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1 rly View puliction on (issue nd pge nuers not yet ssigned; citle using Digitl Oject Identifier DOI) phys. stt. sol. () 9 (008) / DOI 0.00/ Theroluinescence glow-pek shpe ethods sed on ixed order kinetics George Kitis * Reuven Chen nd Vsilis Pgonis 3 Aristotle University of Thessloniki Nucler Physics Lortory 544 Thessloniki Greece Ryond nd Beverly Sckler School of Physics nd Astronoy Tel-Aviv University Tel-Aviv Isrel 3 Physics Deprtent McDniel College Westinster MD 57 USA physic sttus pplictions nd terils science Received 8 Septeer 007 revised 8 Deceer 007 ccepted 3 Deceer 007 Pulished online 9 Mrch 008 PACS Kn Sq * Corresponding uthor: e-il gkitis@uth.gr Phone: Fx: The pek shpe ethods used in theroluinescence (TL) theory to evlute the ctivtion energy re sed on first nd second order kinetics equtions. For the interedite kinetic orders the pek shpe ethods re sed on the epiricl theory of generl order kinetics. In the present work we derive pek shpe ethods sed on the physiclly eningful ixed order kinetics odel. The derived equtions re tested for their ccurcy nd re copred with other pek shpe ethods existing in the TL literture. 008 WILY-VCH Verlg GH & Co. KGA Weinhei Introduction The sic phenoenologicl Theroluinescence (TL) theory predicts two expressions for the description of experientl TL glow-peks. These expressions re derived fro the first order kinetics theory of Rndll nd Wilkins [] nd fro the second order kinetics theory of Grlick nd Gison []. However there re ny reported experientl glow-peks with shpe tht does not correspond to first or second kinetics order ut rther to kinetics order etween the two. A purely epiricl theory to descrie these interedite kinetics order ws suggested y My nd Prtridge [3]. Chen et l. [4] suggested the use of ixed order kinetic theory to descrie these cses of interedite kinetic order. The shpe of TL glow-pek plys n iportnt role in sic reserch nd in TL pplictions. In the cse of sic TL reserch it is the sis of iportnt nd convenient ethods for clculting the trpping preters of distinct energy levels within the crystl. These ethods re sed on esureents of few points on the glow-pek s shown in Fig.. rly work concentrted on the developent of convenient expressions for clculting the ctivtion energy of trpping levels [5 7]. The ter pek shpe ethod is reserved in the TL literture for such ethods lthough there re other ethods for finding which re lso sed on the glow-pek shpe (i.e. curve fitting ethods) [8 9]. Chen [0] surized ll pre-existing ethods nd gve detiled ethodology for deriving the coefficients of the expressions for first nd second order kinetics. For the cse of interedite kinetic orders the pek shpe ethod coefficients were lso evluted y Chen [] y (i) clculting the coefficients for first order kinetics (ii) clculting the coefficients for second-order kinetics nd (iii) using liner interpoltion ethod to otin expressions for the interedite kinetic orders s function of the syetry fctor which ws found to e etween 0.4 nd 0.5 for first nd second order kinetics. Recently Kitis nd Pgonis [] provided theoreticl foundtion for the Chen expressions for interedite kinetics y using the epiricl odel of generl order kinetics respectively. In the current sttus of TL theory the region of interedite kinetics order is descried y epiricl odels. Sunt et l. [3] tested the cpility of the ixed nd generl order kinetics to fit synthetic glow-peks derived fro vrious phenoenologicl odels such s the one trp nd one recointion centre odel (OTOR) non interctive ulti trp syste (NMTS) nd interctive ultitrp syste (IMTS). They found tht the ixed order kinetics odel fits the NMTS nd IMTS glow-peks uch ore successfully thn the generl kinetics order. On the other hnd the ixed order kinetics fits poorly the TL 008 WILY-VCH Verlg GH & Co. KGA Weinhei

