o1,cm-t)

I

.

5500(¢11; I ) i

, ~'l.°°(c"Ft)

I

5000

I0000

f~o oyz + I

I

I

15000 (ten'l)

20000

25000

50000

FIG. 6.9. Spectra of divalent rare-earth ions in CaF2 at room temperature. The bands shown involve f--+ d transitions. The positions off--+ f transitions as observed in fluorescence of four of the ions are also shown. (After McClure and Kiss. t88))

is the difference between the centers of the configurations. The second represents the difference due to the Russell-Saunders splitting of the configurations. Values of the parameters in the expressions EF k for the f " configurations (EFk = CoFo + C2F2 + C4F4 + C6F6) can be obtained by linear interpolations of the values given by Runciman and Wybourne (98) for Pr 3÷ and Tm 3÷. The values used for the divalent ion of an atomic number Z were those for the trivalent ion (Z + 1). The spin-orbit parameters ~(4f), known accurately for the trivalent rareearths, change little with the state of ionization, since the major contribtttions to this effect occur near the nucleus. (99) The value of ~(5d) for Ce 3 + is 996 c m - 1 and that for H f 3÷, 1220 c m - 1, so there is only a gradual increase of ~(5d) in the rareearths. Since S' and S are both of the order of 103 c m - 1, the difference (S' - S) will not be much larger than 102 c m - 1. The spin-orbit parameter thus appears to be relatively unimportant in causing the variations of excitation energy a l o n g the series. The crystal-field splitting acts noticeably on the 5d electron. For the cubic configuration, the E state of the d electron will be lower, and the excitation

Rare-earth impurities in alkaline-earth halides

195

energy is reduced by 6Dq in the crystal as compared to the free ion. In manyelectron ions, however, the crystal-field splitting will compete with RussellSaunders splitting, and the effective reduction in excitation energy will be less significant. From an analysis of the spectrum of Yb 2+ in CaF 2, Struck (1°°) found Dq ~ 1100 cm -1. It was assumed <88) that Dq decreases gradually throughout the series, as Dq for Ce 2 + was estimated (ss) to be about 1500 cm-1. It appears, therefore, that the crystal-field splitting also does not contribute significantly to the variation in excitation energy along the series. The calculated excitation energies [eqn. (6.22)] are compared with experimental observations in Fig. 6.10. The fit from Pr to Eu and from Dy to Yb is I ~C~

25 20 7

I

I

I

I

~ I

Excitation Energy f¢ - fn-i d for divalent rare earthl

/o

t° I

~ ii

o0b$ x Calc

I I I I

o

15

I

tI

iI II I

o

,'-~ t,o /

p-x-4

II I

so

o

/ o oo/ X"x"X

/i ° o !

I

,/ I t

-IC -15

Ii Is I I I I i Xl I I I I I Ce Nd Sm Gd Dy Er Yb Lo Pr Pm Eu Tb Ha Tm Lu I

El

FIG. 6.10. Comparison of calculated and observed onset of absorption in t h e f --, d transitions of divalent rare-earth ions. (Alter McClure and Kiss. (88))

reasonably good. It may thus be concluded that the interpretation of the spectra as arising from 4f---, 5d transitions is supported. The failures at La 2÷, Ce 2÷, Gd 2÷, and Tb 2÷ suggest that the f " - l d configuration rather than f " is the ground state. The ground state of Gd was shown by Albertson <1°1' lO2) to arise from 4f75d6s 2, and that of Gd ÷ to arise from

4fV5d6s. A further support for this interpretation is evident in the observation of Franck-Condon vibronic structures superimposed on the broad absorption bands of the divalent lanthanide ions. In type (ii) transitions, the potential well is identical for both the excited and ground states since f-electrons are not involved in chemical bonding. This leads to the A n = + 1 vibronic selection rule. In type (i) transitions, however, two different potentials are encountered, since the vibrational wavefunction for the excited state is no longer orthogonal to that of the ground state. This will be responsible for the Franck-Condon structures with multi-phonon transitions. The existence of vibronic structures in the broad

196

FRANCIS K. FONG

band absorption due to divalent rare-earths has been suggested by various authors.(lo3, 104) Although the present work is concerned with alkaline-earth halides, the observation of vibronic structures in divalent rare-earths is best exemplified by the spectra measured by Bron employing alkali halides as hosts. ~1°4) The absorption spectrum of Sm 2÷ in KC1 measured at 4.2°K by Wong t1°5) is shown in Fig. 6.11. The even spacing of the vibronic bands is clearly illustrated.

