4 Oscillator Strength of Iodine Atoms
4.2 M ETHODS AND R ESULTS
4.2.2 Determination of emission temperature in the EDL
245 250 255 260 265 0,7
0,8 0,9 1,0
(9.8±0.4)*1010 atoms*cm-3 vertically translated by +0.45
(4.6±0.4)*1011 atoms*cm-3
(2.2±0.4)*1011 atoms*cm-3 vertically translated by +0.25
effective optical density ln(I 0/I)
side arm temperature in K
Figure 4.4 The effective absorbance Aeff=ln(I0/I) for six different absorber concentrations (iodine atoms [I]) in the vessel was determined as a function of side arm temperature under otherwise constant conditions.
For clarity the six traces are shown three at a time. In Fig 4b the lower two curves are translated vertically by +0.25 respectively +0.45 for clarity.
where ν0 is the frequency of the line, kB Boltzmann's constant, T the temperature of the gas, m the mass of the emitting atom in kg, and c the velocity of light. Pressure broadening was calculated via the impact approximation, which yields a Lorentz line profile of width (wavenumber):
π m T k 8 c
A n c
v A n c τ
ν 1
δ~ B
coll
L ⋅
⋅ ⋅
= ⋅
⋅
≈ ⋅
≈ ⋅ (4b)
τcoll is the time between collisions according to gas kinetic theory. It is approximated using the number density of gas particles, n, the cross section for collision, A, and v as the mean particle velocity in a gas with Maxwell velocity distribution.
The overall pressure in the reaction vessel (bathgas with I2) was maintained at about 17 mbar corresponding to ≈4⋅1017 molecules⋅cm-3, which is roughly six times larger than what is found for the gas in the lamp. The vessel temperature was maintained at room temperature 293 K.
Even though this temperature is significantly lower than the temperature in the lamp, the higher concentration led to a larger Lorentz line width of 0.0024 cm-1 (293 K) in the vessel.
Yet again, similar as in the lamp, the corresponding Doppler line width was more than one order of magnitude larger (0.06 cm-1 at 293 K). Therefore both, the emission line profile and the absorption line profile were clearly Doppler dominated.
However, the Doppler width of the emission line in the EDL was larger than that of the absorption line by a factor of approximately 1.5 to 2. Therefore it is expected that the absorbances calculated according to Aeff=ln(I0/I) and plotted versus iodine concentration N in the reaction vessel do not follow the proportional relationship postulated in (3) despite the fact that self-absorption is already minimised and assumed negligible.
The data plotted in Fig. 4.5 clearly displays this expected behaviour: The deduced effective absorbances show increasingly reduced values for higher concentrations and clearly deviate from proportional behaviour. They are not proportional to the absorber concentration N in the vessel. In view of finally deducing the oscillator strength and in order to understand and analyse this behaviour quantitatively, the concept of relative absorption Q=(I0–I)/I0 rather than absorbance A=ln(I0/I) will be used in the following. These quantities are related according to Q=1-exp(-A).
0,0 1,0x1012 2,0x1012 3,0x1012 0,0
0,5 1,0 1,5 2,0
measured (calibration) curve Beer-Lambert
optimal effective optical density ln(I0/I)
iodine concentration in atoms*cm-3
Figure 4.5 The optimal effective absorbance Aeff=ln(I0/I) for T < 245 K is plotted as a function of absorber concentration in the reaction vessel. The straight line represents a Beer-Lambert or an at least proportional relationship between absorbance and absorber concentration. The measured data displays increasingly reduced values for large absorber concentrations deviating from a proportional behaviour.
Relative absorption Q can be expressed in terms of Doppler profiles of emission and absorption lines (see [Mitchell and Zemansky 1972] for a detailed derivation of equations (5) to (7)):
( ) ( ) ( )
( )
ν νΦ
ν
⋅ ν
−
⋅ ν Φ
− ν ν Φ
− =
=
∫
∫
∫
d
d ) L k exp(
d I
I Q I
line 0 line
0 line
0
0
0 , (5)
where
( )
δν
ν
−
⋅ ν
⋅
−
⋅ Φ
= ν Φ
2
em , D
0 0
0 exp 4 ln2 (6a)
represents the emission Doppler profile of maximum intensity Φ0, width δνD,em and centre ν0,
( )
δν
ν
−
⋅ ν
⋅
−
⋅
= ν
2
abs , D
0
0 exp 4 ln2
k
k (6b)
designates the absorption Doppler profile of maximum absorption k0 at the line centre ν0 and of width δνD,abs and where L in (5) is the optical path length in the absorbing medium. A variable α is introduced (see (7) and (9)) to express the dependence of Q on emission and absorber temperature. It is defined as the ratio of the Doppler width in emission and the Doppler width in absorption and can be expressed in terms of the two absolute temperatures:
abs em abs
, D
em , D
T
= T δν
= δν
α (7)
This notation is used in listing the “low temperature limits” from the six data sets expressed in terms of relative absorption Q(α) in Table 4.1 along with the corresponding absorber concentration N.