2 physic sttus G. Kitis et l.: Theroluinescence glow-pek shpe ethods sed on ixed order kinetics peks produced y the OTOR odel. Sunt et l. [3] ttriuted this result to the fct tht the OTOR odel is too siple nd physiclly unrelistic. Their net conclusion ws tht ixed order kinetics is clerly superior to generl kinetics order in the description of experientl TL glowpeks. The i of the present work is to otin new pek shpe ethods for evluting the ctivtion energy which re sed on the physiclly eningful odel of ixed order kinetics. Anlyticl ixed order TL expressions The ixed order kinetics eqution s given y Chen et l. [4] (see lso Chen nd Kirsh [8] Chen nd McKeever [9]) is: dn IT ( ) =- = nn ( + C) s exp Ê - ˆ dt Ë where n (c 3 ) is the concentrtion of electrons in trps (ev) is the ctivtion energy T (K) the teperture nd C is the concentrtion of electrons trpped t soe kind of deep trps. The solution of q. () for liner heting rte β contins the well known exponentil integrl which cn e pproxited y the following syptotic series: T Ú T0 () exp Ê - ˆ dt ª T exp Ê - ˆ Ë Ë n n- Ê ˆ Â (-) n! () Ë n= where T is lrger fro T 0 y t lest few degrees. By considering for the ske of siplicity only two ters of the ove syptotic series the qusi-nlyticl solution of q. () is: with Ê ˆ F( T) Ë ( F( T )-α ) ( ) = αcs exp - IT È Cs FT ( ) = exp exp Ê ˆ Í - ( - ) Í β Ë Î where α = n 0 /(n 0 + C) s = s/(n + C) = / N (c 3 ) is the concentrtion of ville electron trps n 0 (c 3 ) the concentrtion of trpped electrons nd s (s ) the frequency fctor. The condition for xiu is found y equting the derivtive of q. (3) to zero i.e. β F Cs exp Ê ˆ F Á = - -α Ë where T is the teperture t glow-pek xiu intensity nd F the vlue of F(T) t T = T. (3) (4) (5) F(T) (T) TL Teperture ( K ) I τ ω T T T I / =/ω Figure (online colour t: Geoetricl chrcteristics of single glow-pek. 3 Geoetricl chrcteristics of single glowpek The pek shpe ethods re sed on certin chrcteristics of single glow-pek shown in Fig. nely the pek xiu teperture T nd the tepertures t hlf xiu TL intensity T nd T t the low nd high teperture side of the glow-pek respectively. These quntities re used to define further the widths ω = T T = T T nd τ = T T s well s the syetry fctor of the glow-pek = /ω. The derivtion of the existing pek shpe ethods is sed on the so-clled tringle ssuption which cn e expressed in three different wys ech one leding to n individul fily of pek shpe ethods. In the for given y Chen [0] these re ω I = Cω (6) β n0 I = C (7) β n τ I = Cτ (8) β ( n0 - n) with n = Ú I d t (9) t where I is the pek xiu intensity n the high teperture hlf integrl of the glow-pek nd C ω C nd C τ re quntities which chrcterize the degree y which the re of single glow-pek pproches the re of tringle. These quntities were found to vry extreely slowly for given kinetic order nd for very wide rnge of kinetic preters ( s) nd re clled pseudo-constnts. These pseudo-constnts vry only little for ll glow-peks derived using ctivtion energies in the region of 0..6 ev nd frequency fctors in the region of s [0]. 008 WILY-VCH Verlg GH & Co. KGA Weinhei

3 Originl Pper phys. stt. sol. () (008) 3 Their vlues cn e estited s function of the preter α y producing synthetic glow-peks using q. (3). In order to derive the pek shpe foruls using qs. (6) (8) one hs to evlute the ters I /n 0 n /n 0 nd I /n fro the nlyticl TL expressions. 