0.7

0.6

0.5

0.4 O.D. 0.5

0.2

0.1

0

I 5800

I 5900

I 6000

I 6100

I 6200

I 6500

I 6400

6500

Wavelength

FIG. 6.11. Absorption spectrum of Sm 2 + in KCI at 4.2°K. (After Wong and Fong. (los})

The presence of Franck-Condon transitions thus confirms the validity of type (i) absorption in divalent rare-earths. Furthermore, the 4f-5d rather than 4f-6s transitions appear to be responsible for the absorption since the estimated oscillator strength (1" ~ 10-2-10 -3) for 4f-5d transitions agree with those measured by Wood and Kaiser. (sT) In addition to 4f-5d bands, the possible existence of charge-transfer bands in divalent rare-earths has been investigated by Jorgensen in complexed ions. (1°6) In our present case, charge transfer bands should arise from the excitation of valence band electrons into the unfilled 4f-orbitals. Details in this aspect have yet to be investigated. The divalent lanthanide ions in alkaline earth halides normally possess cubic symmetry. Whereas 4f-5d electric-dipole transitions are parity-allowed according to the Laporte rule, 4f-4f transitions are usually of magnetic-dipole origin. Copious quantities of radiation may be pumped via the allowed 4f-5d transitions which subsequently are transferred to the metastable 4f" levels from

Rare-earth impuritiesin alkaline-earthhalides

t97

which fluorescence processes can take place. Fluorescence in divalent lanthanides is characterized by narrow lines extending into the infrared region, and has been reviewed by Wybourne(69~in his recent book on the spectra of rare-earths. The utilization of the sharp line internal 4f-4f fluorescence for laser action has recently been reviewed by Birnbaum. (1°7~ In closing the present section, it is appropriate to quote :(69) "The study of the spectroscopic properties of the divalent lanthanides has only just begun. Their successful preparation now opens up the possibility of preparing crystals that are doped in other unusual valence states. Among the possibilities should be monovalent lanthanides, divalent actinides and trivalent protactinium, and thorium." VII. THERMOLUMINESCENCE IN IRRADIATED RARE-EARTH-DOPED CaF2 7.1. Energy Storage and Thermoluminescence The behavior of a trivalent rare-earth-doped alkaline-earth halide when it is irradiated with ionizing radiation and subsequently heated is identical to that of a phosphor. The subject of phosphors and phosphorescence centers around the concept of electron traps, and has been thoroughly investigated by, among many others, Randall and Wilkins(l°s) and Hoogenstraaten.~1°9~

conduction bond

--

3_

P~

--I --

~ • t+

P

valence

bond

/,/,/,/,/,/,/,/,/,/,/,/,/,/ FIG.7.1. Schematicrepresentationof a phosphor.

Impurity ions or lattice defects in the lattice of an ionic crystal give rise to localized states such as t ÷, t- and p (Fig. 7.1) which may occur in the forbidden band gap. An electron may be raised into the conduction band from the valence band, p or t- by absorption of radiation of sufficiently high energy resulting

198

FRANCISK. FONG

in photoconduction (Section II). The levels t ÷ are empty states, and may trap electrons that have been excited into the conduction band. If the trap levels, t - , are close to the conduction band, electrons from t- may be excited by thermal motion into the conduction band. In a phosphor, the excited electron returns to the ground state of a luminescence center which is usually a trappedhole center: it first drops from the conduction band to p*, an excited state of p, and then from p* to p. The second transition, p* ~ p, is associated with the emission of light. In our example, the trivalent rare-earth impurities are incorporated i n t o an alkaline-earth lattice. The unassociated R 3+ ions (see Sections IV and VI), being electron traps, give rise to empty states, t +. When the crystal is irradiated with ionizing radiation such that R 3÷ ions are reduced to R 2 ÷ ions (associated with the levels t-), a corresponding number of hole centers must also be created elsewhere in the lattice. These hole-centers give rise to recombinationluminescence centers, which are associated with the levels p. During the irradiation, the traps are continuously being filled and emptied, as a continuous process of hole-electron recombination is taking place, which corresponds to the photo-excitation of a phosphor. After the irradiation, a certain fraction of the traps are filled, the degree of filling being radiation-dosage dependent (Section 6.4a). The trapped electrons may subsequently be thermally released with the emission of light characteristic of the R a+ ions (Section 6.4B). The trapping of electrons and holes in the radiation-induced reduction of rareearths exemplifies the metastable separation of charges resulting in the storage of energy, which can be readily released by means of thermal excitation. In the following discussions, our attention will be focused upon Dy3+-doped CaF2 crystals. 7.2. Glow Curves and Pulse Annealin9 The one important factor affecting the intensity of the thermoluminescence is the rate, r~, of escape of the trapped electron, which is rl = ~,n,-,

(7.1)

where nt- is the concentration of the filled traps, and cq = v exp (-E,-/kT),

(7.2)

Et- being the energy of activation of the trapped electron. An important method of evaluating E,_ is the glow-curve method, which was first described by Urbach (11°) over three decades ago, and also more recently by Hoogenstraaten. (1°9) The method consists of heating the sample up in the dark at a uniform rate and observing the changing intensity of the thermoluminescence.

