1953 dexter theory

8/3/2019 1953 Dexter Theory

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836 L. F. H. BOVEYI t is tempting to represent the data on CDaH andCHaD by slightly different values of ro(C- H) and

ro(C-D) (but neglecting the change in angle), as wasattempted in the case of the methyl halides by Milleret alP However, here again a surprisingly large difference has to be assumed: one finds ro(C-H) = 1.09724A,ro(C-D)=1.09021A. At the same time the moment ofinertia of CH, would have to be 5.3719X 10-40, which

17 Miller, Aamodt, Dousmanis, Townes, and Kraitchman, J.Chern. Phys. 20, 1112 (1952).

T H E J O U R N A L OF C H E M I C A L P H Y S I C S

deviates widely from the observed value. Clearly the effect of zero-point vibration cannot be taken into accountin such a simple way.The author wishes to thank Dr. A. E. Douglas forsuggesting the problem and for giving experimental andtheoretical help. He is grateful to Dr. G. Herzberg formuch helpful discussion. The cooperation of Dr. J. A. B.Nolin and Dr. L. C. Leitch in supplying the deuteratedmethyl halides and of Dr. F. P. Lossing in analyzingthe gases by mass-spectroscopic methods is also greatlyappreciated.

V O L U M E 2 ! . N U M B E R 5 MA Y . !953A Theory of Sensitized Luminescence in Solids

D. L. DEXTER*Metallurgy Division, Naval Research Laboratory, Washington, D. C.(Received November 3, 1952)

The term "sensitized luminescence" in crystalline phosphors refers to the phenomenon whereby an impurity ( a c t i ~ ~ t o r , or emitter) is enabled to luminesce upon the absorption of light in a different type ofcenter (sensitizer, or absorber) and upon the subsequent radiationless transfer of energy from the sensitizer!o the activator. The resonance theory of Forster, which involves only allowed transitions, is extended tomclude transfer by means of forbidden transitions which, it is concluded, are responsible for the transferin all inorganic s y s t e m ~ yet investigated. The transfer mechanisms of importance are, in order of decreasingstrength, the overlappmg of the electric dipole fields of the sensitizer and the activator, the overlapping ofthe dipole field of the sensitizer with the quadrupole field of the activator, and exchange effects. Thesem e c ~ ~ n i s ~ s w i J ~ give rise to "sensitization" of about loa-l04, 102, and 30 lattice sites surrounding eachsensitizer m typical systems. The dependence of transfer efficiency upon sensitizer and activator concentrations and on temperature are discussed. Application is made of the theory to experimental results on inorganic phosphors, and further experiments are suggested.

I. INTRODUCTIONONE of the important problems of solid state physicsis that of a description of the processes whichfollow the absorption of visible or ultraviolet light by aninsulating crystal. Of the several possible generalprocesses, luminescence has received the most experimental attention, although unfortunately there seemsto be no simple crystal yet investigated for whichluminescence associated with the undistorted latticehas been unambiguously demonstrated.! Consequentlymost experimental work on luminescence has dealt withsystems consisting of a host lattice and one or moretypes of impurity atoms.2.3 The theoretical work in thisfield has not kept up with experiment for two mainreasons: (1) The development of the field of luminescence, because of its commercial significance, has been* Present address: Institute of Optics, University of Rochester,Rochester, New York.

I The edge emission of CdS [F . A. Kroger, reference 31, C. C.Klick, Phys. Rev. 89. 274 (1953)J may prove to be an exampleof true lattice emission. A number of more complicated inorganiccrystals, CaMoO. and CaWO., for example (see references 8 and12), apparently luminesce even in the absence of imperfections, asdo a number of organic crystals, e.g., anthracene and napthalene.2 P. Pringsheim, Fluorescence and Phosphorescence (IntersciencePublishers, Inc., New York, 1949).3 H. W. Leverenz, Introduction to Luminescence of Solids (JohnWiley and Sons, Inc., New York, 1950).

guided largely by practical considerations, with a resulting reduction in the attention directed towards development and investigation of the simplest possible luminescent systems, which might be more easily interpreted.(2) The theoretical problems of luminescence are exceedingly difficult to treat even in their simplest form, notonly because they involve the simultaneous interactionsamong radiation, matter and phonons, bu t also because the specific details of the wave functions are offirst-order importance. The experimental results onluminophors consisting of electronically similar impurity atoms in the same host lattice and of the sameimpurity atom in similar host lattices demonstrate thestrong influence of the details of the coupling betweenthe impurity atom and its surroundings and thusemphasize the "many-body" nature of the problems tobe solved.A branch of luminescence investigated in recent years,namely, "sensitized luminescence,"'-13 appears at first,4 S. Rothschild, Physik. Z. 35, 557 (1934) j 37, 757 (1936). Schulman, Evans, Ginther, and Murat a, J. Appl. Phys. 18, 732(1947).6 J. B. Merrill and J. H. Schulman, J. Opt. Soc. Am. 38, 471(1948).7 H. C. Froelich, J. Electrochem. Soc. 93, 101 (1948).8 F. A. Kroger, Some Aspects of the Luminescence of Solids(Elsevier Publishing Company, Inc., Houston, Texas, 1948).

wnloaded 02 Sep 2011 to 168.131.116.105. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissio


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T H E O RY OF SENSIT IZED LUMINESCENCE IN SOLIDS 837perhaps, to be even more complicated than the morecommon forms. That this is not the case is, as we shallsee, a result of the difference in the questions that wemay ask regarding the various processes.According to common usage, sensitized luminescencerefers to the process whereby an impurity atom (activator) having no appreciable absorption band in a givenregion of the (visible or uv) spectrum is made to emitradiation upon excitation in this region as a result ofabsorption by and transfer from another impurity atom(sensitizer) or from the host lattice. We shall refer tothese distinct processes as impurity-sensitized and hostsensitized luminescence, respectively.We may now see the relatively simple nature of theproblems to be investigated. I f we accept the experimental facts regarding the positions and strengths ofthe absorption band of the activator and the emissionband of the sensitizer, we may isolate the most fundamental problem peculiar to sensitized luminescence asfollows: What is the mechanism of the process by whichthe excitation energy is transferred from sensitizer toactivator, and what is its efficiency as a function of temperature and of activator and sensitizer concentrations?That is, we may in this case avoid all of the difficultproblems of common luminescence and concentrate onthose of sensitized luminescence alone.The purpose of this paper is to examine the abovequestions regarding sensitized luminescence in insulating crystals on the basis of a resonance theory of ene;gytransfer. Such a theory does not involve transitions toor from the conduction band and hence does not takeaccount of photoconductivity. Other theoretical workon sensitized luminescence in solids includes Mott andGurney'sl4 suggestion that electrons and holes whichare freed by absorption of light at the sensitizer travelthrough the conduction and valence bands, eventuallyto recombine at the activator. Such processes, involvingphotoconductivity, are not included within the presenttheory.A resonance theory of energy transfer is of course notnew. Franck, Cario, and many others have demonstrated the mechanism in gases,1s and a quantum theorywas given by Kallmann and F. London. 16 Transfer hassimilarly been shown in solutions containing organicmolecules,1s and both classicaP7 and quantum-mechanicaP8,19 theories have been formulated for transfer

