Some problems of the emission spectral analysis theory

Spectrochumca ACM. Vol. 338. pp. 577 to 590 Pergamon Press Ltd 1978.PrInted m Great Britain

Some problems of the emission spectral analysis theoryjS. Institute

for Spectroscopy,

L.

MANDELSHTAM

USSR Academy

of Sciences, Troitsk,

(Received

3 March

Moscow

obl., 142092, U.S.S.R

1977)

Abstract-Over the past 50 years spectral analysis has grown from a semi-empirical method to an independent discipline based on a well-developed theory and having the disposal of various experimental techniques. What is the aim of the spectral analysis theory? Spectral analysis, unlike, for example, classical chemical gravimetric methods, is a relative method. The measured analytical signal TL is connected with the quantity to be determined, i.e. the element content (C) in the analysis sample by calibration with reference samples. The author believes that the spectral analysis theory should be aimed not at establishing this connection with such an accuracy that it would allow in the end to abandon the reference samples, but at elucidating the role of those physical factors which form the basis of all the links of the spectral analytical procedure, so that a rational choice can be made for obtaining optimum results. It is from this point of view that the problems of atom excitation and ionization, intensity of spectral lines, detection limits and entry of sample into the excitation source are schematically discussed.

analysis as a method for the determination of the chemical composition of a substance exists for more than 100 years. In this period its development was long and dramatic. Shortly after the works of KIRCHHOFF and BUNSEN, spectral analysis was used in astrophysics, chemistry, mineralogy, metallurgy, etc. However, it was soon discovered that spectral analysis was not so “simple” as it seemed at first sight. When flames, arcs and sparks were used, the spectra of most elements were found to be so complex that their practical application turned out to be difficult. It was also found that the spectrum of one and the same element could undergo a radical change when passing from one excitation source to another, and even with one source the intensity of some spectral lines displayed uncontrolled variations. Some elements were determined with very high sensitivity, sometimes even too high, so that it confused the chemists of that time; some other elements, though they were present in the sample in large amounts, could not be detected at all. These problems limited the application of spectral analysis even to qualitative work, while the possibility of quantitative spectral analysis, in spite of a certain success, was disputed. KAYSER [I] in his fundamental work wrote : “In these conditions there are few grounds to consider that in future the spectroscopic method will find wide application for qualitative analysis” and further: “Summing up all the above mentioned studies I come to the conclusion that quantitative spectral analysis has turned out to be impracticable”. BOHR’S works in 1913 and further development of the quantum theory provided a reliable physical foundation for spectral analysis, but as before it was almost completely ignored in practice. In 1924 there appeared a paper by LOWE [2] “A forgotten method of quantitative spectral analysis” and in 1925 a paper by GERLACH [3] “On the correct realization and interpretation of quantitative spectral analysis”. However, back in 1926, KONEN [4] in his speech “On the state of quantitative spectral analysis” at the meeting of the German Society of Naturalists and Physicians said: “Nothing remains for me but to confirm the conclusion made many years ago by KAYSER and me that quantitative

SPECTRAL

t Dedicated

to the memory

of Professor

HEINRICH

KAISER.

[I] H. KAYSER, Hmdhuch der Spectroskopie Bd. 5, S. 12. 27. Hirzel, Leipzig (19101. [2] F. LOWE, Z. Teck. Phys. 5, 567 (1924). [3] W. GERLACH, Z. Anory. Chem. 142,,383(1925). [4] H. KONEN, Nnturuissensch. 14, 1 I08 (1926). 577

