J . Phys. Chem. 1984,88, 2281-2293 4.0
4.5 100 mM F -
N I
5
o I mM
cl.
IO IO 01 IO
CI-
mM mM mM mM
3.5
3.0
2.5
3.5
3.0
2.5
HYDROQUINONE
Cl0,01-
W 0 40
dH
Y
m
0 x
0301
0 20
L 0. I O F 0.00
4.5
4.0
-LOG C(M) Figure 3. r vs. log C curves for hydroquinone adsorbed onto clean Pt from solutions containing halide. The solid lines connect experimental points and do not assume any theoretical fit. TABLE 111: Competitive Adsorption between I- and 1.4-Naohthohvdroauinone concn: mM NHQ I0.100 0.5 0.100 1.0 0.124 0.5 0.124 1.0 0.240 0.5 0.244 1.0 0.576 0.5 0.578 1.0
r:
nmol cm-2
"Q 0.108 0.086 0.158 0.125 0.223 0.183 0.400 0.348
1-
0.698 0.910 0.569 0.834 0.569 0.761 0.268 0.477
upy
predom
molecule 61.5 61.1 54.4 38.9 38.4 31.8 32.1 30.3
orientnd
"Q
flat (q") qIo $0
tilted tilted vertical (2,3-q2) 2,3-q2 2,3-a2
"The solutions contained 1 M HC104. bThe average relative standard deviations in r were k3%. uNHQ was obtained from eq 3 with uI = 14.1 A2. dThe calculated uNHQ values for $lo and 2,3-q2 orientations are 69.9 and 29.0 A2, respectively.'a,d siderations are indicated. If surface attachment is viewed as a balance between aromatic-to-metal electron donation (u-bonding) and metal-to-electron back-donation (a-back bonding)," an increased negative charge on the electrode would weaken the aromatic-Pt interaction. It may be mentioned that Pt(0) complexes
2287
are usually stabilized by ligands which alleviate charge buildup on the zerovalent metal." Table I1 summarizes results on H Q displacement by halides. F did not displace H Q from the surface. Comparison of data for C1- and Br- media demonstrates that $-HQ is more strongly adsorbed than q2-HQ. At a given Br- concentration, only a slight increase in Ardes was noted as the reaction time was increased, but no such variation was observed for C1-. Competitive Adsorption between Aromatic and Halide. r vs. log C curves for HQ on clean Pt in solutions which contain halide are shown in Figure 3: (i) The adsorption and reorientation of H Q was unhindered by F.(ii) Dilute (CO.l mM) C1- did not affect formation of v6 or q2 adsorbed species, but completion of the q2 oriented layer was slightly retarded. (iii) The effect of 1 mM C1- was the same as that of 0.1 mM Br-; although l7 was significantly lower at all concentrations, the overall shape of the adsorption isotherms remained unchanged. (iv) Except in the transition region, the effect of 10 mM C1- was not too different from that of 1 mM C1-. The adsorbability of C1- is such that it can retard reorientation but not null out q2 adsorption. (v) In the presence of 1 mM Br-, r was lowered dramatically, but the decrease in r was not as severe at high than at low concentrations of aromatic. The data in Figures 1 and 3 show a number of similarities: The general shapes of the isotherms suggest that, at the C1- and Bractivities studied, adsorption of H Q at C0.5 mM leads to flat orientations, adsorption of > 1.O mM to vertical structures. Data on competitive adsorption between NHQ and dilute acidic I- are summarized in Table 111. The rate of N H Q adsorption is at least equal to that of I-, since N H Q adsorption was substantial.lb The influence of NHQ/I- concentrations on adsorbed \amounts was as expected. It is of interest to note that, at 0.12 mM, NHQ adsorption in the presence of 1.0 mM I- yielded partially reoriented intermediates; at higher N H Q concentrations, adsorption was in the v2 orientation even for low I- activities. In contrast to the other halides, I- is strongly coordinating toward the Pt surface to bring about molecular reorientations even at comparatively low aromatic concentrations. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the Air Force Office of Scientific Research for support of this research. Registry No. Pt, 7440-06-4; F, 16984-48-8;C1-, 16887-00-6;Br-, 24959-67-9; I-, 20461-54-5; HQ, 123-31-9; NHQ, 130-15-4.
