# J Orzechowski, A Mandowski - Numerial verification of certain scaling property relating to thermoluminescence peaks - страница 1

ВІСНИК ЛЬВІВ. УН-ТУ

Серія фіз. 2009. Bun. 43. С. 143-149

VISNYKLVIV UNIV. Ser. Physics. 2009. Is. 43. P. 143-149

PACS number(s): 78.60.Kn

NUMERIAL VERIFICATION OF CERTAIN SCALING PROPERTY RELATING TO THERMOLUMINESCENCE PEAKS

J. Orzechowski1, A. Mandowski2

1Stanislaw Staszic High School ul. Szkolna 12, 27-200 Starachowice, POLAND 2Institute of Physics, Jan Diugosz University Armii Krajowej 13/15, 42-200 Czestochowa, POLAND e-mail: a.mandowski@ajd.czest.pl

The phenomenon of thermoluminescence (TL) is described in terms of complex kinetic models. However, it can be shown, that TL curves exhibit several interesting scaling and invariance properties. One of them relates to the dependence of the shape of TL curves on heating rate. We suggest the conservation of a simple integral defining a part of TL peak area with respect to variable heating rate. The hypothesis is verified numerically for various parameters of traps and recombination centres.

Key words: thermoluminescence, dielectrics, traps, recombination.

The simple trap model (STM) is considered as the basic theoretical model for analyzing thermoluminescence (TL) and related phenomena. The model assumes spatially uniform distribution of separate traps and recombination centres (RCs). Charge carrier transitions taking place during heating occur via conduction band after thermal release. The kinetics of trapping and recombination is governed by the following set of differential equations [1]:

-щ = nivi expf-kp-1- ncAi

-m s = Bsmsnc,

k p

Z ms = Z ni + nc + M,

s=1 i=1 i=1..p, s=1..k, (1a) (1b) (1c)

where Ni, ni, and ms denote the concentrations of trap states, electrons trapped in 'active' traps and holes trapped in recombination centres, respectively. M stands for the number of electrons in the thermally disconnected traps (deep traps), i.e. traps that are not emptied during the experiment. Ai and Bs stand for the trapping and recombination probabilities, respectively, and v is the frequency factor. Luminescence intensity is proportional to (-m), i.e.

J dm dt '

(2)

© Orzechowski J., Mandowski A., 2009

TL spectrum consists usually of a series of peaks attributed to different trap levels of the material. Typical TL experiment is performed with linear heating rate scheme, i.e. the temperature T = T0 + pt, where T0 is the initial temperature, t denotes time and p is the heating rate.

The basic set of equations (1) has no analytical solutions in general case. However, some intriguing features may be observed while considering TL curves for various parameters. The simplest experimentally controllable parameter is the heating rate p. For higher heating rates TL peak shifts toward higher temperatures. Typical example of this kind is presented in fig. 1.

1.0

0.8 0.6

пі.

j=! 0.4

0.2

0.0

1.0 0.8

0.6

.та.

j=! 0.4

0.2 0.0

1.0 0.8

0.6

.та.

j=! 0.4

0.2 0.0

HI I I—I—I—I—I—I—I—I—I—I—I I I I Г

300

350

400

450

500

\ІІІІІг^іIIIIIIIIn

350

400

450

500

550

600

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

400 450 500 550 600 650 700 750

Temperature T[K]

Fig. 1. Set of TL curves calculated for different heating rates в

The set of TL curves was calculated for a complex trap cluster system. The heating rates were varied over four orders of magnitude (from p=10-2 to p=102 K/s) and the peak maximum shifts about 200 K, however its shape seems to be unchanged. This behaviour suggests the existence of a scaling property which origin is currently not known. To study these geometric features we considered the following integral:

Skp=[f J(P, 0 dt = в J(P, T) dT , (3)

where 0 <є<1 and tf\Tf,Tf denote positions on horizontal axis, on both sides of the TL peak where the intensity falls down to e-th part of the maximum intensity, i.e.:

J (P,te(1) ) = J (P,tf ) = (Jmax, (4) J (P, T((1) ) = J (P, T((2) ) = (Jmax. (5)

We can suppose that when the scaling is proportional, the integral (3) should not depend on the heating rate p. To check this hypothesis we performed a series of numerical calculations. The basic set of equations (1) was solved using numerical procedures for stiff differential equations. Then, the position of TL peak maximum (Tmax,Jmax) was determined. Then, for a given parameter e (0 < e < 1), two points T(1)

and T(2> (or t(> and tf) were found according to eqs. (4) and (5). Finally, the integral

(3) between the two points was calculated. Results are shown in fig. 2.

d

CD CO

_l I— 0.9 0.8 0.7 0.6 0.5 300 00 OOC

A ti AAi

8 = 0.1

ООО о о oo

8 = 0.3

e = 0.5

e = 0.7

I I 11llllj—I I I ll

T

I 11 I llllj—I 11II

0.01 0.1 1 10

heating rate p [K/s]

100

1

Fig. 2. Relative area under TL curve calculated for different heating rates в and various e according to the eq. (3). The calculations were performed for the following parameters: E = 1eV, B = 1012 cm3s-1, M = 0 , N = 2 • 1015 cm-3, v = 1010 s-1 and r = A/B = 0,01

1.006

1.004

cd 1.002 _l I—

cd >

'■4—'

co cd

01

co

cd

co

_l I—

cd >

'■4—'

co cd

a:

1.000

0.998

1.006

• • • є = 0.1

о о о є = 0.3

а а а є = 0.5

д д д є = 0.7

П0 = 1; r = 0.01

о

. о а а да. а

---------------

а»» о* ••

д

-Щ]—I I 11 llllj—I I 1111111—I I 11 llllj—I I 11 llllj

0.01 0.1 1 10 100

heating rate p [K/s]

1.004

1.002

1.000

0.998

0.996

• • • є = 0.1

n = 1; r = 1

о о о є = 0.3

а а а є = 0.5

д д д є = 0.7

m mm moo

• • ні »» ■■•

д

0.01 0.1 1 10

heating rate p [K/s]

1

100

a er

ra

_i I—

e

>

co

e

a:

1.004

1.002

1.000

0.998

0.996

П0 = 1; r = 100

• • • є = 0.1

о о о є = 0.3

а а а є = 0.5

А А А є = 0.7

ТП|—I I II llllj—I I I lllllj—I I 11 llllj—I I 11 llllj

0.01 0.1 1 10 100

heating rate p [K/s]

Fig. 3. Relative area under TL curve calculated for different heating rates в and various є according to the eq. (3). Results for full initial filling of traps n0 = 1 and various retrapping coefficients. Other parameters the same as for fig. 2

a er

ra

_i I—

e

>

co

e

R

1.004

1.002

1.000

0.998

• •

•є = 0.1 По =

0.5; r = 0.01

О о

є = 0.3

▲ ▲

а є = 0.5

a a

а є = 0.7

ДДД а

ОДА

їй яшл ▲а

• •

88Г

•• ••• -в-^- -

о

2°

в

ДА

д

▲

a

a

a er

ra

_|

e

> со

e

R

d

a er

ra

_і

e

>

co

e

R 1.006 1.004 1.002 1.000 0.998 0.996

1.010 1.008 1.006 1.004 1.002 1.000 0.998 0.996 0.994

ТЛ]—I I 11 llllj—I I 11 llllj—I I 11 llllj—I I 11 llllj

0.01 0.1 1 10 100

heating rate p [K/s]

П0 = 0.5; r = 1

• • «є = 0.1

о о о є = 0.3

а а а є = 0.5