578

S. L. MANIIFI.SHTAM

spectral analysis in the sense of a theoretically grounded branch of science does not exist as yet”. Nevertheless, KONEN considered the situation at the moment rather optimistically. He expressed the hope that the latest success in the study of the elementary excitation processes for atoms and molecules and the use of thermodynamic principles, which proved their value, for example, in Saha’s theory and turned out to be so fruitful in astrophysics, would be applicable also to spectral analysis. Then a turning-point came: owing to the works of many scientists spectral analysis began to develop and came into use at a more and more rapid pace not only in science but in industry as well. At this first stage of development, emission spectral analysis was still considered as an empirical method based only on some general conclusions of the theory of atoms. By the present time as a result of fifty years’ work of numerous scientists spectral analysis has become an independent discipline based upon a well-developed theory and having the disposal of various widely-developed experimental techniques. A brief characterization of some theoretical problems in spectral analysis is given below. Naturally, it does not pretend to be complete. We should first ask ourselves: what is to be understood by the “theory of spectral analysis”? What is this theory aimed at and what questions must it answer? The spectral analysis procedure, in principle, consists of the following links : transition of atoms of the element under study in the sample into the gaseous state in the excitation source C + I ; transition of a certain number of these atoms into the excited state .,t ---f,I ‘*; transition of the excitation energy into the spectral line radiation . I ‘* + I; selection of the radiation of the line we are interested in from the total radiation of the source by the spectral instrument and its transformation by the recording system into a measurable signal from the line under study I + T,,. Each procedure of spectral analysis can thus be written in symbolic form as follows : C+.I’+.I’*+I-+T,,. In practice the directly measured quantity-the signal from the line TL (density in case of photographic recording, photomultiplier photocurrent in case of photoelectric recording, etc.) is connected empirically with the quantity to be determined, i.e. the concentration C of the analyte in the sample, by means of reference samples. The theory of spectral analysis is aimed at determining the connection between TL and C by considering physical and chemical processes which take place in the whole analytical sequence. Here the following should be said : the author believes that it is not reasonable to impose on the spectral analysis theory the requirement that it should establish the relation between TL and C with such quantitative accuracy that it would be possible in the end to exclude the reference samples and be guided only by the theoretical relationship for the determination of the required value C from the measured value TL. This occurs, for example, in classical gravimetric chemical analytical methods where from the weight of the substance formed as a result of the analytical procedure, the concentration of the element in the sample is calculated. In spectral analytical methods the processes which form each practical procedure are very complex, the values of numerous atomic constants which determine the measured signal (transition probabilities between energy levels, crosssections of atomic excitation, diffusion coefficients, etc.) are inevitably inaccurate. Therefore, the author considers such a requirement to be unwise in principle. Reference samples should retain their role even at a full development of spectral analysis theory. In this sense spectral analysis should be considered as a reldue method. The author sees the purpose of spectral analysis theory in the determination of the physical factors that underlie each of the above mentioned links and to such an extent that the picture would be complete enough to clarify the quantitative role of each factor. Such a theory would allow us to control the analytical procedures by a rational choice of optimum conditions for performing spectral analysis. It is from this point of view that the consideration is presented below. There is not much essentially new as compared to

Some problems

the account systemization

of the emission

given earlier [5]. The main of previous considerations.

2. EXCITATION

AND

spectral

purpose

IONIZATION

analysis

is some

theory

further

519

development

and

OF ATOMS

Let a unit volume of the source luminous cloud contain Iv atoms of the analyte. Elementary processes which lead to the excitation of ground state atoms and the quenching of excited atoms-the link .A* + .,1‘* -are in reality rather varied. These are collisions of atoms with free electrons and heavy particles-atoms, ions, and molecules of the gas in the source cloud, as well as, in some cases, chemical reaction with the participation of atoms _,1/^. With spectral analysis in practice we are usually interested in resonance lines of atoms, corresponding to the transition from the nearest excited level to the ground state. In sources with a not very low concentration of electrons, for example, in an arc and spark plasma, the basic processes which lead to the excitation of resonance levels-and also to atom ionizationmPare collisions with electrons, collisions with heavy particles are less effective [6]. For the passing of an exciting particle near an atom to have the character of an inelastic collision required for excitation or ionization of the atom, it is necessary, within the scope of Bohr’s model. that the residence time of the exciting particle 5 near the atom is much smaller than the revolution period of the electron in its orbit T. Otherwise, the collision has an adiabatic character: the field acting upon the bound electron varies slowly and the electron orbit is only deformed. After the passing of the exciting particle the bound electron remains at its orbit and there is no energy transfer. Since r z o/t; where p is the gas-kinetic radius of the atom and u the velocity of the electron, and T 2 27&/E, where E is the excitation or ionization energy and /I Planck’s constant, the condition for an exciting or ionizing collision is pE/2nrll < I. It means that for excitation or ionization to take place, the velocity of the exciting particle L‘must be high (or E must be very low). Assuming p z 10-a cm, E 2 4 eV, we obtain L‘2 lOa cm/s. If the average energies of electrons and atoms are equal (which corresponds to the equality of electron and atom temperatures). the relative number of heavy particles, say with an atomic weight of 50, having the same velocity, for example. at 7 = 5000 K is IO5 times less than the relative number of electrons with the same velocity. Thus, if the degree of ionization of the plasma is not less than 10~4~10~‘. then electron collisions make the main contributions to excitation and ioniration.

