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The application of interactive controlled tempering for deduction of energy variable or distributed desorption processes
Vacuum/volume 42/number 3/pages 195 to 198/1991
0042-207X/91S3.00+.00 ,:~'~1990 Pergamon Press plc
Printed in Great Britain
The application of interactive controlled t e m p e r i n g for deduction of energy variable or distributed desorption processes G C a r t e r , Department of Electronic and Electrical Engineering, University of Salford, Salford M 5 4WT, UK and D K Daukeev,
Institute of Nuclear Physics, Kazakh Academy of Sciences, Alma-Ata, Kazakhstan, USSR
received 2 December 1989
The processes of serial and parallel desorption are considered when the system is exposed to either prescribed tempering and desorption rate is measured or desorption rate is prescribed and tempering is interactively controlled. It is shown that both methods can lead to accurate determination of desorption activation energy variation with population in the serial mode, but only close approximations to the initial population-activation energy distribution in the parallel mode. The exception is the prescribed exponential tempering function which allows exact determination.
1. Introduction Thermal evolution spectrometry has been, for m a n y years, and continues to be a powerful a n d useful m e t h o d for determining the energies of binding of gases to solids. In the usual manifestation of this technique the solid is heated (tempered) in a prescribed, controlled m a n n e r as a function of increasing time, and the rate of evolution (desorption) of gas is continuously monitored. A great deal of theoretical analysis has been devoted to describing the evolution rate b e h a v i o u r for a variety of system parameters. These include evaluation of evolution rates for processes with a single, unique activation energy ~'2 in which the desorption step is either first or higher order in atomic concentration a n d where the temperature, T, is raised, with time t, according to the function d T / d t = rt, TP where r v is a c o n s t a n t and p = 0 or 2 ~: or p - l ~. Studies have also been made for systems where the activation energy is distributed :'4 or variable 5. If the activation energy is distributed this implies the existence of a p o p u l a t i o n of either discrete or c o n t i n u o u s nature sorption centres from which evolution occurs in parallel whereas if activation energy is variable it implies a dependence of activation energy on total i n s t a n t a n e o u s p o p u l a t i o n a n d evolution occurs as a serial process. Investigations have also been made of systems where diffusion processes 67 occur as a precursor to evolution. In all of these systems substantial a t t e n t i o n has been devoted ~ 5.s t e to the d e c o n v o l u t i o n of m e a s u r e d evolution rates to determine system p a r a m e t e r s such as activation energy and its p o p u l a t i o n distribution or variation. As already noted such studies prescribe the tempering function ( d T / d t ) a n d the measured variable is evolution rate, p. Recently, however, Daukeev el al t: have a d o p t e d an alternative a p p r o a c h in which p was initially prescribed (a c o n s t a n t in the Daukeev el al study 12) and the tempering rate d T / d t was interactively controlled to m a i n t a i n this prescription. It was shown that the t e m p e r a t u r e / t i m e behav-
iour could then be used to determine activation energy variation in a serial evolution system. In the present c o m m u n i c a t i o n this concept is extended and generalized to show how activation energy variation can be determined exactly for serial evolution processes a n d approximately for parallel evolution processes. A l t h o u g h analysis is possible for more complex systems, such as higher order dependence on i n s t a n t a n e o u s p o p u l a t i o n or precursor diffusion before desorption, it is more straightforward to consider single step desorption a n d first order dependence in p o p u l a t i o n density. This is the system considered here.
2. Serial desorption processes In such a system the i n s t a n t a n e o u s desorption rate, p, may be represented, at an i n s t a n t a n e o u s p o p u l a t i o n density, n, by the relationship dn
17
P = -- d t
z
(1)
where the relaxation time constant, r, is given by a Frenkel type relation,
£(n) r = re exp k T i t )
(2)
where % is an inverse attempt frequency of order 10 ~ s a n d E ( n ) is the desorption activation energy which is a function of i n s t a n t a n e o u s p o p u l a t i o n density, n. C o m b i n i n g equations (1) and (2) yields the result p =
"
r0
exp
(
-
T(O /
.