199

Rare-earth impurities in alkaline-earth halides

The theoretical shape of the low curve for a sample with one type of trap depends on the kinetics of the luminescence decay. In the case of first-order kinetics, the glow curve is the solution I(T) of the following differential equations :

I(T) -

dn t _

dt

- ~an,- = vn,- exp ( - E , - / k T ) ,

and

(7.3) dT

-8

dt where I(T) is the luminescence intensity at temperature T and fl is the heating rate. F r o m eqns. (7.3), drl t-

n,-

_

vEt-

v

fl e x p ( - E , - / k T ) d T

x dx

= --~e-

~ x 2,

(7.4)

where x = Et-/kT. Letting xp ( - ¢ ¢ - 2a~) - q~(x)

and

vEt- - 2

we obtain n,- = no exp [-2(p(x)]

(7.5)

where n o is the concentration of filled traps at T -- 0. The glow intensity I(T) may now be written as I(T) = v n , - e x p ( - x ) =

vnoex p [ - x

- 2p(x)],

(7.6)

and I(T) may be conveniently plotted against increasing values of T, p(x) being a tabulated function. The glow intensity I(T) for second-order kinetics can also be readily derived from the differential equations I(T) -

dn,- _ ~av exp(-E,-/kT);dT = fl, d~ot2M:n~clt

(7.7)

where ~2 and 0~3 (see Section 7.3) are constants of the recombination of an electron with an empty trap, and that of an electron with a luminescence center, respectively; and M is the sum of the concentrations of t ÷ and t - , a constant. Letting 2' = v'Et-/flk and v' = (~3no/ct2M)v, I(T) = v'no [1 + 2'~o(x)] -2 exp ( - x ) .

(7.8)

200

FRANCIS K. FONG

The theoretical glow curves arising from eqns. (7.6) and (7.8) are illustrated in Fig. 7.2. Although the maximum is at nearly the same temperature, the secondorder glow peak is markedly broader than the first-order case. I.I Er = 0.5 eV I[

\~

/

//

:

/I

-o.,

----- second order

I \

/",,

i/

"

11 200

220

firet order

- -

I\

/S,.",Y"

\,~

, 240

I

I 260

i I I~ 280 300 Temperoture T(=K]

i

i 320

J

340

FIG. 7.2. Theoreticalglow curves of a phosphor with one type of trap. (AfterHoogenstraaten.(t09))

Glow curves o f C a F 2 : 10-1%DyF3 and CaF 2 : 1 0 - 1 % T m F 3 samples gammairradiated for 115 min at room temperature have been measured.* It has been found that the Dy glow curve (Fig. 7.3A) peaks at 112 ° with satellite peaks at 71 °, 152 °, 160 °, and 213°C, and that the Tm glow curve (Fig. 7.3B) consists of only one peak at 211°C for the same temperature range investigated. Tlae presence of five bursts of luminescence glow in the Dy sample suggests that there are at least five different types of traps. Correlation between certain glow peaks and changes in the absorption spectrum of the sample may be established by a pulse annealing technique. Figures 7.4 and 7.5 show that the same bands are observed for all the coloring experiments carried out on CaF 2 :Dy a+ samples except for the one in which the sample was heated in the presence of Ca vapor at 850°C for a prolonged period before it was allowed to cool slowly. Two groups of absorption bands are induced: bands due to electronic 4f l° ~ 4f95d 1 transitions (Section 6.5) of divalent dysprosium ions in cubic symmetry peaking at 910, 715, 572, 458, and 314 m/~, and the short wavelength bands peaking at 280.3, 271.3, 263.0, and 244.0 m,u which are designated as the DI, D2, 03, and D 4 bands, respectively, for convenience of discussion. The Dy 2+ bands occur in a spread of 600 mB whereas the D1, D2, Do, and D 4 bands occur in a comparatively narrow * The author wishes to acknowledge the assistance of Mr. F. Krajenbrink in making these measurements.

Rare-earth impurities in alkaline-earthhalides

201

range of approximately 80 m/~. The absorption of an irradiated CaF2 :Tm sample is also shown in Fig. 7.4G. In addition to the Tm 2÷ 4f 11 ~ 4f~°5d 1 bands, it is apparent that the D4 band (244 m/a) is also present. The D1, D2, and D3 bands, on the other hand, are absent. The massive u.v. absorption in Fig. 7.4B is indicative of colloidal particle formation in the sample, which

70C

Dysprosium

60C

/!

40C 50C

~20C IOC

0

200

400

600

800 I000

1200 1400 1600

1800 2000 2200 2400

time(sec)

B

250"

i

200"

time(uc)

FIG. 7.3. Glow curves of gamma-irradiated (2 hrs.) CaF2 : (A) with Dy ions, (B) with Tm ions. Warmingrate was 0.1 °/see; thermoluminescencewas observedat 4800 A for (A) and at 4250 A for (B)

occurs in fluorite crystals that have been allowed to cool slowly after having been additively colored. TMs2) From Fig. 7.5A and Fig. 7.4, it is apparent that the D4 band is appreciably less intense in the additively colored sample than the irradiated samples. Since no hole-centers exist in additively reduced samples

202

FRANCtS K. FONG

,

zzo

/"-",,

,35p~o 'z~,o'2~,o'2~o'36o'3~'o'34o 46o'66o'86o'1ooo Wavelength

X

(mp.)

FIG. 7.4. Coloration by gamma-irradiation: (A) before gamma-irradiation; (B)-(F) CaF 2 : (0.l real. %) DyF 3 samples irradiated for (B) 55, (C) 115, (D) 165, (E) 300, and (F) 850 rain, respectively; ((3) CaFz: (0.l real. %) TmF3 sample irradiated for 120

min. All samples were 0.05 in. thick. Note the change in scale for wavelengths longer than 360 m/~. (Section 6.4B), we m a y conclude from the above that all the observed bands are due to the excitations of trapped electrons. The results of pulse annealing experiments m a y be summarized in Fig. 7.6. Each of the temperatures employed, 70 °, 110 °, 145 °, 160 °, 200 °, and 229°C, corresponds approximately to a glow peak. Pulse annealing up to 70 ° (Fig.