For simplicity the atoms will be considered to have only two states--an and an excited (1) state. For the excitation rate one can write:

unexcited

(0)

(1) Here the first term gives the number of exciting collisions with electrons which transfer the atom from state 0 to state 1; the second term is the number of quenching collisions with electrons, transferring the atoms from state 1 to state 0; the third term is the number of atom transitions 1 + 0 due to spontaneous emission of quanta hv. In Equation (1) ool is the excitation cross-section, u the velocity of the exciting electrons; o’io the cross-section for the de-exciting (superelastic) collisions with electrons, and v’ the velocity of the electrons after the exciting collisions; < > brackets mean averaging over the electron velocities: (au> = ja(v)f(o)vdv [7]. In the arc and spark plasma there exists a Maxwellian electron velocity distribution with parameter T,, i.e. the electron temperature of the plasma. In practical cases of interest the plasma is in a stationary state, i.e. dA”T/dt = 0. For an arc discharge it is clear without any special explanations; for a spark discharge this state is achieved in a time z lo-‘OS, so that the spark plasma is practically also stationary [7]. Then from (1) we obtain .A

.f

=

,$

‘o

I .(ao12!> 1;(cT;oc’) + AlO

Izo. Akad. Nuuk SSSR. Ser. Fiz. 26, 848 (1965). [6] A. D. SAKHAROV. Ix. Aktrd. Ntruk SSSR. Ser. Fiz. 12. 372 (1948). [7] N. K. SUKHODREV, Twtly f'iz. Iw. P.N. Lehrdelo 1.5, 123 (1961); Trtrns. P.N. Lrbrdec Sci. USSR, Vol. 15, Part 3. Consultants Bureau/Plenum. New York (1962).

[S] S. L. MANDELSHTAM,

P/ZJX Insr., Acud.

580

S. L. MANDELSHTAM

with the Klein-Rosseland of detailed balancing :

relationship between cro1 and oiO as follows from the principle

GJo1u)_ !he-E,‘kTe Gflov’) .40

(3)

where g1 and go are the statistical weights of the levels 1 and 0, and E, is the energy of the excited level 1. Introducing the ratio of radiation and radiationless quenching transition rates R=---

AIO Gflov’>

we obtain for the number of excited atoms in state 1 per unit volume:

In the limiting case of high A’,, we have ~~+~(cr~~v’) $ Alo and expressions (2) and (5) convert into the Boltzmann formula :

m_gJ No

e

- E,IkT<,

90

In the second limiting case, i.e. at low electron density, U~;(~;o~‘) 6 Alo, so

(8) the so-called coronal model.? If the spectral lines belong to ions, No in the given formulae should be replaced by A’-io-the concentration of ions of the relevant element in the ground state. For determining .h i0 we consider similarly the processes of atom ionizations by collisions with electrons. Let us assume that for a given element only neutral and singly-ionized atoms are present. One can write for the ionization rate:

where the first term gives the rate of ionizing collisions with electrons, the second term the rate of recombinations in a so-called triple collision (an ion and two electrons), and the third term the rate of photo recombinations in a double collision (an ion, one electron and one photon). g,, (T: and a: are cross-sections for corresponding processes. As in the case of excited atoms, also for ionization, as a rule, virtually stationary states are realized [7]. Then from (9) we obtain :

Cross-sections

0, and ai are related by the principle

co,.> (do;&> where E; is the ionization

of the detailed

2g,o (2mMY _~_

go

balancing:

Ed

L,,,.T,,

h’

energy

t This name originates from the fact that similar conditions were first observed in the solar corona. At present it is found that the coronal model is realized also in some laboratory sources, for example, in the plasma of a high-temperature laser and vacuum sparks when exciting the spectra of highly ionized atoms [S]. [8]

S. L. MANDELSHTAM,

Proc. 14th Co//. Spectrosc.

Int.. Dehrecen

1967, Vol. 1, p. 109, Hilger,

London

(1968).

Some problems

We again introduce

of the emission

the ratio of photo and radiationless

spectral

analysis

recombination

581

theory

rates:


Q=-

(12)

((T:u;u;)

and obtain

the general

expression JV,’ 2g,, (2am,kTJ3’* _,+/iO= ~ h3 I + QIJ’, go

In the limiting

(13)

case of high JV, we have _+;(a:u;u:>

and expression

JYOe-V’*.

