This report presents measurements of the {sup 4}G{sub 7/2} emission for 26 nd-doped materials. A model of nonradiative decay based on dipole-dipole energy is developed and found to be supported by the data

Bibeau, C.

Payne, S.A.

We present measurements of the 4G7/2 emission lifetime for 26 Nd-doped materials. A model of nonradiative decay based on dipole-dipole energy transfer is developed and found to be supported by our data

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UCRL-JC-1ZS623 Rev 1PREPRINTPicosecond Nonradiative Processes in Neodymium-dopedCrystals and Glasses Mechanism for the Energy Gap LawC. BibeauS. A. PayneThis paper was prepared for submittal to theOptical Society of AmericaAdvanced Solid State Lasers and Radiative Processes and Dephasingin Semiconductors Topical MeetingsCoeur d’Alene, IDFebruary 2-4,1998September 29,1997–—DISCLAIMERThis document was prepared as an account of work sponsored by an agency ofthe United States Government.  Neither the United States Government nor theUniversity of California nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the University of California.  The views and opinions of authorsexpressed herein do not necessarily state or reflect those of the United StatesGovernment or the University of California, and shall not be used for advertisingor product endorsement purposes.Picosecond Nonradiative Processesin Neodymium-doped Crystals and Glasses:Mechanism for the Energy Gap Law*Camille Bibeau and Stephen A. PayneUniversity of CaliforniaLawrence Livermore National LaboratoryP.O. BOX 808 L-441Livermore, CA 94550Tel. (510) 422-7798, Fax (510) 423-6195AbstractWe present measurements of the 4GTnemission lifetime for 26 Nd-doped materials. A model of nonradiative decaybased on dipole-dipole energy transfer is developed and found to be supported by our data.*This work was performed under the auspices of the Division of Materials Science of the OffIce of Basic EnergySciences, U. S. Department of Energy, and by the Lawrence Liverrnore National Laboratory under contract W-7405-ENG-48.Picosecond Nonradiative Processesin Neodymium-doped Crystals and Glasses:Mechanism for the Energy Gap Law*Camille Bibeau and Stephen A. PayneUniversity of CaliforniaLawrence Liverrnore National LaboratoryP.O. BOX 808 L-441Liverrnore, CA 94550Tel. (510) 422-7798, Fax (510) 423-6195IntroductionIt has been extensively verified in the past that thenonradiative decay rate between rare earth energy levelsis predominantly determined by the energy gap and theparticular host medium. The energy gap law isembodied in the simple expression for the nonradiativerate:WM =(1 / To)exp(-p. p), (1)where % and ~ are constants characteristic of the hostmedium, and p (=AE / hv-) is the number of phononsneeded to bridge the gap, The expression hv~a isrelated to the highest phonon frequency of the hostmedium and AE is the energy gap between thepopulated rare earth state and the next energeticallylower level. The constant, ~, is related to the details ofthe electron-phonon coupling. In nearly all casesreported in the literature, eqn. (1) is regarded asadequately describing the measured decay rates overseveral orders of magnitude [1-3].In the present paper we report the nonradiativedecay rates of the 4G7n state of Ndw in 26 differentcrystals and glasses, with the goal of providing a database relevant to the relaxation rate of the 4111nstatewhich has a similar energy gap as the 4GT~state [4].The 411,n state of Nd3+ is a particularly important onefrom a practical point of view, since it can potentially“bottleneck” during lasing and give rise to transientabsorption at the laser wavelength (constituting loss).Much of the motivation for establishing thiscorrelation is that emission lifetimes are much simplerto measure and can be widely applied to numerous Nd-doped crystals and glasses, while the pump-probetechnique used to directly assess the 411in decay time iscomplex to setup and execute.The relevant energy levels are depicted in Fig. 1,where the ‘c5n,17n, and ~1In lifetimes are identified onthe diagram along with the 532 nm pump wavelengthand the -600 nm detection