On the Pre-Exponential Factor Comparing in Thermoluminescence (TL) Theory

Transcription

1 Open Acce Library Journal On the Pre-xponential Factor Comparing in hermoluminecence (L) heory ugenio Chiaravalle 1, ichele angiacotti 1, Claudio Furetta 2, Giuliana archeani 1, ichele omaiulo 1 1 Centro di Referenza Nazionale per la Ricerca della Radioattività nel Settore Zootecnico-Veterinario, Itituto Zooprofilattico Sperimentale della Puglia e della Bailicata, Foggia, Italy 2 Independent Reearcher, Rome, Italy mail: c.furetta@alice.it Received 3 ay 214; revied 13 June 214; accepted 25 July 214 Copyright 214 by author and OALib. hi work i licened under the Creative Common Attribution International Licene (CC BY). Abtract he aim of the work ha two folder: the firt i to apply numerical and approximated method to olve the integral comparing in the thermoluminecence theory, i.e. exp d. he econd k point i to compare the influence of the two different way of calculation on the value of the preexponential factor. Keyword hermo Subject Area: Applied Phyic, Condened State Phyic 1. Introduction he thermoluminecent intenity, I, a a function of temperature, i obtained developing the rate-equation decribing the different L kinetic procee, i.e. the Randal-Wilkin or firt order kinetic [1], the Garlick- Gibon model related to the econd order [2] and the general order kinetic introduced by ay-partridge [3]. he equation decribing the different procee are the following: - for firt order I( ) = n exp exp exp d k β k (1) - for econd order How to cite thi paper: Chiaravalle,., angiacotti,., Furetta, C., archeani, G. and omaiulo,. (214) On the Pre- xponential Factor Comparing in hermoluminecence (L) heory. Open Acce Library Journal, 1: e618.

2 2 n I( ) = ( n ) exp 1 + exp d k β k (2) where = N. - for general order, 1< b < 2, the L intenity i given by ( b 1) I( ) = n exp 1 + exp d k β k (3) ( b 1) where = n. In the previou equation the meaning of the ymbol i the following: - n (cm 3 ) i the numbert of electron contained in the electron trap at temperature, - N i the trap concentration, - (ev) i the energy depth related to the trap below the bottom of the conduction band, - k (ev K ) i the Boltzmann contant, - β i the linear heating rate d d. t he quantity, which appear in the cae of Randal-Wilkin equation, i called frequency factor and it ha a precie phyical meaning: it hould repreent the number of time for econd that a bound electron interact with the lattice phonon, multiplied a tranition probability factor. In the cae of Garlick-Gibon econd order model and ay-partridge general order, the quantitie and are called pre-exponential factor. All equation contain the integral F(, ) = exp d k (4) which cannot be olved analytically. he aim of thi paper ha two folder. he firt one i to compare the olution of (4) obtained by the approximation method and by it numerical olution. he econd apect i to invetigate the influence of the two different way of calculation on the value of the pre-exponential factor. 2. Integral Approximation A method which i uually followed for evaluating the value of the integral i by integration in part, when the lower limit of integration i intead of. So, a good approximation i provided by the aymptotic erie he value of (4) i then given by n k n (5) k k n= 1 ( ) = = ( ) 2 b b 1 F, exp d exp 1 n! exp d = F, F, k ( ) ( ) F, conidering =, i negligible F, the right hand ide of quation (5) can be conidered to repreent the integral value from Since F(, ) i a very trong increaing function of, (, ) compared to (, ) a well. In the mot practical cae, a good approximation of the integral i given by the econd order approximation, I2, of quation (5) [4] [5]: 2 k 2k F(, ) = exp 1 k quation (6) ha been calculated a a function of and. he range of the activation energy wa from.5 ev to 2.25 ev in tep of.25 ev and the temperature wa ranging from 3 to 5 K. able 1 give the data obtained. I2 indicate the econd order approximation. (6) OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

