Evidence of a New Spectral Line Intensity Formula for Optical Emission
S T E N Y N G S T R O M * and B O T H E L I N t Swedish Institute of Space Physics, P.O. Box 812, S-981 28 Kiruna, Sweden
Experimental evidence of a new intensity formula in optical emission spectroscopy is presented. The evidence is based on two different methods of data analysis. The mathematical foundation of analysis of the fluc- tuations in data of repeatedly measured line intensities is presented in detail. This method applied to data from various spectral lamps yields strong support for the new formula. By another more straightforward method, tabulated standard values of absolute intensities of a great num- ber of spectral lines are shown to be in excellent agreement with the new formula. It is suggested that the two methods of analysis can be useful in spectrochemical research for identification of spectral lines and determination of atomic transition rates. The significance of the new formula is discussed in view of Planck's blackbody radiation law.
Ever since about 1920, the Boltzmann distribution law has been employed to describe the steady state of a sam- ple in optical emission spectroscopy experiments. This description yields a spectral line intensity formula which is used by analytical spectroscopists in theoretical ap- plications. This "old formula" is thus based on the as- sumption that the intensity of an atomic spectral line is proportional to the Boltzmann population number of the initial (upper) energy level and the probability rate for the transition to the final (lower) energy state (see, e.g., chapter IV in Herzberg's book1).
The old formula is conventionally written ]1 I~ , = C,~, g i e x p ( - E ~ / k T . e x p ( - E ~ / k T ) (1)
where the indices a, m, and n stand for the spectroscopic sample element and upper and lower energy level, re- spectively. The statistical weight of level "i" is gi, and the photon energy is
ht, mn= E~ - E~ ~- E~n n. (2)
The factor C~, in Eq. I is assumed to include all the proportionality factors affecting the measured intensity I~,. It depends on the various instrument parameters, the concentration of sample atoms and electrons, the absorption, the above-mentioned transition rate, etc. For further details in this context we refer the reader to works by Barnett et al. 2 and Suckewer)
A common practice among analytical atomic spectros- copists was to employ the internal standardization tech-
Received 7 May 1990. * Author to whom correspondence should be sent. "~ Present address: STRI AB, P.O. Box 701, 771 01 Ludvika, Sweden.
nique presented in 1925 by Gerlach. 4 This method is based on study of the intensity ratio of pairs of spectral lines in the sample. Later on, researchers tried to refine Gerlach's original approach in various ways. One way was to employ the intensity formula expressed by Eq. 1. This approach is thoroughly discussed in Ref. 2. It turns out that researchers have experienced serious drawbacks when interpreting their data by means by Eq. 1. It has been necessary to enforce certain "rules of thumb" in order to select good line pairs and discard bad line pairs. Good line pairs are consistent with Eq. 1; the bad ones are not.
By the year 1910, experimental physicists agreed that Planck's blackbody radiation law gave a very good de- scription of the emission spectrum from hot metallic bod- ies. It was established from measurements in the infrared spectral range that the temperature-dependent factor in the radiation intensity formula was Planck's factor [exp(h~/kT) - 1] -1 and not Wien's factor e x p ( - h v / k T ) (for references, see notes to chapter V in Kuhn's bookS). In fact it seems that Langley's work in the mid 1880s had already resolved the Wien-Planck issue in favor of Planck's factor (see chapter I in Ref. 5). Especially in- teresting are Langley's curves of the spectrum of the sun. The average spectral distribution of radiation from the sun at least qualitatively demonstrates the presence of the Planck factor.
Today one finds in most textbooks on astronomy that the authors present methods of analysis of spectra from stellar photo-spheres based on average intensity distri- butions, with temperature dependence given by a Planck factor. Since the basic physics of a stellar photo-sphere is the same as the physics of a laboratory spectral lamp, we are apparently facing a conflict between the analytical spectroscopist's formula and the astronomer's Planck- type formula. Both formulae are supposed to describe fundamentally identical physical systems.
We close our rough historical survey of radiation in- tensity formulae by noting that in 1926 an English group of analytical spectroscopists published a book, s in the seventh chapter of which they applied a Planck formula for spectral lamp radiation. The basis for their data anal- ysis was not reported to yield inconsistent results.
The work to be presented in this paper originates from a certain discovery made a few years ago by one of us (B.T.) while working with analytical spectroscopy at the Swedish Institute for Metals Research in Stockholm. Data obtained in using a new, computerized, extremely ver- satile spectrometer system were suited for the study of simultaneous intensity fluctuations of a large number of spectral lines in a sample. After application of the formula shown in Eq. 1 for the data it turned out that the sub-
stitution of E,~ by E,~, from Eq. 2 led to a much more consistent organization of the intensity fluctuation data. This method of analysis was applied to specially mea- sured fluctuation data. The results of that work are pre- sented in Ref. 7.