Absorber Concentration N [1011 atoms⋅cm-3]
Relative Absorption Q(α) [dimensionless]
0.98 ± 0.1 0.2607 ± 0.0002
2.2 ± 0.4 0.4423 ± 0.0004
4.6 ± 0.4 0.6082 ± 0.0005
9.0 ± 0.6 0.7517 ± 0.0004
13.8 ± 0.6 0.8002 ± 0.0004
28.0 ± 1.0 0.8498 ± 0.0005
Table 4.1: Relative absorption Q(α) at 183.038 nm as a function of absorber concentration N.
The instrument response function and the fact, that the line profiles are not spectrally resolved are taken into account in (5) by integrating over the line profiles according to (1). Note that the intensity I measured with absorbing atoms in the vessel is expressed in terms of the incident emission line profile. This is multiplied by a Beer-Lambert factor, which is modulated by the profile of the absorbing line k(ν). The absorber line profile according to equation (6b) contains the maximum value k0, which is proportional to the absorber concentration N in the vessel:
N
k0 ∝ (8)
After inserting the expressions (6a) and (6b) into (5) a relationship between the experimentally measured integrated intensities I and I0 and (via k0 and relationship (8)) the absorber concentration N is defined. Using a series expansion equation (5) can be transformed into [Mitchell and Zemansky 1972]:
( ) ( )
∑
∞=
+
α
⋅ +
⋅
⋅
⋅
= −
= − α
1
n 2
n 0 1 n
0 0
n 1
! n
L k 1 I
I ) I
(
Q (9)
If α were known, this expression could then be solved for k0 using Q(α) and L known from the experiment. Yet by reversing the argument we numerically solve equation (9) for k0 using
1. the measured Q(α)=(I0–I)/I0 and L for the different concentrations in the vessel and 2. a value for α dependent on an assumed (trial) Tem ( Tabs being known)
and then check for the proportionality postulated by relationship (8). If the values resulting for k0 are not proportional to the known absorber concentration then the value assumed for the emission temperature is not correct. On the contrary: The better the proportionality between k0 and N, the better the emission temperature is estimated.
800 900 1000 1100
0,980 0,985 0,990 0,995 1,000
923 K ± 50 K
correlation coefficient squared
temperature in K
Figure 4.6 The square of the correlation coefficient achieved for a linear fit k0=σ⋅N (with σ as proportionality constant) is plotted against assumed emission temperature. At T=923 K it clearly reaches a maximum indicating that the proportionality is best and, therefore, the best estimate for the emission temperature is reached. The accuracy is conservatively estimated to ±50 K yielding Tem=(923 ±50) K.
This analysis was performed for various values of emission temperatures. The degree of proportionality was estimated by performing a linear fit according to k0=σ⋅N (with σ as proportionality constant) and by taking the square of the correlation coefficient that resulted from the fit as a measure for proportionality. For Tem=(923 ±50) K the best correlation was achieved with a maximum correlation coefficient squared of 0.9983. Within the error limits of
±50 K the square of the correlation coefficient significantly decreased to 0.997 (see Fig. 4.6).
In Fig. 4.7 k0 is plotted versus concentration for this optimal estimate as well as for two different values that were assumed for emission temperature in order to further illustrate the method.
0,0 1,0x1012 2,0x1012 3,0x1012
-0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30
k0(923K)
k0(573K) vertically translated by -0.05
k0(1073K) vertically translated by +0.05
peak absorbance k 0 at 183.038 nm
iodine concentration in atoms*cm-3
Figure 4.7 The maximum of the absorption profile, k0, as determined from the measured integral intensities is plotted versus absorber concentration in the reaction vessel. Depending on the source temperature chosen in the determination of k0 the data for k0 displays a different curvature. The best estimate for the source temperature is determined by minimising the deviation from proportionality. In addition to the best fit result two other cases at different temperatures are shown to illustrate the method.
The latter are vertically translated by +0.05 respectively –0.05 for clarity.