350 36O

~

2~o' 24o' 2~o z~o'36o' 31o' 3~o' ~o' 6~o 8~o ,ooo WavelengTh X (rn~)

FIG. 7.5. Additive col0ration of CaF2 :Dy 3+ crystals: (A) 15 min treatment at 850°C followed by slow cooling (0.05 in. thick); (B) 20 min treatment at 850°C followed by slow cooling after (A). The sample for (A) was approximately four times as thick as that for (B).

Rare-earth impuritiesin alkaline-earthhalides

203

7.6B) cause a slight bleaching of Dy 2+ bands and a considerable growth in the D bands. At 110° (Fig. 7.6c) the Dy 2+, D1, D2, and D3 bands are extensively bleached as the sample crystal acquires a faint pink color due to absorption in a band peaking at 500 m/~ (presumably the RO band, see Section 6.4A). At 145 ° (Fig. 7.6D) and 160° (Fig. 7.6E) (corresponding to the glow peak at 152°), the predominant changes in the absorption spectra are the diminution of the D4 band and the growth of the RO band. At 200° (Fig. 7.6F) all the bands are uniformly bleached, and, finally, at 229 ° the predominant feature is the bleaching of the RO band.

B

F

,

,

,

,

,

,

?~o 3,so

zzo 240 260' zBo'300' 3z0'340 Wavelength),(mF)

,-

-~-~:,

~60 600 800',0o0

FIG. ?.6. Pulse annealing of gamma-irradiated (2 hr) CaF 2 :Dy 3+ crystal: (A) after irradiation; (B)-(G) warmed momentarily to (B) 70 °, (C) l l 0 °, (D) 145°, (E), 160°, (F) 200, and (G) 229~C. Dotted curves under (G) show graphic resolution of the D1, D2, D3, and D4 bands in an estimate of relative intensities of these bands.

From the above results, it is clear that at least four types of color centers exist in gamma-irradiated dysprosium samples: (1) Dy 2+ centers consisting of electrons trapped in the 4f shell of Dy 3+ ions occupying substitutionally Ca 2+ sites in a cubic environment; (2) D centers (associated with D1, D2, and D 3 bands) that differ in thermal behavior from Dy 2+ centers, (3) D 4 centers that differ in thermal behavior from both the Dy 2+ centers and the D centers, and (4) RO centers that are formed by heating the irradiated sample to temperatures above 110°C, and bleached at temperatures above 200°. The absence of the D1, D2, and D 3 bands in gamma-irradiated CaF 2:Tm samples (Fig. 7.4) indicates that they are characteristic of electrons trapped at sites consisting of Dy 3+ ions and some defect aggregates that distort the cubic symmetry of the Dy ions. The presence of the D4 band in both the Dy and Tm samples suggests t h a t t h e excitation need not be characteristic of the 4f-5d transition

204

FRANCIS K. FONG

of the tripositive ions. In the case of the D centers, two possibilities occur: either there is only one type of center that gives rise to all the three (D1, D2, and D3) bands, or different types of centers exist giving rise to the various bands individually. In view of the similar thermal and formation behavior of these three bands, it appears that they are indeed due to one type of center in much the same way that the various visible absorption bands are due to the same Dy 2+ centers. The D 4 band, on the other hand, differs from the other D bands not only in its thermal behavior, but also in its formation. It is apparent, for example, that the D4 band is less prominent in an additively colored sample than it is in an irradiated sample (Figs. 7.3 and 7.4). It appears certain that the various possibilities of charge compensation of tripositive ions in CaF2 must be responsible for the existence of these various types of color centers. The simultaneous growth and decay of various traps as evidenced in the pulse annealing experiments strongly suggests that electrons are involved in the thermally activated charge-transfer process. The high temperature Dy glow peak (at 213°C) corresponds to a process whereby all traps are emptied, and is the only one that can be attributed to thermal hole ionization, if it exists. It may be of significance that the Tm glow peak occurs at 211°C which is indistinguishable from the high temperature glow maximum at 213°C for Dy. It is to be expected that an electron trapped at an inner f orbital of a Tm 3+ ion requires a higher ionization potential for a change in electronic configuration from 4f 13 --+ 4f 12 than a corresponding change from 4f 1° --+ 4f 9 in the case of dysprosium (Section 6.3). It is, therefore, not surprising that lower temperature glow peaks are absent in the case of thulium. 7.3. The Isothermal Decay of Thermoluminescence Due to the presence of several closely spaced Dy glow bursts, it is not possible to determine the order of decay from the shape of the glow curve (Fig. 7.2A).t Instead, it is necessary to derive the kinetic order from isothermal decay studies. For the sake of simplicity, only the Dy 2+ centers are considered. By thermal excitation, the trapped electron on a Dy 2+ center (correspond to t- in Fig. 7.1) may be freed (presumably into the conduction band): tr 1

Dy 2+ ~ Dy 3+ + e-

(i)

The free electron e- can either be recaptured by a Dy 3÷ ion (t + in Fig. 7.1): ~2

Dy 3+ + e- --* Dy 2 ÷

(ii)

1" The broad tail of the single Tin glow curve (Fig. 7.3B), on the other hand, apparently resembles that of the theoretical second-order glow curve (Fig. 7.2).