One of us (S.Y.) suggested on theoretical grounds that spectral line intensity fluctuations be analyzed by em- ploying Eq. 1 in comparison with
I~. = C~.exp[-(J" + E,~.)/kT]. (3)
Here J" is the ionization energy of sample element "a". The analysis clearly indicated that the data correlated much better with Eq. 3 than with Eq. 1. Examples of this result are given in Ref. 8. These findings were first reported in Ref. 9 and presented in 1984 at the FACSS XI conference in Philadelphia? ° The outline of a theory of a new spectral line intensity formula was also put forward at this conference.** The new formula, now hav- ing a Planck factor, is expressed by
I,~. = C~.exp(-Ja/kT) • [exp(h~'m./kT) - 1]-L (4)
The detailed theoretical derivation of this intensity for- mula is given in Ref. 12. It is important to note that the thermodynamic variable T in the intensity formulae is the internal electron temperature of atomic elements and thus may not be directly related to the gas temperature of the spectroscopic lamp. The definition of temperature T relates a specific value of T to every degree of freedom in a macroscopic physical system.
Earlier published work concerning our intensity for- mula (most recently in Ref. 13) is now supplemented with underlying details of our methods of analyzing spec- tral line intensity data.
In the second part of this paper we will present the method of fluctuation analysis of spectral line pair in- tensities. The main point of this method is that it allows direct comparison of the temperature-dependent expo- nential terms in Eq. 1 and Eq. 3. Here we assume that Eq. 3 is a good approximation of Eq. 4 (i.e., kT << E~°).
In the third part of this paper, we present the exper- iments and the data analyzed according to the method described in part two. In the fourth section, we present the theoretical method of analyzing absolute spectral line intensity measurements with our new intensity formula (Eq. 4) as a basis. This method is applied to data from standard intensity tables in the following section. The main contents of the fourth and fifth parts of this paper have been reported earlier in Ref. 14. In the conclusion, we discuss the results. We claim that the experimental evidence strongly supports our proposal of a new spectral line intensity formula.
THEORETICAL PRINCIPLES OF FLUCTUATION ANALYSIS
Let us consider simultaneous measurements of the in- tensities I,~. and I~z of two spectral lines in a spectroscopic experiment. The physical state of the incandescent sam- ple is stationary (the spectral lamp is shining steadily), and the performance of the apparatus is stable during the experiment. By these suppositions we assume--as is common practice--that the set of repeatedly measured
values of the ratio I~./I~, is centered around an average value and sampled from a Gaussian type of distribution.
If Eq. 3 is used to describe line intensity, we get for the ratio
a b I~./Ikt = (C~../C~,)" exp( -D (E)/k T), (5) D(E) = J" - ~ + E~, - Eft. (6)
We shall now derive a relation for fluctuations of the quantities in Eq. 5. We shall especially consider devia- tions from the average values and use the symbol A for these. Thus, for example, AT will denote a fluctuation from the average atomic electron temperature T.
At first we note that Eq. 5 has a structure similar to I = A .B(x). We have A / = A(A.B) = (A + AA)(B + AB) - A . B , i.e.,
A(A.B) = AA(B + AB) + A.AB. (7)
Now B + AB = B(x + Ax), and by means of the mean value theorem of differential calculus we have AB(x) = Ax .B'(x + qAx) with 0 -< q -< 1. Thus Eq. 7 leads to the rule
Let us apply Eq. 8 to Eq. 5 with T-* as the variable x. We can write the result in the form
A(I~J/~z) _ A( C~./C~i) -D(E) AT-' • e k a b ImJIkz C~n/C~z
D (E) DiE) AT-' AT-I"e-q'-T- (9)
k
We choose the line ratio so that D(E) _> O.