(3)
If the temperature, T, is increased with time, the desorption rate
195
G Carter a n d D K Daukeev . Deduction of desorption energy variation
p varies and the residual population n decreases to
i
n0-
t
p dt )
where n~, is the initial population density. This may be determined by desorbing all gas and so p and n may be measured for all T (and 1). Consequently if 5, according to equation (3), log p/n is plotted as a function of 1/T, the behaviour of E(n) is immediately recovered from the slope of the curve at each n value. It is notable that the form of the measured function is independent of the form of variation of T with time. On the other hand it is possible ~= by appropriate interactive control of the tempering function T(t) to prescribe the experimental behaviour of p ( t ) - p(T). For example '2, if T~(t) is controlled such that it(t) is maintained constant, e.g. p(t) = ~,, then, when a given residual population, n, remains undesorbed,
may be noted that the method of extraction of E(n) from observed data is identical to the earlier case where T(t) was prescribed. The essential difference in the two approaches is that in one the tempering function is predefined and controlled and the parameter log p(t)/n varies and is measured whilst in the other the parameter log p(t)/n is predetermined and controlled and the tempering function must be consequentially varied and measured. 3. Parallel desorption processes
As in the case of serial desorption, the rate of desorption from centres with activation energy E, when the system is exposed to a tempering schedule T(t) is given by equation (3) with E replacing E(n). For an initial population density, N,, in such centres the formal solution to equation (3) is :
kT(7)
,,
~-i =
"ro
exp
(5)
and T~U~) represents the temperature T~ at time t. when n is attained. If the study is repeated with an identical initial population, no, but with a different, but constant, desorption rate, ~_~, then when the undesorbed population is again n, exp
exp
(9)
kru)
or
~lt~
~, =
dt
(4)
\ Tl(tl) ]
where n=n0
,
Z
(6)
p ( T , t) = N 0 e x p ro
(
--E
kT(t) -
£'
dT!
,,
dt % ' d T "exp
(
E
kT(t)
)1 (10)
where T,~ is the initial temperature. In a parallel desorption system the atoms are distributed, initially, amongst centres of different activation energies. If n.(E)($Eis the initial population density in centres with activation energies in the range E ~ E+SE, the desorption rate from such centres is given, from equation (10), by:
where
n=n,
~:t2
(7)
pE(T,t)
n°cSE exp
,d,
and Te(te) is the temperature (and time) when n is again attained. Combining equations (4) (7) yields the result
logl2=E(n){] t, Te(te)
l }. Tl(/i)
(8)
Consequently by determining the times t and t2 and corresponding temperatures Tt and T2 at which the same residual populations n arc achieved using differing tempering schedules, T(t), but which both result in two different but constant desorption rates ,:~ and ,~:, equation (8) can be used to determine E(n) wllues as a function of n. This procedure can be generalized by requiring that, by appropriate variation of T(t), p(t) can be controlled to be a prescribed function of time. Ifp(t) is prescribed, then at any residual population,
n
no-
f'
dtp(t), the p a r a m e t e r Iog~ p(t)
0
I1
is then also a prescribed function of time for all values of n. Measurement of the variation of I/T(t) with time necessary to maintain this prescription and determination of the slope of the log,,
p(t) ;' 1 n /' r ( t )
function defines the E(t) or E(n) function. In this general case it 196
kr(t)-
dT
TO
r0 d T exp
(
-/,@(t)
'
(11)
The total, observable, instantaneous desorption rate from all cenlres is then obtained by integration over the whole energy spectruln.
p ( T , t ) = ro
"r0
) d E ' n o ( E ) • exp
-
kT(t)
"
-,, d T
(12)
The exponential term in equation (12) is seen to be in the nature of a convolution function s " which in general, is a function of E a n d T(or t), i.e. A(E, T). The effect of this convolution function is, essentially, to map or transform, the energy distribution ['unction no(E), into the observed desorption rate temperature domain. The precise form of this mapping depends upon the form of the tempering function, dt/dT, and the bchaviour of A(E, t) has been determined ~ t t lk)r dt/dT - rp ~• T ~' ['or values o f p = 0, I and 2, i.e. for a set of prescribed tempering functions. The general lk)rln of A(E, T) lbr a given E, and allp, is a constant (equal to unity) for T < T,,,, a rapid decline through T = T,,, towards zero R)r T > T,,. The temperature 7",,, is, in fact, that at which lb.(t, T), in equation (11), possesses a maximum value.
