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Thermal resolution of multiple order desorption energy spectra
Thermal Resolution of Multiple Order Desorption Energy Spectra G. CARTER Department
of Electrical
Engineering,
Liverpool
University,
England
(Received 23 November 1962 : accepted I February 1963)
The desorption of adsorbed atoms, of different energies of desorption, from a surface is investigated when the surface is exposed to a time varying temperature schedule. Any orders of desorption reaction is supposed, and the two specific heating schedules, the reciprocal temperature and linear temperature/time sweeps are considered. The ability of these schedules to allow resolution of the desorption from two types of adsorption site of slightly different adsorption energy is computed. Introduction
In a previous cornmunicationt (subsequently referred to as I) the author has discussed desorption of adatoms from a solid surface, under the influence of an increasing temperature function applied to the surface. Adatoms bound to the surface with varying binding energies were considered and the ability of linearly increasing temperature/time and reciprocal temperature/time sweeps to resolve individual energy adatoms was investigated. In I it was assumed that no interaction between the various energy states occurred so that the depletion of each type of site could be described by the reaction kinetics for this site only. Further it was assumed that only first order kinetics operated, so that desorption was described simply by the population density of the filled sites. In the present work the analogy between optical line spectra and desorption energy spectra is extended to include any order of reaction. The resolution for this general case is obtained for the above temperature/time functions and the two important cases of 1st and 2nd order reactions studied. Theory. In I it was shown that if either of the prescribed temperature was imposed on the surface, the maximum desorption rates for sites of binding energy Qi and Qi + A Qi The times occurred at different times tm and tm + At,. at which the desorption rates where e-1 of the maxima were also deduced (te and te + d tJ and a criterion for resolution of the two peaks was proposed. The criterion was deduced assuming that two energy peaks would be completely resolved if the higher exponential rate of the lower energy sites occurred earlier in time than the lower exponential rate of the higher energy sites. Reference to I and Fig. 1 indicates that this criterion is essentially At,,t > Atme + Atmteg, and it was shown in I that this criterion is essentially similar to the Rayleigh criterion for the resolution of two optical spectral lines. Before considering the resolution of two energy states, when desorption from these states is of order x in the adatom concentration, it is necessary to consider some properties of such desorption, as below.
tm
te’
te
tm’
s-d,, < > Lit e’m’ < < A,,’ > Temperature
or time
FIG. 1
For desorption of order x from sites of concentration the governing rate equation has the form
n,
(1) where A is some unspecified rate constant. A is in general, related to the energy of activation desorption by a Frenkel type of equation, A
=
Ao exp
- gT (
a9
>
for
G.
90
where Q is the energy of activation the absolute surface temperature.
for desorption
CARTER
and T is
If the temperature, when the desorption rate is e-1 of the maximum rate, is written Te, then the preceding equation reduces, on taking logarithms to
Equation (1) is integrable to 1
=
Re7)m(2)
_ /TIAoexP(-
nX-1
Again, differentiation leads to d%
- &))] .
(*;")(exP(gm
(8)
Y, then Equ. (8) becomes
A
+ .x Q
0
dt2
logei;-
of Equ. (1) and some manipulation,
=
RT2
i
.
dT
1+ Y =
=
2:
nx-lj
exp(-&)I
-(1-x)exp
Y) 1
. (9)
Yl =
-1ogu)x -1.8
Y2
Kx
=
where K is a constant
N 1.
(10)
Yi corresponds to the temperature Tel below the rate maximum temperature Tm and Y2 to the temperature Tez above T,,,.