Consider now a fluctuation with A(C%JC~t) > 0 and AT-* < 0, i.e., AT > 0. This surely describes a fluctuation for which A(I~.H~) > 0. We obtain then by Eq. 9
,A(I~./I~), < [,A(C~n/C~), b - ] Imn/lh~ C~./C~t + --k--" I AT-11
"eD-~ )''IzLT-'I ( 1 0 )
after using AT -1 = - OAT -11 and making q = 1. Consider next a fluctuation "opposite" to the previous
one, i.e., A(C,~./C~) < 0 and AT -~ > 0(AT < 0). This signifies A(I,~.//~t) < 0. We also demand that the absolute values of A(C,~./C~) and AT-* be the same as in Eq. 10. Equation 9 now leads to
,A(I~./I~t), > [,A(C~n/C~,), D(E) ] I~ . / I~ t - C~,./C~ + T " I AT-1I
-D(E) • e - - ' ~ " I~T- ' I (11)
Let us introduce the notation R(Q) = I AQ I/Q, meaning the relative fluctuation of Q. The results in Eqs. 10 and 11 can then be comprised in the equation
a b R(Imn/Ih,) = R(C~./C~t) + D(E).k -~[AT-*[, (12)
which is valid within the limits given upon multiplying the right-hand side by the error factor F(±) ,
APPLIED SPECTROSCOPY 1567
F( ± ) = exp( ± D ( E ) k -~ . [AT -1 [) 1 ± D ( E ) k -~. [AT-a[. (12')
Assuming that the new intensity formula shown in Eq. 3 is valid, we have thus found that Eq. 12 should describe the experimental data for the cases when
A(C,~n/C~).AT > 0. (12")
As mentioned before, we assume that the set of line intensity ratios has been obtained by simultaneously measuring the two lines a great number of times during steady experimental conditions. Let us consider the data set as being sampled from a normal statistical distri- bution. Then a mean value and a standard deviation a exist. The probability for a measured value to differ more than 2a and 3a from the mean value is 5% and 0.25%, respectively. Hence one can conclude that the chance of the occurrence of an extreme, isolated data point is very small. Such a point is easily recognized and will not be included in our data set.
This reasoning may perhaps be strengthened by the following heuristic argument: The registration of a spec- tral line intensity is made during a certain time interval. A data point is thus a time-averaged value. Since one may expect that the physical system attains an extremely fluctuated state for a very short time, it seems likely that such a state is never detected.
Let us then try to fit our data to Eq. 12. A mean value for the measured line intensity ratio is determined and the data are represented in terms of relative fluctuations from the mean value. Equation 12 is a statement con- cerning the cases when A(C,~n/C~) and AT are either both positive or both negative. The simplest way to select points in our data set that would correspond to this re- quirement is obviously to choose the maximal relative fluctuation in the data. This experimental data value, henceforth called the R value, is thus identified with the mathematical quantity R(I,~,/I~). We add the remark that one may alternatively select the maximal value of other high percentiles of the data (and also mean values of those-- the "upper percentile average") to satisfy the requirements of the linear relation expressed by Eq. 12.
If we have data from the simultaneous intensities of n lines measured repeatedly, we can form 1/2n(n - 1) line pair ratios. The R values of the intensity ratios are then plotted against their corresponding values of D(E) in an RD plot. Suppose that the R values of the C-factor ratios R(C~n/C~z) are all equal and that the maximal intensity ratios are attained at the same value of ]AT -1 [. According to Eq. 12, the points in the RD plot would then lie along a straight line with inclination coefficient K = k- i [AT- i [ , and the line would cut the R-axis at
a b R (Cmn/Ck~). The RD plot would demonstrate evidence for the validity of our new intensity formula. Even if R(C,~n/C~t) and I AT -~ I were strictly constant, as we have supposed, the data points might not be exactly on line in the plot. They might have a spread in the R-axis direction limited by the error factor F ~ 1 _+ KD according to Eq. 12'. The point of this observation is that if KD is small (as compared to unity) we would certainly be able to draw a straight line representing the data on an av- erage.
Suppose now that we can actually construct a straight
line graph in the RD plot of an experimental data set. This should be possible when the values of ~ and R(C,%JC~t) have small relative deviations from well-de- fined mean values. In fact, as will be reported in the next section, our experimental data yield in every case a straight line in the RD plot.
In order to explain this, we first discuss the C-factor fluctuations. The biggest sources of these are most likely "bulk" effects, common to all spectral lines. They ema- nate from the apparatus and from sample quantities such as density and pressure. Such effects can be assumed to lead to practically the same maximal relative fluctuation of the C-factor ratios.
There are, however, certain exceptions. If the C factors contain very diverse functions of fluctuating quantities, some relative C-factor ratio fluctuations may be much larger than others. This seems to be the case for auroral oxygen lines. 15 The inelastic electron scattering cross sec- tions enter the C:s of these emissions in atomic spin-flip and electron energy-dependent functions. Fluctuations in the electron density and energy distribution may then cause the observed large differences of R(CT~,/C~) in au- roral emissions. In this paper we shall only treat cases where the differences in R(C~,/C~z) may be assumed to be small.
Let us next discuss possible differences in the values of [AT-l[ giving us the various values of R(I,~n/I~). The mean value theorem of differential calculus yields
IATI [AT-i[ = 0 < q < 1. (13)
T 2 1 + q
We now use Eq. 13 to express I AT -11 in the inequalities expressed by Eqs. 10 and 11. The value of IATI is not the same in our two cases, AT -1 = ± I AT -~ I. On trial, one easily finds that by enlarging the error bounds given by F (± ) in Eq. 12' one can introduce an average relative temperature fluctuation R ( T ) to cover both cases. For- mally we then write A T / T = ± R ( T ) when using Eq. 13 for the two cases. The point of this approach is that we may consider R ( T ) as the average representative of the electron temperature fluctuation of every atomic element in the sample.