Generalized Methods for Determining Thermal Activation Energies: Applied to Bromosodalite Witold P. Maszara* and Lee T. Todd, Jr. Department of Electrical Engineering, University of Kentucky, Lexington, Kentucky 40506 (Received: July 11, 1983)
The evaluation of thermal activation energies of electron traps located in the bandgap of crystalline materials with absorption bands in the visible spectrum is discussed. Relations between the activation energy and various parameters of the thermal decay curves are derived. Both isothermal and constant rate thermal bleaching experimentsare used to determine the activation energies of samples with established order of bleaching kinetics as well as samples with undetermined order of kinetics. The methods were applied to bromosodalite to determine an activation energy of 1.45 eV. Sodalite samples were found to obey bleaching kinetics of the intermediate order between first and second.
Introduction Basic principles of kinetics regarding the decay of luminescence in crystals were described in several early monographical works.'-* *On leave from Technical University of Wroclaw, Poland. 0022-365418412088-2287$01SO10
Two differeht mechanisms-monomolecular (kinetics of the first order) and bimolecular (kinetics of the second order)-were an(1) H. W. Leverenz, "An Introduction to Luminescence of Solids", Wiley, New York, 1950.
0 1984 American Chemical Society
I
2288 The Journal of Physical Chemistry, Vol. 88, No. 11, 1984
Maszara and Todd, Jr.
alyzed by many authors, and experimental thermoluminescence curves were utilized to evaluate activation energy and other parameters of electron traps. An exhaustive review of those was given in ref 6. Lushchik7 reported one of the first attempts to adopt those methods for evaluation of activation energy by thermal bleaching of color centers. It was focused on strictly first- or second-order kinetics cases using a constant rate of temperature increase. This paper proposes a broader approach to thermal bleaching as a tool for activation energy evaluation. A method employing data obtained from both linear temperature increase and isothermal bleachings is presented. Solutions are derived for the cases of undetermined order of bleaching kinetics as well as for those exhibiting exactly first or second order. Constant Rate Thermal Bleaching Method Proportionality of the number ni of electrons trapped at centers of a given type i to the optical absorption in the crystal enables one to use the latter in an attempt to determine the activation energy ET of thermal release of electrons or holes from traps using the relations ni(T) and ni(t)lT=m,,st( T = temperature, t = time). The number of released electrons per unit time is given in ref 7 for the case of only one trap depth
--dni = dt
+
urn
P’n’urn ut(Ni- ni)
(1)
where u, is the effective recombination cross section, u, is the effective capture cross section of the center, Ni is the number of trapping centers, n is the number of unoccupied recombination sites, and p , is the probability of thermal release of the electrons:
Pr =Po exp(
-2)
dn _ _dn_ d-T -- -Pdn d T dt
dT
Inserting eq 2 and 2a into eq 1 and rearranging, one obtains
- ni(u, - ut) + Niut dn urn12
exp(
=
P
limiting cases: (1) first-order kinetics where the recombination probability, A,, is much higher than the retrapping probability, A,, and (2) second-order kinetics where A, is much higher than A,. Using simplifying assumptions, he calculated expressions which are given by ET =
F)d T -ET
kL’P Tk22 8k In 2
-
2kTk2
-
Solving eq 5 for ET, one obtains r
By obtaining data from two bleaching experiments with different rates P1 and P2, eq 6 yields eq 7 , which can be used to calculate ET [compare with similar method used by A. N. Booth for thermoluminescence maxima (Can. J. Chem., 32,214-15 (1954))J. ET = -k
TklTk2
Tk2
(2) N. F. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals”, 2nd ed., Clarendon Press, Oxford, 1950. (3) G. F. J. Garlick, “Luminescent Materials”, Clarendon Press, Oxford, 1949. (4) D. Curie, “Luminescence in Crystals”, Wiley, New York, 1963. (5) P. Goldberg, “Luminescence of Inorganic Solids”, Academic Press, 1966. (6) C. S. Shalgaonkar and A. V. Narlikar, J . Mater. Sci., 7, 1465-71 (1972). (7) C. B. Lushchik, Sou. Phys.-JETP (Engl. Transl.), 3, 390-99 (1956).
