Nuclear Instruents and Methods in Physics Research B 262 (27) 33 322 NIM B Bea Interactions with Materials & Atos www.elsevier.co/locate/ni Peak shape ethods for general order theroluinescence glow-peaks: A reappraisal G. Kitis a, *, V. Pagonis a Aristotle University of Thessaloniki, Nuclear Physics Laoratory, 5424 Thessaloniki, Greece Physics Departent, McDaniel College, Westinster, MD 257, United States Received 9 May 27; received in revised for 7 May 27 Availale online 9 June 27 Astract This paper presents a reappraisal of the well known peak shape expressions for calculating the activation energy in a theroluinescence (TL) glow-peak. This study leads to new insights as to the eaning of the coefficients used in the original equations. The reappraisal leads to new equations for the coefficients of the peak shape expressions which contain the general order paraeter, instead of the experientally deterined geoetrical shape factor which is used in the original equations. Previously only the coefficients for first and second order kinetics were deterined on the asis of existing theory and the coefficients for interediate kinetics order were deterined epirically using a linear interpolation extrapolation ethod. In the present work the iproved peak shape coefficients are evaluated in analytical for as a function of the kinetic order, y using the general order kinetics expression for the TL intensity. The intrinsic errors in the newly derived expressions for are evaluated and their relevance to experiental work is discussed in detail. A ethod for a further iproveent of the accuracy of the peak shape ethods is suggested. Ó 27 lsevier B.V. All rights reserved. PACS: 78.6.Kn; 29.4.Wk; 6.82.Pv; 87.66.Sq Keywords: Theroluinescence; Peak shape ethods; Activation energy; General order kinetics; Triangle assuption. Introduction The shape of a theroluinescence (TL) glow-peak is the asis of iportant and convenient ethods for calculating the trapping paraeters of distinct energy levels within the crystal. These ethods are ased on easureents of a few points on the glow-peak, shown in Fig.. arly work on the developent of such expressions concentrated on the developent of convenient expressions for calculating the activation energy of trapping levels [ 3]. The ter peak shape ethods is reserved in the TL literature for such ethods, although there are other ethods for finding which are also ased on the glow-peak * Corresponding author. Tel./fax: +3 23 99875. -ail address: gkitis@auth.gr (G. Kitis). shape (i.e. curve fitting ethods) [4,5]. The seinal work y [6] is a reference point in the derivation of the peak shape ethods. [6] suarized all pre-existing ethods and gave a detailed ethodology for deriving the coefficients of the expressions for first and second order kinetics only. The coefficients used in the peak shape ethods for the interediate kinetic orders were also evaluated y [7] y (i) evaluating the coefficients for first order kinetics, (ii) calculating the coefficients for second order kinetics and (iii) using a linear interpolation extrapolation ethod to otain expressions for the interediate kinetic orders as a function of the syetry factor l g, which was found to e etween.42 and.52 for first and second order kinetics respectively. The expressions derived y have een used extensively over the last 35 years. Although the interediate 68-583X/$ - see front atter Ó 27 lsevier B.V. All rights reserved. doi:.6/j.ni.27.5.27
34 G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 TL Intensity, I..8.6.4.2. 32 36 4 44 Teperature ( K ) kinetics order expressions are in fact approxiations, they proved to e very accurate in practice. However, their success has not een explained on a theoretical asis efore. Moreover, the characteristic paraeter of the kinetic order is copletely asent fro oth the derivation ethod and the final expressions for. The ai of the present work is (i) to provide a theoretical foundation to the peak shape ethods coefficients as a function of the kinetics order, (ii) ased on this theoretical foundation, to explain why the original expressions for interediate kinetic orders are so accurate, (iii) to evaluate the intrinsic errors in the newly derived expressions for and discuss their relevance to experiental work and finally (iv) to suggest a ethod for further iproveent of the accuracy of the expressions... Analytical TL expressions The