Our data analysis formula (Eq. 12) will thus be ex- pressed by
R ( T ) R(I,~n/I~t) = R(C~n/C~,) + D(E) . ~ , (14)
F( ± ) = e +-D(E)~T~'(1-R(T)) -2
1 ± D ( E ) - ~ - ~ ( 1 - R ( T ) ) -2, (14')
and F (±) given by Eq. 14' is now the factor that gives the error limits of the right-hand side of Eq. 14. Equation 12" still applies.
Let us now derive the result we should expect if we were to use the intensity formula expressed in Eq. 1 instead of that in Eq. 3 for our data analysis. The error analysis for arbitrary fluctuations now becomes much more complicated owing to the partition function factor
1568 Volume 44, Number 9, 1990
,sf
o--e v
A m
C
oE i , , , , -0 v
10
5 • •
I I % 1 2
D (E) (eV) FIG. 1. Plot of maximal relative fluctuations R of line intensity ratios I~. /I~t versus photon energy difference D ( E ) = hvm, - h~k~. The ICP sample was dissolved steel (NIST 363) and the six Fe(I) lines 323.6, 357.0, 386.0, 392.2, 427.1, and 516.7 nm were measured ten times, yield- ing data of the fifteen line pair ratios.
[ ~g /exp( -E~ /kT) ] -1. For the purpose of comparing the i
old and new intensity formulae, it suffices to consider only small maximal relative fluctuations, as the results in the following section will show. We shall then use the approximation expressed by
R(Q) = - ~ = I d In Q I.
That is, the relative fluctuation corresponds mathemat- ically to the absolute value of the logarithmic differential. We apply this to the expression for I~,/I~ that one ob- tains in using Eq. 1. This results in the analysis formula
a b R(I~n/I~,) = R(Cmn/Ck,) + D ' ( E ) ' - - R(T)
k T ' (15)
(16) D'(E)= - E ~ + E b + E~m- E~
and
The order of magnitude of the error factor for the right- R(T)
hand side of Eq. 15 is roughly i ± D ' ( E ) . - ~ - , and Eq.
12" is again a subsidiary condition for selecting data. On comparing Eq. 15 with Eq. 14, we find that they
differ only by the definitions of D(E) and D'(E) in Eqs.
15
E
oE o - . , 0
10
%
5 _o
0 0
• o •
OO
I ! 2
D ' (E) (eV) Fro. 2. The same fluctuation data as in Fig. 1 plotted against upper level energy difference D ' ( E ) = E m - Ek.
6 and 16. Our method of fluctuation analysis has thus provided a means of direct comparison of the two inten- sity formulae. We plot the maximal relative fluctuations of measured spectral line intensity ratios against D(E) in one diagram and against D'(E) in another. An inten- sity formula with a correct 1/kT-dependent exponential is expected to yield a linear graph according to the results of this section as expressed in Eqs. 12, 14, and 15.
FLUCTUATION DATA ANALYSIS OF EXPERIMENTS
In this section we shall employ the analysis method presented in the previous section to experimental data obtained a few years ago at the Swedish Institute for Metals Research, Stockholm. The equipment used in this research made it possible to develop a measurement tech- nique that later on appeared to be suited for the fluc- tuation analysis of the preceding section.
The measurements were made with a so-called IDES system, an image dissector Echelle spectrometer coupled to a computer. 16 IDES is equipped with a high-resolution, stigmatic and coma-corrected, high-speed Echelle spec- trograph with the wavelength range of 200 to 800 nm. The computer registers the photon count on the pho- tocathode for a great number of selected spectral lines. In the work to be reported here, the computer was pro- grammed to measure the top of each spectral line for 0.1 s. This was done in 15 sequences so that each spectral line intensity was measured for 1.5 s. This registration procedure yields improved time averaging in the sense that rapid, extreme intensity fluctuations are washed out,
D ( E ) ( e V ) FIG. 3. Plot of fluctuation data R versus D ( E ) = J ° - j b + E ~ . -
EAt (difference of ionization energy plus photon energy) from fifteen steel samples used in a SCHC lamp. Spectral line measurements of the elements Ag, As, Ti, A1, P, Mo, Co, Sn, Si, Cu, V, Mn, S, Cr, Ni, Fe, and Ca provided intensity data for the plot. The line intensity measure- ments were repeated 75 times.
and all the recorded intensity values may be considered as being simultaneously obtained in one spectrometer run. The spectrometer was routinely run numerous times at each spectral sample measurement. The data obtained in these runs could then be used with great confidence to perform the fluctuation analysis presented in the sec- ond part of this paper.