4 Derivtion of the pek shpe ethods 4. Derivtion of I /n 0 By considering q. (3) for T = T we otin Ê ˆ F I = αc s exp Á-. (0) Ë ( F -α ) Using the condition for the xiu given y q. (5) nd tking into ccount tht α = n 0 /(n 0 + C) one otins fter soe lger I F = ( -α ) β. n F - α 0 () 4. Derivtion of n /n nd I 0 /n Solving q. () one otins n + C ÈCs Ê ˆ α = exp exp ( ) n Í β Á- - Î Ë = F () fro which fter soe lger it is found tht n n -α = =. F -α 0 (3) According to Hlperin nd Brner [7] the quntity n /n 0 which represents the rtio of the high teperture hlf integrl of glow-pek to the totl integrl is the integrl syetry fctor µ g of glow-pek which differs very slightly fro the coonly used geoetricl syetry fctor defined ove. By dividing q. () y q. (3) one otins I F = β. n F (4) 4.3 Method sed on the totl width (w) Fro q. (6) one otins I Cω =. (5) n0β ω Using q. () one otins F - ω = Cω. (6) ω -α F Fro q. (3) it is found tht + ( g -). α µ F = µ g (7) Tking into ccount q. (7) q. (6) ecoes ω = cω ω (8) where F cω = Cω. µ F (9) g 4.4 Method sed on the high teperture hlf-width (d) Fro q. (7) one otins I C =. (0) nβ Using q. (4) with c = c () F = C. () F 4.5 Method sed on the low teperture hlf-width (t) qution (8) cn e re-written s n0 τ I - =. (3) n Cτ β n Replcing the ters I /βn nd n 0 /n fro q. () nd (3) respectively one otins fter soe lger F - F τ = Cτ. (4) τ -α F Tking into ccount q. (7) q. (4) ecoes τ = cτ. (5) τ With - F cτ = Cτ. (6) µ F g 5 Results 5. Nuericl vlution of geoetricl chrcteristics The geoetricl chrcteristics of the glowpeks were evluted y siulting synthetic glow-peks using q. (3). The syptotic series of q. () used to derive q. (3) ws not used since the softwre used contins the exponentil integrl s uilt-in function. The nuericl siultion of synthetic glow-peks ws perfored y using very rod regions of the trpping preters in order to cover s ny prcticl cses s possile. The ctivtion energy ws vried etween 0.7 ev nd ev the frequency fctor etween 0 7 s nd 0 s. The ixed order preter α ws vried fro 0.0 to WILY-VCH Verlg GH & Co. KGA Weinhei

4 physic sttus 4 G. Kitis et l.: Theroluinescence glow-pek shpe ethods sed on ixed order kinetics Syetry fctors C ωτ C C ω C τ Mixed order α Figure (online colour t: Syetry fctors () Geoetricl syetry fctor nd () integrl syetry fctor µ g s function of the ixed order preter α. All the siultions were perfored for n 0 = N i.e. with the electron trp in sturtion. The cse of n 0 N will e studied in Section 5.8. Fro ech synthetic glow-pek the following quntities re evluted: (i) The geoetricl quntities shown in Fig. (ii) the integrl syetry fctor (iii) the tringle ssuption pseudo-constnts C ω C nd C τ nd (iv) the fctor (F )/F. In order to ensure n ccurte deterintion of the vrious preters very sll teperture step is necessry. In the present siultion the TL intensity ws evluted every 0.0 K. 5. Syetry fctors Fro the siulted synthetic glow-peks oth the geoetricl syetry fctor nd the integrl syetry fctor were evluted. Their ehviour s function of the ixed order preter α is Mixed order α Figure 3 (online colour t: Tringle ssuption pseudo-constnts s function of the ixed order preters α. shown in Fig.. The in difference etween nd is tht the geoetricl syetry fctor is sturted for α eyond 0.9 wheres the integrl syetry fctor is not. Their nuericl vlues re listed in Tle. 5.3 vlution of C w C d nd C t The ehviour of C ω C nd C τ given y qs. (6) (8) otined fro the siultion is shown in Fig. 3 wheres their nuericl vlues re listed in Tle. Looking t the results of Tle one cn see why they re clled pseudo-constnts. For given ixed order preter α their vrition over ll ( s) vlues used is 0.4% 0.34% nd 0.67% respectively. The preters C τ cn e considered s pseudo-constnt even over ll the vlues of the ixed order preter α. 5.4 Prcticl fors of the pek shpe equtions The derived pek shpe ethods given y qs. (8) () (5) re not pplicle to experientl glow-peks. The reson Tle Geoetricl ( ) nd integrl ( µ ) g syetry fctors for vrious vlues of the ixed-order preter α. rrors in (%) µ g (.7%). The tringle ssuption pseudo-constnts C ω C nd C τ for vrious vlues of ixed order preters α. rrors in C ω (0.4%) C (0.34%) C τ (0.67%). The vlues of the pek shpe ethods coefficients c ω c nd c τ for vrious vlues of ixed order preters α. rrors in c ω (.%) c (0.9%) c τ (.3%). α µ g C ω C C τ c ω c c τ WILY-VCH Verlg GH & Co. KGA Weinhei