general order kinetics equation is [4,5,8]: IðT Þ¼ dn dt ¼ s n ; ðþ where n (c 3 ) is the concentration of electrons in traps, (ev) is the activation energy, the kinetic order, not necessarily or 2, s ( 3() s ) the pre-exponential factor and T (K) the teperature. It is noted that this is an epirical equation which has een found useful in representing adequately experiental TL glow curves in various aterials. The solution of q. () for 5 and a linear heating rate is [5]: I ¼ n s þ ð Þs n T dt Z T The condition for axiu TL intensity is found y equating the derivative of q. (2) to zero to otain T T T 2 /2 μ g = δ/ω Fig.. The characteristics points on a TL glow-peak, which define the peak-shape paraeters. τ ω δ : ð2þ þ ð Þs n ¼ s n Z T T dt ; ð3þ where T is the teperature at glow-peak axiu intensity. The integral on the left-hand side of q. (3) can e approxiated using a certain nuer of ters of the asyptotic series: Z T X T dt ffi T exp n¼ n ð Þ n n! Usually q. (4) is a very good nuerical approxiation, and only the first two ters of the series need to e taken into account, so that q. (3) yields ¼ n s ð þð ÞD Þ; ð4þ ð5þ where T is the teperature at glow-peak axiu intensity, and D =2 /. 2. Geoetrical characteristics of a single glow-peak The peak shape ethods are ased on certain characteristics of a single glow-peak, shown in Fig., the peak axiu teperature T and the teperatures at half axiu TL intensity T and T 2 at the low and high teperature side of the glow-peak respectively. These quantities are used to define further the widths x = T T 2, d = T 2 T and s = T T as well as the syetry factor of the glow-peak l g = d/x. The derivation of the existing peak shape ethods are ased on the so called triangle assuption, which can e expressed in three different ways, each one leading to an individual faily of peak shape ethods. In the for given y [5,6] these are x ¼ C x ; n d ¼ C d ; n s ðn n Þ ¼ C s; with Z n ¼ I dt; ð9þ t where is the peak axiu intensity, n the high teperature half integral of the glow-peak, t the tie at peak axiu intensity and C x, C d and C s are quantities which characterize the kinetic order. These quantities were found to vary extreely slowly for a given kinetic order and for a very wide range of kinetic paraeters (,s) and are called pseudo-constants. These pseudo-constants vary little for all ð6þ ð7þ ð8þ
G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 35 glow-peaks derived using activation energies in the region..6 ev and frequency factors in the region etween 5 and 3 s [6]. Their values can e estiated as a function of the kinetic order y producing synthetic glow-peaks using q. (2). In order to derive the peak shape forulae using qs. (6) (8), one has to evaluate the ters /n, n /n and /n fro the analytical TL expressions. 3. Derivation of /n, n /n and /n 3.. Derivation of /n Fro q. (2) considering I(T )= we otain ¼ n s þ ð Z Þs n T n T dt : ðþ Using the exact condition for the TL peak axiu given y q. (3), we have after soe algera: I h i ¼ n s e n : ðþ Using the approxiate condition for the TL peak axiu given y q. (5), yields: " # ¼ 2 ; ð2þ n þðþd and after soe algera one otains ¼ 2 ð3þ n þðþd and finally,! ¼ : ð4þ n þðþd 3.2. Derivation of n /n and /n Solving q. () for n we have Z n Z dn T n n ¼ s T dt : Fro q. (5) after soe algera we otain n ¼ n þ ð Z Þs n T dt T ð5þ : ð6þ Using the exact condition for the axiu given y q. (3) one otains for the ter n /n : n n ¼ s n 2 exp : ð7þ Using the approxiate condition for the axiu given y q. (5) one otains n ¼ : ð8þ n þðþd According to Halperin and Braner [3] the quantity n / n, which represents the ratio of the high teperature half integral of a glow-peak over its total integral is the geoetrical syetry factor, let us say l g of a glow-peak, which differs very slightly fro the coonly used syetry factor l g defined aove. According to Halperin and Braner [3] glow-peaks with a syetry factor around ( + D )/e are of first order and glow-peaks with syetry factor ( + D )/2 (see q. (8), for = 2) are of second order. Oviously the syetry factor of interediate kinetic orders can e otained fro q. (8) for given values of and D. For a given order of kinetics, the syetry factor is not a fixed nuer ut rather a very slowing varying quantity with the paraeters (, s), i.e. a pseudo-constant. Historically, the notation l g was used for the syetry factor n /n and not for the ratio d/x. However, in this paper we use the coonly used notation of l g = d/x. By dividing q. (4) over q. (8) one otains n ¼ : ð9þ 4. Methods ased on the total width (x) q. (6) can e re-written as ¼ C x n x : Using q. (4) we otain ¼ n þðþd ð2þ! ; ð2þ and finally coining qs. (2) and (2) we have 2 ¼ C x þðþd x : ð22þ Since ( )D is less than unity, the following approxiation, which introduces an error of less than %, can e used ð þðþd Þ ffi þ D ¼ þ D ; ð23þ ¼ C x þ D x ; ð24þ or ð þ D Þ¼C x x ; ð25þ