9 (&u’),

Q/~I; < I

(13) takes the form of the Saha formula Jlr,

=

2%

Jlr_,

to

e

(2nm.kT.)3’2 ~0

e_E,kT

h3

’

(14)

*.

90

In the second limiting

case of low JV, we have Q/J(re g 1 and

J’ lo = i.e. the expression

corresponding

(UiU) ~ (OUI)

0

to the coronal

= 1 2gi0 PTNP go Q

model.

h3

_Noe-Er~‘Tc,

. I& and .~1b are connected

“Vi0 + J+“0 Z J1” where JV is the total number

of atoms of the element

under consideration

(15)

by (16)

per cm3.

It is seen from the foregoing that for the calculations of JVY and Jlrio, in general, the cross-sections of elementary processes contained in R and Q, and the values of Alo have to be known. Cross-sections and A values are given in Refs [9-121. To estimate R and Q one can use the approximations [8] R x Q x 1013 E3TI12

(17)

where E is the excitation energy E, or ionization energy Ei; E and T, are expressed in eV. It follows from this that for the values E, z 34 eV, Ei z 10-15 eV and T, z 0.54 eV (500&40000 K), usually realized in spectroandlytical practice, the Boltzmann and Saha expressions are realized for Me 1 1015 cm-3 and the conditions of the coronal model for Ne s 1013 cmm3. It means that for the spark discharge plasma (.Ne z 10” cme3) the Boltzmann and Saha formulae are met and for the hollow-cathode plasma type those of the coronal model. As to the plasma of the arc discharge AT, z 10’4-1015 cmP3; apparently, as a rule, the Boltzmam-Saha formulae seem to be applicable; however, some small deviations are possible. The plasma model described by (1) and (9) is a rather rough one, i.e. the so-called two-level approximation, since in (1) only direct and reverse transitions between the ground state and the resonance level of the atoms considered are taken into account, and in (9)-only direct and reverse transitions between the atom ground state and the ionization limit. Actually, one should consider direct and reverse transitions between other atomic levels as well as ionization and recombination processes with the participation of excited levels. In this case cross-sections of collisions increase rapidly with decreasing AE-the difference in the energies of states between which the transition occurs, and the cross-sections of radiation processes rapidly decrease. Therefore, for a given electron concentration N, and temperature T,, for some low levels the condition of the coronal model can be realized, and for higher levels-the Boltzmann-Saha conditions. Further, it should be noted that all the expressions include only the electron temperature. The value of the gas temperature is not contained in these expressions, as in (1) and (9) we meant by v and u’ the velocities of electrons, i.e. atoms were considered to be motionless. If we assume that atoms are moving, then by L’and tl’ Probabilities, N.S.R.D.S.-N.B.S., c91 W. L. WIESE, M. W. SMITH and B. M. GLENNON, Atomic Transition Vol. I. U.S. Department of Commerce, Washington D.C. (1966). W. L. WIESE, M. W. SMITH and B. M. MILES, Atomic Transition Probabilities, N.S.R.D.S.-N.B.S., Vol. II. U.S. Department of Commerce, Washington D.C. (1969). [lOI C. H. CORLISS and W. R. BOZMAN, Experimental Transition Probabilities for Spectral Lines of Seaenry Elements, N.B.S. Monograph 53. U.S. Department of Commerce, Washington D.C. (1962). [Ill L. A. VAINSTEIN, I. I. S~BELMAN and E. A. YUKOV, Secheniya Vozbuzhdeniya Atomov i Ionol; Elektronami. Nauka, Moskva (1973). c121I. I. SOBELMAN, Vuedeniye u Teoriyu Atomnykh Spektrov, Nauka, Moskva (1977).