9 Th. P. J. Botden and F. A. Kroger, Physica XIV, 553 (1948).10 F. A. Kroger, Physica XV, 801 (1949).11 Schulman, Ginther, and Klick, J. Electrochem. Soc. 97, 123(1950).12 Th. P. J. Botden, Philips Research Repts. 6, 425 (1951).13 J. Franck and R. Livingston, Revs. Modem Phys. 21, 505(1949). This paper contains references to work on organic systems.14 N. F. Mott and R. W. Gurney, Electronic Process in IonicCrystals (Oxford University Press, New York, 1940), p. 207.15 For references to the original work see, for example, references2 and 13.16 H. Kallmann and F. London, Z. Physik. Chem. B2, 207(1929).17 J. Perrin, Compt. rend. Paris, 184, 1097 (1927).18 F. Perrin, J. phys. et radium 7, 1 (1936).19 Th. Forster, (a) Ann. Physik 2, 55 (1948) j (b) Fluoresezenz

processes involving allowed transitions. Investigationsof the propagation of excitons in insulators have likewise been based on resonance theories,2 as has a theoryof the annihilation of F centers by excitons.21Previous theoretical work has been concerned withresonance between two allowed (electric dipole) transitions. In the common organic systems the transitionsinvolved are allowed, and the significance of Forster'swork19 has been generally recognized by workers in theorganic field. However, one of the principal reasons forthe interest in sensitized luminescence in inorganicsolids has been that resonant transfer can be obtainedbetween an allowed transition in the sensitizer and aforbidden transition in the activator. That is, an activator with a suitable emission spectrum (for example,manganese with a forbidden transition) can be made toluminesce after energy transfer from an absorbingsensitizer, although the activator will not absorb theenergy directly.22 Clearly a description of forbiddentransi tions is essential.Several authors have suggested, on the basis of theabove analogies in gases and liquids, that resonant transfer occurs in the systems they have investigated.4- 13Some of these writers state that the transfer probabilityis determined by the overlapping of the wave functionsof the sensitizer and activator9-11 and that only whenactivator and sensitizer occupy adjacent lattice sitesmay transfer occur.9,10 Inspection of the above-mentioned resonance theories shows that overlap integralsbetween activator and sensitizer occur only in thenormalization of the wave functions and are of minorimportance. Furthermore, in the case of impuritysensitization exchange intervals (which depend partially on overlap) are unimportant when strongelectric dipole transitions are allowed. Previous treatments of sensitized luminescence therefore have notincluded exchange effects and are concerned withthe overlapping of the "near-zone" electric fields, andnot the wave functions. I t will be shown that exchangeeffects become important for magnetic dipole (andhigher order) transitions.Section II contains discussions of the physical modelemployed here and of the procedure for normalizationof the wave functions. Sections III, IV, and V treatelectric dipole-dipole, dipole-quadrupole, and exchangeeffects and electric dipole-magnetic dipole transitions,respectively. The above all refer to impurity-sensitization, whereas Sec. VI discusses host-sensitization.organische Verbindungen (Vandenhoeck and Ruprecht, Gottingen,1951); (c) Z. Elektrochem. 53, 93 (1949).

20 The latest paper on this subject, W. R. Heller and A. Marcus,Phys. Rev. 84, 809 (1951), contains references to other previouswork.21 D. L. Dexter a,nd W. R. Heller, Phys. Rev. 84, 377 (1951).22 A second practical benefit of sensitization comes from thedouble degradation of the absorbed energy, as a result of a Stokes'shift in both the sensitizmg and activating center. For example,the Hg 2537 line can be used for excitation, and emission can beobtained in the viSible, as in the ordmary fluorescent lamp. (Inthis example it is probable that internal transitions to lower excited states of the Mn ion are responsible for part of the shift.)



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838 D. L. DEXTERThe transfer probabilities are evaluated from opticaldata in Eqs. (16), (17), (25), (26), and (30). SectionVII contains a discussion of the application of thetheory to experiment.

II. NORMALIZATION PROCEDUREFor the next four sections we shall be concerned withimpurity sensitization in an insulating crystal. Thenatural physical model is that of an atomic fraction Xs ofsensitizing impurity atoms or ions arranged at randomon the suitable lattice sites of the crystal, withoutappreciable mutual interaction, and a fraction Xa ofactivators similarly arranged. For the present we shallassume weak concentrations X8 and Xa, such that thetransfer probability from one sensitizer to another isnegligible, such that the probability of the formation ofnew crystalline phases is negligible, and such that effectsdepending on pairs or clusters of activators are unimportant. Interactions of the lattice vibrations with thesensitizer and with the activator will be accounted forexperimentally. Interactions between impurities andthe charged bodies of the lattice will be taken into account by means of a suitable .dielectric constant. Afurther restriction is the assumption that no photoconductivity occurs. Clearly none of these assumptionsis universally valid, and certain additional effects willbe discussed later.The entire transfer process consists of five stages:(1) absorption of a photon of energy "-Eo by the sensitizer, (2) relaxation of the lattice surrounding the sensitizer by an amount such that the available electronicenergy in a radiative transition from the sensitizer isE 1


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T H E O R Y OF S E N S I T I Z E D L U M I N E S C E N C E IN SO L ID S 839press the probabilities that S is in the particular energystate denoted by W.' and that A is in the state Wa.Therefore consider

(2)

For the final states we nonnalize the wave functions onan energy scale as follows:1 fW+


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840 D . L . DEXTERdipole-dipole interaction, and the second the dipolequadrupole interaction; higher order interactions, including quadrupole-dipole, have been omitted. (Weshall assume throughout that the sensitizer has anallowed transition, so that the dipole term for S givesthe largest contribution.) In Eq. (6) r8=Lmr8.m refersto all the electrons on S, measured from its nucleus, andsimilarly with ra= Lnra. n; e is the magnitude of theelectronic charge.Inserting the first bracket of Eq. (6) into Eq. (5), weobtainP.a(dd) = (27r/1i) L L (e4/K2R6)(g.'ga)-1