205

Rare-earth impurities in alkaline-earth halides

or recombine with a trapped hole center p : (iii)

~3

e- +p~e-p

The recombination process (iii) apparently occurs with the emission of light. Since only Dy 3 ÷ emission is observed, it is possible that the luminescent center p consists of a Dy 3+ ion. Annihilation of the trapped hole is accompanied by the excitation of the Dy 3+ ion, the return of which to the ground state is responsible for the observed luminescence, L. Differential equations governing such a thermally activated charge transfer process may be written: and

L = - d p / d t = ~3(e-)(p),

(7.9)

d ( e - ) / d t = ~l(Dy 2÷) - ct2(Dy3+) (e -) - eta(e-) (P).

(7.10)

Since the sum of (Dy 2+) and (Dy 3+) must be equal to some constant M, M -- (Dy 2+) + (Dy3+),

(7.11)

and since the sample must maintain charge-neutrality, (p) = (Dy 2÷) + (e-),

(7.12)

the quantities (Dy 2 +) and (Dy 3 +) may be expressed in terms of M, (p) and (e-). If thermally ionized electrons are immediately recaptured by hole centers and traps (i.e. ~2 >> Ctx '~ °t3) such that (e-) ~ 0 and d ( e - ) / d t ~ 0, the steady state approximation becomes valid. Thus, d(e-) -- 0 = a t l ( D y 2+) - ~ 2 ( D y 3 + ) (e - ) - ~ 3 ( e - ) (P), dt

(7.13)

and Ctl(Dy2+) (e-) = ~2(Dy3+ ) + 0t3(p).

(7.14)

From eqn. (7.14) the expression for the decay process is obtained: O~lO~3(Dy 2 +) (/7)

~l~3(p) 2

L = - d(p)/dt = 0~3(e-) (17) --- tx2(Dy 3+) 4- ~3(P) ~ (0~3 - tx2)(P) 4- o~2M s i n c e (Dy 2÷) ~ (p) for (e-) differential eqn. (7.15),

(7.15)

0 according to eqn. (9). The solution of the

where (P)o is the value of (p) at t = 0, consists of two special cases. For 0~2 a second order rate equation is obtained: d(p)/dt

=

°~1°~3

,

\2

-- ~x2M Ip) ,

=

~3,

(7.17)

206

FRANCIS K. FONt

and for ~x2

=

0, a first order rate equation is obtained: d(p)/dt = - ~ I(P).

(7.18)

Unless 0(2 = 0, however, the thermoluminescent process will have an apparent kinetic order of 2, since for small values of (p)/M, the first term of eqn. (7.16) will always be more important than the second. For this reason, but for the very unusual condition that ~2 = 0, the second order description [eqn. (7.17)] should prevail, from which we obtain, in anticipation of the experimental observations to be discussed shortly,

dL/dt

-

2~10(3 d~) -~

(p)

~_~)½L3/2

= -2

(7.19)

t4¢ 194.1 "C 12C

IOC •

74C

•

°

~ *c

~ sc 4( 2C time

(see..)

,~r,c 6 '~ii 60 ~

162.5" a156*C 149°C

time

146"C

J41"C

[see..) so

~

~ 2o

ioa.c

-

I

a'~

FIG. 7.7. Decay plots of gamma-irradiated (2 hr) CaF 2 :Dy a÷ showing dependence of (l/L)* vs. t in the temperature range of 55°-194°C.

Rare-earth impurities in alkaline-earth halides

207

The solution of eqn. (7.19) may readily be given as

\~2M} J

\~x~31

(7.20)

which gives the square root of 1/L as a linear function of time t. From eqn. (7.20) we expect to observe, if the thermoluminescent process obeys a second-order law [eqn. (7.17)], the empirical relationship:

(1)~ = a + bt,

(7.21)

where b =

(~10~3~ ½ = ( 0~3 ~ V½exp(-E,-/2kT),

(7.22)

in which, as in eqn. (7.2), ~1 = v exp (-Et-/kT). Decay curves of thermoluminescence due to gamma-irradiated CaF 2 :Dy(II + III) samples have been measured at 22 different temperatures in the range of 55°-190°C. The square root of the reciprocal luminescence was indeed found to vary linearly with time according to eqn. (7.21), as is shown in Fig. 7.7. From Fig. 7.7, it is observed that change of slope in the rate plots occurs at temperatures around 75°-85 °, 105°-108 °, and 118.5°-137 °, which correspond approximately to three of the major glow peaks. We may thus conclude that the linearity of the (l/L) ~ vs. t plots and the changes in slope at temperatures corresponding to glow peak temperatures support the above interpretation of second-order kinetics. The derivation for eqn. (7.20) above is based upon the assumption that the various types of traps are independent of one another. Since electron transfer from one type of trap to another has been observed (Fig. 7.6), however, we are faced with a somewhat more complicated situation. Letting t - and t ÷ be all the filled and empty traps other than Dy 2÷ and Dy 3÷, we have, instead of the reactions (i), (ii), and (iii) above, the following reactions : Dy 2+ ~ Dy 3+ + e-

(i')

e- + t + ~ t-

(ii')

and %

e- + p ~ e - p . Equations (7.11) and (7.12) must likewise be rewritten as: M = (Dy 2+) + (Dy 3+) + (t+),

(iii') (7.11')

and (p) = (Dy 2+) + (e-) + (t-).