wavelength. One of thedetails that must be handled in the numerical analysis ofthe data is the effect of the overlapping -600 nmemission arising from the 4Gwz, 2G7n states on themeasured results (i.e. T5n), as depicted in Fig. 1. Thenumerical analysis and resulting fits to the data will bediscussed.4~312TIICf,fiC*532 nm ResonantPump emissionat -600 nm154+741J13/211/2B‘cl,~912Fig. 1 Energy levels and transitions for Nd3+ExperimentalThe picosecond emission lifetimes [4] weremeasured using the time-correlated single photoncounting system together with a Coherent mode-lockedlaser producing -90 psec pulses at 76 MHz and thendoubled to 532 nm. The sample emission was detectedat -600 nm with a monochromator followed by amultichannel plate photomultiplier (MCP-PMT). The(deconvoluted) temporal resolution of the data is about50 psec. For the case of the fluoride samples, thelifetimes of -10 nsec were too long to be measured withthe time-correlated photon-counting apparatus. Tohandle these longer lifetimes, we employed aconventional set-up consisting of a Q-switchedNd:YAG laser (12 nsec pulsewidth), along with amonochromator, oscilloscope, and MCP-PMT.ResultsThe results of the emission lifetime experiments arecontained in Table 1. Based on an analysis of the bestfit to the data, the uncertainty in the value of the ~7nlifetimes is A7VZ(<1 ns) = + 50 ps for data withlifetimes less than 1 ns and A77n(> 1 ns) = + 200 ps fordata with lifetimes greater than 1 ns.Many of the trends that can be gleaned from thedata in Table 1 are expected. For the case of thefluorides, the zTn values are the longest, being 4,000-41,000 psec. This is anticipated, since their phononfrequencies are the lowest among the hosts listed, being<600 cm-’. Whereas the phosphate and silicate glassesare in the range of 150 to 250 psec owing to the ratherhigh vibrational frequencies of the SiOi and P04 anions.It is noteworthy that comparisons of previous ~VZmeasurements in the literature are satisfactory,including [5]: 1400 versus 1090 psec for YA1OS;370versus 200 psec for Y#@lz; and 8400 psec versus9100 psec for LiYF4; 56,400 psec versus 41,000 psecfor LaF3.Table 1. Emission lifetimes of the 4GTnexcited state(T7JName Formula T7n,psecPhosphateglassesLG-750APG-1m-xAPG-2LG-812SilicateglassesLG-660LG-650Sol-gelVanadateYV04TungstateCaW04OxideYALOGSGGYAGGGGLLGGApatitesC-FAPS-FAPC-VAPS-VAP------FluoridesZBLANYLFNa3Sc2L13F12KY3F,0YF,p@5+/d@3+modifiersPz05+fluorides+modifiersSi02+A1203+modifiersSi02YV04CaW04YAl~Gd3SczGa301zYsAl@,zGd3Ga501zLa3Lu2Ga301zCaJP04)3FSr5(POJ3FCa5(V04)3FSr3(V04)3FSrs(VO&ClZrF4-BaF2-AIFYLaF3-NaFLiYF4Na3Sc2L13FlzKYJFIOYF3228215210150-1400215210245190510109071520053012007017520033038018,00091004000900022,000LaF3 L*3 41,000Analysis and DiscussionAs mentioned earlier, measurements of the 4GTnpopulation decay time have long been suspected to besimilar in magnitude to that of the4111n level of Nd3+.Using the4111n lifetimes from a direct measurement in[7] we find that the 4111n and 4G7n lifetimes arecorrelated with each other to within about a factor oftwo, across a variety of different host media. Thisconcurrence is reasonable, since some differences in theenergy gap and perhaps the electron-phonon couplingmay be expected, in addition to the experimental andanalytical uncertainties. Furthermore, due to the smalldifference in the size of the energy gap for a given hostmedium, this data offers some experimental validationthat the specific characteristics of the electronic states(symmetry, crystal field interactions, spin, etc.) do notstrongly influence the nonradiative decay rate.,o~10 m lam Iccco loomFig. 