36 G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 Tale The nuerical values as a function of the kinetic order of the part f a () of the coefficients in peak shape ethods expressions given y qs. (26), (28) and (34) f x ()= /() f d () = f s () =( /() ) s ðþ ¼ =ðþ =ðþ. 2.72..778.582. 2.853..753.6275.2 2.986.2.786.679.3 3.7.3.87.754.4 3.2467.4.847.758.5 3.375.5.875.8.6 3.52.6.92.842.7 3.628.7.928.887.8 3.7528.8.953.928.9 3.8768.9.97.96 2. 4. 2. 2. 2. and finally y solving for, 2 ¼ C x x 2 : ð26þ The last equation shows that the coefficient of the ter =x consists of two ters, C x and f x () = /(). The quantity C x is the respective triangle assuption pseudoconstant which has to e evaluated y siulation, whereas the values of f x () depend on the kinetic order and are listed in Tale. This equation is of the sae for as the well-known expression, ut it contains explicitly the kinetic order and is derived here fro analytical expressions, while previously it was otained using a linear interpolation extrapolation technique. 5. Methods ased on the high teperature half-width (d) q. (7) can e re-written as ¼ C d n d and y taking into account q. (9) one otains ð27þ " # s ¼ C s þðþd s : ð3þ Taking into account qs. (23), (3) ecoes "! # s ¼ C s þ D s : ð3þ By using the approxiation elow, which introduces an error of less than %. =ðþ ffi =ðþ ð D Þ: ð32þ þ D q. (3) after soe anipulation gives: s ¼ C s 2 s Cs s 2 : ð33þ s The first ter in this equation is the doinant ter in deterining the value of s, while the second ter represents a sall correction. If the s appearing in the second ter of the right hand side of q. (33) is replaced y the first ter of the right hand side of q. (33), then with a loss of less than % in accuracy, q. (33) can e written as! s ¼ C s s =ðþ ð2 =ðþ Þ: ð34þ The first coefficient in the last equation consists of two ters, C s and f s ()=( /() ). The ter C s is the respective triangle assuption pseudo-constant which has to e evaluated y siulation, whereas the values of f s () are fixed nuers for a given. The evaluated values of f s () are listed in Tale. Siilarly the second coefficient s () =( /() )/( /() ) is a fixed nuer for given kinetic order and its values are also listed in Tale. 7. Nuerical siulation ¼ C d d : ð28þ The coefficient of the ter =d consists of two ters, C d and f d ()=. The ter C d is the respective triangle assuption pseudo-constant which has to e evaluated y siulation, whereas the values of c d () are fixed nuers for a given. The evaluated values of f d () are listed in Tale. 6. Methods ased on the low teperature half-width (s) q. (8) can e re-written as n ¼ s : ð29þ n C s n Replacing the ters /n and n /n fro qs. (9) and (8) respectively, one otains after soe algera The nuerical siulation of synthetic glow-peaks was perfored y using very road regions of the trapping paraeters, in order to cover as any practical cases as possile. For the sake of siplicity the Rasheedy version [9] of the general order equation is used in which n = N, one can use directly the value of s in s. The activation energy region was varied etween.7 ev and 2 ev in steps of. ev (4 values). The frequency factor was inserted in the for s = exp(a) with A ranging etween 6 and 46 in steps of (3 values), covering a region etween 7 and 2 s. Fro all possile glow-peaks corresponding to the (,s) pairs in these wide regions of (,s), only those having their glow-peak axiu teperature in the practical range 2 8 K were considered. The kinetic order was varied fro. to 2 in steps of.. The general order expression given y q. (2), does not hold for =. However, it holds for values of which are very