3 able 1. of the quation (6). emperature [K] I2 (, =.5) I2 (, =.75) I2 (, = 1) I2 (, = 1.25) I2 (, = 1.5) I2 (, = 1.75) I2 (, = 2) I2 (, = 2.25) Numerical Solution of the Integral he integral (4) ha been alo calculated, a a function of and, by mean of the atlab program. Special attention ha been paid at the low temperature region where the exponential aume initially very low value and then it change very fat a the temperature increae. he lower limit of the integration interval i the mallet floating point number, i.e. realmin in atlab, while the upper limit value change from 3 K to 7 K. he integration limit ha been divided in two ubinterval: the firt for 4 K and the econd for > 4 K in order to define the firt region where the integration function aume very low value and increae trongly with the temperature. In the following the numerical olution of the integral (4) i indicated by Q (, ). able 2 how the data obtained. 4. Condition at the aximum Conidering the equation for general order kinetic, the condition at the maximum i obtained by quation (3) a following. he logarithm of I( ) i: b ( b) ln I = ln n ln 1 + exp d k b 1 β (7) k then, the condition at the maximum i obtained quoting it derivative to zero: ( ) ( ) ( I) n ( b ) ( b ) 2 d ln b = 1 + exp d exp = d k b β k β k = OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

4 able 2. Calculated value of Q (, ) for given value of and. emperature [K] Q= (,.5) Q= (,.75) Q= (, 1) Q= (, 1.25) Q= (, 1.5) Q= (, 1.75) Q= (, 2) Q= (, 2.25) from which we obtain ( b ) 2 k 1 b n exp = 1+ exp d β k β (8) k From the lat equation it i poible to obtain the expreion for the pre-exponential factor. Rearranging quation (8), we obtain: or, conidering = n 2 k exp ( 1) b b exp d 1 k k = n β β 2 k exp ( 1) b b exp d k k = β β Uing the econd order approximation and =, we get from quation (9) (9) (1) OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

5 and finally or, conidering = n, 2 k exp 1 k 2k ( b ) = 1+ n β β exp k 2k ( b ) = nok β exp k 2k ( b ) = k expreed in ec. Oberving the previou equation, it i clear that both the heating rate, β, and the initial doe, n, are multiplication factor: for thi reaon it i poible to compare directly the quation (1) and (13) and to find a poible difference in the pre-exponential factor value when it i obtained by econd order approximation, 2, i.e. quation (13), or uing the numerical olution of the integral,, in quation (1). In both cae only one parameter ha been varied in each imulation, i.e. the kinetic order b, the heating rate β, the temperature at the maximum or the initial delivered doe n. able 3-6 how the frequency factor a a function of the kinetic order b. he activation energy change from.5 ev to 2. ev in tep of.5 ev. and β have been kept contant and the value are 5 K and 1 K/ec repectively. Figure 1-4 how the behaviour of a a function of the kinetic order b for variou value of the activation energy, i.e. equal to.5 ev, 1. ev, 1.5 ev and 2. ev; and β are kept contant, repectively equal to 5 K and 1 K/ec. able 3. Frequency factor a a function of the kinetic order b, for a value of the activation energy =.5 ev. ha been evaluated by mean of numerical integration, quation (1), 2 by mean of the econd order approximation, quation (13); 2/ i the ratio of the two value. and heating rate β are repectively fixed to 5 K and 1 K/. Activation nergy =.5 ev b (kinetic order) able 4. Frequency factor a a function of the kinetic order b, for a value of the activation energy = 1. ev. ha been evaluated by mean of numerical integration, quation (1), 2 by mean of the econd order approximation, quation (13); 2/ i the ratio of the two value. and heating rate β are repectively fixed to 5 K and 1 K/. Activation energy = 1 ev b (kinetic order) (11) (12) (13) OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

6 able 5. Frequency factor a a function of the kinetic order b for a value of the activation energy = 1.5 ev. ha been evaluated by mean of numerical integration, quation (1), 2 by mean of the econd order approximation, quation (13); 2/ i the ratio of the two value. and heating rate β are repectively fixed to 5 K and 1 K/. Activation energy = 1.5 ev b (kinetic order) able 6. Frequency factor a a function of the kinetic order b for a value of the activation energy = 2. ev. ha been evaluated by mean of numerical integration, quation (1), 2 by mean of the econd order approximation, quation (13); 2/ i the ratio of the two value. and heating rate β are repectively fixed to 5 K and 1 K/. Activation energy = 2. ev b (kinetic order) x Activation nergy = Kinetic order b Figure 1. Behaviour of a a function of b for a given activation energy =.5 ev. OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