2 Comparison of the 4111nnonradiative relaxationtime from [4] and the 4GTnemission lifetime.Another theory of nonradiative decay derives themulti-phonon rate on basis of energy-transfer theory, asoriginally described by Forster [6] and Dexter [7].With the assumption that dipole-dipole interactionsdominate the process, and that the phonon absorption~@) k a slowly VW@2g fU33CtiOt3 compared to theemission cross section spectrum Ca(k), we derive that:(m) ‘tin “where V~in is the minimum volume that is non-absorbing in nature (centered on the rare earth ion), n isthe refractive index, and c is the speed of light. Thisexpression essentially emerges from the well-knownForster-Dexter spectral overlap integral between theemitting and absorbing species, and a volumeintegration over a uniform acceptor (phonon)concentration, (a derivation similar to other reports inthe literature [8-1 l]). It is also noteworthy however, thatthe multi-phonon absorption spectrum is commonlydescribed with an exponential expression expression[12,13]:aph=Aho,t~x~(-yhOsthv~NchM),(3)where U* is the phonon frequency, ~Chw is acharacteristic phonon frequency of the host material,and ~,1 and Ah.,t are other host-dependent constants.So if we combine eqns. (3) and (4), we can suggest analternative route to deriving the form of the energy gaplaw:where we have identified ~phas o~~r The MO$I parameterturns-out to be similar for many crystals, for instancebeing in the range of 4-5 for alkali and alkaline-earthhalide crystals [13]. The main point to note regardingeqn. (4) is that the exponentiated factor in squarebrackets only contains information concerning the hostmedium, while the rare earth properties are exclusivelyrepresented in the pre-exponential factor- implying thatwe may expect reasonably good adherence to the formof the energy-gap law, eqn. (l),We can explore the validity of eqn. (4) by inputtingreasonable estimates for the terms in the pre-exponential factor and deducing the magnitude of thisconstant. Using vW”= 5 x 10X cm3, Aho,t= 20,000 cm-1 (average from ref. [13]), and cs.~*Al=l.8 x 10-~ cm2(calculated from Judd-Ofelt theory using the average ~Jparameters of YAG and YLF [14]), we obtain a value of0.6 x 1012see-*,or G = 1.7 psec, which is defined fromeqn. (1) as:(m) ‘WnWe are now in a position to compare thiscalculation to the data in Table 1, where we use theenergy gap (&n) and highest phonon frequencies(hv_) to calculate number of phonons p= AETn/ hvm.If we also group all of the phosphate glasses into asingle datum, and all of the silicates into a second one,then there are potentially ten usable points. Finally, ifthe LaFJ result is eliminated because of its stronglynonexponential character, the results of the exercisemay be displayed as shown in Fig. 3, where we haveplotted the nonradiative decay time, ~Tn, against thenumber of phonons, p. The data is then fitted to thereciprocal of the energy gap law with the result of thenumerical fit yielding ~ = 3.3 psec and ~ = 3.1. We arevery encouraged by this result because it may becompared with theoretical calculation of G = 1.7 psecnoted above in connection with eqn. (5). Now, if weassociate ~0$~with ~, and h%~ with h~max> the p = 3.1value from Fig. 3 appears to be within the range of whatone may expect from the energy-transfer theory ofnonradiative decay encompassed in eqn. (4), since thephonon spectra of many materials is characterized byyho,t= 4-5.The concurrence of these different approaches inestimating m is remarkable, and is an implicit statementof the robustness of the energy gap law in general, aswell as good support for the energy-transfer mechanismof nonradiative decay./YLF\c:?o:\\/YALosilkat~G18ss ● ●/wPhosphate Glass, YAG\C-FAP0 1 2 3Number of PhononeFig. 3 Plot of the nonradiative decay time against thenumber of phonons needed to bridge the gapAcknowledgmentsWe would like to thank Christopher Marshall, GaryWilky, Howard Powell and Bill Krupke for theirsuggestions throughout this project.*This work was performed under the auspices of theDivision of Materials Science of the OffIce of BasicEnergy Sciences, U. S. Department of Energy, and bythe Lawrence Livermore National Laboratory undercontract W-7405 -ENG-48.[1] H. W. Moos, J. Luminesc. 1/2 (1970) 106.[2] W. D. Partlow and H. W. Moos, Phys. Rev. 157(1967) 252.[3] L. A. Riseberg, W. B. Gandrud, and H. W. Moos,Phys. Rev. 159 (1967) 262.[4] C. Bibeau, Ph. D. Dissertation, “Evaluation of the4111nterminal level lifetime for several neodyrnium-doped laser crystals and glasses,” April 25, 1995.[5] T. T. Basiev, A. Yu. Dergachev, Yu. V. Orlovskii,and A. M. Prokorov, J. Luminesc. 53 (1992) 19.[6] T. Forster, Z. Naturf. 