I F

xf Ef WaPa(wa)f w.'p.'(w.')x l (r.)(ra)-3r.)R)ra)R) 12, (7)

where each matrix element, such as (r 8), should bewritten as a function of two of the energy parameters, as(r8(w.', w.'-E'sf8' Note that the dimensions of (r)2are length squared per unit energy; note also the familiar R-6 dependence found in the van der Waals' energy.In Eq. (7) we have neglected a,ll terms associated withthe antisymmetrization of the wave functions. I f welook at the interaction terms before expanding, we findwe must compute the matrix elements of terms of theform e2/R, e2/1 R+ral and e2/1 R+ra-r8 1. Neither ofthe first two terms gives a nonzero matrix element inview of our neglect of overlap effects, i.e., the first twoterms contain two and one factors, respectively, of theform fifia* 2)ifis (2)dr2' The third term gives rise toEq. (7) and also to typical exchange integrals of theform -e 2fifi,'(1)*ifia(2)*(1/r12)ifi.(2)ifia'(1). Such termswill be discussed in Sec. V; they are unimportantwhen dipole-dipole interactions are present. I t shouldalso be noted that each matrix element (r)=fifi'*rif;drvanishes in the event that the electron spin flips duringthe transition. Spin-flip transitions will likewise be discussed in Sec. V.We now average the absolute square of the matrixelement in Eq. (7) over all possible orientations of Rand obtain(I(rs)(ra)- (3/R2) r8) R)ra) R) 12)AV

= (2/3) 1r8) 121 (ra) 1 , (8)where

x{f dw.' p.'(w.') 1 r 8 (w.', we' - E 12}X {f WaPa(wa) 1(ra(W a, wa+E 12}. (9)

An evaluation of these matrix elements as a functionof energy is of course the foremost problem of luminescence, a problem which, as we have stated, we do notpropose to discuss here. Knowing the wave functions,it would be a trivial matter to evaluate the integrals;not knowing them, it is expedient to evaluate the matrixelements directly from experiment. These same matrixelements determine oscillator strengths, absorption co-efficients, and decay times, all of which are measurablequantities. We shall now express Eq. (9) in terms ofthese experimental parameters.The probability of a spontaneous radiative transition of an isolated atom from a state i to a state f isgiven by26

(10)

where c is the velocity of light. For an atom in a crystalline medium this expression is to be changed in severalways. First, the energies of the initial and final states,because of vibrations of the surroundings, are not welldefined, so that we introduce a probability functionp'(w'), as ill Sec. II , and normalize the final states onan energy scale. Second, we must change the expressionto take account of the dielectric properties of the medium. Since the transition probability is proportional tothe square of the matrix elements of the electric fielddotted into the electric moment at the atom, the ex-pression on the right-hand side of Eq. (10) is to bemultiplied by the square of the ratio of the fieldwithin the crystal to that at an isolated atom, thatis, by (SciS)2. The average value of Se within thecrystal and the value of S in a vacuum are to bechosen so as to correspond to the same photon density. Further, the transition probability is proportionalto the density of states in momentum space, or to k2dk,where k is the propagation vector of photons in themedium. Since k is increased in a medium by the factorn (index of refraction), (10) is also to be multiplied bythe factor n3

In view of the above arguments, the probability ofemission of a photon of energy E is

X !1(r,/(w',W'-E1 2P'(w')dw', (11)where the sum is over all transitions, and the degeneracyfactor g' is introduced for the same purpose as in Eq. (5).The shape of the function A (E) is given by the emission spectrum of the atom, and the integral fA (E)dEis equal to the decay constant of the level, 1/r. Let the

25 See, for example, N. F. Mott and I. N. Sneddon, Wave M e-chanics and Its Applications (Oxford University Press, New York,1948), p. 253 if ; note that there are several misprints on p. 257.



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T H E O R Y OF S E N S I T I Z E D L U M I N E S C E N C E IN S O L I D S 841normalized function feE) represent the observed shapeof the emission band, so that f f(E)dE= 1. Then fromEq. (11) we have the result,

L Lf 1(r,j(w', w'-E)1 2p'(w')dw' j

Equation (12) is of course general and refers to eitherS or A. The left-hand side of Eq. (12), which refers toSin Eq. (9), may be evaluated experimentally by m ~ a s uring the decay time and emission spectrum of S In acrystal containing a low concentration of S as its onlyimpurity. Note that we assume no interactions betweenS and any other impurity in the derivation of Eq. (12).I f interactions occur and change the band shape ordecay time or if any competing processes occur toaffect the decay time, Eq. (12) is not applicable. Ingeneral, Eq. (12) will be applicable if the concentrationx. is weak and if the quantum yield for luminescenc: ofS is unity. I f for some reason Eq. (12) cannot be apphed,the left-hand side may be evaluated through absorptionexperiments as follows:The usual expression for the Einstein B coefficient26for induced transitions in an isolated atom is

In the event that the absorption band becomes broad,we may normalize the wave functions as previously andspeak of the induced transition probability per unitenergy range. Including the effect of the changed electric field in the medium, we obtain

The absorption cross section then becomes

x 1(r,j(w, w+E) > 2, (13)where the factor n appears because we have dividedthe transition probability B by a photon flux v= cln.The absorption cross section, which will not in generalbe greatly distorted except by very large concentrations,can be measured as the ratio of the absorption coeffi-cient (cm-I ) to the number density of impurities in themedium (cm-3) . Let us introduce therefore the normalized function F(E) such that u(E)=QF(E), where

fF (E) dE = 1. Then we obtain from Eq. (13) the result,L L: Ip(W)dWI (r,Aw, w+E)12, j

3hcg ( 0 )2- QF(E).47r2e2nE 0 e (14)By applying this equation to the activator, if it has anobservable absorption band, we may evaluate the remaining matrix elements in Eq. (9). Q=fu(E)dE ismeasured as the area under the absorption band.Equation (14) may also be useful for the evaluationof the sensitizer's matrix elements in the event that thequantum yield of S when present by itself is less thanunity, or when the decay rate either cannot be measured or is not a simple exponential function. In suchcases, we may still measure the (normalized) shape of theemission band f.(E) bu t may express the "strength" ofthe line in terms of the absorption area, Q . Thus we obtain the relation,L L Ip s ' w.')dw.' 1 r,j(w.', W.' - E) 12t8 18

3g shc (0)2= - Q.f.(E).4re2nE 0e (15)I t should be borne in mind that this last result is onlyapproximate; the quantity Q. is determined by thewave functions of S when its surroundings are in thepositions characteristic of the unexcited state, whereas7 . should be computed using wave functions in the"relaxed" lattice. Since these wave functions will ingeneral be different in the two configurations of thelattice, the transition strength as measured by Q. will beinexact.Making use of Eqs. (12), (14), and (15), we may express Eq. (9) as