G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 37 close to. It was found that for =.5 the general order expression given y q. (2) approxiates the TL intensity for first order kinetics accurately, up to the fifth significant digit []. Therefore the value of =.5 was chosen to represent the case of first order kinetics through the general order kinetics expression. In total, 42 synthetic glow-peaks were produced for each one of the kinetic orders. The ean values evaluated and listed in susequent sections are the average of ore than 4 values. In order to ensure an accurate deterination of the various paraeters, a very sall teperature step is necessary. In the present siulation the TL intensity was evaluated every.5 K; this corresponds to.2 5 points eing evaluated for each glow-peak. 7.. valuation of C x,c d and C s Fro qs. (26), (28) and (34) it is seen that in order to evaluate the coefficients in front of the peak shape ethods, the values of the pseudo-constants C x, C d and C s ust e found. For the siulation of synthetic glow-peaks the exponential integral appearing in q. (2) was not approxiated y the usual asyptotic series, since the software used contains this integral as a uilt-in function. The values of the pseudo-constants C x, C d otained fro the siulation are listed in Tale 2, whereas Fig. 2 shows their ehavior as a function of the syetry factor. The presentation of these pseudo-constants as a function of the syetry factor is preferale to using the kinetic order, since the syetry factor is an experientally easured quantity. During the siulations descried in this paper it is possile to perfor two tests. The first test concerns qs. (22), (28) and (3). This test evaluates the output values of activation energy using the values of the pseudo-constants C x, C d, C s and T evaluated within the sae siulation. The ai here is to test the accuracy of the aove equations. The results showed that the output activation energies values coincide with the input values, ensuring the correctness of these equations. The second test concerns qs. (22) and (3). For the derivation of these quations two approxiations were used, given y qs. (23) and (32). The second test, therefore, evaluates output values of the activation energies using the individual values of the pseudo-constants C x and C s evaluated during the siulation, aiing to estiate the influence of these approxiations. The agreeent etween the input and output values is estiated y the error in out in C ω, C τ, C δ.97.92.87.82.4.42.44.46.48.5.52 Syetry factor, μ g, which is shown in Fig. 3. The left-hand side figure concerns the x expression descried y q. (22) and the right-hand side of the figure concerns the s expression fro q. (3). Note that each one of the figures contains 32 (,s) pairs. As it is seen fro the figures, in the ajority of cases the errors are less than.5% for the case of x and less that.4% for the case of s. These sall percent errors represent the influence of the approxiations given y qs. (23) and (32). 7.2. valuation of c x,c d,c s and s C Cτ ω C δ Fig. 2. The triangle assuption pseudo-constants C x, C d and C s as a function of the syetry factor. The general for of the peak shape ethods given y [7,5] is a ¼ c a a a ð2 Þ; ð35þ where the index a stands for x, d and s. Tale 2 The values of the triangle assuption pseudo-constants C x, C d and C s resulting as the ean values of all (,s) pairs fro the siulations along with the ean values of syetry factor, l g, as a function of the kinetic order l g C x C d C s..485 ±.34.934 ±.29.986 ±..876 ±.47..4339 ±.36.95 ±.25.9655 ±..8745 ±.45.2.444 ±.36.99 ±.2.952 ±.8.878 ±.44.3.4556 ±.37.96 ±.7.9374 ±.2.883 ±.4.4.4666 ±.37.928 ±.4.925 ±.24.8842 ±.4.5.476 ±.38.8992 ±.9.933 ±.28.887 ±.4.6.4854 ±.37.8955 ±.6.922 ±.3.8894 ±.39.7.4939 ±.4.896 ±.3.896 ±.3.898 ±.38.8.59 ±.37.8877 ±.2.888 ±.35.8937 ±.36.9.594 ±.37.8837 ±.5.8724 ±.38.8957 ±.35 2..567 ±.3.8796 ±.8.8635 ±.4.8946 ±.33
38 G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322.2.9 ( - ω ) /.8.4 ( - τ ) /.6.3...2.6 2. Kinetic Order,.2.6 2. Kinetic Order, Fig. 3. The error x;s of activation energies x and s as a function of the kinetic order. represents the input values whereas the x,s values were evaluated through qs. (22) and (3), respectively. [6,7] evaluated the coefficients c a and a for first and second order kinetics and used an interpolation extrapolation ethod to evaluate the values for interediate order kinetics. His final expressions are c x ¼ 2:52 þ :2ðl g :42Þ x ¼ ; c d ¼ :976 þ 7:3ðl g :42Þ d ¼ ; c s ¼ :5 þ 3ðl g :42Þ s ¼ :58 þ 4:2ðl g :42Þ: ð36þ ð37þ ð38þ C ω 3.4 3.2 3. 