S. L. MANDELSHTAM

582

the relative velocities of the electron and the atom should be understood. If atoms possess a Maxwellian bution of velocities with temperature T., then in all the expressions instead of z we will have [6,7] :

distri-

It is easily seen, however, that 0 zz T,; therefore, the intensity of lines is determined by the electron temperature. As to the temperature of the atoms, practically in a plasma in a stationary state with I i, > 10’4~10’5 we have T, z 7,.t However, if the degree of plasma ionization is very low, as it is the case, e.g. in the flame, the main contribution to the atom excitation is made by collisions with heavy particles [14] and in the expressions for levels population the electron temperature T, is replaced by the gas temperature 7, [l5, 161. It occurs also when the value E is very small. As shown in [l7], in a plasma with a low electron density hollow cathode. reduced-pressure high-frequency discharges, etc.) the Boltzmann distribution of populations (with T= To) is found for very closely spaced levels of multiplet terms. A similar phenomenon is observed for the distribution of populations of higher levels in the arc plasma of noble gases [18]. It is due to the fact that atoms of noble gases have metastable levels where considerable concentrations of excited atoms are accumulated. Finally, plasmas are known in which the population of excited states and ionization of atoms exceed the values that correspond to the Boltzmann-Saha expressions. This is the case, for example, in the now widespread high-frequency inductively coupled plasma and this departure from LTE is probably due to the influence of metastable argon atoms [l9]. Owing to a large variety of elementary processes leading to atom excitation and ionization, such cases are difficult for generalization.

3. SPECTRAL

LINE

INTENSITIES

Consider now the link NT + I. Let the luminous cloud of the source have the thickness 1along the line of sight. We denote by I, the intensity of the line (the energy radiated by unit volume per unit solid angle per second in the frequency band dv within the limit of the line profile Av). For the increase of the line intensity over the length of the layer dl we can write : dl, = hv@,XT - b,k’,,l, + 6;J”T) dv dl (19) where a,,, by, bk are the probability densities of spontaneous stimulated emission which are related to appropriate Einstein

s

a, dv = AI,,,

and between

themselves

absorption by

and

b;. dv = III0

b, dv = BoI, s AV

AV

emission, coefficients

s Av

by

t The fulfillment of the Boltzmann and Saha formulae should not be identified--as it is sometimes donewith local thermodynamic equilibrium (LTE). The Boltzmann-Saha formulae are fulfilled, if the number of direct elementary processes of particle collisions equals the number of reverse processes of this kind, i.e. if the conditions of detailed balancing for the given type of interaction are met. In this case the Kirchhoff law for the lines is also followed (for the continuous spectrum a Maxwellian velocity distribution is also required [13]). Thermodynamic equilibrium implies conditions at which detailed balancing takes place for all kinds of particle interactions both with one another and with photons, while T, = T, = 7. In this case, in addition to the Boltzmann and Saha formulae, the Planck formula for the radiation density also applies. Approximate local thermodynamic equilibrium represents conditions in which the Boltzmann-Saha formulae are obeyed for a given small region of an inhomogeneous source, whereas the radiation and absorption processes are not balanced and the Planck formula is not fulfilled [ 15, 161. [13] [14] [15] [16] [17] [18] [I93

S. L. MANDELSHTAM and N. K. SUKHODREV, lzv. Akad. Nauk SSSR, Ser. Fiz. 19, 1 I (1955). C. TH. I. ALKEMADE, Proc. 10th Co/@. Spectrosc. Inr. Maryltrnd 1962, p. 143. Spartan Books, Washington D.C. (1963). P. W. J. M. BOUMANS, Theory ofSpectrochemica/ Excifation. Adam Hilger, London (1968). P. W. J. M. BOUMANS, Excitation qf Spectra. Analytical Emission Spectroscopy. (Edited by E. L. GROVE), Part II, Chap. 6, p. I. Dekker New York (1972). E. N. PAVLOVSKAYAand I. V. PODMOSHENSKII,Opt. Spektrosk. 23,873 (1967). V. YA. ALEXANUROV, D. B. GUREVICH and I. V. PODMOSHENSKII,Opt. Spektrosk. 26,36 (1969). P. W. J. M. BOUMANS and F. J. DE BOER, Spectrochirn. Acftr 32, 365 (1977).