49 (1949) 321.[7] D. L. Dexter, J. Chem. Phys. 21 (1953) 836.[8] N. T. Timofeev and E. B. Sveshnikova, Opt.speCtrOSC. (USSR) 54 (1983) 595.[9] V. L. Ermolaev and E. B. Sveshnikova, J. Luminesc.20 (1979) 387.[10] V. L. Ermolaev and E. B. Sveshnikova, Chem.Phys. Lett. 23 (1973) 349.[11] E. B. Sveshnikova, N. T. Timofeev, and V. M.Zolotarev, Izvestiya Akademii Nauk SSSR. SeriyaFizicheskaya 44, (1980) 722.[12] T. C. McGill, R. W. Hellwarth, M. Magir, and H.V. Winston, J. Phys. Chem. Solids 34 (1973) 2105.[13] L. L. Boyer, J. A. Barrington, M. Haas, and H. B.Rosenstock, Phys. Rev. B 11 (1975) 1665.[14] R. P. Bauman and S. P. S. Porto, Phys. Rev. 161(1967) 842.Technical Information Department  • Lawrence Livermore National LaboratoryUniversity of California • Livermore, California  94551

Picosecond nonradiative processes in neodymium-doped crystals and glasses: mechansim for the energy gap law

UCRL-JC-128623 Rev 2 PREPRINT Picosecond Nonradiative Processes in Neodymium-doped Crystals and Glasses: Mechanism for the Energy Gap Law C. Bibeau S. A. Payne This paper was prepared for submittal to the 13th Topical Meeting on Advanced Solid-State Lasers Coeur d’Alene, ID February Z-4,1998 March 2,1998 Thisisapreprintof apaperintendedforpublicationinajoumalorproceedings. Since changes may be made before publication, this preprint is made available with the understanding that it will not be cited or reproduced without the permission of the author. DISCLAIMERThis document was prepared as an account of work sponsored by an agency ofthe United States Government.  Neither the United States Government nor theUniversity of California nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the University of California.  The views and opinions of authorsexpressed herein do not necessarily state or reflect those of the United StatesGovernment or the University of California, and shall not be used for advertisingor product endorsement purposes.Picosecond Nonradiative Processes in Neodymium-doped Crystals and Glasses: Mechanism for the Energy Gap Law* Camille Bibeau and Stephen A. Payne University of California Lawrence Livermore National Laboratory P.O. Box 808 L-441 Livermore, CA 94550 Tel. (5 10) 422-7798, Fax (5 10) 423-6195 Abstract We present measurements of the 4G~jl emission lifetime for 26 Nd-doped materials. A model of nonradiative decay based on dipole-dipole energy transfer is developed and found to be supported by our data. Introduction It has been extensively verified in the past that the nonradiative decay rate between rare earth energy levels is predominantly determined by the energy gap and the particular host medium. The energy gap law is embodied in the simple expression for the nonradiative rate: W,,, = (llQexp(-ad, (1) where ra and p are constants characteristic of the host medium, and p (=AE / hu,& is the number of phonons needed to bridge the gap, The expression hu,,, is related to the highest phonon frequency of the host medium and AE is the energy gap between the populated rare earth state and the next energetically lower level. The constant, a, is related to the details of the electron-phonon coupling. In nearly all cases reported in the literature, eqn. (1) is regarded as adequately describing the measured decay rates over several orders of magnitude [l-3]. In the present paper we report the nonradiative decay rates of the 4G7,2 state of Nd3+ in 26 different crystals and glasses, with the goal of providing a data base relevant to the relaxation rate of the 4111c? state which has a similar energy gap as the 4G7,2 state [4]. The 41iic? state of Nd3+ is a particularly important one from a practical point of view, since it can potentially “bottleneck” during lasing and give rise to transient absorption at the laser wavelength (constituting loss). Much of the motivation for establishing this correlation is that emission lifetimes are much simpler to measure and can be widely applied to numerous Nd- doped crystals and glasses, while the pump-probe technique used to directly assess the 41i112 decay time is complex to set up and execute. The relevant energy levels are depicted in Fig. 1, where the 5512, ~712, and 2ii,2 lifetimes are identified on the diagram along with the 532 nm pump wavelength and the -600 nm detection wavelength One of the details that must be handled in the numerical analysis of the data is the effect of the overlapping -600 nm emission arising from the 4G5,2, 2G7n states on the measured results (i.e. r.