2.8 Using the values of the pseudo-constants C x, C d and C s, which were evaluated in the siulation and are listed in Tale 2, the new values for the coefficients c a and a are evaluated through qs. (26), (28) and (34). Their values as a function of kinetic order are listed in Tale 3. The new values of c a and a are copared with the original expressions of in Figs. 4 7. In these figures it is seen that the new values of the coefficients evaluated in the present work are slightly non-linear as a function of the syetry factor. On the other hand the values of the coefficients derived y lie on a straight line which is very close to the non-linearity otained in the present work. The differences are due to the interpolation extrapolation ethod used during their evaluation. The agreeent in Tale 3 The net values of the peak shape ethods coefficients as a function of the kinetic order c x c d c s s. 2.4845 ±.79.986 ±..4955 ±.8.582. 2.66 ±.7.62 ±..533 ±.79.6275.2 2.746 ±.6.44 ±.22.5683 ±.79.679.3 2.8244 ±.53.286 ±.26.63 ±.73.754.4 2.93 ±.45.295 ±.34.6329 ±.74.758.5 3.348 ±.3.3699 ±.42.663 ±.75.8.6 3.36 ±.2.4435 ±.48.696 ±.74.842.7 3.2347 ±.9.557 ±.5.794 ±.73.887.8 3.334 ±.6.5872 ±.63.7453 ±.7.928.9 3.4259 ±.9.658 ±.72.7645 ±.69.96 2. 3.584 ±.32.7387 ±.8.7952 ±.66 2. 2.6 2.4.4.42.44.46.48.5.52 Syetry factor, μ g Fig. 4. The net values of the peak shape ethods coefficients c x of the x ased ethod, as they were evaluated y and in the present work, as a function of the syetry factor. C δ.8.6.4.2..8.4.42.44.46.48.5.52 Syetry factor, μ g Fig. 5. The net values of the peak shape ethods coefficients c d of the d ased ethod, as they were evaluated y and in the present work, as a function of the syetry factor.
τ G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 39.8 2 C τ.7.6.5 ( - ω ) / - -2.4.4.42.44.46.48.5.52 Syetry factor, μ g -3..2.4.6.8 2. Fig. 6. The net values of the peak shape ethods coefficients c s of the s ased ethod, as they were evaluated y and in the present work, as a function of the syetry factor. 2..9.8.7.6.5.4.42.44.46.48.5.52 Syetry factor, μ g Kinetic Order, Fig. 8. The error x of the activation energies x as a function of the kinetic order. are the input values, whereas the x values were evaluated through q. (35) using the coefficients and the newly otained coefficients, which are listed in Tale 3. The horizontal line passes syetrically through the results of the present work. At each kinetic order the errors values of 37 siulated (, s) pairs are plotted. the present study follows the open curve as a function of the syetry factor and give results passing through the average line, whereas the coefficients are a strictly linear function of the syetry factor and follow a paraolic shape. Fig. 9 shows the results concerning the d ethod. The ehavior is very siilar to the previous case. Fig. shows the results concerning the s ethod. The horizontal lines at.5% to.5% define a region where the two sets of coefficients give the sae accuracy. Looking at Fig. 6 one can see that oth sets of coefficients follow a Fig. 7. The net values of the peak shape ethods coefficients s of the s ased ethod, as they were evaluated y and in the present work, as a function of the syetry factor. these figures explains why the expressions developed y give very accurate values for the trapping paraeters. 7.3. Coparison of the newly otained expressions with those of The application of the general for q. (35) is now possile using the c a, a values shown in Tale 3 for each kinetic order. Under this situation a coparison is possile of the resulting values fro q. (35) using oth the newly otained values of c a, a and those of derived fro qs. (36) (38) using the syetry factors of Tale 2. The coparison is perfored y siulating synthetic glow-peaks using the sae (, s) regions descried aove. Fig. 8 shows the results concerning the x ethod. The horizontal line passes through an average value of the results in the present study. In order to understand this ehavior one has to look at Fig. 4. The coefficient c x of ( - δ ) / 2-2 -4..2.4.6.8 2. Kinetic Order, Fig. 9. The error d of activation energies d as a function of the kinetic order. The are the input values, whereas the d values were evaluated through q. (35) using the coefficients and the newly otained coefficients, which are listed in Tale 3. The horizontal line passes syetrically through the results of the present work. At each kinetic order the errors values of 37 siulated (, s) pairs are plotted.