(4)
(second order) 8k where Tk is the temperature at which the number of captured electrons n k decreases to half of its initial value, nklT=T,,= no/2, and 8k corresponds to the slope of the bleaching curve at T = Tk (Figure 1). This particular method requires a knowledge about the type of kinetics actually obeyed by a given sample. Although being relatively simple, the above method does not solve the most general case when A, is of the order of A,. A method to calculate ET without assuming strictly first- or second-order cases can be derived using the same starting point as Lu~hchik‘s.~At T = Tk, eq 3 becomes
(3)
Lushchik’ showed that it is possible to evaluate ET using the data from the bleaching characteristic n/no =f(T)ISPconst for two
(first order)
and ET =
where the exponential factor is the Maxwellian distribution of thermal energies of electrons in the traps; po is called the“escape frequency”; k is Boltzmann’s constant; and T i s temperature in kelvin. It is assumed that, during the coloration process, electron-hole pairs are created. One member of the pair creates a color center when captured at a structural defect; the other member remains as a localized charge on one of the atoms in the lattice and serves as a recombination site. Hence, in an equilibrium state n = n,. This relation will also hold in a dynamic process of bleaching, since the bleaching rate is much slower than the rate of recombination of single electrons due to the very short lifetime T of electrons released into conduction band (Le., at any instant of time we can practically assume that all electrons either are in traps or have already recombined at recombination sites). The rate of change of unoccupied recombination sites may be expressed in terms of P, the constant rate of temperature increase during bleaching, as dt
Temperature
Figure 1. Normalized constant rate thermal bleaching curves for only one trap depth and two different heating rates.
-
Tkl
In
’klP2
6k281
(7)
This equation permits the calculation of ET regardless of the kinetics involved. If there are several traps of differing depths, electrons freed from the ith level during the bleaching process may be recaptured by the same kind of trap as well as by any of the other traps. If their depths are significantly different, then in the temperature region in which the electrons are being released from levels i7 dni
pini
urn + j>i Cut,(N,
- nj)
Thermal Activation Energies of Bromosodalite
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 2289
"O
p = Const
L
0
>- 0 . 5 C
C
TK2
KI
0
TK3
'S2
'SI
Temperature
Time
traps of significantly different depths.
Figure 3. Normalized isothermal bleaching curves and associated parameter t,.
Using the procedure employed for the case of a single trap depth, we obtain
(ni/no= 1 / 2 ) is measured in each case (Figure 3). Equation 10 then has the form
Figure 2. Constant rate thermal bleaching curve for a phosphor with
ET = k
(;
TkilTki2 '3kilP2 In -Tkil '3ki2P1
Tki2
where Tki and s k i are as shown in Figure 2. For the deepest level, j = i and &T,~(N~ - nj) = 0. Therefore, eq 8 reduces to the case of single trap depths and eq 1 applies. Thus, in principle, one should be able to obtain all trap depths from quite complex bleaching curves. Isothermal Bleaching This method is introduced somewhat analogously to the isothermal decay method described in ref 3 for luminescence. Again, for the case of a single trap depth, eq 1 is rearranged into the form ni(u, - ut)
+ Niu,
- 1) In
UtNi 21 + a,= p o t s exp
( i2,) --
(13)
Using eq 13 for two different bleaching temperatures, one obtains ET
TslTs2 ts2 kIn Tsl - Ts2 tSl
where tsl and ts2 are as shown in Figure 3. In the case of several nonoverlapping trapping levels, eq 14 will still hold, providing bleaching is performed at the temperature at which the given ith level of traps is activated (compare with analogous problem for constant rate thermal bleaching treated previously).