G Carterand D K Daukeev: Deduction of desorption energy variation For all p values T,, is approximately linearly ~'2 related to the corresponding energy E whilst for p = 1 the linear relationship is exact 3. The temperature or time width, over which rapid decline in A(E, T) occurs may also be determined ~ '~ and an example of the behaviour of A(E, T) is shown in Figure 1 for p = 1 which leads to the exponential tempering function T = T Oe"', where a is a constant, The temperature narrowness, AT, of A(E, T) compared to T,~ is notable and this is a quite general result for all tempering functions and energy, E, values. If the tempering function is prescribed then A(E, T) can be evaluated ~'9 either analytically or numerically for all E and T and inserted into equation (12). If, then, initial forms of the no(E) function are assumed the integral of equation (12) can be performed and the predicted form of p(T, t) behaviour determined. This can then be compared with the measured p(T, t) behaviour and if discrepant, the form of no(E) adjusted to attempt better match. This procedure may be iterated until best fit is achieved and an optimised no(E) function is determined. Alternatively, detailed analysis s9 shows that A(E, T) can, for all p, E and T be given rather precise analytic form so that if, then, no(E) is also assumed to have analytic expression (such as a power series Z,K,E'), the integral of equation (12) can be evaluated analytically (e.g. as a power series Y~L,T~). Comparison of the measured p(T, t) behaviour with the predicted behaviour then allows evaluation of the L~ values and then the K~ values to which they are closely related ~~. This procedure can be undertaken with reasonable accuracy forp = 0 and 2 with total precision forp = 1 (refs 8, 9) since in this latter case A(E, T) relaxes to the form A(E/kT). If no(E) is then defined by a power series expansion the predicted p(T, t) is an exact analogue in a power series expansion in temperatureSC This case is unique 9 and is the only tempering for which A(E, T) is the form A(E/kT) which allows precise specification of the p(T, t) function for assumed forms of no(E). In all other conditions the specification of p(T, t) is less exact so that comparisons with measured p(T, t) data allow only good approximations to the no(E) function. Clearly, therefore, the no(E) function can be derived with more or less precision, by comparing predicted and measured p(T, t) functions provided that the tempering function is initially prescribed or defined. The question is now, if p(T, t) is prescribed, can the interactively
controlled behaviour of the tempering function be used to deduce the no(E) population? From the preceding discussion the answer is, with analytic precision, no! Fundamentally the reason is that, only for the exponential tempering function, p = I and T = To e"', can the convolution function be written in the form A(E/kT), so that ifn0(E) is written as no(E/kT" kT), the integral of equation (12) can be given analytic form as a function of k T. No other tempering function allows this precise mapping from the initial population-energy domain to the desorption rate temperature domain so that no other arbitrary, but interactively controlled, tempering function will allow precise deconvolution. However a reasonable first approximation to the n(E) function can be obtained by this method in the following manner. Since the general nature of the A(E, T) function is of rapidly declining form over a small temperature range it is possible to approximate it as either '3'4 a step-down function from unity to zero at T,, or a linearly declining ramp function ~9 from unity to zero, centred on 7",,,and of width AT. The former approximation is also illustrated in Figure 1 and has been shown s'9 to be an acceptable approximation provided that the energy 'width' of the no(E ) distribution does not exceed about an order of magnitude between the lowest and highest energies. This approximation is equivalent to assuming that all depopulation occurs instantaneously and uniquely at a temperature T,,(E) corresponding to its associated activation energy E. Analysis has shown' 3 that relatively independent of the form of the tempering function, TIn(E) and E a r e closely linearly related (exactly forp = I) where E
Tm(E)-(log~rlo)k-30
E
k.