32 G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 2 h Dx ¼ ðdt 2 Þ 2 þð DT Þ 2 i 2 ; ð4þ Dd ¼ ½ðDT 2 Þ 2 þð DT Þ 2 Š 2 ; ð42þ ( - τ ) / - -2 parallel ehavior, which is reflected exactly in the ehavior of Fig.. The reason is that except in the central coon region, the accuracy of the coefficients is elow the dearkation line of.5% and the accuracy of the newly derived coefficients is aove the dearkation line of +.5%. 7.4. Accuracy of the peak shape ethods The peak shape expressions found in this paper estiate the activation energy with an accuracy etter than.5% in the ajority of (,s) pairs. However, this accuracy is otained using a siulation where the teperature is evaluated with an accuracy of.5 K. The situation, however, is very different in experiental situations, where the experiental accuracy in the teperature easureent ay e larger than K. The experiental quantities which have to e experientally easured in order to evaluate the activation energy are T, T, and T 2. The errors of these quantities are propagated to the errors of through q. (35). The total proale error of a function depended upon x i quantities, U = f(x i )is[]: " DU ¼ #f 2 dx þ #f 2 dx 2 þþ #f # 2 2 dx n #x #x 2 #x : n Then the relative error will e: r ¼ DU U :..2.4.6.8 2. Kinetic Order, Fig.. The error s of activation energies s as a function of the kinetic order. are the input values, whereas the s values were evaluated through q. (35) using the coefficients and the newly otained coefficients, which are listed in Tale 3. The horizontal lines pass through the ±.5% errors containing values of oth sets of coefficients. At each kinetic order the errors values of 37 siulated (, s) pairs are plotted. ð39þ ð4þ Applying the error propagation forula, q. (39), the total errors for x, d, s and l g will e, respectively, Ds ¼ ½ðDT Þ 2 þð DT Þ 2 Š 2 ; ð43þ " Dl g ¼ d 2 Dd þ Dx # 2 2 x d x : ð44þ On the other hand the total error of a derived y applying q. (39) to q. (35) is, " D a ¼ a ð a D Þ Dc 2 a þ DT 2 c a 2T # þ ð 2 a D Þ Da 2 2 þ ðd D a Þ 2 ; ð45þ a where D =2 / a. The coefficient paraeters c a and a are included, since they are coposite paraeters consisting of fixed and dependent quantities. The coefficients c a in q. (35) consist of two ters, the ter C a fro the triangle approxiation, whose error is estiated fro the siulation and of the ter f a (), which is a function of the kinetic order. The error of c a is therefore Dc a = f a ()D- C a. The values of DC a for x, d and s are listed in Tale 2.In the case of the siulations the error D a is zero, since a depends only on and therefore it is a fixed nuer for a given order of kinetics. However, this ter is included ecause in s s ethod the s is expressed as function of the syetry factor, which introduces an additional source of error. It is usually said [5] that the s ased ethods yield ore accurate results than the other peak shape ethods. By exaining q. (45), a first estiate aout the accuracy of each of the x, d and s ethods can e otained. The difference in the accuracy of the 3 peak shape ethods coes fro the first and third ters of q. (45). The ost accurate ethod should e the one for which these two ters are iniized. In the case of the d ethod, which does not contain the ter a,( a = ), these ters have their largest value and therefore this ethod is expected to e less accurate than the other two expressions. On the other hand a = for the x ethod and greater than for the s ethod. Therefore, the two ters are iniized for the s ethod which is expected to e the ost accurate. Suarizing, the relative accuracy of the three ethods is as follows: D s < D x < D d : ð46þ However, this is an ideal case and holds only when, additionally, Ds/s < Dx/x < Dd/d. Furtherore, the experiental glow-peaks are not usually clean peaks. In ost cases satellites ay e present, which can easily e therally pre-cleaned at the lower part of the peak and not at the high teperature side. The order of accuracies under these circustances is inverted.