Here we shall present the results of the analysis of two experiments where very stable light sources were used. In the first experiment, an ICP unit from Plasma Therm provided the data2 ,17,1s This ICP has an automatic tuning system and was combined with a highly efficient nebu- lizer and a cyclone spray chamber. The plasma column was very stable with very good control of sample con- centration.
In Figs. 1 and 2 we show the results of our first fluc- tuation analysis for ten repeated runs of side-on ICP intensity measurement of six Fe(I) lines. The dissolved steel sample was specified according to NBS (now NIST) 363 (1 g/100 mL).
In Fig. 1 we have drawn the straight line to which the data points apparently correlate. The point pattern from the same data in Fig. 2 does not show any linear structure. Hence the data from this ICP experiment support our new intensity formula rather than the old formula.
A
V
oE v
14
12
o° •
10
Oo ** . , . • oo
8 " : - "
f, :~.'~ . • • • " , o% • .o •
6 ":"< '£ '" =,
. . . ; . , , •
'- t .¢ ,." b = , ~ , • •
4 ;....
2
O Q
O
0 0 0
% L 8
D ' ( E ) ( e V ) FIG. 4. The same fluctuation data as in Fig. 3 plotted against D ' ( E )
(approximated by the upper level energy difference E,~ - E~ according to the argument given in the text).
In the second experiment a single-chamber hollow cathode spectral lamp (SCHC) was used. TM This lamp has a water-cooled electrode. The solid metal sample is a cylinder of 4 mm inner diameter and 29 mm length. It is flushed with neon and works at a discharge current of 0.5 A. The plasma column along the sample cylinder axis becomes very stable and intense. Measurement condi- tions in the direction of this axis are very favorable. The spectral lines are very sharp, and background effects are negligible.
Here we present the data analysis of one SCHC ex- periment in which a great number of lines from 17 ele- ments could be identified and measured simultaneously with the use of the versatility of the IDES system. The spectral line intensity measurements were repeated five times for each one of fifteen steel samples. The samples had been prepared with varying concentrations of me- tallic elements. The SCHC lamp was run with the same discharge current and cooling efficiency for the different steel samples. In this way 75 intensity data values of every line were obtained, with fluctuations of concentra- tion and possibly temperature deliberately introduced. Our equations for the relative fluctuations of physical entity ratios are applicable to this data set since the equations are valid for arbitrarily large fluctuations of the entities.
1570 Volume 44, Number 9, 1990
The fluctuation analysis of the data set yields the R D and R D ' plots shown in Figs. 3 and 4, respectively.
In plotting the data according to Eq. 15, in Fig. 4 we have neglected the Boltzmann average energy differences E b - E" in D ' ( E ) as given by Eqs. 16 and 17. The dif- ferences were found to be 0.1 eV at most from estimates based on tabulated values of partition functions in Ref. 20. This lack of precision does not affect the irregular "cloud" shape of the point pattern in Fig. 4. The cor- relation of the data to Eq. 15 is obviously very poor (correlation coefficient r = 0.29). In contrast to this re- sult, we see that the fluctuation data of this experiment fit very well to a straight line in Fig. 3 (r = 0.90). If we consider this fit as conclusive evidence for the validity of Eqs. 12 or 14, we can determine the mathematical error factor (Eqs. 12' or 14'),
F ( ± ) = 1 ± D ( E ) . O . 1 / e V . (18)
The C-factor ratio fluctuations can then also be esti- mated from the vertical spread of the points around the straight line. Figure 3 yields
R(C,~n/C~) = 3.1 ± 1.6%. (19)
The SCHC spectral line intensity data plot in Fig. 3 and the relations in Eqs. 18 and 19 thus demonstrate excel- lent support for our new intensity formula and our fluc- tuation analysis method. The conjectures concerning the fluctuations of temperature and C factors made in the section above are all very well confirmed. Similar evi- dence from measurements of spectra using several other types of lamps is presented in Refs. 8, 21, and 22.
INTENSITY OF ELECTRIC DIPOLE EMISSION LINES
In this section we shall discuss the theoretical possi- bility of testing the validity of our new spectral line in- tensity formula
I ( h v ) = C ( h v ) . e -J/kT" [e hp/kT - - 1] -1 (20)
by using data of absolute intensity values. Such a pos- sibility presents itself if the v-dependence of C(hg) is explicitly expressed. Let us assume this dependence is given by
C(hv) = KX-" = Kg"/c" (21)
for the average intensity distribution of electric dipole emission lines from one atomic element in a spectroscopic measurement.