After integration of both sides (n = no at t = 0) we have
Combined Isothermal and Constant Rate Thermal Bleaching Method Another method for evaluating the activation energy, ET, of an electron in a trap can be derived when both isothermal and constant rate thermal bleaching techniques are combined together. Integrating both sides of eq 3, we have
When the second term of this equation is negligibly small, A, >> A,, we have first-order kinetics and the relation In (nilno)= f ( t ) will be linear. Also, since pi is always positive, ut has to be smaller than u,. Measuring the slopes of the isothermal bleaching curves for similar samples bleached at two different temperatures T,, one can evaluate the activation energy ET of the trap
where the right-side integral of eq 3 was solved with the help of the approximation
-dni
n?u,
= p i dt
(9)
where ml and m2 are the slopes of the cuurves In (nilno) = f(t) for bleaching temperatures Ts, and Ts2,respectively. When the kinetics are of the second order, the first term in eq 10 is negligible and the reaction l / ( n / n o ) = f(t) is linear and ET is given by
near T = Tk where nilno = l I 2 . Comparing eq 15 with eq 13 for isothermal bleaching at Ts = Tk, we notice that the left sides of both equations are identical (assuming identical samples, Le., parameters ut, u,, Ni, no, and po remain unchanged). We obtain
and finally where 1, and I2 are the slopes of the curves l/(n/no) =At)for temperatures Tsl and Ts2,respectively. Hence, we arrived at the equation of the same form as in the case of the first-order kinetics (11). Next, consider the general case where none of the terms in eq 10 can be neglected and the process is neither first nor second order. Again, two identical samples are bleached at two different temperatures. The time, ts, corresponding to the point where the number of color centers has dropped to half of its initial value
k Tk2
ET =
Pt,
Again it can be shown that (17) holds also for the case of several nonoverlapping trapping levels. Theoretical Summary Let us check whether the above methods yield indeed the correct answers and how accurate they are.
2290
The Journal of Physical Chemistry, Vol. 88, No. 1 1 , 1984
Maszara and Todd, Jr. TABLE I order of kinetics first
p,
K/min 4.5 21 internicdiatc 4.5 21 second 4.5 21
Tk, K 488.0 510.7 546.6 575.7 568.3 598.3
hk, K
18.6 20.3 56.4 64.0 36.9 41.1
-
eq 4 1.59 1.60 (0.66) (0.64) (1.08) (1.08)
.ET, eV
cs 4a.
(2.20)a (2.22) (0.91) (0.89) 1.5 1.5
.-
.
__ -
cq 7
I1.4l
i1,32 11.40
a Results in parentheses are given for comparison. They are obtained when the equation referring to wrongly assumed type of kinetics is used.
T E M P E R A T U R E
(OK)
Figure 4. Theoretical constant rate bleaching curves for first, interrne-
diate, and second order of kinetics. Three different cases of kinetics are examined: the first, second, and intermediate orders. Theoretical curves are plotted, and the activation energy ET is evaluated from the graphs with all applicable methods introduced above. In all three cases, two basic arbitrary assumptions are made. The value of "escape frequency" po is set equal to 10l2s-l, and ET = 1.4 eV. Furthermore, for every case of kinetics the constant rate bleaching curves are constructed for two rates P1 = 4.5 and Pz = 21 K/min. The isothermal characteristics are calculated and graphed for two temperatures T,, such that T, = Tk, where Tk)s are those obtained from previously plotted constant rate curves. This is to provide the match of isothermal and constant rate bleaching data for the combination method calculation of ET. The first-order kinetics is represented by -dn,/dt = pini (18) This is obtained from eq 1 when high recombination probability is assumed; i.e. urni >> ut(N - ni) For the constant rate of thermal bleaching p, eq 18 can be modified into
and after integration of both sides
n = exp[ no
-;Jl -2) exp(
(20)
dT]
where no is an initial concentration of color centers at the temperature To (in our case r 3 0 0 K). The two plots of eq 20 for two different p's are evaluated with numerical methods and are given in Figure 4. The second order of kinetics is approached with the assumptions urni > ni