(13)
The significance of the convolution process, in terms of the stepdown nature of the convolution function resulting from the tempering schedule now becomes transparent. As temperature is increased instantaneous depopulation of centres occurs as through a narrow window with each temperature corresponding to a unique activation energy. From equation (13), if temperature increased from T to T+6T, centres depopulate completely over an energy range
E 30k
to
E+6E -30k
In this temperature (and energy) range a total desorption of
no(E)6E occurs, so that the rate of desorption, p, is equivalent 1.0----
~
to
4-
(E)6E no n-~
5t
Tm
where 6t is the incremental time for the incremental temperature increase. Consequently
0,5
dE dT no(E).3ok dT. p = ~,,(E) d r " d , -~ dt 1 435
1 /,,50
1 /.,65
T(K)
1 /,'.80
II
/,,95
1~
510
I 525
Figure 1. The behaviour of the convolution function with increasing temperature T for the exponential tempering schedule: dT/dt = r~T. Parameters:E-~ 1 . 2 5 e V ; r = 2 x 1 0 2s ~ ; r 0 - 10 13s.
(14)
In this level of approximation, the desorption rate at temperature T, and equivalent activation energy 30kT, is linearly related to the initial population density no(E) and the tempering rate dT/dt. If the parameter p(dT/dt) ~ is plotted, from experimental measurements, as a function of temperature then this yields immediately 30k no(E) as a function of E. Clearly either 197
G Carter a n d D K Daukeev : Deduction of desorption energy variation
the tempering function, and hence dT/dt, can be prescribed and p(T, t) measured or p(T, t) can be prescribed and dT/dt interactively controlled and measured. In both cases the parameter p(dT/dt) ' is determined to deduce n0(E). Although the method is only approximate it will usually be good enough to give a clear indication of no(E) over most of the energy domain whether prescribed tempering or prescribed desorption rate conditions are applied. Full precision can only be obtained however by studies employing the prescribed exponential tempering function. In all cases it should be recognised that some imprecision will occur at the lower and upper limits of the energy (and equivalent temperature) domain since desorption rates will respectively rise and fall rapidly with increasing temperature and be relatively difficult to measure or control precisely.
it must be observed that the prescribed desorption rate method requires interactive feedback control l¥om observed desorption rate/prescribed desorption rate differences to mediate the tempering function. The prescribed tempering schedule method requires internal feedback control in the heating system only from intercomparison of measured and prescribed tempering functions. The present analysis has only been intended to bE preliminary to indicate the general application, and limitations, of both control approaches. It is intended to explore, in greater detail, the relative levels of accuracy attainable by both methods and to include studies of multiple order desorption processes and precurso, diffusion. These investigations together with considerations of system design to meet the necessary control criteria will be communicated subsequently.
4. Discussion and conclusions It has been demonstrated that information on the activation energy variation with population density in serial desorption processes and the initial population distribution with activation energy in parallel desorption processes can be determined by either prescribing the tempering function and measuring dcsorption rate or by prescribing the desorption rate and interactively controlling and measuring the tempering rate. In the serial dcsorption process determination of the energy variation is exact (to within the limits of experimental error) in both prescriptions but, in the parallel desorption process, determination of the initial population distribution is only approximate in both cases except for the prescribed exponential tempering function mode. Both techniques are technically achievable ~2s t~.~2 and preliminary experiments in the second author's laboratory have demonstrated feasibility but without, as yet, detailed data whilst
198
References i G Carter, Vacuum, 12, 245 (1962). 2p A Redhead, Vacuum, 12, 203 (1962). ;G Carter, W A Grant. G Farrell and J S Colligon. l'acmmz, 18, 263 (1968). 4W A Grant and G Carter, Vacuum. 15, 13 (1965). G Carter, D G Armour and P Bailey, Vacuum, 33, 317 (1983). ('G Farrell, W A Grant, K Erents and G Carler, Vacuum, 16, 2t~5 (1966). "G Carter, Vacuum, 26, 329 (1976). s G Carter, P Bailey. D G Armour and R Collins, Vacuum, 32, 233 ( 1~)g2). "G Carter, P Bailey and D G Armour, Vacuum, 34, 797 (1984). ~')G Carter, P Bailey and D G Armour, Vacmm~,34, 8()I (1984). J~G Carter, D G Armour. P Bailey and G Kiriakidis. Vacuum. 35, 39 (1985). 12D K Daukeev, Zh R Zhotobaev and 1 V Hromushin..ll
0042-207X/91S3.00+.00 ,:~'~1990 Pergamon Press plc
Printed in Great Britain
The application of interactive controlled t e m p e r i n g for deduction of energy variable or distributed desorption processes G C a r t e r , Department of Electronic and Electrical Engineering, University of Salford, Salford M 5 4WT, UK and D K Daukeev,
Institute of Nuclear Physics, Kazakh Academy of Sciences, Alma-Ata, Kazakhstan, USSR
received 2 December 1989
The processes of serial and parallel desorption are considered when the system is exposed to either prescribed tempering and desorption rate is measured or desorption rate is prescribed and tempering is interactively controlled. It is shown that both methods can lead to accurate determination of desorption activation energy variation with population in the serial mode, but only close approximations to the initial population-activation energy distribution in the parallel mode. The exception is the prescribed exponential tempering function which allows exact determination.