We then imagine a diagram in which such intensity values have been plotted against v (or X) and where we have drawn the best fitting smooth curve to represent the average intensity distribution. This curve should then fit well to the graph of
I (hg) = K c - n g n e - J / k T [ e h~/kT - - 1] -1 (22)
for some value of n. The dominant term for the y-dependence of I over the
whole spectral range is the Planck factor [e h~/kT - 1]-k In the IR limit hv << k T , the Planck factor becomes approximately equal to [hv / kT] -~. In this limit we get from Eq. 22
In I = const. - (n - 1)ln X. (23)
Hence a plot of IR intensity data versus In~ may yield the value of n that fits Eq. 23 best. Examples of such plots in the following section show that the value n = 2 gives a good description of electric dipole emission line intensity.
This value of n can also be justified by considering the fact that the dominating y-dependent factor in C(hg) is the transition rate for emission of a photon hr. For an electric dipole transition from state [i> to state If>, the accompanying rate of energy emission by the atom is proportional to hf~ 4" [ < f [ _r [ i > [ 2 (see any standard textbook). The dipole term [ < / [ r [ i >[2 is the square of a length characterizing the transition i -~ f. If we further limit the consideration to transitions without spin-flips, i.e., AS = 0, the transition may be thought of as a pure spatial jump by the electron in a quasi-classical picture. This suggests that the characteristic length measure is Xn, implying that [ </[ _r [ i > [ 2 ~ X~ and thus
C ( h v ) - - K X - 2 = Kg2/c 2. (24)
If we introduce 0 = k T , we can transform Eq. 20 into
l n I = l n K - l n X 2 - J / O - l n ( e h~/°- 1) which we shall use in the form
hv ln(IX 2) = In K - J/O - - -
0
+ 0 _ e_h~/0)/" m
• [1 ~ ln(1 ~ (25)
This equation should be interpreted as an "average order of magnitude" relation which, for fixed In K, is supposed to describe the "smooth" average intensity distribution of electric dipole lines with AS = 0 for one atom in a spectroscopic sample. The differences in values of In K for different atoms in the same sample are assumed to be determined mainly by the relative orders of magni- tude of the various concentrations of respective atoms.
In the UV limit hv >> 0 the intensity distribution from an atom will from Eq. 25 be given by
ln(IX 2) = const. - hg/O. (26)
In plotting ln(IX 2) versus hv for UV lines, one should thus obtain a straight line if Eq. 26 is valid. From this plot a value of 0 can be determined. This 0-value can then be
0 used to plot ln(IX 2) versus hv[1 + ~ ln(1 - e-he/°)] to
check the validity of Eq. 25. The maximum value of I is attained when
hv = hymn = 1.68(=1.5936 . . . 0). (27)
If we have a spectrum with data points densely distrib- uted over the whole spectral range, a plot of I versus hv will yield the position of Iron and the value of 0 is deter- mined by Eq. 27. This method together with the method based on Eq. 26 thus provides a double check for the atomic electron temperature 0 in most cases.
At I ~ we now have, according to Eqs. 25 and 27,
ln(Im~X2m~) = In K - 1.4 - 1.6J/hg=~. (28)
If we have a spectroscopic sample with known concen- trations of various atomic elements, we can analyze the correlation of the measured line intensity to Eq. 28. All
APPLIED SPECTROSCOPY 1571
tnl
" ~ n=3 i I i
. . . . . . . . . . ~ Io 1'1 I~x Fro. 5. Logarithmic plot of average C(I) electric dipole line intensities I versus wavelength ~ in the range 7000-26000 A from a table in Ref. 23. The solid line is determined by least-squares fit to the points. The dashed lines are graphs of Eq. 23.
the differences in values of In K will be determined ac- cording to our earlier assumptions, and for J we insert tabulated values of ionization energy.
We make I versus h~ plots from which we determine h~m~ (if possible) and then obtain by plotting l n ( I ~ ; ~ J K ) versus 1.6 J/h~,m~ a test of Eq. 28. If the plotted points correlate to a straight line, we have ob- tained evidence for the factor e -J/hr in our intensity for- mula. In the next section we shall show that data from standard tables of absolute intensity values yield excel- lent support for the intensity formula expressed in Eq. 20 with C = K~ -2.
ANALYSIS OF ABSOLUTE INTENSITY DATA
We shall now report results of applying the method of analysis developed in the previous section. Data of spec- tral line intensities will be taken from tables by Striganov and Sventitskii 23 and by Meggers et al. 24 The spectral line intensity data of these tables were obtained by reg- istration on photographic plates. The degree of black- ening of an exposed plate was scanned by photometric measurement of light passing through the plate in a sec- tion determined by a "window" slit. Depending on how the fixed slit width is chosen, a certain bias may occur in intensity data, of either broad or narrow lines. With modern photon-counting registration systems, this sort of bias is eliminated in spectroscopic measurements. With this remark in mind we shall use the table data as ex- perimental values of spectral emission intensity in units of energy per unit area per unit time.