&, as depicted in Fig. 1. The numerical analysis and resulting fits to the data will be discussed. 4F3/2 07/z c5J2 532 nm Pump li 0. 1512 Resonant emission at -600 nm 41.J 1312 1112 912 Fig.1 Energy levels and transitions for Nd3’ Experimental The picosecond emission lifetimes [4] were measured using the time-correlated single photon counting system together with a Coherent mode-locked laser producing -90 psec pulses at 76 MHz and then doubled to 532 nm. The sample emission was detected at -600 nm with a monochromator followed by a multichannel plate photomultiplier (MCP-PMT). The (deconvoluted) temporal resolution of the data is about 50 psec. For the case of the fluoride samples, the lifetimes of -10 nsec were too long to be measured with the time-correlated photon-counting apparatus. To handle these longer lifetimes, we employed a conventional set-up consisting of a Q-switched Nd:YAG laser (12 nsec pulsewidth), along with a monochromator, oscilloscope, and MCP-PMT Results The results of the emission lifetime experiments are contained in Table 1. Based on an analysis of the best fit to the data, the uncertainty in the value of the ~7/2 lifetimes is A~712 (<l ns) = + 50 ps for data with lifetimes less than 1 ns and Ar7,2 (>I ns) = + 200 ps for data with lifetimes greater than 1 ns. Many of the trends that can be gleaned from the data in Table 1 are expected. For the case of the fluorides, the ~71~ values are the longest, being 4,000 - 41,000 psec. This is anticipated, since their phonon frequencies are the lowest among the hosts listed, being <600 cm-‘. Whereas the phosphate and silicate glasses are in the range of 150 to 250 psec owing to the rather high vibrational frequencies of the Si04 and PO4 anions. It is noteworthy that comparisons of previous T,,~ measurements in the literature are satisfactory, including [5]: 1400 versus 1090 psec for YAIO,; 370 versus 200 psec for Y3Al50t2; and 8400 psec versus 9100 psec for LiYF,; 56,400 psec versus 41,000 psec for LaF3 Table 1. Emission lifetimes of the 4G7j2 excited state correlated with each other to within about a factor of two, across a variety of different host media. This concurrence is reasonable, since some differences in the energy gap and perhaps the electron-phonon coupling may be expected, in addition to the experimental and analytical uncertainties. Furthermore, due to the small difference in the size of the energy gap for a given host medium, this data offers some experimental validation that the specific characteristics of the electronic states (symmetry, crystal field interactions, spin, etc.) do not strongly influence the nonradiative decay rate. Name Phosphate Formula P205+A1203+ v12, psec glasses LG-750 APG- 1 APG-x APG-2 LG-812 Silicate glasses LG-660 LG-650 Sol-gel Vanadate YVOj Tungstate CaWO, Oxide YALO GSGG YAG GGG LLGG Apatites C-FAP S-FAP C-VAP S-VAP ------ Fluorides ZBLAN YLF NasSc2LisFr2 KY~FIo YF3 modifiers PzOs+fluorides+ modifiers Si02+A1203+ modifiers SiOz YVO4 CaW04 YAlO, Gd3Sc2Ga3012 Y3A15012 GWWb LasLu2GasOr2 ZrF4-BaFs-AlF3- LaFs-NaF LiYF, Na3Sc2Li3Fr2 KY~FIo YF3 228 215 210 150 -1400 215 210 245 190 510 1090 715 200 530 1200 70 175 200 330 380 18,000 9100 4000 9000 22,000 Lab LaF3 41,000 Analysis and Discussion As mentioned earlier, measurements of the 4G7,2 population decay time have long been suspected to be similar in magnitude to that of the 41tt,z level of Nd3+. Using the 4111,2 lifetimes from a direct measurement in [7] we find that the 41ttn and 4G7,~ lifetimes are lMCO)m LG~Y~(RS~+WEGLASS) i IO...,,..,, 10 lcn 1003 lwo3 lo3003 3,, Lifetime @s) Fig. 2 Comparison of the 41 i tj2 nonradiative relaxation time from [4] and the 4G,,2 emission lifetime. Another theory of nonradiative decay derives the multi-phonon rate on basis of energy-transfer