=
{exp(-fF)-
+log,(l
Equation (9) can be solved for Y graphically by assuming integral values of x and plotting both sides of Equ. (9). For values of x between 2 and 10 (a sufficiently wide range for all practical purposes) it can be shown that Y can be adequately represented by
where the suffix refers to the value at rate maximum. In order to proceed further it is necessary to consider the exact form of the temperature/time function. Initially we shall consider the reciprocal case 1/T = a + bt, where a = 1/TO the reciprocal initial temperature and b is a constant. This allows integration of Equ. (2) to 1 i
;ri1 10s.;
dt
This leads, as in I, to the condition for maximum rate
, (4)
Thus Atme + Atmfer
and condition (3) becomes 1
xn,x-1
on substitution
A,R
(5)
bQ
Since 1/Tm is generally < 1/TO the exponential term involving TO can be neglected in Equ. (4) and substitution in Equ. (5)
Identification of this result with Equ. (7), then deduction of the expression for the resolution
It1=
{log,(-RA,n@/bQ)
Q
AQx
leads to the rate maximum condition -AORnOX-l
bQ
(6)
’
If sites of infinitesimally higher desorption energy Q + dQ are considered, then differentiation of Equ. (6) with respect to time, allows, as in I, an evaluation of the time difference &?lm~ between the peak rates for these two site states, i.e. At,,,,)
of
Differentiation of Equ. (4) allows expression of the desorption rate from energy Q sites at a time t as
w
This simplifies for x = 2, to
Q AQz I-I
+ 1 -2 cash-1 de
log,(-RA,n,/bQ)
=
--.~~ ~~~~- 4 coshmi de
eQIRT~
=
AolRTm2noX-~ ~~~~ ---~
dn
>t
*
9
As a result of a particularly straight-forward expression for Atme + Atm~e~. Analysis of the linear temperature/time sweep is not as amenable to tractable analysis as the above, but by making the assumptions as in I, that Tr,, 2: Te,* one deduces expressions analogous to Equ. (7) and Equ. (9) above
(7)
=
+ 1} -Kx
Kx + loglox + 1.8
allows
bQ
,
and
=
1+ Y = Thus in terms of the maximum desorption
(1 -x) exp( - Q/RT)
x/x-1
-xw4-Q&IRTm) *The assumption
of T, N T, is correct
within
log,{:
-(liT)($)2
exp Y]
.
(14)
As in I, these expressions are not readily soluble and even graphical analysis is tedious. However, as noted in I, the general behaviour of the Resolution, as a function of the variable parameters involved, follows the pattern of the reciprocal temperature function. This is evident from Equ. (14) where the approximation Te E Tm would equate
rate,
-exp( - Q/RT,)
x
10
per cent up to x = 5, for all values of
b and Q.
91
Thermal Resolution of Multiple Order Desorption Energy Spectra Equ. (9) and Equ. (14). Consequently it is felt unproductive to pursue this analysis further but to note that results obtained for the reciprocal case will largely apply to the linear temperature function. Investigation of Equ. (12) for the resolution, reveals that as anticipated, and indicated in I for the first order case, the resolution decreases both as the desorption energy Q, of the sites increases, and also as the heating rate (described by - 1lb) increases. However as the initial site population n, increases, the resolution also increases (except for the first order case where the resolution is population independent). Investigation of the maximum rate shows that this also increases as n, increases, hence one obtains an unusual physical situation in which both the resolution and sensitivity can be increased simultaneously. It is difficult to predict the behaviour of the resolution with increasing x, the order of reaction. According to Equ. (12) the resolution, in general, increases with x, but simultaneously the rate constant A0 will generally decrease as x increases which induces a reduction in the resolution. Consequently it is not possible to specify the actual form of the resolution variation with x since a compensating effect arises from the variation of A,. A final interesting point may be deduced from the equation for the maximum desorption rate, for on substitution of
Equ. (6) one derives, for the maximum rate dn dt,=I-I
bQno R
1 X/X-’ 0 x-
(It is readily checked that this concurs with the result for x = 2inI.) It is thus seen that the maximum rate dependence upon the initial sorbed quantity cannot be utilised to determine the order of reaction, since this is a linear dependence for all x (even for x = unity from I). However the temperature at maximum rate dependence is a function of x (see Equ. (6)) and hence decision to the reaction order may be reached. We conclude that the temperature displacement schedule is a convenient method for resolving multiple desorption However it is energy spectra for all reaction orders. difficult to predict the value of the resolution for increasing reaction orders since the rate constant is rarely known. In order to achieve adequate resolution of a given region of an energy spectrum it is necessary to suitably control the heating rate. References 1 G. Carter. Vacuum. 12(S). 11962). 245. 2 P. A. Redhead, V&z&‘@4), (i962), 203. 3 P. A. Redhead, Private communication.