G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 32 During practical applications the coefficients of the peak shape ethods are considered to e constants which do not affect the error evaluation of the activation energy. However, as was shown in the present work, these coefficients are not constants ut rather pseudo-constants which carry their own intrinsic errors. Therefore, any faily of peak shape ethods will possess an intrinsic error coing fro the accuracy of its coefficients and this error is in addition to any errors deriving fro the experiental accuracy of the variales T, T and T 2 appearing in these expressions. In order to explore this intrinsic error let us assue that the errors in T, x, s and d, i.e. the quantities DT and Da are zero. In this case q. (45) gives: D a a ¼ð a D Þ Dc a c a : ð47þ The relative error in q. (47) is an intrinsic iniu liit which depends upon the error in the triangle assuption pseudo-constants C a, (i.e. C x, C d and C s ), the values of which are evaluated fro the siulation with qs. (6) (8). It ust also e noted that this liit holds also in the case where the coefficients c a are expressed as a function of l g. The reason is that the syetry factor for a given order of kinetics is not a fixed nuer ut rather is also a pseudo-constant as seen fro q. (8). The general ehavior of the error in q. (45) is siulated for D x, D d and D s. In order to siplify the siulation it is assued that: (i) DT = DT 2 = DT and (ii) the error DC a of all C a is taken equal to a typical value of.4% otained during the siulations. The results are shown in Fig.. The straight lines are the ovious results otained when the first ter of q. (45) (or q. (47)), which represents the liit discussed aove, is oitted y setting DC a =. However, taking into account the first ter of q. (45) with a typical value of DC a =.4%, then Fig. shows that the error in can not e less than.35.4%, up to DT =. K. ( Δ / ).. Intrinsic liit δ τ ω.. ΔT ( K ) Fig.. The straight lines correspond to the errors otained fro q. (45) when the first ter, which represents the intrinsic error liit (also q. (47)), is oitted, i.e. DC a =. The arrow shows the intrinsic error liit of.35.4%, otained, when the first ter of q. (45) is taken into account considering DC a =.4%. δ Another oservation is that the accuracy of x ethod is etter than that of the s ethod aove. K. This can e understood fro the first assuption DT = DT 2 = DT stated aove. Since x is alost doule the value of s, the value of Dx/x is uch lower than the value of Ds/s and therefore the error in x will e lower. If one wants to evaluate the error in when using the coefficients, one ust take into account that in this case the ters c a, a are expressed as a function of l g,as seen in qs. (36) (38). By applying the error propagation forula given y q. (39) it is found that Dc x =.2 Dl g, Dc d =7.3Dl g, Dc s =3Dl g and D s = 4.2 (2 )Dl g. The ost practical result, however, which eerges fro the aove error analysis is that the errors in increase linearly with DT. Therefore, the experiental error in teperature ust e lowered as uch as possile. In practice it is rather easy to achieve a very low error in T, although this is not usually done during experiental work. First of all one has to achieve a very good theral contact etween the saple and the heating eleent and to use a low rate of heating, not greater than K/s, in order to avoid possile teperature lag effects. The crucial point to consider is the sapling tie in photon counting systes, which ust e set as short as possile. For exaple in the RisøTL/OSL reader having an Analog to Digital Converter (ADC) of 2 3 channels, a TL readout up to 4 C with K/s gives a teperature increent, DT of.45 K. As was discussed aove, the intrinsic error liit of.35.4% in is set y the accuracy of the triangle assuption pseudo-constants C x, C d and C s which are evaluated y the siulation. The intrinsic error liit under discussion shown in Fig. is evaluated y considering an average relative error of.4%. Although this error is very low when one considers the very road (,s) regions used, it can e lowered even ore as follows. Let us assue for exaple that one applies one of the peak shape ethods using the newly otained coefficients and evaluates the trapping paraeters ± D and s ± Ds. ven if one has extreely low errors in T, one can not achieve accuracy etter than the intrinsic error liit of.35.4% evaluated aove. It is very easy, however, to decrease this error liit y re-evaluating the triangle assuption pseudo-constants in an (,s) region restricted within the liits ± D, s ± Ds otained. In fact, since the triangle assuption pseudo-constants vary extreely slowly, a single siulation using the experientally otained and s values, is enough to otain a new highly iproved pseudo-constant value C a. Then using q. (47) one can find that the intrinsic error liit of.35 4% could e decreased y at least a factor of five, with a corresponding increase in the accuracy of. The aove error evaluation approach is necessary if one wishes to gain all the enefits of the peak shape ethods when they are applied to experiental results. However, there is another field of research where the aove approach should e very useful. This is the field of siulations involving TL odels and TL effects (i.e. super-linearity,