1. Introduction Thermal evolution spectrometry has been, for m a n y years, and continues to be a powerful a n d useful m e t h o d for determining the energies of binding of gases to solids. In the usual manifestation of this technique the solid is heated (tempered) in a prescribed, controlled m a n n e r as a function of increasing time, and the rate of evolution (desorption) of gas is continuously monitored. A great deal of theoretical analysis has been devoted to describing the evolution rate b e h a v i o u r for a variety of system parameters. These include evaluation of evolution rates for processes with a single, unique activation energy ~'2 in which the desorption step is either first or higher order in atomic concentration a n d where the temperature, T, is raised, with time t, according to the function d T / d t = rt, TP where r v is a c o n s t a n t and p = 0 or 2 ~: or p - l ~. Studies have also been made for systems where the activation energy is distributed :'4 or variable 5. If the activation energy is distributed this implies the existence of a p o p u l a t i o n of either discrete or c o n t i n u o u s nature sorption centres from which evolution occurs in parallel whereas if activation energy is variable it implies a dependence of activation energy on total i n s t a n t a n e o u s p o p u l a t i o n a n d evolution occurs as a serial process. Investigations have also been made of systems where diffusion processes 67 occur as a precursor to evolution. In all of these systems substantial a t t e n t i o n has been devoted ~ 5.s t e to the d e c o n v o l u t i o n of m e a s u r e d evolution rates to determine system p a r a m e t e r s such as activation energy and its p o p u l a t i o n distribution or variation. As already noted such studies prescribe the tempering function ( d T / d t ) a n d the measured variable is evolution rate, p. Recently, however, Daukeev el al t: have a d o p t e d an alternative a p p r o a c h in which p was initially prescribed (a c o n s t a n t in the Daukeev el al study 12) and the tempering rate d T / d t was interactively controlled to m a i n t a i n this prescription. It was shown that the t e m p e r a t u r e / t i m e behav-
iour could then be used to determine activation energy variation in a serial evolution system. In the present c o m m u n i c a t i o n this concept is extended and generalized to show how activation energy variation can be determined exactly for serial evolution processes a n d approximately for parallel evolution processes. A l t h o u g h analysis is possible for more complex systems, such as higher order dependence on i n s t a n t a n e o u s p o p u l a t i o n or precursor diffusion before desorption, it is more straightforward to consider single step desorption a n d first order dependence in p o p u l a t i o n density. This is the system considered here.
2. Serial desorption processes In such a system the i n s t a n t a n e o u s desorption rate, p, may be represented, at an i n s t a n t a n e o u s p o p u l a t i o n density, n, by the relationship dn
17
P = -- d t
z
(1)
where the relaxation time constant, r, is given by a Frenkel type relation,
£(n) r = re exp k T i t )
(2)
where % is an inverse attempt frequency of order 10 ~ s a n d E ( n ) is the desorption activation energy which is a function of i n s t a n t a n e o u s p o p u l a t i o n density, n. C o m b i n i n g equations (1) and (2) yields the result p =
"
r0
exp
(
-
T(O /
.