In order to analyze the tabulated data we used the device of first splitting the spectral energy range into intervals. The widths of the intervals were 0.5 eV in the optical and UV regions and 0.2 eV in the IR region. The average intensity values of electric dipole lines with AS = 0 were determined from the table for each interval. Of course, no data point was obtained in intervals void of lines. The midpoints of the intervals then defined the value of hv or ~ for the points along the abscissae in our diagrams.
In a series of six figures we present analysis results based on tables in Ref. 23. The averaged values of In I in the IR region are plotted against In X in Fig. 5 for C(I) and in Fig. 6 for Cs(I). By the method of least-squares
Inl
x
x x
0 . . . . ~ ' ; ' 1; ' 1'1 ' 1'2 rn
Fro. 6. Logarithmic plot of average Cs(I) electric dipole line intensities I versus wavelength ~, in the range 400-7400 nm from a table in Ref. 23. See caption of Fig. 5 for explanation of lines.
we determine the best straight line to represent the data in the two figures. We see that in both cases the most probable linear relation is
In I = const. - In
which by Eq. 23 leads to n = 2; i.e., the relation expressed by Eq. 24 is established.
As an example of plotting ln(Ik 2) versus hv we show in Fig. 7 the result for C(I) intensity data. The straight line in Fig. 7 obviously fits the data in the UV limit. According to Eq. 26 the inclination of the straight line is - 0 -1. From the figure, 0 = 3.3 eV was thus obtained for the C(I) experiment. This method also unambigu- ously determines 6 for the Cs(I) data. The value of hvm~ can in both cases be found by graphically determining /max. The double check for 0 by Eq. 27 yields consistent results for these spectral data.
The final plot of ln(Ik 2) to test Eq. 25 is shown for C(I) data in Fig. 8 (the same as fig. 1 of Ref. 14) and for Cs(I) data in Fig. 9. These two data sets differ very much in spectral range and values of 0.
In Fig. 10 we show again (see Ref. 14) the result of analyzing He(I) intensity data. Notice that in this case we have a large spectral range gap with no data. The method of double check for 0 cannot be used. Since there
FIG. 7.
25-
20-
1 5 -
I n ( I . X 2 )
e l e
• : : . .
I I I 5 10 15 h~ (eV)
Plot of ln(I '~ 2) versus hv of C(I) intensity data from Ref. 23.
1572 Volume 44, Number 9, 1990
s 1 in ( ] . ~ 2 ) . e = 3 . 3 e V
/
T5"
- 5 5 10 h~ 1 1 . ~ I n l l - ; g l ) leVI I I I I
o'.s ; ~ ~ ~ ~ ~ ~ ~ ~ ,0 ,', ,'2,~ ~(,v)
Fro. 8. The data of Fig. 7 plotted against h~[1 + ~ In(1 - e h~/0)].
The curve In k 2 and the position of in/~., serve to illustrate the pro- cedure of plotting described in the text. (The figure is reprinted from Ref. 14.)
are only three UV da ta points, the initially de te rmined value of 0 by Eq. 26 is uncertain. In a case like this, one therefore has to make several final plots of ln(Ik9 with trial values of 0. F rom the resulting graphs one can judge which choice of 8 yields the best fit of the data to Eq. 25. Figure 10 thus shows tha t a sat isfactory value of can be de termined even in a case where trial and error mus t be applied.
Figures 8-10 thus demons t ra te very strong evidence for the s ta tement tha t the average intensi ty of electric dipole spectral lines of an a tom is described by the ex- pression
p2 [ em'/hT I = S . ~ - 1] -~. (29)
T h e propor t ional i ty factor B in Eq. 29 is equal to K e-J/hr according to our new intensi ty formula. Evidence for this is obta ined by analyzing spectral intensi ty data in Ref. 24 of s tandardized arc exper iments in which the concentra t ion was set a t the same value for seventy anal- ysis elements in s tandard electrodes.
On the basis of assumptions in the previous section we may argue tha t the order of magni tude of K is the same for all analysis lines in this da ta set. Thus Eq. 28 should apply, with In K having the same constant value
-2 -I I i o1 01.2
Fro. 9.
[ n l . X 2
8 = 0.66 eV
I 2 3 4 hv(1* ~ In(1-eT})(eV) I I , ~ I 0.5 1 2 3 6 h'~ (eV)
Plot of Cs(I) intensity data tabulated in Ref. 23.
In [I.A2)
25
O : 4.3eV
20
-15 - I0 -5 5 10 15 20 hv ( I . , ~ - In ( 1 - e g l ) (eV) 51 J I I I i i i i i i i i [
n~
FIG. 10. Plot of He(I) intensity data tabulated in Ref. 23. (The figure is reprinted from Ref. 14.)
for these spectral data. In order to have evidence for our in tensi ty formula,
ln(Im~ki~) = const. - 1.6J/h~,m~ (30)
mus t hence be obta ined from the data. We make plots of I versus h~ for seventeen elements
in the manner described above and de te rmine I=~ and h ~ m a x = 1.68. Th e resulting values of 0 and tabula ted values of J are given in Table I.