(7.12')

208

FRANCIS K. FONG

Equation (7,14) now becomes • l(Dy 2+) (e-) ~ (~3 - ~2)(P) + 0t2[M + (t-)]"

(7.14')

assuming that ~t2 ~ ~ . From eqns. (7.14') and (7.12'), we obtain the expression for the decay process : ~x~a(P) [(P) - (t-)] L = -d(p)/dt = ga(e-)(p) ~ (0t3 _ g2)(P) + ~2[ M + (t-)]'

(7.15')

which does not give rise to simple-order kinetic equations unless (p) >> (t-) < M. Except for the low temperature data corresponding to the first glow peak at 71°C, this condition could conceivably be valid. From Fig. 7.3, it is clear that the deeper traps corresponding to the higher temperature glow peaks are far less numerous than those corresponding to the lower temperature glow peaks, particularly that at 112°C, since the number of traps should be approximately proportional to the intensity of the glow burst. Alternatively, we may assume that charge transfer between different types of taps, though observable, is of negligible importance in the determination of the kinetic order, and that different types of traps by and large behave independently of one another. In this case, eqns. (7.9) to (7.20) are valid. 7.4. Activation Energies in the Thermoluminescence Process For the first order decay law [eqn. (7.18)], the trap depth is related to the temperature T6 of maximum glow by tx°a)

{ ln(kT~/_flE,-)~ er = kT In s 1 + In s J

(7.23)

where fl is the warming rate and s = v, the frequency factor of the thermal ionization process. Knowing Et-, it is possible to estimate the escape factor v. In the case of second-order decay, however, the description is somewhat more complicated. The quantity s in eqn. (7.23) is now defined as ,09) ,=

P )ol

L~2MJ

(7.24)

From simple statistical considerations, (a 11)

v = ½ct2Nc = ct2(2nm*kT/h2) a/2

(7.25)

where Nc = 2(27~m*kT/h2) a/2 is the effective density of states of the conduction band, m* being the effective mass of the carrier. It follows from eqns. (7.24) and (7.25) that s=~

x ctaNc(p)o M

(7.26)

Rare-earth impurities in alkaline-earth halides

209

where the weakly temperature-dependent quantity Nc can be considered as a constant for a given host material and a3 is characteristic of the recombination hole center. Since the term In (kTg/flE~_) is nearly constant for the range of variation in Tg/E,_, and s in eqn. (7.26) is constant provided that the ratio (p)o/M is of the same order of magnitude for the various types of centers, the trap depth E,- should vary linearly with the peak maximum TG for the set of glow peaks observed at a constant value of fl, i.e.

E t_ = Ck ~ In s ; C = constant

(7.27)

Evaluation of the trap depth E t- from the temperature TG of maximum glow according to eqn. (7.27) alone, however, is not feasible due to a lack of kn0wledge of the quantities s and C. The values of E~ for the first glow peak and s, however, can be estimated as follows. The glow intensity at a given instant can be written after eqn. (7.17) as L(T)2~

, v e x p ( - E t - / k T ) ~ c o n s t a n t ' e x p ( - E t /kT), for

(7.28)

T
A. F. WELLS, Structural Inorganic Chemisto,, Oxford University Press, London, 1950, 2nd ed. H. BRAEKKEN,Z. Krist. 83, 222 (1932). K. PRZIBRAM, Irradiation Colours and Luminescence, Pergamon Press Ltd., London, 1958. N. F. MOTT and R. W. GURNEY, Electronic Processes in Ionic Crystals, Oxford University Press, London, 1948, 2nd ed. J. FRENKEL,Zeits. F. Physik, 35, 652 (1926). C. WAGNER and W. SCHOTTKV, Zeits. f phys. Chem. (B), 11, 163 (1930). W. JOST, J. Chem. Phys. 1, 466 (1933); Trans. Faraday Soc. 34, 860 (1938). R. H. FOWLER and E. A. GUGGENHHM, Statistical Thermodynamics, Cambridge, 1939, ch. XIII. C. TUBANDT, H. R~INHOLD and G. LIEBOLD, Zeits.f. anorg. Chem. 197, 225 (1931). C. TUBANDT and S. EGGERT, Zeits.f. anorg. Chem. 110, 196 (1920). C. TUBANDT, Zeits.f. anorg. Chem. 115, 105 (1921). A. B. LXDDIARD,Handbuch der Physik 2,11,Springer Verlag, Berlin, 1957, p. 245.