(3)
If the temperature, T, is increased with time, the desorption rate
195
G Carter a n d D K Daukeev . Deduction of desorption energy variation
p varies and the residual population n decreases to
i
n0-
t
p dt )
where n~, is the initial population density. This may be determined by desorbing all gas and so p and n may be measured for all T (and 1). Consequently if 5, according to equation (3), log p/n is plotted as a function of 1/T, the behaviour of E(n) is immediately recovered from the slope of the curve at each n value. It is notable that the form of the measured function is independent of the form of variation of T with time. On the other hand it is possible ~= by appropriate interactive control of the tempering function T(t) to prescribe the experimental behaviour of p ( t ) - p(T). For example '2, if T~(t) is controlled such that it(t) is maintained constant, e.g. p(t) = ~,, then, when a given residual population, n, remains undesorbed,
may be noted that the method of extraction of E(n) from observed data is identical to the earlier case where T(t) was prescribed. The essential difference in the two approaches is that in one the tempering function is predefined and controlled and the parameter log p(t)/n varies and is measured whilst in the other the parameter log p(t)/n is predetermined and controlled and the tempering function must be consequentially varied and measured. 3. Parallel desorption processes
As in the case of serial desorption, the rate of desorption from centres with activation energy E, when the system is exposed to a tempering schedule T(t) is given by equation (3) with E replacing E(n). For an initial population density, N,, in such centres the formal solution to equation (3) is :
kT(7)
,,
~-i =
"ro
exp
(5)
and T~U~) represents the temperature T~ at time t. when n is attained. If the study is repeated with an identical initial population, no, but with a different, but constant, desorption rate, ~_~, then when the undesorbed population is again n, exp
exp
(9)
kru)
or
~lt~
~, =
dt
(4)
\ Tl(tl) ]
where n=n0
,
Z
(6)
p ( T , t) = N 0 e x p ro
(
--E
kT(t) -
£'
dT!
,,
dt % ' d T "exp
(
E
kT(t)
)1 (10)
where T,~ is the initial temperature. In a parallel desorption system the atoms are distributed, initially, amongst centres of different activation energies. If n.(E)($Eis the initial population density in centres with activation energies in the range E ~ E+SE, the desorption rate from such centres is given, from equation (10), by:
where
n=n,
~:t2
(7)
pE(T,t)
n°cSE exp
,d,
and Te(te) is the temperature (and time) when n is again attained. Combining equations (4) (7) yields the result
logl2=E(n){] t, Te(te)
l }. Tl(/i)
(8)
Consequently by determining the times t and t2 and corresponding temperatures Tt and T2 at which the same residual populations n arc achieved using differing tempering schedules, T(t), but which both result in two different but constant desorption rates ,:~ and ,~:, equation (8) can be used to determine E(n) wllues as a function of n. This procedure can be generalized by requiring that, by appropriate variation of T(t), p(t) can be controlled to be a prescribed function of time. Ifp(t) is prescribed, then at any residual population,
n
no-
f'
dtp(t), the p a r a m e t e r Iog~ p(t)
0
I1
is then also a prescribed function of time for all values of n. Measurement of the variation of I/T(t) with time necessary to maintain this prescription and determination of the slope of the log,,
p(t) ;' 1 n /' r ( t )
function defines the E(t) or E(n) function. In this general case it 196
kr(t)-
dT
TO
r0 d T exp
(
-/,@(t)
'
(11)
The total, observable, instantaneous desorption rate from all cenlres is then obtained by integration over the whole energy spectruln.
p ( T , t ) = ro
"r0
) d E ' n o ( E ) • exp
-
kT(t)
"
-,, d T
(12)
The exponential term in equation (12) is seen to be in the nature of a convolution function s " which in general, is a function of E a n d T(or t), i.e. A(E, T). The effect of this convolution function is, essentially, to map or transform, the energy distribution ['unction no(E), into the observed desorption rate temperature domain. The precise form of this mapping depends upon the form of the tempering function, dt/dT, and the bchaviour of A(E, t) has been determined ~ t t lk)r dt/dT - rp ~• T ~' ['or values o f p = 0, I and 2, i.e. for a set of prescribed tempering functions. The general lk)rln of A(E, T) lbr a given E, and allp, is a constant (equal to unity) for T < T,,,, a rapid decline through T = T,,, towards zero R)r T > T,,. The temperature 7",,, is, in fact, that at which lb.(t, T), in equation (11), possesses a maximum value.