5 Originl Pper phys. stt. sol. () (008) 5 (F ) / F Syetry fctror Figure 4 (online colour t: Reltion etween the fctor (F )/F nd the syetry fctors nd µ. g is tht the ter (F )/F cnnot e evluted experientlly wheres the tringle ssuption pseudoconstnts C ω C nd C τ re very difficult to evlute experientlly. The pek shpe ethods given y qs. (8) () (5) cn e rought into useful prcticl for y either expressing the ter (F )/F or the totl coefficients c ω c nd c τ given y qs. (9) () (6) s function of experientlly esured quntities like nd. These ttepts re descried elow. 5.5 Replcing the ter (F +α)/f The ter (F )/F is coon fctor in the three filies of pek shpe ethods given y qs. (8) () (5). By evluting this ter for wide rnge of preters nd s it ws found tht this fctor depends nerly linerly on the syetry fctors nd s is shown in Fig. 4. It is cler tht the liner correltion is excellent etween this ter nd the integrl syetry fctor µ g wheres in the cse of the geoetricl syetry fctor there is devition fro linerity s the ixed order preter α pproches. So this reltionship cn e written s F F = µ - (7) g where nd re constnts. Therefore the pek shpe ethods cn tke the generl for C ( ) ω = ω ω µ g - (8) = C ( µ g -) (9) - µ τ = Cτ g τ µ g - ( ). (30) Tle Nuericl vlues of the liner coefficients of qs. (30 3) s function of the ctivtion energy. (ev) ± ± ± ± ± 3 ± ±6.780 ±.7093 ± ±.384 ±6.93 ± ±6.784 ±.7083 ± ±.404 ±7.004 ± ±6.83 ±.7070 ± ±.47 ±7.065 ± ± ±.75 ±9. ±.4488 ±7.49 ±.005. ±6.976 ±.749 ±9.406 ±.4758 ±7.795 ±.07. ± ±.770 ± ±.4977 ±7.6 ± ±6.99 ±.76 ±9.473 ±.54 ±7.606 ± ±7.079 ±.770 ±9.40 ±.56 ±7.94 ± ±7.047 ±.706 ±9.587 ±.556 ± ±.05.6 ±7.066 ±.783 ±9.694 ±.553 ±7.350 ±.05.7 ± ±.73 ± ±.578 ±7.384 ± ±7.004 ±.700 ±9.780 ±.5774 ± ± ±7.40 ±.797 ± ±.5846 ±7.48 ±.0645 en ± ±0.6 ±.755 ±0.005 ± ±0.34 ±.5003 ±0.07 ±7.87 ±0.6 ±.068 ±0.03 The replceent of the ter (F )/F cn e tested using the vlues of C ω C nd C τ fro Tle. It ws found tht the liner coefficients nd depend slightly on the kinetic preters nd s. Therefore they were studied seprtely s function of the ctivtion energy. In ddition the en vlues for the whole ctivtion energy region (0.7 ev) were lso clculted. Their vlues s function of the ctivtion energy s well s their en vlues re listed in Tle. As it is seen fro Tle lthough the ctivtion energy vries fro 0.7 ev to.9 ev the coefficients nd vry y less thn % nd 0.3% respectively. However even this low vrition cn cuse serious increse in the error of. The results re shown in Fig. 5. Figure 5() shows the errors in evluted y q. (3) using the respective coefficients fro Tle. On the other hnd Fig. 5() shows the errors in evluted y q. (8) when using the en vlues of the coefficients fro Tle. The error in is sustntilly incresed when going fro Fig. 5() to 5(). The sitution is siilr for the sed nd the τ sed ethods. Fro the ove results it is concluded tht the ter (F )/F ws successfully pproxited y q. (7). The ppliction of qs. (8) (30) to experientl glow-peks requires lso functionl reltion etween the tringle ssuption pseudo-constnts nd the integrl syetry fctor µ. However s is shown in Fig. 3 it is rther coplicted for C ω C nd rther siple for C τ. A functionl reltion ws not necessry since the pek shpe ethods in their for given y qs. (8) (30) cn e pplied y using the tulted vlues of the relevnt preters s function of fro Tles nd. 5.6 vlution of c w c d nd c t The vlues of the coefficients c ω c nd c τ given y qs. (9) () (6) respectively re listed in Tle. The ove estited vlues of the pek shpe ethod coefficients should e useful for prcticl ppliction if n WILY-VCH Verlg GH & Co. KGA Weinhei