3h4c4Qa ( 0 )4I f 8 (E)F a E)Psa(dd) - dE (16)47rR6n47 K!0 e E4with absorption data for A ~ n d emission data for S,or asP .a(dd) 3 h c 2 Q a Q . ( ~ ) ( ~ ) 4 f f 8 ( E ) F aCE)dE, (17)47r3n2R6 go' K!0 e E2making use also of absorption data for S. We couldsimilarly use an emission time 7 a to express the linestrength for the activator, but if A has an allowedtransition as assumed, Qa can be easily measured.The transfer probability is thus given by the strengthsof the individual transitions in terms of decay times 7and absorption band areas Qand by the energy overlapof the absorption and emission bands of A and S. Thesequantities are all directly measurable optically exceptfor the ratio 0/0 e We know that the average field



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842 D. L. DEXTER

1.0

0.80706t

~ 0 . 5 -...;:.1 ~ 0 . 4

0302

100

FIG. 1. Quantum yield for dd transfer as a function of the reduced concentration y of the act ivator. The upper abscissa represents the numerical value of the atomic concentration Xa in atypical case.strength within the medium is reduced in the ratio(8./S)2=1/K, and we shall set the quantity (S/KiSc)4equal to unity in the following numerical evaluation.The foregoing discussion of the dipole-dipole transfer, based on a physical model similar to that ofForster,19 has proceeded in a manner similar to, althoughsomewhat more general than, the development by him.I t has been included here in detail for the sake of completeness and for the sake of clarity in the treatmentsof transfer by forbidden transitions.

Let us evaluate Eq. (16) for a typical pair of impurities. With good overlap we may expect F a (E) to overlap f.(E) with an overlap integral of "'-'1/(3 ev), E tobe about 5 ev, n4",6, and Qa",-,lo-ls cm2 ev. Then P. abecomes (27/R)6 (1/r.) , where R is expressed inAngstrom units. In order that appreciable transferoccur before S luminesces, PsaTa must be >1 , ortransfer will occur if S and A are separated by notmore than about 25A. For T8 '-'10-8 sec and for S and Ain adjacent positive ion sites in NaCI (R=3.97A), thejump time would be of the order 10-12 or 10-13 sec.The experimentally significant quantity is the quantum yield for transfer, T/T. I f we neglect nonradiativetransitions in S and also neglect transfers from S to S,this quantity becomes T/T=P,aTa/(1+P.aTs). Defining

v= (4/3)lIR3, we obtain TiT=INv2+{32, where {3 is defined through the relation P,a={32/Tv2. In the typicalcase just discussed, {3 is 8.26X10-20 cma, and TiT as defined is the quantum yield for transfer from an excitedcenter S to an activator at distance R= (3v/41r)I. Toobtain the average yield fiT in a crystal, we require theintegral over-all space of TiT times the probability thatthe nearest activator is at distance R. Let C+ be thenumber density of lattice sites of the type that can ac-

commodate A, and let Va be the volume excluded by thepresence of the sensitizer. Introducing the parametery=xaC+{3, we find the average yield to be given byfiT=yexaC+vo{y-l[e-xaC+v"-1J+Ci(y) siny

-Si(y) cosY+(1r/2) cosy}, (18)where Ci and Si are the cosine- and sine-integral functions. For concentrations xa < 1 percent (y


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THEORY OF SENSIT IZED LUMINESCENCE IN SOLIDS 843The temperature dependence of the transfer probability can be inferred from Eqs. (16) and (17). Ingeneral the strengths of the transitions as indicated by

Qa and Ts or Q. are roughly independent of temperature. Thus the main temperature dependence occurs inthe amount of overlap between js and Fa. Both of thesefunctions broaden with temperature, and in the eventthat they overlap perfectly at all temperatures thetransfer probability will decrease with increasing temperature. However, if their centers are well separated,the broadening associated with increasing temperature will increase the overlap and hence the transferprobability. At low temperatures the broadening isnegligible as compared with natural, zero-point widths,and the transfer probability should be independent oftemperature. A further effect is the possible shift of thecenter of one function relative to the other, an effectwhich may either increase or decrease the probability.Such effects as these will vary from one case to the next,and are extremely difficult to calculate accurately fromfirst principles.Kroger8 has suggested that energy transfer be visualized on a "configurational coordinate" diagram, inwhich the energy of the center is plotted in the configuration space of the nuclear coordinates of the latticeneighbors. He presents such a diagram (in two dimensbns) showing both the energy of a sensitizer and thatof an activator as a function of the same configurationalcoordinate, and concludes that energy transfer can occuronly at the point in configuration space where the curvescross, and that this point in general corresponds tospecific excited vibrational states for both centers. Hefurther concludes that the transfer process requires anactivation energy and that the probability of transfercontains an exponential dependence on the reciprocaltemperature, P"-'exp-W./kT, where W. is the energydifference between the crossing-point and the minimumof the curve representing the excited sensitizer atom.(See reference 8, p. 210.) These ideas have been developed in further detail by Botden.12

In two dimensions such a diagram is of course almostmeaningless as is recognized by the authors. Specificallythe one coordinate, if important for the sensitizer, willin general not be important for the activator, so that,whereas the sensitizer curve may be a rapidly varyingfunction of the coordinate, the activator function willnot; and the curves will in general not intersect. Also,the generalized energy surfaces for the two centerswhen plotted simultaneously in a multidimensionalspace will in general intersect at an infinite numberof points with energy parameters, W., ranging fromvery close to zero (depending on the separation) up toinfinity. Furthermore, these points do not represent theconfigurations for which transfer may occur.With suitable modification, however, an interpretation based on configurational coordinate diagrams maybe useful in some instances. The only restriction on the