Th e plot of ln(Im~ki~) versus 1.6J/hvm~(=J/O) is pre- sented in Fig. 11. Equa t ion 30 is represented by the s traight line in the plot. We see tha t the correlat ion of the da ta points to this line is very good.
Suppose tha t the deviation of the da ta points f rom the line in Fig. 11 is mainly in the vertical direction. This deviation is at most 0.25. Reference to Eq. 28 then shows tha t the relative deviation of K, and hence of concen- trat ion, is at most 25%. Differences in volatil i ty could very well have caused such a deviation of analysis ele-
2.5•-" 'In (Ir.ax "k2max) ' ' ZO~
1.5 I I ~
1.0 •
0.5
0 2'5 3.0 3'.5 4[0 1.6 "J/h~mo x
FIG. 11. ln(/maxki,.) plotted against 1-6J/hvma, = J/8 for the elements of Table I. Intensity data are taken from tables in Ref. 24. (The figure is reprinted from Ref. 14.)
A P P L I E D S P E C T R O S C O P Y 1 5 7 3
TABLE I. Ionization energy J and electron temperature 0 of seventeen elements (based on intensity data in Ref. 24).
Ele- ment Cs Na Ba Li Ca Yb Sc Cr Ti Sn Mo Mn Ag Ni Fe Co Pt
ment concentration in the arc. (The authors of Ref. 24 estimate the error in the intensity data to be 15-25%.)
Experimental error could thus account for the major part of the deviation of the data from our Eq. 30. We may claim that Fig. 11 provides tremendously strong evidence for the factor e -J/kr in our electric dipole emis- sion line intensity formula
p2 [ e h v / h T I = K e -JI'T ~ - i] -I. (31)
C O N C L U S I O N
By two different methods of analysis we have pre- sented strong experimental evidence for the validity of a new spectral line intensity formula.
The fluctuation analysis method described in the sec- ond section of this paper could be worked out successfully because of the exponential functions in the intensity for- mulae. This method can easily be generalized to apply to data of any exponential distribution (for example Maxwell-Boltzmann particle distributions). Defying in- tuition, it seems that the worse the data fluctuations, the better the corresponding R D plots for measurements of exponential distributions.
If our intensity formula is valid, the R D plot could be very useful in optical spectral analysis when there are difficulties in identifying an emission line because of in- terference or uncertainty of term schemes. A correctly resolved line should fit to the R D graph of the intensity data in the manner shown by the plots described in the third part of this paper.
The results of the following two sections suggest a possibility of improving measurements of atomic tran- sition rates in a spectroscopic lamp. Suppose a plot of electric dipole lines of an atom is obtained according to the rules that yield Figs. 8-10. The straight line in the plot defines, by Eq. 25, the value of In K, and KX -2 multiplied by a constant yields the average transition rates. If the absolute intensity values of all lines are plotted in this graph, the vertical deviations of these points from the straight line yield the proportionality factors for the transition rates, as compared to the av- erage transition rates.
In the Introduction, we discussed spectral line inten- sity formulae in view of Planck's blackbody radiation law. Let us now conclude that discussion by a simple comparison of our intensity formula
1(1,) = C ( v ) e -J/hr" [e h~/hT - 1] -1
with the blackbody intensity distribution
2 h p 3 r = _ _ |eh~,/hT - - 1 | - 1 . E(p)
C 2 L J
(32)
(33)
In Eq. 32, I(v) is given in units of energy per unit time per unit area, and in Eq. 33, E(p) is expressed in energy per unit time per unit area per frequency unit.
If our I(v) is supposed to be the intensity of a line of width Av, the expressions in Eqs. 32 and 33 yield
I(~) = D.E(v).A~ (34)
with D = C(~,)e-J/kTc2/2hv3A~, .
The form of the spectral line intensity formula ex- pressed in Eq. 34 suggests that the number D may be considered the product of two factors--one factor being the emissivity of the spectral sample in the frequency interval Ap at temperature T, the other factor being a normalization number (determined by experimental pa- rameters and measurement procedure). In other words, the new spectral line intensity formula can be said to describe the spectral sample as a blackbody viewed through a screening filter.
ACKNOWLEDGMENTS
We express gratitude to Prof. Bengt Hultqvist for his encouraging support of our work. One of us (S.Y.) acknowledges a leave of absence granted by the Local School Board of G~illivare, and thanks Prof. Hultqvist for kind hospitality at the Swedish Institute of Space Physics. Miss Birgit Sandahl (Malmberget) is thanked for very helpful lin- guistic criticism during the preparation of this paper.
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