of Electrical
Engineering,
Liverpool
University,
England
(Received 23 November 1962 : accepted I February 1963)
The desorption of adsorbed atoms, of different energies of desorption, from a surface is investigated when the surface is exposed to a time varying temperature schedule. Any orders of desorption reaction is supposed, and the two specific heating schedules, the reciprocal temperature and linear temperature/time sweeps are considered. The ability of these schedules to allow resolution of the desorption from two types of adsorption site of slightly different adsorption energy is computed. Introduction
In a previous cornmunicationt (subsequently referred to as I) the author has discussed desorption of adatoms from a solid surface, under the influence of an increasing temperature function applied to the surface. Adatoms bound to the surface with varying binding energies were considered and the ability of linearly increasing temperature/time and reciprocal temperature/time sweeps to resolve individual energy adatoms was investigated. In I it was assumed that no interaction between the various energy states occurred so that the depletion of each type of site could be described by the reaction kinetics for this site only. Further it was assumed that only first order kinetics operated, so that desorption was described simply by the population density of the filled sites. In the present work the analogy between optical line spectra and desorption energy spectra is extended to include any order of reaction. The resolution for this general case is obtained for the above temperature/time functions and the two important cases of 1st and 2nd order reactions studied. Theory. In I it was shown that if either of the prescribed temperature was imposed on the surface, the maximum desorption rates for sites of binding energy Qi and Qi + A Qi The times occurred at different times tm and tm + At,. at which the desorption rates where e-1 of the maxima were also deduced (te and te + d tJ and a criterion for resolution of the two peaks was proposed. The criterion was deduced assuming that two energy peaks would be completely resolved if the higher exponential rate of the lower energy sites occurred earlier in time than the lower exponential rate of the higher energy sites. Reference to I and Fig. 1 indicates that this criterion is essentially At,,t > Atme + Atmteg, and it was shown in I that this criterion is essentially similar to the Rayleigh criterion for the resolution of two optical spectral lines. Before considering the resolution of two energy states, when desorption from these states is of order x in the adatom concentration, it is necessary to consider some properties of such desorption, as below.
tm
te’
te
tm’
s-d,, < > Lit e’m’ < < A,,’ > Temperature
or time
FIG. 1
For desorption of order x from sites of concentration the governing rate equation has the form
n,
(1) where A is some unspecified rate constant. A is in general, related to the energy of activation desorption by a Frenkel type of equation, A
=
Ao exp
- gT (
a9
>
for
G.
90
where Q is the energy of activation the absolute surface temperature.
for desorption
CARTER
and T is
If the temperature, when the desorption rate is e-1 of the maximum rate, is written Te, then the preceding equation reduces, on taking logarithms to
Equation (1) is integrable to 1
=
Re7)m(2)
_ /TIAoexP(-
nX-1
Again, differentiation leads to d%
- &))] .
(*;")(exP(gm
(8)
Y, then Equ. (8) becomes
A
+ .x Q
0
dt2
logei;-
of Equ. (1) and some manipulation,
=
RT2
i
.
dT
1+ Y =
=
2:
nx-lj
exp(-&)I
-(1-x)exp
Y) 1
. (9)
Yl =
-1ogu)x -1.8
Y2
Kx
=
where K is a constant
N 1.
(10)
Yi corresponds to the temperature Tel below the rate maximum temperature Tm and Y2 to the temperature Tez above T,,,.