322 G. Kitis, V. Pagonis / Nucl. Instr. and Meth. in Phys. Res. B 262 (27) 33 322 pre-dose effect, etc.) y the nuerical solution of the differential equations governing the TL process [5,]. During such siulations the quality of the resulting TL glow-peak can e continuously tested using the peak shape ethods. It ust e noticed, however, that the derivation of the peak shape ethods is ased on the Randall Wilkins [2] and Garlick Gison [3] kinetic odels, for which the quasiequiliriu condition is satisfied. In the case of general order kinetics of May and Partrige [8] the respective equation can not e derived fro the original set of siultaneous equations descriing the TL process. However, since the general order equation coincides with second order for = 2 and approxiates first order for.5, it is assued that it is an equation for which the quasi-equiliriu condition is satisfied too. Therefore, these odels are expected to descrie accurately only those siulated glow-peaks for which the quasi-equiliriu condition is satisfied. In cases deviating fro the quasi-equiliriu conditions, one ight expect that the output activation energy values evaluated y the peak shape ethods will deviate fro the input values used in the odels, in a anner depending upon the degree of satisfaction of the quasi-equiliriu conditions. The present work indicates another very effective way of testing glow-peaks resulting fro the nuerical solution of the differential equations, y evaluating the equivalent analytical expressions and aking a direct glow-peak coparison. This can e done as follows. Once the glow-peak is derived during the siulations, its syetry factor n /n is evaluated as the ratio of the high teperature half glow-peak over the total integral of the glow-peak. Then using q. (8) the corresponding kinetic order is iteratively evaluated, to an accuracy depending upon the nuer of ters in the asyptotic series and the analytical glow-peak is evaluated using q. (2). A final reason why the aove suggested approach is ideal for coparing glow-peaks derived fro the analytical TL expressions with those derived fro the nuerical solution of the differential equations, is that in the siulations one can control the teperature to any desired accuracy. 8. Conclusions The peak shape ethods for general order kinetics are reappraised in detail. The peak shape ethod coefficients are given a theoretical foundation on the asis of analytical expressions as a function of the kinetic order. Furtherore the very good accuracy of the approxiate coefficients is easily understood y inspection of Figs. 4 7. A nuerical siulation of synthetic glow-peaks was perfored in an activation energy region etween.7 ev and 2 ev in steps of. ev (4 values), a frequency factor region etween 7 and 2 s (3 values) and kinetics order etween and 2 ( values). The newly derived peak shape ethod coefficients contain the general order paraeter, instead of the experientally deterined geoetrical shape factor which is used in the original equations. The errors of the peak shape ethods are studied in detail and an error evaluation procedure is descried, which allows one to gain all the enefits of the peak shape ethods when they are applied to experiental results. A ethod for further iproveent of the accuracy of the evaluation of the activation energy is also suggested. References [] L.I. Grosswiener, J. Appl. Phys. 24 (953) 36. [2] C.B. Luschik, Dokl. Akad. Nauk, S.S.S.R (955) 64. [3] A. Halperin, A.A. Braner, Phys. Rev. 7 (96) 48. [4] R., Y. Kirsh, Analysis of Therally Stiulated Processes, Pergaon Press, 98, p. 67. [5] R., S.W.S. McKeever, Theory of Theroluinescence and Related Phenoena, World Scientific, 997. [6] R., J. Appl. Phys. 4 (969) 57. [7] R., J. lectroche. Soc. 6 (97) 254. [8] C.. May, J.A. Partrige, J. Che. Phys. 4 (964) 4. [9] M.S. Rasheedy, J. Phys.: Condens. Matter 4 (993) 633. [] V. Pagonis, G. Kitis, C. Furetta, Nuerical and Practical xercises in Theroluinescnce, Springer, 26, p. 8. [] P.R. Bevington, Data Reduction and rror Analysis for the Physical Sciences, McGraw Hill, 969, p. 5. [2] J.T. Randall, M.H.F. Wilkins, Proc. Phys. Soc. London 84 (945) 39. [3] G.F.J. Garlick, A.F. Gison, Proc. Phys. Soc. London A 6 (948) 574.