Some problems

of the emission

spectral

analysis

theory

583

In (19) the first term gives the number of spontaneous radiative transitions, the second term is for the number of absorptive transitions and the third term for the number of stimulated radiative transitions per second. Integrating this expression with respect to dl, we obtain the following expression for the integral intensity over the whole profile of the line emerging from the sourcel’ : I = I8

s

TV [l _ e-(bv-b:J’:I-V,)-VY,J]

(20)

dv

where 1 27rv3 I$ = __ c2 (1 + R/. 1 e)ehv’kTt,- 1

(21)

differs by the coefficient 1 + R/-l; at the exponent from the Planck expression for the black-body radiation intensity corresponding to a Boltzmann distribution of excited levels. If the concentration of atoms No in the cloud of the source is sufficiently low A‘,,l< 1, the exponent in (20) can be expanded as a series and, using the relations between Alo, Bol and Blo, we obtain

In this case the line intensity I is proportionalto the atom concentration of the given element in the source cloud J$‘“~.In the general case given by (20) we have a different situation. When analyzing the dependence of I and No in this case, it should be borne in mind that the absorption coefficient in the exponent k,, = h,, - bl./+ ‘T/<1 b z b,. is a function of frequency. In other words, for example, the central parts of the line profile are subjected to greater self-absorption than the wings, etc. Thus, the dependence of the line integral intensity I on No, given by (20) is determined by the dependence of k, on v. The latter is determined by the spectral line profile. Theoretical and experimental studies of the spectral lines profiles have made a considerable progress. In the arc plasma the line broadening is due to the Doppler broadening and impact broadening. The latter is formed by Van der Waals interaction by collisions of radiating atoms with the gas atoms and molecules, and by the Stark shift of levels by collisions of radiating atoms with the electrons. There exists a sufficiently well-developed so-called “non-stationary” theory of spectral line broadening which permits the calculation of line broadening with an accuracy sufficient for practical purposes [20]. Calculations show that at the dispersion shape of spectral lines, i.e. profiles corresponding to the impact mechanism of broadening, expression (20), which links the line intensities with the atom concentration of a given element in the luminous cloud of the source, is well approximated by log I = bl log ,I ‘o + u1

(23)

where bl 5 1; the values bl and u1 remain constant at the variation of _,l/‘owithin relatively wide limits [21]. The whole foregoing consideration referred to the case where T,, M,, de0 and other plasma parameters are constant within the volume of the luminous cloud. In reality this is, naturally, not the case. If the gradient of temperature and of atom-concentration from the source axis to the periphery is accounted for, it is in some cases possible to obtain t Strictly speaking the terms B ,,i. I J and Bit,. 1’71should be introduced in (1) and equations (1) and (19) should be integrated jointly (the transport equation). In practice, however, these terms play a very negligible role in (1) since the concentration of excited atoms is regulated by the electron impacts. Therefore, the separation of equations (1) and (19) is justified. [20] [21]

1. 1. SOBELMAN, An Introduction to the Theory of‘ Atomic Spectrcr, Pergamon S. L. MANDELSHTAM. VretleniyrI'Spektrd~~yi htrliz. OGIZ Gostekhizdat.

Press, Oxford (1973). Moscow (1946).