transfer is that the virtual photon emitted by the sensitizer be equal in energy to that absorbed by the activator. That is, if one plots the difference between theenergies of the excited electronic state and of the groundelectronic state of the sensitizer center, and simultaneously the analogous energy difference for the activator, in the generalized coordinate space, transfer mayoccur with appreciable probability, according to theFranck-Condon principle, only at the intersections ofthese generalized surfaces of energy difference. Againthere will in general exist an infinite number of suchintersections; these intersections appear for all values ofthe energy W s as defined above.For example, if the low temperature emission band ofS occurs at higher energy than the absorption band ofA and if the centers are sufficiently far apart that thecoordinates important to A do not greatly influence S,transfer may occur from S to A with no appreciablethermal activation Ws of S, bu t with activation only ofA. The probability of achieving one of the possible configurations for transfer may in some cases be roughlyproportional to exp- Wa/kT*, where T*=8 cothO/T isthe "effective temperature" which takes account,through the Debye 8, of the zero-point broadening ofthe levels.26 T* and hence P sa are independent of T forlow T.Transfer will also be possible for configurations corresponding to "activation energies" of all higher valuesthan the minimum. Further, if S and A are close together, the minimum activation energy may be eithergreater or smaller than the above. In so far as the exponential dependence on W/T * is valid, the value for Wwill be the smallest amount of thermal energy neededto bring S and A into resonance. The energy W a, ifpresent at A, may be sufficient to bring about resonance,whereas an energy W. may be required at S. In generalWa and W. will not be equal, and if S and A are notidentical atoms, it is not possible a priori to say whichof W sand Wa is the smaller. I f S and A are identicalatoms, as in S -S transfer, the least thermal energysufficient to bring about resonance will usually be required at the excited atom, since the energy of the morespatially extended excited atom is not as strong a function of neighboring displacements as is that of theground state. To repeat, in so far as W /kT* representsthe temperature dependence of p. a, W will be thesmaller of W. and Wa. In the absence of detailed calcutions we may not say which is the smaller. I t shouldfurther be borne in mind that there exist many moreconfigurations of all higher values of energy for whichtransfer may occur, so that the approximation by ac;ingle activation energy is expected to be valid onlyover a limited temperature range where W/kT* is 1.These complicated averaging processes over activationenergies are taken account of empirically in Eqs. (5),

26 K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940).



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844 D. L. DEXTER(16), and (17). A transfer probability with simple exponential dependence on the reciprocal absolute temperature, exp(-W/kT), over a wide range of T, whereW is independent of concentration Xa, seems extremelyunlikely in view of the above considerations, whateverthe interpretation given to W. A simple dependence onthe effective temperature of the form P 8a"-'exp-W /kT*could best be sought in systems of low concentrationXa and x. having well-separated low temperature emission and absorption bands. I t should be emphasizedthat all of the above considerations refer to the transfer probability itself and not to the quantum yield forthe transfer process. The quantum yield will of courseexhibit other temperature variation depending on nonradiative transitions within S.An assumption that has been made throughout theforegoing treatment is that the presence of the activatordoes not alter the shape or position of the emissionspectrum of the sensitizer. That this assumption is notprecisely valid is clear both experimentallylO and theoretically. The reason for the changes in the spectra isbased on the difference in the interaction between theactivator and the sensitizer and that between the ionnormally occupying the activator site and the sensitizer.This difference is of the same order of magnitude as eachinteraction alone. Each interaction is the differencebetween the energy perturbation to the ground stateand that to the excited state of S by the other ion.Again the difference is of the same order as each energyperturbation. Thus the shift in the peak of the emissionband of S must be of the same order as, or less than, thevan der Waals' interaction between S and A at distanceR. This energy is of the order of I:.E,,-,Eaaa./Rs, whereE is the energy of the transition and aa and as are thepolarizabilities of A and S. Since a is of the order 10-24cm3, it is clear that for separations R larger than nearestneighbor separations the perturbation by van der Waalsforces is negligible. I f A and S are nearest neighbors inthe NaCl latt ice, the relative energy shift I:.E/E is of theorder (2.8)-S,,-,2X 10--3 Such a shift would be of importance in the case of very poor overlap between absorption and emission bands, but not otherwise. Theperturbation for nearest neighbors and for nearest likeneighbors might in some cases be somewhat larger thanthe above estimate as a result of exchange effects, particularly in crystals of high dielectric constant, but itseems safe to treat these perturbation effects on thespectra as of second-order importance.IV. ELECTRIC DIPOLE-QUADRUPOLE INTERACTION

To determine the transfer probability when S makesan allowed transition and A makes a forbidden, quadrupole transition, we insert into Eq. (5) the absolutesquare of the matrix element 1(H1)12 computed fromthe second bracket of Eq. (6). Before inserting 1(HI) 12however we average over all orientations of R. This

process results in

4+-[ 1(XaYa) 12+ 1(YaZa) 12+ 1(ZaXa) 12J38

- - [ (Xa2 ) (Ya2) 1+ 1(Ya2 ) (Za2) 121+1 (Z})(Xa2)IJ}, (21)

where, as before, we omit exchange terms. The usualexpression for the quadrupole transition probabilitiesare given in terms of the double dot product of thedyadic271(N)/2= 1(X2) 12+ (y2) 12+ (Z2) 12

+21 (xy)12+21 (YZ)12+21 (ZX)12.By summing 1(HI) 1A,2 and 1(N a) 12 over all D and Sstates we obtain the numerical relation between theirmatrix elements, ge4a(I (H 1)[2)A,=-1 (r.)1 2 1(N a ) 12, (22)4R8 K2

where a= 1.266. Thus we obtain the t ransfer probabilityfrom Eqs. (5) and (22):

x{fdw.'p.'(w.') 1 r8(w.', w.' -E ) 12}X {f WaPa(Wa) 1 Na(Wa, W a+E) 12 }. (23)

We may relate the quantit ies in the brackets in Eq. (23)to either absorption or emission data as in Sec. III.The first quantity, referring to the sensitizer, can beexpressed either as in Eq. (12) or as in Eq. (15), depending on whether or not the decay time T. is knownand meaningful. The quadrupole matrix elements shouldproperly be expressed in terms of absorption curves,bu t unfortunately there is little hope of obtaining suchdata for the following reasons. The strength of a quadrupole transition is ordinarily weaker by a factor of(a/X)2 than an electric dipole transition, where a is theradius of the atom and X the wavelength of the ab-

27 E. U. Condon and G. H. Shortley, The Theory of AtomicSpectra (Cambridge Umversity Press, Cambndge, 1935), p. 96.



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THEORY OF SENS IT IZED LUMINESCENCE IN SOLIDS 845sorbed light, or is weaker by a factor ,,-,10-7. Since theabsorption coefficient resulting from a ten percent impurity with an allowed transition is of the order 1()4cm-t, ten atomic percent of the activator with a forbidden, quadrupole transition will have an absorptioncoefficient of only 10-3 per cm and thus will not bemeasurable. We may relate the matrix elements to theemission band, however, since decay times of the order(1o-S)/ (10-7) =0.1 sec can easily be observed. Thespontaneous emission probability27 is related to / (N) /2through the relation,

(24)where we have included the effects of the medium andof broadening as in Eq. (11). Using Eqs. (22), (12), and(23), we find