theory, as originally described by Forster [6] and Dexter [7]. With the assumption that dipole-dipole interactions dominate the process, and that the phonon absorption cc,,,(h) is a slowly varying function compared to the emission cross section spectrum cre,(h), we derive that. (nn) ‘min where Vmin is the minimum volume that is non- absorbing in nature (centered on the rare earth ion), n is the refractive index, and c is the speed of light. This expression essentially emerges from the well-known Forster-Dexter spectral overlap integral between the emitting and absorbing species, and a volume integration over a uniform acceptor (phonon) concentration, (a derivation similar to other reports in the literature [8-l 11). It is also noteworthy however, that the multi-phonon absorption spectrum is commonly described with an exponential expression expression [ 12,131. aph = Ahost “P( - Yhost hv,char), (3) where upt, is the phonon frequency, &hnr is a characteristic phonon frequency of the host material, and yhosl and Af,ost are other host-dependent constants. So if we combine eqns. (3) and (4), we can suggest an alternative route to deriving the form of the energy gap law. where we have identified Z)ph as Ugap and &har as Urnax (maximum phonon frequency), the . The nlost parameter turns-out to be similar for many crystals, for instance being in the range of 4-5 for alkali and alkaline-earth halide crystals [13]. The main point to note regarding eqn (4) is that the exponentiated factor in square brackets only contains information concerning the host medium, while the rare earth properties are exclusively represented in the pre-exponential factor - implying that we may expect reasonably good adherence to the form of the energy-gap law, eqn. (l), We can explore the validity of eqn. (4) by inputting reasonable estimates for the terms in the pre- exponential factor and deducing the magnitude of this constant. Using Vnlin = 5 X 1O-24 Cm3, Ahost = 20,000 cm- i (average from ref. [13]), and a,,*Ah=1.8 x 1O-25 cm2 (calculated from Judd-Ofelt theory using the average RI parameters of YAG and YLF [ 14]), we obtain a value of 0.6 x 10” sec.‘, or r. = 1.7 psec, which is defined from eqn. (1) as: 2c /J 3 oemdh l/T, q - W-d 2 ‘min A host ’ (5) We are now in a position to compare this calculation to the data in Table 1, where we use the energy gap (AE& and highest phonon frequencies (hv,,& to calculate number of phonons p= AETj2/ hUmax. If we also group all of the phosphate glasses into a single datum, and all of the silicates into a second one, then there are potentially ten usable points. Finally, if the LaF3 result is eliminated because of its strongly nonexponential character, the results of the exercise may be displayed as shown in Fig. 3, where we have plotted the‘nonradiative decay time, ~712, against the number of phonons, p. The data is then fitted to the reciprocal of the energy gap law: ‘I; nr = To exp(aM (f-3) with the result of the numerical fit yielding z. = 3.3 psec and a = 3.1. We are very encouraged by this result because it may be compared with theoretical calculation of r. = 1.7 psec noted above in connection with eqn. (5). Now, if we associate Yhost with a, and huChar with bnax~ the a = 3.,1 value from Fig, 3 appears to be within the range of what one may expect from the energy- transfer theory of nonradiative decay encompassed in eqn. (4), since the phonon spectra of many materials is characterized by Yh,,st = 4-5. t / ‘C-FAP 11 I I 0 1 2 3 4 Number of Phonons Fig. 3 Plot of the nonradiative decay time against the number of phonons needed to bridge the gap. In referring back to the more general energy-transfer- based description of nonradiative decay relating to eqn. (2), we can apply another test of this model by plotting W,,, of the rare-earth ion against the absorption coefficient c$h of the host and fitting these data to the simple expression: w,,, = ka,,, (7) where k can be theoretically calculated and related to ~~ by: 2c /J k = 1/ (Ahost *To) = ---+ oemdh ‘min (8) (nn> based on eqns. (2) and (5). In Fig. 4 we have chosen to focus this analysis on data derived exclusively