which lead to
--dn, dt
urn2 =Pi-
atN For a constant rate of bleaching, P, the equation is modified into
After integration
n = no
[+ 1
'fiL;exp( Path'
-$)d T ]
-1
(25)
Unlike in the first order of kinetics relation, this response depends on several material parameters. The initial number of color centers no and number of trapping centers as well as capture and recombination effective cross sections are present in the final form of the equation. To satisfy the conditions 21 and 22, it was assumed that ut/u, = 10 and N/no = 10. Equation 25 is then computed by using numerical methods and graphically illustrated in Figure 4 for the case of two b's. The intermediate order of kinetics can be represented by another modification of eq 1
where strong domination of the retrapping process over the recombination process (ut/u, = 100) was assumed. It is also assumed that the initial number of color centers is very high and equal to the number of traps ( N = no). There are, of course, other choices of the values of material parameters that lead to the intermediate order of kinetics, e.g., u, > ut and N > no. Integration of eq 26 leads to the relation
which, again, can be evaluated numerically. It is plotted in Figure 4. For all three cases of kinetics the values of Tkand ak were found from each curve in Figure 4. The activation energies ET were calculated from eq 4, 4a, and 7. Their values along with the values of (3, Tk,and 8 k are given in Table I. The overall rate of decoloration of the sample depends, as it was mentioned before, on both the recombination and retrapping processes and the relative significance of their occurrence. Examination of eq 1 leads to the conclusion that the retrapping process hampers the bleaching (attained through recombination of freed electrons); hence, the samples with the same activation energy but higher retrapping probability have to be bleached longer. In the case of constant rate bleaching that means that higher temperatures will be reached before decoloration is completed. This effect is indeed observed in Figure 4. The results in Table I clearly indicate that eq 3, in which no particular order of kinetics is assumed, yields the value of activation energy in a good agreement with assumed 1.4 eV in all three cases of assumed order of kinetics of bleaching. The values of ET calculated with the help of Lushchik's eq 4 and 4a are also correct wherever applicable. The matching of the value of E,, calculated for the sample of unknown kinetics, with one of those obtained from eq 4 or 4a would indicate the order of kinetics observed (first or second, respectively). It should be pointed out that the values of ET obtained for intermediate order of kinetics (see second row in Table I) with the help of eq 4 and 4a fall far from the actual value of activation energy. This suggests a rather short span of validity of these two equations. An attempt to use either of these to approximate energy for the sample exhibiting intermediate order of kinetics may result in a significant error. Knowing that the order of kinetics of thermal bleaching for single trap depth materials is determined
The Journal of Physical Chemistry, Vol. 88. No. 11, 1984 2291
Thermal Activation Energies of Bromosodalite
09
-
1-51
order
inlermadiote order
07
---
06
Il-nd order
n.
IN-nd o r d e r
20
15
01
0
0
2
4
6
8 T I M E
12
IO
14
16
2
0 ‘
K‘
4
6
T I M E
(minl
Figure 5. Theoretical isothermal bleaching curves for first, intermediate, and second order of kinetics.
12
10
8
18
14
16
18
(min)
Figure 7. Theoretical isothermal bleaching curves for second-order kinetics. The intermediate order is shown for comparison.
TABLE I1 order of kinetics
ts3
m,
T,, K
min
min-’
488.0 510.7 inter546.6 mediate 575.7
3.35 0.78 4.16 0.92
0.208 0.91
first
second
I, min-’
eV vmET,
11.38
11.40
11.41
0’229
568.3 4.36 598.3 1.05
0.966
11.39
}1.1.41
TABLE 111 order of kinetics first Figure 6. Theoretical isothermal bleaching curves for first-order kinetics. The intermediate order is shown for comparison.