1357rali 9csga'( S ) 4jJs(E)Fa(E)P.a(dq) = - - dE. (25)4n6R sT sTa ga K1Sc ESI f T . is not obtainable or significant, we may use Eq.(15) instead of Eq. (12), with the result

135ah6c6 Q.P.a(dq) = -47rn4Rs Ta

Again it should be recalled that these line strengths aregiven only approximately by Q. and Ta for the samereason as that given following Eq. (15); that is, thestrengths should be properly expressed in terms of T .and Qa if these parameters were known. Note the inverse eighth-power dependence on R.The ratio of the dipole-quadrupole transfer probability to the dipole-dipole probability is found from Eqs.(25) and (16) to be (45a/4r) (nA/R)2Ta(d)/ra(q), wherewe have made use of Eqs. (12) and (14). Now a quadrupole radiative transition [compare Eqs. (24) and(12)J has a probability of the order (a/X)2 times thatfor a dipole transition. Inserting Ta(d)/Ta(q)"-' (a/X) 2,we find P.a(dq)/P.a(dd) to be of the order (a/R)2, whichamounts to only an order of magnitude or so reductionfor close neighbors. We see therefore that resonancetransfer can easily occur from a sensitizer to a nearbyactivator if the latter has a quadrupole transition in thesuitable frequency range; on the other hand, directabsorption of radiation in the activator occurs with aprobability only "-'10-7 of that in an impurity with anallowed transition.The dependence on concentration is found as in thelast section. We define l ' through the relation P.a(dq)

FIG 2. Quantum yield for dq transfer as a function of the reduced concentration y of the activator, as in Fig. 1. Typicalatomic concentrations are shown on the upper abscissa scale.Note the more rapid variatIOn of fiT' with concentration in the dqcase than in the dd case.

00 e- l l tfiT'=y i - - d t .o 1+t S/ 3

(27)

(27')The integral has been evaluated numerically as a function of y and is shown in Fig. 2. Since concentrationsXa higher than 1 percent may be of interest in the dqtransfer,- it is not permissible to replace exp (xaC+vo)by unity in the general case, and consequently fiT is nota function of y alone in part of the range of interest.Figure 2 shows therefore the function fiT' (y), and forhigh concentrations (xa>1 percent), fiT will require theindicated corrections. I f we accept fiT' = t as thecriterion for appreciable transfer, we find again thaty=0.65 determines a critical concentration, xa*=0.65/I'C+. In a typical case [Eq. (25)J we may expect E=5ev, n6=13, Ta=O.1 sec, ga'/ga=5, fJ.(E)Fa(E)dE= 1/(3 ev), and C+= 2.25X 1022 cm-3, so that l ' becomes3XI0-21 cm3 and Xa is 0.96XI0-2. Thus S may be expected to sensitize about 100 sites. As is to be expectedfrom the stronger R dependence in the dq case, fiT' is asomewhat more rapidly varying function of Xa in thedq case than in the dd case. The discussion of the temperature dependence given in the last section is alsoapplicable to dq transfer.One final remark should be made regarding the function Fa (E), the absorption shape function for A. Thisfunction cannot be determined directly from experi-



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846 D. L. D E X T E Rment because the absorption cannot be observed. Someidea as to its shape may be inferred in some cases fromthat of the observable emission band, bu t the energyof its center will still be unknown. The calculation ofF a (E) is a difficult task, bu t some qualitative successmay be expected from application of the methods ofLax, Huang and Rhys, Williams, and others.28

V. EXCHANGE EFFECTS AND ADDITIONALFORBIDDEN TRANSITIONS

We shall be concerned in this section with effects ofthe electron spin and shall therefore write out explicitlythe matrix element of HI in terms of the wave functions1/;(r, 0')= lP(r)x(O'), where x(O') are the spin wave functions.(HI)=f 1P.'*(rl) lPa*(r2)H IlPs(rl) lPa'(r2)

X x.'*(O'lh* a(0'2h8(0'Iha'(0'2)-f 1P.'*(rl) lPa*(r2)H IlPa'(rl) 1P8(r2)X xs'*(O'lha *(0'2ha'(0'Ih.(0'2)'The first term in (HI) is the Coulomb term with whichwe have been concerned in the last two sections. Sincein the approximation of Eq. (6) HI does not operate onthe x's, we see the previously mentioned selection ruleagainst spin-flip transitions, i.e., unless x. ' = x. andXa=Xa', the first integral vanishes. Thus all our resultsso far are applicable only to transitions in which all thespins are unchanged. The second integral is an exchangeintegral with HI= e2/ "'12. Ignoring the x's for the moment, we see that this integral merely represents theelectrostatic interaction between the two charge cloudsQ' (rl) = 1P.'*(rl) lPa' (rl) and Q(r2) = lPa*(r2) 1P8 (r2) j sinceeach function IP dies off exponentially with distancefrom S or A, it is clear that each product Qwill be verysmall throughout all space unless S and A have smallseparation. I f the separation is small, both functions Qwill be sizeable in the same region of space, namely,between S and A where '12 is small j thus the integralfQ ' (rl) (1/'12)Q(r2)drI2 may well be sizeable (for smallR) even though the overlap integrals fQ(r)dr, whichenter in normalization, are negligibly small.

Let us now look at the selection rules in the exchangeintegrals. Unless X.'=Xa' and Xa=X8 the integral vanishes. However x' is not necessarily equal to x, so thatthe spin functions on both atoms may change simultaneously. I f such a transition is forbidden on S (or A)by the Pauli principle, however, the spin function onA (or S) may not change either.Other selection rules occur in the exchange integralsover the spatial variables, bu t in each case they forbidonly specific transitions and do not preclude all transitions in any symmetry class. Thus, although all Cou-28 See M. Lax, ]. Chern. Phys. 20, 1752 (1952) for furtherreferences.

lomb integrals may vanish, exchange will allow transfer to and excitation of A by all types of allowed andforbidden transitions, including, for example, L = 0 toL=O transitions.