210

FRANCIS K. FONG

13. R. W. URE, JR., J. Chem. Phys. 26, 1363 (1957). 14. A. TAYLORand E. BARSIS, Report No. 306, Materials Science Center, Cornell University, 1965 (unpublished). 15. A. SMEKAL,Handbuch der Physik 24/2, 881 (1933). 16. E. KOCH and C. WAGNER,Zeits. f phys. Chem. (B) 38, 295 (1937). 17. C. WAGNERand P. HANTLEMANN,J. Chem. Phys. 18, 72 (1950). 18. H. W. ETZEEand R. J. MAURER,J. Chem. Phys. 18, 1003 (1950). 19. Y. HAVEN,'Rec. Tray. Chim. 69, 1259, 1471, 1505 (1950). 20. H. KELTINGand H. WITT, Z. Physik 126, 697 (1949). 21. J. TETLOW,Z. physik. Chem. 195, 197, 213 (1950); Ann. Physik 5, 63 (1949). 22. H. EYRING,J. Chem. Phys. 3, 107 (1932). 23. GLASSTONE,LAIDLERand EYRING, The Theory of Rate Processes, McGraw-Hill, New York, 1941, pp. 184-191. 24. G. S. RUSHBROOKE,Introduction to Statistical Mechanics, Oxford University Press, London, 1960. 25. C. WERT and C. ZENER, Phys. Rev. 76, 1169 (1949). 26. C. ZENER,Imperfections in Nearly Perfect Crystals (edited by W. Schokley), John Wiley, New York, 1952, p. 289. 27. Y. HAVENand J. H. VAN SANTEN,Philips Research Repts. 7, 474 (1952). 28. U. CROATTOand M. BRUNO,Gazz. Chim. ItaL 78, 95 0948). 29. U. CROATTOand M. BRUNO,Gazz. Chim. Ital. 74, 246 (1946). 30. C. TUBANDT,Handbuch der Experimentalphysik 12 (1). 383 0932). 31. E. ZINTL and A. UDGARD,Z. anorg, u. allgem. Chem. 2~._, 150 (!939): 32. A. MEYER, Gazz. Chim. ItaL 73, 199 (1943). 33. J. SNORTand R. RoY, J. Phys. Chem. 67, 1860 (1963). 34. O. STASlWand J. TETLOW,Ann. Phys. 1, 261 0947). 35. D. C. STOCKBARGER,J. Opt.Soc. Am. 39, 731 (1949). 36. H. BONTINCK,Physica 24, 650 (1958). 37. A. D. FRANKLIN,J. Chem. Phys. Solids26, 933 (1965). 38. K. THARMALINGHAM,Thesis, University of Reading, 1961; J. Phys. Chem. Solids 24, 1380 (1963); 25, 255 (1964). 39. M. BORNand J. E. MAYER,Z. Phys. 75, 1 (1932). 40. R. W. PONE, Proc. Phys. Soc. 49 (extra part), 3 (1937). 41. F. SEITZ,Revs. Mod. Phys. 18, 384 (1946); 26, 7 (1954). 42. J. H. SCHULMANand W. D. COMPTON,Color Centers in Solids, MacMillan, New York, 1962. 43. A. KENNGOTT,S. B. Akad. WiNs. Wien 11, 12 (1853). 44. H. STEINMETZ,Z. Kristallogr. 58, 330 (1923); 61, 380 (1925). 45. E. REXER,Z. Kristallogr. 78, 251 (1931). 46. M. BERTHELOT,C. R. Acad. Sci. (Paris) 143, (1906). 47. F. HENRICH,A. angew. Chem. 33, 5, 13, 20 (1920). 48. T. PEARSALL,J. Roy. Inst. 1, 77, 267 0830); Ann. Chem. (Phys.) 49, 337, 1832 (1831). 49. M. BLEAR,S. B. Akad. WiNs. Wien, IIa, 132, 45 (1923). 50. L. WOHEER,Z. anorg. Chem. 47, 364 (1905). 51. H. HABERLANDT,S. B. Akad. WiNs. Wien, IIa, 141, 441 (1932); 142, 29 (1933). 52. E. MOLLWO,Nachr. Ges. WiNs. G6ttingen 6, 79 (1934). 53. A. SMAKULA,Phys. Rev. 77, 408 (1950). 54. F. LOTY, Z. Physik 134, 596 (1953). 55. F. K. FONG and P. N. YOCOM,J. Chem. Phys. 41, 1383 (1964). 56. J. H. SCHULMAN,R. J. GINTHERand R. D. KIRK, at. Chem. Phys. 20, 1966 (1952). 57. H. ADLERand I. KRETA,Oesterr. dkad. Wissen., Math. Naturu. Klasse, Abt. II, 166 (1957). 58. H. BONTINCK,Physica 24, 639 (1958).