=
{exp(-fF)-
+log,(l
Equation (9) can be solved for Y graphically by assuming integral values of x and plotting both sides of Equ. (9). For values of x between 2 and 10 (a sufficiently wide range for all practical purposes) it can be shown that Y can be adequately represented by
where the suffix refers to the value at rate maximum. In order to proceed further it is necessary to consider the exact form of the temperature/time function. Initially we shall consider the reciprocal case 1/T = a + bt, where a = 1/TO the reciprocal initial temperature and b is a constant. This allows integration of Equ. (2) to 1 i
;ri1 10s.;
dt
This leads, as in I, to the condition for maximum rate
, (4)
Thus Atme + Atmfer
and condition (3) becomes 1
xn,x-1
on substitution
A,R
(5)
bQ
Since 1/Tm is generally < 1/TO the exponential term involving TO can be neglected in Equ. (4) and substitution in Equ. (5)
Identification of this result with Equ. (7), then deduction of the expression for the resolution
It1=
{log,(-RA,n@/bQ)
Q
AQx
leads to the rate maximum condition -AORnOX-l
bQ
(6)
’
If sites of infinitesimally higher desorption energy Q + dQ are considered, then differentiation of Equ. (6) with respect to time, allows, as in I, an evaluation of the time difference &?lm~ between the peak rates for these two site states, i.e. At,,,,)
of
Differentiation of Equ. (4) allows expression of the desorption rate from energy Q sites at a time t as
w
This simplifies for x = 2, to
Q AQz I-I
+ 1 -2 cash-1 de
log,(-RA,n,/bQ)
=
--.~~ ~~~~- 4 coshmi de
eQIRT~
=
AolRTm2noX-~ ~~~~ ---~
dn
>t
*
9
As a result of a particularly straight-forward expression for Atme + Atm~e~. Analysis of the linear temperature/time sweep is not as amenable to tractable analysis as the above, but by making the assumptions as in I, that Tr,, 2: Te,* one deduces expressions analogous to Equ. (7) and Equ. (9) above
(7)
=
+ 1} -Kx
Kx + loglox + 1.8
allows
bQ
,
and
=
1+ Y = Thus in terms of the maximum desorption
(1 -x) exp( - Q/RT)
x/x-1
-xw4-Q&IRTm) *The assumption
of T, N T, is correct
within
log,{:
-(liT)($)2
exp Y]
.
(14)
As in I, these expressions are not readily soluble and even graphical analysis is tedious. However, as noted in I, the general behaviour of the Resolution, as a function of the variable parameters involved, follows the pattern of the reciprocal temperature function. This is evident from Equ. (14) where the approximation Te E Tm would equate
rate,
-exp( - Q/RT,)
x
10
per cent up to x = 5, for all values of
b and Q.
91
Thermal Resolution of Multiple Order Desorption Energy Spectra Equ. (9) and Equ. (14). Consequently it is felt unproductive to pursue this analysis further but to note that results obtained for the reciprocal case will largely apply to the linear temperature function. Investigation of Equ. (12) for the resolution, reveals that as anticipated, and indicated in I for the first order case, the resolution decreases both as the desorption energy Q, of the sites increases, and also as the heating rate (described by - 1lb) increases. However as the initial site population n, increases, the resolution also increases (except for the first order case where the resolution is population independent). Investigation of the maximum rate shows that this also increases as n, increases, hence one obtains an unusual physical situation in which both the resolution and sensitivity can be increased simultaneously. It is difficult to predict the behaviour of the resolution with increasing x, the order of reaction. According to Equ. (12) the resolution, in general, increases with x, but simultaneously the rate constant A0 will generally decrease as x increases which induces a reduction in the resolution. Consequently it is not possible to specify the actual form of the resolution variation with x since a compensating effect arises from the variation of A,. A final interesting point may be deduced from the equation for the maximum desorption rate, for on substitution of
Equ. (6) one derives, for the maximum rate dn dt,=I-I
bQno R
1 X/X-’ 0 x-
(It is readily checked that this concurs with the result for x = 2inI.) It is thus seen that the maximum rate dependence upon the initial sorbed quantity cannot be utilised to determine the order of reaction, since this is a linear dependence for all x (even for x = unity from I). However the temperature at maximum rate dependence is a function of x (see Equ. (6)) and hence decision to the reaction order may be reached. We conclude that the temperature displacement schedule is a convenient method for resolving multiple desorption However it is energy spectra for all reaction orders. difficult to predict the value of the resolution for increasing reaction orders since the rate constant is rarely known. In order to achieve adequate resolution of a given region of an energy spectrum it is necessary to suitably control the heating rate. References 1 G. Carter. Vacuum. 12(S). 11962). 245. 2 P. A. Redhead, V&z&‘@4), (i962), 203. 3 P. A. Redhead, Private communication.