7 5.6 x Activation nergy = Kinetic order b Figure 2. Behaviour of a a function of b for a given activation energy = 1. ev. 9.2 x Activation nergy = Figure 3. Behaviour of a a function of b for a given activation energy = 1.5 ev. he following Figure 5 how the frequency factor a a function of for a value of the activation energy, i.e. =.5 ev. Figure 6 how and 2 calculated uing quation (9), a a function of the initial charge concentration n ; the other parameter, i.e., b, and β are contant. Figure 7 how the and 2 value a a function of variou heating rate, from 1 K/ec to 5 K/ec in tep of 1 K/ec. 5. Higher Order Approximation Kinetic order b In thi paragraph higher order approximation are preented and the reult are compared with the econd order approximation. OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

8 1.34 x Activation nergy = Kinetic order b Figure 4. Behaviour of a a function of b for a given activation energy = 2. ev [K] Figure 5. Frequency factor a a function of the temperature of the maximum. he activation energy value ha been fixed at.5 ev and the kinetic order at 2. range from 3 to 7 K. By mean of the aymptotic erie, the expreion of become: + n β k n+ 1 k = e ( b) ( 1) ( n+ 1! ) k n= 1 (14) where the econd order approximation i obtained for n = 2 ; the third order approximation for n = 3, and o on. In particular, the third order expreion i the following OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

9 , doe n [cm -3 ] Figure 6. he pre-exponential factor " a a function of the doe n evalueted numerically and by mean of the econd order approximation. Activation energy i fixed at.6 ev, the temperature of the maximum i fixed at 5 K and the heating rate at 1 K/. " i expreed in [ cm 3(b-1) ]. 14 x β [K/] Figure 7. Frequency factor a a function of the heating rate β. he activation energy i fixed at.6 ev and the temperature of the maximum,, at 5 K. and the fourth order approximation i given by β 2k k 3= e ( b) 6 k k 2 β 4 = e ( b) k 2 3 k 2k k k (15) (16) OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

10 b 1 Conidering = n, the expreion for the exponential factor in higher order approximation are obtained. he calculated value are reported in able 7 a a function of the kinetic order; the other parameter are contant a before. Figure 8 how the frequency factor evaluated according to the variou approximation ued. It eem evident that the increae of the order of approximation doe not produce a real benefit in the final reult; a very little difference in the reult can be oberved a the kinetic order increae from 1 to Concluion Figure 1-4 how that the diagreement between the data, obtained uing the numerical evaluation and the one calculated with a econd order approximation, become larger a the value of the kinetic order, b, increae; the difference i more evident for low value of the activation energy, i.e. =.5 ev ; thi difference tend to diappear a the activation energy increae: at = 2. ev no difference can be oberved. he behaviour of the frequency factor a a function of the peak temperature at the maximum,, i given in Figure 5 and Figure 6: the numerical evaluation give the ame reult of the numerical approximation in the range of ued temperature: from 3 K to 7 K and an activation energy value of.5 ev, and from 45 K to 7 K with an activation energy equal to 1.5 ev. able 7. numerical evaluation; 2 econd order approximation; 3 third order approximation; 4 fourth order approximation; 2/, 3/, 4/ are repectively the ratio between econd order approximation 2 and, third order approximation 3 and and finally fourth order approximation 4 and. Activation energy =.6 ev b (kinetic order) x hird order approximation Activation nergy = Kinetic order b Figure 8. Frequency factor a a function of the kinetic order, evaluated uing different method. OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618

11 No difference can be oberved between numerical and econd order calculation for the pre-exponential factor a a function of the irradiation doe (Figure 7). A difference appear for high value of the heating rate (Figure 8); the difference increae a the value of the kinetic order increae too. Reference [1] Randall, J.. and Wilkin,.H.F. (1945) Phophorecence and lectron rap. Proceeding of the Royal Society, A184, 366. [2] Garlick, G.F.J. and Gibon, A.F. (1948) he lectron rap echanim of Luminecence in Sulphide and Silicate Phophor. Proceeding of the Phyical Society, 6, [3] ay, C.. and Partridge, J.A. (1964) hermoluminecent Kinetic of Alpha-Irradiated Alkali Halide. he Journal of Chemical Phyic, 4, [4] Chen, R. and ckeever, S.W.S. (1997) heory of hermoluminecence and Related Phenomena. World Scientific, Singapore City. [5] Chen, R. and Kirh, Y. (1981) Analyi of hermally Stimulated Procee. Pergamon Pre, Oxford. OALibJ DOI:1.4236/oalib Augut 214 Volume 1 e618