Rare-earth impurities in alkaline-earth halides

211

A. SMAKULA,Phys. Rev. 91, 1570 (1953); Z. Physik. 138, 276 (1954). S. BARILE,J. Chem. Phys. 20, 297 (1952). V. M. GOLDSCHMIDT,Geochemische Verteilungsgesetze der Elemente 8, 69 (1926). K. PRZIBRAM,S. B. Akad. Wiss. Wien, IIa, 147, 261 (1938). W. J. HUMPHREYS,Astrophys. J. 20, 266 (1904); 22, 157 (1905). G. WILD, S. B. Akad. Wiss. Wien, Ila, 146, 479 (1937). W. J. SCOULERand A. SMAKULA,Phys. Rev. 120, 1154 (1960). J. R. O'CONNER and J. H. CHEN, Phys. Rev. 130, 1790 (1963). L. PAULING,J. Am. Chem. Soe. 69, 542 (1947). C. K. JORGENSEN,J. lnorg. Nuclear Chem. 1,301 (1955). B. G. WY~OURNE, Spectroscopic Properties o[' Rare Earths, Interscience Publishers, John Wiley, New York, 1965. 70. L. B. ASPREYand B. B. CUNNINGHAM. Prog. Inorg. Chem. 2, 267 (1960). 71. B. B. CUNNINGHAM,Int. Congr. Pure Appl. Chem. 17, 64 (1959). 72. F. BASSANIand F. G. FUMI, Nuovo Cim. 11,274 (1954). 73. F. K. FONG, P. N. YOCOMand H. L. PINCH, unpublished results. 74. National Bureau of Standards Circular 500. 75. M. A. EL'YASHEVtCH,Spectra qfthe Rare Earths 2, Translation Series, United States Atomic Energy Commission Office of Technical Information, AEC-tr-4403, 1953. 76. C. E. MOORE. Circular of National Bureau of Standards, No. 467, June 15, 1949. 77. H. L. PINCH, J. Am. Chem. Soc. 86, 3167 (1964). 78. P. BRAUER,Z. Naturforsch. 6a, 561 (1951). 79. B. BLEANEY,P. M. LLEWELLYN,and D. A. JONES, Proc. Phys. Soc. (London) B69, 858 (1956). 80. J. M. BAKER,W. HAVESand M. C. M. O'BRIEN, Proc. Roy. Soc. A254, 273 (1960). 81. E. FRIEDMANand W. Low, J. Chem. Phys. 33, 1275 (1960). 82. M. DVIR and W. Low, Proc. Phys. Soc. (London) 75, 136 (1960). 83. W. HAYESand J. W. TWlDELL, J. Chem. Phys. 35, 1521 (1961). 84. H. R. LEwis and E. S. SABISKY,Phys. Rev. 130. 1370 (1963). 85. W. HAYES, G. D. JONES, and J. W. TWlDELL, Proc. Phys. Soc. (London) 81, 371 (1963). 86. F. D. S. BUTEMENT, Trans. Faraday Soc. 44, 617 (1948). 87. D. L. WOOD and W. KAISER,Phys. Rev. 126, 2079 (1962). 88. D. S. MCCLUREand Z. J. KIss, J. Chem. Phys. 39, 3251 (1963). 89. Z. J. K~ss, Phys. Re;,. 127, 718 (1962). 90. P. P. FEOFILOVand A. A. KAPLYANSKH,Opt. i Spektr. 12, 493 (1962). 91. P. GORLICH, H. KARRAS,K. KOHNE, and R. REIMANN,Sitzber. deut. Akad. Wiss. Berlin, KI. Math., Phys. U. Tech. No. 2 (1959); ibid. No. 3 (1960). 92. F. K. FONG, J. Chem. Phys. 41,245 (1964). 93. E. SABISKY,J. Appl. Phys. 36, 802 (1965). 94. Z. J. KISSand P. N. YOCOM,J. Chem. Phys. 41, 1511 (1964). 95. H. GUGGENHEIMand J. V. KANE,Appl. Phys. Letters 4, 172 (1964). 96. F. K. FONG, RCA Rev. 25, 303 (1964); J. Chem. Phys. 41, 2291 (1964). 97. F. K. FONG, Proceedings of the Fourth Rare Earth Con Jerence, Gordon and Breach Scientific Publishers, New York, 1965. 98. W. A. RUNCIMANand B. WYBOURNE,J. Chem. Phys. 31, 1149 (1959). 99. D. S. McCLURE, Advan. Solid State Phys. 9, 400 (1959). 100. C. STRUCK,unpublished results. 101. W. E. ALBERTSON,Phys. Rev. 47, 370 (1935); 52, 644 (1937). 102. W. E. ALBERTSON,M. BRUYNESand R. HANAU,ibid. 57, 292 (1940). 103. Ji D. AXE and P. P. SOROKIN,Phys. Rev. 130, 945 (1963). 104. W. E. BRON and M..WAGNER, Phys. Rev. 139A, 233 (1965). 105. E. Y. WONG and F. K. FONG (to be published). 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69.

212

FRANCIS K. FONG

106. 107. 108. 109. 110. 111. 112.

C. K. JORGENSEN, Mol. Phys. 5, 271 (1962). G. BIRNBAUM,Optical Masers, Academic Press, New York, 1964, Ch. IX. J. T. RANDALLand M. H. F. WILKINS, Proc. Roy. Soc. (London) A184. 365 0945). W. HOOGENSTRAATEN,Thesis, University of Amsterdam, Holland, (1958). F. URBACH, Sits. Bet. Akad. Wiss. Wien, Math. Naturw. Klasse 139, 353 (1930). W. SHOCKLEY,Electrons and Holes in Semiconductors, Van Nostrand, New York, 1950. J. J. HILL and P. SCHWED, J. Chem. Phys. 23, 652 (1955).