6 physic sttus 6 G. Kitis et l.: Theroluinescence glow-pek shpe ethods sed on ixed order kinetics (ev) (ev) Figure 5 rrors in ctivtion energy introduced y the replceent of the ter (F )/F in q. (7). () rrors in the ω sed pek shpe ethods when the coefficients in q. (8) re considered s function of the ctivtion energy. () rrors in the ω sed pek shpe ethods when the coefficients in q. (8) re tken s the en vlues over ll ctivtion energies. nlyticl reltionship s function of or cn efound. Indeed it ws found tht the coefficients c ω nd c re liner functions of the geoetricl syetry fctor s is shown in Fig. 6(). Therefore prcticl for of the ω nd ethods is: ω = ( µ g -) (3) ω = ( 3µ g -3). (3) Unfortuntely in the cse of the τ ethod no good liner reltionship exists etween c τ nd s shown in Fig. 6(). The liner coefficients in qs. (3) (3) re lso ctivtion energy dependent. Their vlues s function of the ctivtion energy s well s their en vlues over the whole ctivtion energy rnge re shown in Tle. As is seen fro the tle lthough the ctivtion energy vries fro 0.7 ev to.9 ev the coefficients 3 nd 3 vry y less thn 4%. However even this vrition cn cuse serious increse in the error of. The results re shown in Fig. 7. Figure 7() shows the errors in evluted y q. (3) using the respective coefficients fro Tle. Figure 7() shows the errors in evluted y q. (3) using the en vlues of the coefficients fro Tle. The error in is sustntilly incresed when going fro Fig. 7() to (). The sitution is siilr for the sed q. (3). At this point it is useful to discuss the ccurcy of the experientl evlution of oth the geoetricl nd integrl syetry fctors nd µ g respectively. Oviously n ccurte deterintion of oth requires clen isolted glow-peks. However in prctice even isolted glowpeks cn hve soe type of shoulders t oth low nd high tepertures. The low teperture shoulders cn e reoved successfully y proper nneling procedure. However this is ipossile for shoulders t the high teperture prt. In these cses of course there is n unvoidle increse of the error which depends on the contriution of the shoulder. At first look it y see tht the influence of the shoulder would e higher on the integrl syetry fctor µ g rther thn on the geoetricl syetry fctor. However the finl ccurcy is ore or less the se ecuse the geoetricl syetry fctor depends on three points of the glow-pek the T T nd T the errors of which re propgted to the vlue of = (T T )/(T T ) wheres the integrl syetry fctor depends only on one point T which is deterined ore ccurtely thn T nd T. 5.7 Properties of the fctor (F +α)/f A very interesting oservtion is tht the vlues of the ter (F )/F which will e clled ix hve role in the ixed order kinetics siilr to the kinetic order of the generl order kinetics. In principle this is concluded fro the following fct. c ω c 3 () c ω c Syetry fctror c τ.6.5 () c τ Syetry fctror Figure 6 (online colour t: () Reltionship etween the totl coefficient vlues of the ω nd sed pek shpe ethods s function of the geoetricl syetry fctor. () Reltionship etween the totl coefficient vlue of the τ sed pek shpe ethod s function of the geoetricl syetry fctor. 008 WILY-VCH Verlg GH & Co. KGA Weinhei

7 Originl Pper phys. stt. sol. () (008) (ev) (ev) Figure 7 rrors in ctivtion energy introduced y the replceent of the ter c ω y its liner dependence on µ. () rrors in the ω sed pek shpe ethods when the coefficients in q. (3) re considered s function of the ctivtion energy. () rrors in the ω sed pek shpe ethods when the coefficients in q. (3) re considered s en vlues over ll ctivtion energies. The fily of pek shpe ethods for the cse of generl order kinetics is given y the eqution [9] G = CG. (33) G The respective ethod for ixed order kinetics is given y q. () which is re-written in the for elow: = C (34) M M ix M where the index M is for ixed order nd the index G for generl order kinetics. Since oth ethods ust give the se i.e. G = M it is esily found tht ix C G M =. (35) C M G Oviously M = G nd the