intermediate second
by the properties of the trap and recombination center as well a s the level of population of the traps with electrons (level of coloration), we can safely claim that the majority of the investigated cases will not fall into the category of strictly first- or second-order kinetics. The above observations lead to the conclusion that the approach proposed by Lushchik and represented by eq 4 and 4a has a relatively limited range of application. Relations between time and the number of color centers during inothermal bleaching at certain temperature T, can also be derived from eq 1. Similarly to the above derivations, three equations and corresponding graphs for n/no vs. t are obtained for three assumed types of kinetics. The equations have the following forms. First order of kinetics:
n = exp[ -pot exp(
-$)]
no Intermediate order of kinetics:
Second order of kinetics:
The same assumptions for q / u r and N / n o were made as for the constant rate bleaching. The plots illustrating the above relations are shown in Figure 5. For the purpose of use of eq 11 and 12 the results representing first and second order of kinetics are shown in semilogarithmic (Figure 6 ) and semihyperbolic (Figure 7) coordinates, respectively. For illustrative purposes, also shown are intermediate order of
K
is, min
P> K/min
ET eV
488.0 510.7 546.6 575.7 568.3 598.3
3.35 0.78 4.16 0.92 4.36 1.05
4.5 21 4.5 21 4.5 21
1.36 1.37 1.37 1.48 1.42 1.40
Tk = T,,
9
kinetics curves. The calculated activation energies, from all available formulations introduced previously, are summarized in Table 11. Again, as in the case of constant rate thermal bleaching, very good agreement between these values and the assumed value of ET = 1.4 eV is achieved. Finally, the combined method, represented by eq 17, can be tested. Using the corresponding data from Tables I and 11, we arrive at the set of values of ET(see Table 111), all closely matching the assumed value. All calculations performed above confirm the correctness and self-consistency of the derived methods and indicate that, given the record of experimental relation between the concentration of color centers and temperature or time, the error introduced by the method itself is confined to a few percent only. This would be further reduced if parameters Tk,6k, and T, were found numerically rather than graphically. Activation Energy of F-Center in Bromosodalite The methods derived above were used to evaluate the thermal activation energy necessary to erase a color center in cathodochromic bromosodalite. The material, known to exhibit only one kind of electron-type color center, the F-center, seems to be perfectly suitable to illustrate the methods developed above. Experimental Section. The material was produced in powder form by using a hydrothermal growth process described elsewhere.* The powders were sensitized by annealing them in vacuum9 at (8) M. S.Perlmutter, L. T. Todd, Jr., and E. F. Farrell, Mater. Res. Bull., 9, 65 (1974). (9) L. T. Todd, Jr., and M. K. Badrinarayan, Am. Ceram. SOC.Bull., 58, 1193-1195 (1979).
2292 The Journal of Physical Chemistry, Vol. 88, No. 11, 1984
-
02 01 0
1
\ \
04
0
,
70
, 90
, 110
1
,
IS 150
~
170
,
,
,
Maszara and Todd, Jr.
T :468'K
w
190 210 230 E50 270 29 T E M P E R A T U R E YC)
05
30
Figure 8. Experimental constant rate bleaching curve and corresponding converted curve.
850 OC in order to create the negative ion vacancies necessary for color center formation. Sensitized powders were deposited on 12.5 X 12.5 X 0.1 mm aluminum plates by using a standard water slurry settling technique.lo The samples were then colored in a demountable cathode ray tube system using an anode voltage of 18 kV. Four different doses of electron beam irradiation (25.6, 64.0, 320, and 1600 pC/cmZ) were used. Bleaching of the colored samples was achieved using both the isothermal and constant rate of temperature increase techniques. The thin aluminum sample plates were chosen to achieve a compromise among low heat capacity, high heat conductivity, and necessary rigidness for safe handling of the samples. The sample plates were attached to a small heater block, with imbedded nichrome heating element. A conducting paste, Omegatherm "201", was used between the heater and sample to ensure good thermal contact. A thermocouple located within the heater such that it touched the back of the sample plate near its center permitted sample temperature measurements with an accuracy of f l K. The heater was mounted in a spectrometer housing, and the samples were illuminated with a white-light source. Light reflected from the sample surface was passed through the monochromator which was set at 5650 A, the peak of the color center absorption. A photomultiplier with an S-20 response was used for signal detection, and the resulting decay curves were plotted on a strip chart recorder. Results. Diffuse reflectivity that was measured during bleaching experiments is not, in general, linearly proportional to the number of color centers. Hence, suitable transformation has to be applied to experimental data. Discussion of such a procedure has been given in ref 11. Shortly, it is a modification of Melamed's diffuse reflectance theory'* where a geometrical factor xu related to an arrangement and size of cathodochromic screen particles is established empirically. The conversion allowed the expression of the decaying coloration in terms of the number of color centers vs. temperature (or time) as required by the methods of evaluation of activation energy. The transformations of the experimental data led to decay characterististics of similar shape to the original ones (reflectivity vs. temperature or time) but with significant shifts toward lower temperatures or shorter times and changed slope. A typical constant rate thermal bleaching curve and its conversed form are given in Figure 8. The isothermal bleaching technique was used to determine the order of kinetics obeyed by the erasure process; however, the results proved inconclusive. That is, no definite first or second type of kinetics was observed. The curves of theoretically transformed data for four differently colored samples bleached isothermally (10) F. Rosebury, "Handbook of Electron Tube and Vacuum Techniques", Addison-Wesley, Reading, MA, 1965. (1 1) W. P. Maszara and L. T. Todd, Jr., to be submitted for publication. (12) N. T. Melamed, J . Appl. Phys., 34, 560, (1963).