G Carterand D K Daukeev: Deduction of desorption energy variation For all p values T,, is approximately linearly ~'2 related to the corresponding energy E whilst for p = 1 the linear relationship is exact 3. The temperature or time width, over which rapid decline in A(E, T) occurs may also be determined ~ '~ and an example of the behaviour of A(E, T) is shown in Figure 1 for p = 1 which leads to the exponential tempering function T = T Oe"', where a is a constant, The temperature narrowness, AT, of A(E, T) compared to T,~ is notable and this is a quite general result for all tempering functions and energy, E, values. If the tempering function is prescribed then A(E, T) can be evaluated ~'9 either analytically or numerically for all E and T and inserted into equation (12). If, then, initial forms of the no(E) function are assumed the integral of equation (12) can be performed and the predicted form of p(T, t) behaviour determined. This can then be compared with the measured p(T, t) behaviour and if discrepant, the form of no(E) adjusted to attempt better match. This procedure may be iterated until best fit is achieved and an optimised no(E) function is determined. Alternatively, detailed analysis s9 shows that A(E, T) can, for all p, E and T be given rather precise analytic form so that if, then, no(E) is also assumed to have analytic expression (such as a power series Z,K,E'), the integral of equation (12) can be evaluated analytically (e.g. as a power series Y~L,T~). Comparison of the measured p(T, t) behaviour with the predicted behaviour then allows evaluation of the L~ values and then the K~ values to which they are closely related ~~. This procedure can be undertaken with reasonable accuracy forp = 0 and 2 with total precision forp = 1 (refs 8, 9) since in this latter case A(E, T) relaxes to the form A(E/kT). If no(E) is then defined by a power series expansion the predicted p(T, t) is an exact analogue in a power series expansion in temperatureSC This case is unique 9 and is the only tempering for which A(E, T) is the form A(E/kT) which allows precise specification of the p(T, t) function for assumed forms of no(E). In all other conditions the specification of p(T, t) is less exact so that comparisons with measured p(T, t) data allow only good approximations to the no(E) function. Clearly, therefore, the no(E) function can be derived with more or less precision, by comparing predicted and measured p(T, t) functions provided that the tempering function is initially prescribed or defined. The question is now, if p(T, t) is prescribed, can the interactively
controlled behaviour of the tempering function be used to deduce the no(E) population? From the preceding discussion the answer is, with analytic precision, no! Fundamentally the reason is that, only for the exponential tempering function, p = I and T = To e"', can the convolution function be written in the form A(E/kT), so that ifn0(E) is written as no(E/kT" kT), the integral of equation (12) can be given analytic form as a function of k T. No other tempering function allows this precise mapping from the initial population-energy domain to the desorption rate temperature domain so that no other arbitrary, but interactively controlled, tempering function will allow precise deconvolution. However a reasonable first approximation to the n(E) function can be obtained by this method in the following manner. Since the general nature of the A(E, T) function is of rapidly declining form over a small temperature range it is possible to approximate it as either '3'4 a step-down function from unity to zero at T,, or a linearly declining ramp function ~9 from unity to zero, centred on 7",,,and of width AT. The former approximation is also illustrated in Figure 1 and has been shown s'9 to be an acceptable approximation provided that the energy 'width' of the no(E ) distribution does not exceed about an order of magnitude between the lowest and highest energies. This approximation is equivalent to assuming that all depopulation occurs instantaneously and uniquely at a temperature T,,(E) corresponding to its associated activation energy E. Analysis has shown' 3 that relatively independent of the form of the tempering function, TIn(E) and E a r e closely linearly related (exactly forp = I) where E
Tm(E)-(log~rlo)k-30
E
k.