The transfer probability by the exchange mechanismmay be written in the formP8a(ex) = (27r/h)Z2f .(E)Fa(E)dE, (28)

where Z2, a quantity that cannot be directly related tooptical experiments, is defined by(29)

Note that the IP'S and Q's are normalized over space(and not on an energy scale) so that Z2 has the dirp.ensions of energy squared. The separation and concen tration dependences are hidden in Z2, which varies approximatelyas Y(e4/,,2R02) exp( -2R /L), where L is an effective average Bohr radius for the excited and unexcitedstates of the atoms S and A, and Y is a dimensionlessquantity 1 which takes account of the cancellation asa result of sign changes in the wave functions. I f SandA occupy nearest like lattice sites in an ionic crystalsuch as N aCl, a typical transfer time would be of theorder 10--10_10--11 seconds j if S and A occupy secondnearest like lattice sites, the probability would be reduced by a factor of the order of 10--2 In insulators withlarge dielectric constants the excited state wave functions may extend over very large volumes, so that Q' issizeable even for R of the order 2 or 3 Ro j neverthelessthe ground-state wave functions are relatively littleinfluenced by the dielectric constant, are much morecompact, and make Qnegligibly small for R,....,2R o. Thusit seems unlikely even in a crystal of large" that exchange may transfer excitation energy farther than theshell of third nearest like lattice sites. The NaCl structure contains, in the first shell, 12, in the second shell,6, and in the third shell, 24 like lattice sites. Thus probably about 40 is an upper limit for the number of sitessensitizable by the exchange mechanism, or in otherwords the exchange mechanism is slightly less efficientthan is dq transfer.Another type of transfer mechanism may be investigated by including magnetic fields in the Hamiltonian.We may thereby obtain transfer by an electric dipolemagnetic dipole (em) process. The probability for thisprocess will be seen to be negligibly small compared tothat for exchange transfer when exchange is allowed bythe spin selection rules.The most important term in HI involving the magnetic field is of the form

eh-H raXVa+iO'a) ,2mc



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THEORY OF SENSIT IZED LUMINESCENCE IN SOLIDS 847where H is the magnetic field at A due to the motion ofcharge at S. The largest term in H is of the order(e/c)RXv./R3, or (eh/mc)RXV./R3. Thus (H 1) hasthe magnitude (eW/2m2c2R2) (V.) or, since


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848 D. L. DEXTERthe nearest W04 sites, so that at higher temperatures(S200K) the quantum yield increases. In order to explain the low temperature behavior Botden postulatesad hoc that in the S - A transfer the Sm is raised to anexcited electronic state exactly in resonance with theW04 transition, from which it jumps to a lower excitedelectronic state and from which finally the Sm emits.An alternative interpretation may be made whichavoids the above assumption and still fits the temperature data: The postulate of more than one excited electronic state is unnecessary since relaxation around theSm will prevent back-transfer, and the assumption ofprecise resonance to explain the absence of a temperature dependence at low temperature is unnecessary sincethermal broadening of absorption and emission lines issmall in comparison with zero-point widths at low T.In the above system the oscillator strength for thetransition in S is small, since the transition probabilityin theW04 group is ",,105 per sec. Thus the second typeof S -S transfer is operative, and as we have seenabove and in Section III, this type of transfer will alsohave a probability independent of T for low T. We mightexpect then a total probability of transfer from the initial tungstate ion to the Sm to depend on the temperature through a factor of the type exp- (W.a+nW )/kT*, where n is the number of S - S transfers required.A simple dependence such as this is to be expected onlywhen complicating factors of the type discussed inSection III are unimportant.

VII. CONNECTION WITH EXPERIMENTSensitized luminescence experiments on the simplesttransfer mechanism discussed above, namely, the ddtransfer mechanism, do not seem to have been reportedin the literature for simple inorganic systems. Probablythis experiment has been neglected for two reasons:(1) I f A has an allowed transition, it can absorb radiation itself and hence requires no sensitizer in order to beexcited. I t should be recalled, however, that the secondreason for the practical interest in sensitization, namely,a double degradation of the energy, is still operative,so that, for example, the 2537 Hg line could be used toexcite visible emission of short lifetime in the event

that two sizeable Stokes' shifts (",,1 eveach) can be employed. (2) In some experimental systems there mightbe difficulty in distinguishing dd transfer from a cascadephenomenon. In the latter process, where a photon isemitted by S and reabsorbed by A, the quantum yieldfor transfer depends not on the concentration Xa alonebu t also on the size and shape of the system. Neglectingdd transfer, we find the quantum yield for transfer bythe cascade mechanism is equal to

where the average is performed over the linear dimension l of the sample. 1-7iT(C) is the probability that a

photon emitted from S will leave the system. We neglect S -S transfer of all types. Note that the criticalconcentration for this process varies inversely as thestrength of the transition in A and inversely as theaverage size of the sample. See Eq. (20) for a comparison with dd transfer. The quantity xaC+l represents thenumber of activators per cm squared projected on aplane perpendicular to the path of the photon. In aspherical system of a given volume containing a givenconcentration Xa the quantum yield will be much largerthan for the same system flattened out into a thin film.We found 7iT(dd) to be independent of the shape of thesample, or of its size. As an example to illustrate theorders of magnitude involved, le t us consider a sphericalsample of 1-mm radius, containing a concentrationXa = 3 X 10-4. Let us further assume the system to becharacterized by those parameters given in Sec. IIIso that 7iT(dd) is equal to t. For convenience let us assume that both !.(E) and u(E) are step functions each1 ev wide, and that they overlap by t ev. These specificassumptions are in agreement with the value of theoverlap integral between emission and absorption assumed in Sec. III. We find 7iT(C) to be ! , as corresponds to the circumstance that practically all of theradiation emitted in a frequency range which can beabsorbed by A is absorbed, and all the rest is transmitted. I f the concentration Xa is increased to 3X10-a,7iT(dd) is increased to almost 1 (see Fig. 1), but 7iT(C)is still!. This example serves to illustrate the differencein the dependence of the two transfer mechanisms uponoverlap of. emission and absorption. I f the emissionband of S were 1 ev wide and the absorption band of Awere 3 ev wide and covered the S emission range, theoverlap integral f !.(E)Fa (E) dE would be unchanged,and hence 7iT(dd) would be unchanged. On the otherhand the quantum yield 7iT(C) would now be >0.99for xa>3.1XI0-5. I f the sphere were flattened out intoa thin film, say 10-4cm thick, 7iT(dd) would still be! forxa=3X 10-4, bu t 7iT(C) would be close to zero regardlessof the particular shape of the overlap integrand.Thus the two distinctions between cascade and resonant transfer occur in the dependence of the cascadetransfer probability on the size and shape of the sampleand on the details of the overlap of the absorption andemission spectra of A and S. Cascade effects can bemade unimportant experimentally by dealing with thinsamples of systems in which the overlapping of spectrais poor. For further discussion of this topic the readeris referred to reference 19(b), (c).A second point should be emphasized in connectionwith the dd transfer: Because of the high efficiency ofthe transfer mechanism, it should be possible to utilizeas a sensitizer an impurity which, even when present byitself, will not luminesce appreciably because of competing nonradiative processes. For example, let thethermal lifetime be denoted by TN , so that the quantum



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THEOR Y OF SENSIT IZED LUMINESCENCE IN SOLIDS 849yield for transfer becomes

The value of the integral is given by Eq. (19) in whichy is to be replaced by y(1+T 8 /rN)-i. Hence if nonradiative processes within S are one hundred times moreprobable than emission from S, a concentration x ten atl;nes l a r ~ e r than .the value of 3 X 10-4 obtained pre-vIOusly wlll result i l l efficient transfer. Thus though S,when present alone in the lattice, may luminesce withan efficiency as low as one percent, it may still serve asan excellent sensitizer for several hundred lattice sites.These considerations have only partial applicability tothe dq and exchange processes. I f T 8 / T N ~ 1 0 2 concen-. 'tratIOns of the order ten percent are required for ap-preciable transfer.