IO
15 T
20 25 30 I M E (min)
40
35
45
50
Figure 9. Experimental isothermal bleaching curves (converted).
TABLE IV
25.6 64.0 320.0 1600.0 (saturated)
4.5 21.0 4.5 21.0 4.5 21.0
468.2 488.2 481.0 503.6 491.6 515.0
71.4 72.4
11.51
71.0 70.6
11.43
4.5 21.0
525.7 549.4
52.3 52.2
67.6 67.7
::::;:"0 ::::0":; :::;:::; ::;:
) 1.43 11.62
TABLE V exposure,
t,,
pC/cmz
T,,K
min
25.6
468.0 488.0 481.0 504.0 492.0 515.0
9.06 1.98 2.16 0.49 0.72 0.18
526.0 549.0
0.12 0.08
64.0 320.0 1600.0 (saturated)
E,,eV cq 14
11.34
11'32 10.44
K/min
ET,~V eq 17
4.5 21.0 4.5 21.0 4.5 21.0
0.46 0.49 2.05 2.12 6.44 6.04
13,
2::i
at different temperatures T, are plotted in semihyperbolic coordinates in Figure 9. Their shape resembles very closely those theoretically calculated for intermediate order of kinetics (Figure 7). In light of these findings no predetermined-type-of-kinetics method could be used in calculations of ET. This excludes eq 4 and 4a for constant rate thermal bleaching and eq 11 and 12 for isothermal bleaching from being used. Activation energies for samples colored with the same amount of electron beam irradiation as those bleached isothermally were determined by using the constant rate of temperature increase technique at two different heating rates, and Pz. Activation energies calculated from eq 7 are summarized in Table IV along and 6k. For comparison reasons, shown with the parameters 0, Tk, along with the above are the results calculated with the help of the known-type-of-kinetics eq 4 and 4a. The values obtained for these two equations fall far from that calculated with eq 7 . A similar situation was observed for the theoretically calculated intermediate-order relations. This result confirms the isothermal bleaching findings that the erasure of F-center coloration in sodalite does not comply with first or second order of bleaching kinetics. The values of activation energy calculated for four differently colored samples are reasonably close to each other and yield the average ET = 1.50 eV with about &7% error. The activation energies for samples identical with those used above were also determined from the isothermal decay curves with eq 14. The results are summarized in Table V. The resulting activation energies are also quite consistent and yield an average value ET = 1.39 eV. This compares favorably with the results obtained using the constant rate approach. The last sample
Thermal Activation Energies of Bromosodalite (exposure 1600 pC/cm) was excluded from the calculation of the average ET. Bleaching half-time t, was short enough to be of the order of time when initial distortion of temperature appears. That is probably a sole factor that greatly contributed to an unreasonably small value of E T . The combined technique, which employs parameters determined from both constant rate and isothermal experiments, was also used, and the results calculated with the help of eq 17 are summarized in Table V. The resulting values of ET are very inconsistent with those previously obtained. The reason for this discrepancy is not yet fully understood. One possible explanation may consider inaccurate temperature calibration for given rates p and for the isothermal process. Whereas certain systematic error can be lessened or eliminated in eq 7 and 14 where data from two samples bleached in the same way are used, mismatched temperatures Tk and T, of constant rate and isothermal bleaching may cause significant error in eq 17. Discussion of Experimental Results. The smooth, monotonically decreasing shape of the constant rate thermal bleaching curves (e.g., see Figure 8) is similar to those theoretically calculated for the case when only one trap depth is assumed to exist in a material. This suggests that there are no other "active" traps capable of interfering with F-center bleaching process. Hence, it justifies the choice of sodalite for verification of the presented theory. During bleaching, electrons are removed from halogen ion vacancies and are either tentatively