tringle ssuption pseudoconstnts lthough not equl differ y less thn 5% so it is concluded tht q. (35) is very close to unity so tht ix. This reltionship ws further studied nd the results re shown in Fig. 9( c). In Fig. 9( ) one cn see the e-hviour of ix s function of the ixedorder preter α nd the syetry fctor µ respectively Activtion nergy (ev) = ev ω n 0 / N Figure 8 (online colour t: rrors in ctivtion energy s function of the trp occupncy n 0 /N. wheres in Fig. 9(c) one cn see the rtio of ix over s function of the ixed-order preter α. In the worst cse shown in Fig. 9(c) the difference etween the two preters is of the order of 5%. The interesting result here is tht the epiricl generl order preter is found to e lost equl to the physiclly eningful preter ix. 5.8 Cse of n 0 N The results presented up to now concerned the cse of sturted trp i.e. n 0 = N. However coplete picture requires to exine the cses of non sturted trp i.e. n 0 N. This cse ws studied for = ev s = 0 s nd n 0 /N vrying y two orders of gnitude etween 0.0 nd. The results concerning the ppliction of the pek-shpe ethods given y qs. (3) (3) re shown in Fig. 8. As cn e seen the dependence on n 0 /N is etween 0% nd.5%. It is oserved tht the dependence of the error on n 0 /N sees to e systetic. It ust lso e noticed tht these systetic vritions enle the iproveent of the results. For exple y inserting fctor of in front of qs. (3) (3) the error will e reduced fro 0.5 % to ±.5%. 5.9 Coprison with other pek shpe ethods The pek shpe ethods derived in the present work will e copred with the existing pek shpe ethods like those of Chen [] nd those suggested y Kitis nd Pgonis []. The coprison ethod dopted hs the following steps. Step : Synthetic ixed order glow-peks re derived using q. (3). Step : The integrl syetry fctor µ g is evluted for ech glow-pek Step 3: Using the vlue of µ g the kinetic order of generl order glow-pek hving the se µ g is evluted y n itertion ethod using the following eqution suggested y Kitis nd Pgonis []: È = Í ( ) Î (36) where = /. The ccurcy of q. (36) depends only on the nuer of ters in the syptotic series pproxi WILY-VCH Verlg GH & Co. KGA Weinhei

8 physic sttus 8 G. Kitis et l.: Theroluinescence glow-pek shpe ethods sed on ixed order kinetics ix () ix. Mixed order α ix () ix ix / Syetry fctor Syetry fctror Figure 9 (online colour t: Behviour of the generl order kinetics nd the ixed order preter ix () s function of ixed order preter α nd () s function of the integrl syetry fctor µ g. (c) The rtio of ix / s function of the integrl syetry fctor µ g. (c) G.O M.O Mixed order α 4 - Chen s Mixed order α Figure 0 () G.O. represents the errors in when generl order pek shpe ethods re pplied to ixed order glow-peks nd M.O. represents the errors in when ixed order pek shpe ethods re pplied to generl order glow-pek. () Chen s represents the errors in when the Chen s pek shpe ethod is pplied to ixed-order glow-peks. rror G.O M.O rror 7-5 Chen s Mixed order α Mixed order α Figure As in Fig. 0 for the ethod. rror τ G.O M.O Mixed order α rror τ Chen s 0.0 Mixed order α Figure Se s Figs. 0 nd for the τ ethod. 008 WILY-VCH Verlg GH & Co. KGA Weinhei

9 Originl Pper phys. stt. sol. () (008) 9 tion of the exponentil integrl given y q. (). For given the kinetic order cn e evluted to ny desired ccurcy depending upon the nuer of itertions. Step 4: Once the kinetic order is evluted then the corresponding generl order glow-pek is derived using the se ( s) vlues. Hving thus oth kinds of glow-peks the pek shpe ethods re pplied to the s following. The generl order pek shpe ethods of Chen [] nd of Kitis nd Pgonis [] re pplied to ixed order glow-peks wheres the ixed order pek shpe ethods derived in the present work re pplied to the generl order glow-peks. The degree to which the vlues found y pek shpe ethods gree with