An experimentally important process has so far beenentirely neglected, i.e., concentration quenching.80a Although the precise mechanism for this process is notunderstood, it is nevertheless clear experimentally thatquenching effects occur for concentrations Xa as largeas those required for efficient transfer by means of thedq and exchange mechanisms. Thus it is impossible inmany cases to isolate experimentally the S -A transfer process from quenching phenomena. This circumstance is a further argument for the investigation ofsimple dd systems in attempting to understand sensitized luminescence in general; such systems might consist of impurities such as TI, Pb, Ag, Cu in the alkalihalide lattices.One of the commonly used luminescent impurities isMn, which exhibits visible emission in a wide variety ofhost lattices,6-11,31 as well as in manganese salts. 8 ,32Klick and Schulman32 have compared the absorptionspectrum of Mn++ in MnCb with the atomic absorphon spectrum, and have correlated the levels in solidswith the corresponding levels in the isolated atoms.This correlation shows a number of spin-flip and quadrupole transitions in the wavelength range of 2370-SSOOA; these transitions are probably those of importance in the process of transfer to Mn when used asan activator. I t seems probable that the Mn ion whenexcited by any of these transitions is able to undergononradiative transitions into the lowest excited electronic state, from which it emits with a large Stokes'shift. I t is difficult in practice to distinguish transfer bythe dq process from that by the exchange process (whichalone may excite the Mn in a spin-flip transition) since30 . A manuscript is in preparation discussing concentration9uenching from t h ~ point. of view of resonance transfer of energy,I.e:, S-S transfer, m ordmary s ~ s t e m s c ? ~ t a i n i n g one impurity.

I t IS c o n ~ l u d e ~ that S-S transfer IS the cntlcal process for systemsof low dIelectriC constant and small Stokes' shift.31 F. A. Kroger, Luminescence in Solids Containing Manganese(Van Campen, Amsterdam, 1940).(1;5ij. C. Klick and J. H. Schulman, J. Opt. Soc. Am. 42, 910

the critical concentration of Mn would be of the sameorder of magnitude for both processes and since theabsorption shape Fa(E), and hence P8a, are unknown.quenchi.ng effects ,:ill also make difficult such a separahon. I f i l l any parhcular case the critical concentrationturns out to be one percent, it would appear safe toassert that transfer is by the dq, rather than the exchange, process; if Xa is> two percent it might a prioribe either mechanism if both types of levels were knownto exist in the general energy region capable of exciton by S. Critical concentrations of Mn of the order ofthree percent have been observed experimentally inCaSiOa:Pb+Mn91 1 and in Ca3(P04)2:Ce+Mn.9 Botden and Kr6ger9 have assumed that transfer may occuro;nly to n e a r ~ s ~ neighbors and have accepted the relatively low cntIcal concentration of Mn as evidence forthe preferential formation of pairs Ce+Mn andPb+ Mn. Since transfer is not restricted to nearestneighbors, however, preferential pair formation is notnecessarily indicated by these experiments.From host-sensitization experiments one hopes toobtain information on several aspects of excitons: (1)From the use of activators with measurable absorption spectra can be determined the amount of Stokes'shift, if any, within the host-sensitizer. Since mostpure lattices do not emit, the above information is notobtainable by direct means. (2) By varying the concentration Xa the diffusion length of the exciton may befound. (3) I f such measurements are made as a functionof temperature, indirect evidence may be inferred as tothe thermal processes of the exciton. (4) Experimentsthat utilize directly the detecting properties of theactivator for excitons may readily be imagined. Forexample, assume that a suitable activator has beenfound, say in the course of experiment (1) above suchthat upon its introduction as an impurity s o m ~ particular crystal becomes luminescent upon irradiation inthe first fundamental absorption band. Then evaporateonto one face of a thin "perfect" crystal a layer of thesame material containing the activator in high concentration, irradiate the other face of the crystal in thefirst fundamental absorption band, and measure theresulting luminescence. For a crystal of thickness muchgreater than 10-6 cm essentially none of the incidentlight will remain unabsorbed, and if luminescence isobserved on the back face (and if photoconductivity isabsent) the excitons must have travelled through thecrystal. Such an experiment is in principle capable ofmeasuring directly the diffusion length for an excitonand if successful would be the first experimental proofof its motion. (The elegant experiments by Apker andTaft33 on alkali halides were performed with a concentration of imperfections so large that, were there reasons to doubt the motion of excitons, the same interpretation21 which has been based on their motion could

33.L. Apker and E. Taft, Phys. Rev. 83,479 (1951), and severalearlier papers.



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850 D. L. DEXTERalso be satisfactorily based on the nearness of everyhalide ion to a vacancy or F center.)The difficulty of the above experiment of course resides in obtaining a "perfect" crystal sufficiently thin.An evaporated film would surely be so full of imperfections that the exciton's diffusion length would be lessthan or about the absorption depth of the incident light.I f the diffusion length D in a good single crystal shouldbe much larger than Heller and Marcus' estimate,2010"-4 em, a crystal could probably be cleaved and annealed successfully, but if D is < 10-4 cm the experiment must await new methods of preparation of perfectthin films. Since the interaction between excitons andionic crystals is large,21,ao the experiment might best beperformed with non-ionic materials.

The most thoroughly investigated systems so far reported are the tungstates of Cd and Zn and the tungstate and molybdate of Ca, all activated with Sm.12The W04 and Mo04 ions are the sensitizers in thesecrystals. The interpretation by Botden and Kroger ofthe temperature dependence has been discussed abovein Section VI, and modifications have been presented.Several tentative conclusions may be drawn from theseexperiments. With a radiative decay time of ",,10-5 secin the W04 ion, or an oscillator strength of the order10-a, the exchange mechanism is probably the operative method for S -S transfer, and in the absenceof relaxation effects, each jump time would be perhaps10-10 sec. However, the tungstate excitation and reflection spectra12 indicate that the tungstate absorptionpeaks at

1953 dexter theory

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