from the YAG, YALO and YLF hosts, because the phonon spectra of these materials have been reported in the literature [15, 16, 171; in addition, the nonradiative decay rates and the associated energy gaps derived from refs. [18, 19, 20, 21, 221 entail a wide variety of different electronic states and rare earth ions (in contrast to the prior exclusive attention given to the 4G7/2 state of Nd”“). While we notice that the scatter in the data of Fig. 4 is substantial, it does span many orders of magnitude and the fitted or experimental value of kXp = 1.0 x lo7 cmisec based on the data is found to be near the independently theoretical value of k = 3 x lo7 cm/set deduced from eqn. (8) for the 4G7,2 state. Employing the experimental value of k in the simple relationship of eqn. (8): 2, = l/(kexp*Ahosc), we find a third estimate for 2, of 5 psec, a value that is in reasonable accord with the theoretically derived value of 1.7 psec from eqn. (5), and the other experimentally determined magnitude of 3.3 psec from Fig. 3 and eqn. (6). I-/ 0 3 0 W M  = lO’(cmlsec) ciph -I 21 lffl ’ ’ ’ ’ ’ ’ ’ ’ ’ 1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 b31~p,(cm ‘)IFig 4. Plot of the nonradiative decay rate of various states of rare earth ions in YAG, YALO, and YLF, as a function of the phonon absorption coefficient of the host at the particular energy gap. It is important to keep in mind that the results from the first two calculations are specific to the 4G7,2 state, while the treatment in the last calculation entails numerous different rare earth ions and states (Fig. 4). The concurrence of these three independent approaches in estimating 1, is remarkable and an implicit endorsement of the usefulness of the energy-transfer- based mechanism of nonradiative decay. Acknowledgments We would like to thank Christopher Marshall, Gary Wilke, Howard Powell and Bill Krupke for their suggestions throughout this project. *This work was performed under the auspices of the Division of Materials Science of the Office of Basic Energy Sciences, U. S. Department of Energy, and by the Lawrence Livermore National Laboratory under contract W-7405ENG-48. [I] H. W. Moos, J. Luminesc. l/2 (1970) 106. [2] W. D. Partlow and H. W. Moos, Phys. Rev. 157 (1967) 252. [3] L. A. Riseberg, W. B. Gandrud, and H. W. Moos, Phys. Rev. 159 (1967) 262. [4] C. Bibeau, Ph. D. Dissertation, “Evaluation of the 41 11,2 terminal level lifetime for several neodymium- doped laser crystals and glasses,” April 25, 1995. [5] T. T. Basiev, A. Yu. Dergachev, Yu. V Orlovskii, and A. M. Prokorov, J. Luminesc. 53 (1992) 19 [6] T. Forster, Z. Naturf. 49 (1949) 321. [7] D. L. Dexter, J. Chem. Phys. 21 (1953) 836. [8] N. T. Timofeev and E. B. Sveshnikov_a, Opt. Spectrosc. (USSR) 54 (1983) 595. [9] V. L. Ermolaev and E. B. Sveshnikova, J. Luminesc. 20 (1979) 387. [IO] V. L. Ermolaev and E. B. Sveshnikova, Chem. Phys. Lett. 23 (1973) 349. [I 11 E. B. Sveshnikova, N. T. Timofeev, and V. M. Zolotarev, Izvestiya Akademii Nauk SSSR. Seriya Fizicheskaya 44, (1980) 722. [12] T. C. McGill, R. W. Hellwarth, M. Magir, and H. V. Winston, J. Phys. Chem. Solids 34 (1973) 2105. [ 131 L. L. Boyer, J. A. Harrington, M. Haas, and H. B Rosenstock, Phys. Rev. B 11 (1975) 1665. [14] R. P. Bauman and S. P. S. Porto, Phys. Rev 161 (1967) 842. [15] V. P. Gaponsev, M. R. Sirtlanov, A. K. Gromov, A. A. Isineev, Proceedings on the International Corlference on Lnsers ‘81 (STS Press, McLean, 1981), p. 763. [16] G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, Phys. Rev. 177 (1969) 1308. [17] E. B. Sveshnikova, A. A. Stroganov, and N. T.Timofeev, Opt. Spectrosc. (USSR) 64 (1988) 43. [ 181 M. J Weber, Phys. Rev. B 8 (1973) 54. [19] A. M. Tkachuk, A. V. Khilko, and M. V. Petrov, Opt. Spectrosc. 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Picosecond nonradiative processes in neodymium-doped crystals and glasses: mechanism for the energy gap law

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Picosecond nonradiative processes in neodymium-doped crystals and glasses: mechanism for the energy gap law

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