the input vlues is esure of the greeent etween generl nd ixed order glow-peks. The results re shown in Figs. 0 for the ω nd τ pek shpe ethods respectively. The sic conclusion fro these results for ll filies of pek shpe ethods is tht when the generl order kinetics pek shpe ethods y Kitis nd Pgonis [] re pplied to ixed order glowpeks they overestite the vlue of in nner depending upon the ixed order preters α. On the other hnd when the ixed order pek shpe ethods re pplied to generl order glow-pek they underestite the in nner depending upon the ixed order preters α. The differences etween generl nd the ixed-order pek shpe ethods reflects exctly qulittively nd quntittively the reltion etween the ixed order ter ix nd the kinetic order shown in Fig. 9(c). The ove reltion etween ixed order pek shpe ethods nd the generl order pek shpe ethod y Kitis nd Pgonis is not followed y the generl order pek shpe ethods of Chen [] in the cse of the ω nd ethods nd it is followed very well y the τ ethod. 6 Conclusions In this pper we derive new pekshpe ethods for evluting the ctivtion energy of TL glow-pek which re sed on the physiclly eningful ixed order kinetics odel. In their originl for given y qs. (8) () (5) the pek shpe equtions re not very useful for nlyzing experientl glow-peks. However this for should e very useful in testing the qulity of glow-peks generted during the nuericl siultion of vrious TL phenoen where ll relevnt preters cn e evluted during the siultion. The reson is tht s it ws shown y Sunt et l. [3] the ixed order kinetics odel is uch ore successful thn the generl order kinetics odel in descriing glow-peks resulting fro physicl odels like NMTS nd IMTS. The pek shpe ethods derived here cn e used for fst nd ccurte pek qulity test within the siultion. The pek shpe equtions given y qs. (8) (30) re expressed s function of the integrl syetry fctor µ g. However in this for these equtions re of liited prcticl use since they contin the tringle ssuption pseudoconstnts C ω C nd C τ. Nevertheless they cn e pplied to experientl glow-pek y using the tulted vlues of C ω C nd C τ s function of µ g fro Tles nd. Finlly the fors given y qs. (3) (3) re useful in prctice since they re expressed s function of the geoetricl syetry fctor. The error in the ctivtion energy is ±.5% when the slight dependence of the coefficients in qs. (3) (3) is tken into ccount (Fig. 7()) nd ±3.5% when it is not (Fig. 7()). It hs een found tht the ter ix = (F )/F of ixed order kinetics is very siilr to the generl order kinetics preter s shown in Fig. 9( ). Fro this siilrity it ws found tht the physiclly eningful ixed order kinetics differs fro the epiricl generl order kinetics preter y less thn 5% in the worst cse s is shown in Fig. 9(c). Finlly the ppliction of the ixed-order pek shpe ethod to generl-order kinetics glow-pek nd the ppliction of the generl-order kinetics glow-peks on ixed order kinetics glow-peks gve errors (see Figs. 0 ) which follow qulittively nd quntittively their ehviour shown in Fig. 9(c). References [] J. T. Rndll nd M. H. F. Wilkins Proc. R. Soc. Lond. A (945). [] G. F. J. Grlick nd A. F. Gison Proc. Phys. Soc. Lond (948). [3] C.. My nd J. A. Prtridge J. Che. Phys (964). [4] R. Chen N. Kristinpoller Z. Dvidson nd R. J. Visoceks Luin (98). [5] L. I. Grossweiner J. Appl. Phys (953). [6] C. B. Lushchik Dokl. Akd. Nuk SSSR 0 64 (955). [7] A. Hlperin nd A. A. Brner Phys. Rev (960). [8] R. Chen nd Y. Kirsh Anlysis of therlly stiulted processes (Pergon Press 98) p. 67. [9] R. Chen nd S. W. S. McKeever Theory of Theroluinescence nd relted phenoen (World Scientific) 997. [0] R. Chen J. Appl. Phys (969). [] R. Chen J. lectroche. Soc (969). [] G. Kitis nd V. Pgonis Nucl. Instru. Methods B 6 33 (007). [3] C. M Sunt W.. F Ayt J. F. D Chuci nd S. Wtne Rdit. Mes (00) WILY-VCH Verlg GH & Co. KGA Weinhei