(13)
The significance of the convolution process, in terms of the stepdown nature of the convolution function resulting from the tempering schedule now becomes transparent. As temperature is increased instantaneous depopulation of centres occurs as through a narrow window with each temperature corresponding to a unique activation energy. From equation (13), if temperature increased from T to T+6T, centres depopulate completely over an energy range
E 30k
to
E+6E -30k
In this temperature (and energy) range a total desorption of
no(E)6E occurs, so that the rate of desorption, p, is equivalent 1.0----
~
to
4-
(E)6E no n-~
5t
Tm
where 6t is the incremental time for the incremental temperature increase. Consequently
0,5
dE dT no(E).3ok dT. p = ~,,(E) d r " d , -~ dt 1 435
1 /,,50
1 /.,65
T(K)
1 /,'.80
II
/,,95
1~
510
I 525
Figure 1. The behaviour of the convolution function with increasing temperature T for the exponential tempering schedule: dT/dt = r~T. Parameters:E-~ 1 . 2 5 e V ; r = 2 x 1 0 2s ~ ; r 0 - 10 13s.
(14)
In this level of approximation, the desorption rate at temperature T, and equivalent activation energy 30kT, is linearly related to the initial population density no(E) and the tempering rate dT/dt. If the parameter p(dT/dt) ~ is plotted, from experimental measurements, as a function of temperature then this yields immediately 30k no(E) as a function of E. Clearly either 197
G Carter a n d D K Daukeev : Deduction of desorption energy variation
the tempering function, and hence dT/dt, can be prescribed and p(T, t) measured or p(T, t) can be prescribed and dT/dt interactively controlled and measured. In both cases the parameter p(dT/dt) ' is determined to deduce n0(E). Although the method is only approximate it will usually be good enough to give a clear indication of no(E) over most of the energy domain whether prescribed tempering or prescribed desorption rate conditions are applied. Full precision can only be obtained however by studies employing the prescribed exponential tempering function. In all cases it should be recognised that some imprecision will occur at the lower and upper limits of the energy (and equivalent temperature) domain since desorption rates will respectively rise and fall rapidly with increasing temperature and be relatively difficult to measure or control precisely.
it must be observed that the prescribed desorption rate method requires interactive feedback control l¥om observed desorption rate/prescribed desorption rate differences to mediate the tempering function. The prescribed tempering schedule method requires internal feedback control in the heating system only from intercomparison of measured and prescribed tempering functions. The present analysis has only been intended to bE preliminary to indicate the general application, and limitations, of both control approaches. It is intended to explore, in greater detail, the relative levels of accuracy attainable by both methods and to include studies of multiple order desorption processes and precurso, diffusion. These investigations together with considerations of system design to meet the necessary control criteria will be communicated subsequently.
4. Discussion and conclusions It has been demonstrated that information on the activation energy variation with population density in serial desorption processes and the initial population distribution with activation energy in parallel desorption processes can be determined by either prescribing the tempering function and measuring dcsorption rate or by prescribing the desorption rate and interactively controlling and measuring the tempering rate. In the serial dcsorption process determination of the energy variation is exact (to within the limits of experimental error) in both prescriptions but, in the parallel desorption process, determination of the initial population distribution is only approximate in both cases except for the prescribed exponential tempering function mode. Both techniques are technically achievable ~2s t~.~2 and preliminary experiments in the second author's laboratory have demonstrated feasibility but without, as yet, detailed data whilst
198
References i G Carter, Vacuum, 12, 245 (1962). 2p A Redhead, Vacuum, 12, 203 (1962). ;G Carter, W A Grant. G Farrell and J S Colligon. l'acmmz, 18, 263 (1968). 4W A Grant and G Carter, Vacuum. 15, 13 (1965). G Carter, D G Armour and P Bailey, Vacuum, 33, 317 (1983). ('G Farrell, W A Grant, K Erents and G Carler, Vacuum, 16, 2t~5 (1966). "G Carter, Vacuum, 26, 329 (1976). s G Carter, P Bailey. D G Armour and R Collins, Vacuum, 32, 233 ( 1~)g2). "G Carter, P Bailey and D G Armour, Vacuum, 34, 797 (1984). ~')G Carter, P Bailey and D G Armour, Vacmm~,34, 8()I (1984). J~G Carter, D G Armour. P Bailey and G Kiriakidis. Vacuum. 35, 39 (1985). 12D K Daukeev, Zh R Zhotobaev and 1 V Hromushin..ll