recaptured by another vacancy or recombine with holes located somewhere in the lattice. The shape of isothermal bleaching curves and a comparison of activation energies calculated with the help of eq 4 and 4a with those obtained from "insensitive" to the order of kinetics eq 7 indicate that sodalite does not exhibit first nor second order of kinetics. Also, if the first order of kinetics were observed, the normalized constant rate thermal bleaching curves n/no vs. T for differently colored samples should be identical according to eq 20 (Le., the bleaching process does not depend on initial coloration of the sample). This is not observed in our experiments. The consistency of results, obtained for differently colored samples, calculated from constant rate thermal bleaching curves (eq 7) and isothermal characteristics (eq 14) supports the correctness of the theoretical method. The intermediate order of kinetics observed indicates that the recombination sites are not in the immediate vicinity of the trapped electrons. If the recombination sites were adjacent to the F-centers, the trapped electrons would favor recombination even when additional empty vacancies were available and the kinetics would be the first order. The "immediate vicinity" of the trap, which consists of four sodium ions surrounding a halogen ion vacancy, is the alkalisilicate framework structure. The fact that first-order kinetics is not observed for sodalite indicates that species other than the framework atoms are associated with the recombination sites. The only other possible locations of the recombination sites in an impurity-free crystal are then the Na4Br groups in the cages different from those of the trap (Na4+) structure itself. This is contrary to the assignment proposed by B a d r i n a r a ~ a nbut ' ~ in agreement with the findings of Denisov et al.l49l5and Faughnan16 which locate the recombination site in a cage with one sodium ion missing. On the other hand, the fact that the second order of kinetics is not observed either, even for (13) M. K. Badrinarayan, Dissertation, University of Kentucky, 1980. (14) R. A . Denisov, V. Denks, and H. Uurike, Tr. Inst. Fiz. Akad. Nauk Esr. SSR, 49, 99, (1979). (15) V. P. Denks, A. E. Dudelzak, Ch. B. Lushchik, T. V. Russ, N. P. Soshchin, and T. I. Trofimova, Zh. Prikl. Spektrosk., 24, 37, (1976). (16) B. W. Faughnan, I. Gorog, P. M. Heyman, and I. Shidlovsky, Proc. IEEE, 61, 927-41 (1973).
The Journal of Physical Chemistry, Vola88, No. 11, 1984 2293 the lightly colored samples (where the significance of the factor ( N - n ) in eq 1 increases) suggest that the value of recombination-effective cross-section ur is significantly larger than the one for retrapping, ut. The modified, more versatile form of the constant rate thermal bleaching method introduced above permits the determination of the thermal activation energy without prior knowledge of the bleaching kinetics. So do the isothermal and combined methods. The average value of ET calculated from all results with the exclusion of the combined method is 1.45 eV. This value represents all samples studied regardless of the level of coloration, which, according to eq 1, affects the rate of decay but not the activation energy. The value of ET is much larger than those reported in the literature.17-19 All three references give a value of ET of the order of 0.6 eV. Although none of them specifies explicitly the type of kinetics considered, it can be concluded from the description given by each author that, at least in the two cases of ref 17 and 19, first-order kinetics was arbitrarily assumed. Neither Faughnan et al.I7 nor Chang and Ontonl8 give details of the experimental procedure that produced their data, leaving a doubt about whether the specific optical properties of the powdered samples were taken into consideration. Todd, Jr.,I9 measured the diffuse reflectance contrast ratio (CR) and, assuming its proportionality with absorption coefficient k for lightly colored samples ( k d