584

S. L.

MANDELSHTAM

more precise expressions for the line intensities but, apparently, this does not give essentially different results from the physical point of view. Usually in a gas dischat’ge we have JV”T < Jlr, and consequently the third term in (19) is considerably smaller than the second one. In some cases, however, conditions can be created so that Jr/-T > MO, i.e. it corresponds to the “negative” temperature in [6]. Using this phenomenon a strong line generation is obtained in lasers. In principle, it is possible to realize this effect also in spectral analysis light sources, so as to obtain considerable amplification of individual analytical lines. However, it seems that no attempts have been made in this direction as yet. As follows from (22) and the more general expression (20), the main parameter that determines the line intensities, and can be controlled by the analyst, is the plasma electron temperature T, which enters into the exponential factor e PEik7’c,.By varying the source parameters, for example, discharge contour capacitance, inductance and resistance in the case of a spark, the current in the case of an arc and so on, it is possible to vary the absolute and relative intensity of lines. An especially strong variation of the line intensity in the case of a carbon arc can be obtained by introducing additives with a low ionization potential [21] into the arc. It should be noted that at sufficiently strong variations of the plasma temperature the line intensity of neutral atoms and successive ions goes through a welldefined maximum value. It is attributed to the joint influence of two factorsmPwith the temperature increase the portion of excited atoms increases due to the factor ePEJkTe, but at the same time the concentration of neutral atoms rapidly decreases, which is followed by a decrease in the concentrations of ions ofincreasing charge, as a result of their successive ionization due to the increase of the factor e@JkTt [21]. The temperature which corresponds to the maximum intensity of each line is called “the norm temperature” [15]. Note that variation of the plasma temperature is usually accompanied by the variation of the electron concentration JV”,, which, in the general case, influences the line intensity. 4. ANALYTICAL SIGNAL Consider now the link I + TL, i.e. the selection of the line of analytical interest from the radiation of the source and the transformation of the line intensity to a measurable signal : this link is provide&by the spectral instrument and the recording system. Here a whole number of problems should be considered theoretically. KAISER considered the problem of the informing power of the spectral instrument : its ability to produce a maximum number of structural details in the spectrum, which means the number of distinguishable values in the signal amplitude and the number of distinguishable frequency bands [22] : Pi = [R log, S In vb/v,] bit, where vb and v, are the upper and the lower limits of the spectral range of the instrument, S is the maximum number of distinguishable steps in the signal amplitude, and R is the instrument resolution. As it follows from this expression the informing power of a spectral instrument is determined primarily by its resolution. The second important point is the signal threshold value which the spectral instrument and the recording system are able to detect. Connected with this is, in particular, the question of the limit of detection achievable with spectral analysis. At present the requirements imposed upon the purity of materials used in different branches of science and technology are becoming more severe. Accordingly, spectroanalysts are facing the requirement upon increasing detection power of spectral analysis. In some fields of spectral analysis application, however, we have already approached the practical limit of detection of conventional methods. Therefore, it is of interest to consider the factors which determine the sensitivity of spectral analysis and, in particular, the theoretically achievable limit of detection. These problems were considered in a number of papers [23-35l.f t The exceptionally high sensitivity of the spectral analysis was noted by KIRCHHOFF and BUNSEN already in their first work by evaporating a small amount of sodium and providing the uniform distribution of atoms by brandishing with a big umbrella [36]. The limit was evaluated as 3 x IO- log, this being very close to the present-day figure. [22]

H. KAISER, Anal. Chenz. 42 (2), 24A (1970).

Some problems of the emission spectral analysis theory

585

Let us define the problem once more. In qualitative spectral analysis the criterion for the presence of the analyte in the sample is that the signal TL measured at that place in the spectrum where the analysis line should occur exceeds the signal of the “background” T,, measured at an adjacent spot of the spectrum or at the same place in the spectrum of the “blank” sample, which does not contain the analyte. Here the “background” signal is understood in the broad sense as the signal produced by the continuous radiation from the excitation source, the light scattered in the optics of the spectral instrument, photoplate fog, dark current of the multiplier, etc. If the element concentration in the sample is sufficiently high, the criterion for the presence of the element in the sample is just the relation TL+B-

Te>O.

If the signals from the line and from the background were constant in time or from one measurement to another, we could, in principle, measure any small increase of the signal TL+ B over Ts and use the criterion (22) for detecting any small amount of the analyte in the sample. In reality, however, signals T L+B and Ts values are subject to fluctuations, i.e. spontaneous, random deviations from their average values. Fluctuations of the measured signals from the line and the background are caused by fluctuations in the radiation of the source and fluctuations introduced by the recording system. A certain portion of these fluctuations is, in principle, inevitable because of the discrete nature of the radiation and the statistical character of the action of recording systems. The presence of these fluctuations sets a fundamental limit to the achievable analytical detection power. When determining very small concentrations of elements, the absolute value of the signals from the line and from the background turns out to be comparable with the magnitude of fluctuations of these signals. In this case we can state that the line is present in the spectrum only if the difference T L+ B - TB exceeds the magnitude of the possible fluctuations of this difference : TL+B - % > KG.+B - Ts).

We can assume that fluctuations of the signals have Poisson distribution then, a practically reliable criterion for the line presence in the spectrum is (X+B - TB)2 3a(Tt+~ - TB)

probability; (24)

where G is the standard deviation of this difference. This criterion was first introduced by KAISER [23] and bears the name of the “KAISER criterion”. It is evident that one should take into account only the uncorrelated parts of fluctuations of the signals TL+~ and TB. If, for example, fluctuations of the excitation conditions occur, which cause the same variations in the line and background intensities, or, for example, the variations in the supply voltage of the two photomultipliers giving the signals TL+ B and Ts, etc., then an appropriate choice of the measuring technique allows neglecting [23] H. KAISER,Spectrochim. Actu 3,40 (1947). [24] H. KAISERand H. SPECKER, 2. Anal. Chem. 149,46 (1956). [25] H. KAISERand A. C. MENZIES,The Limit of Defectionof LIComplete Analytical Procedure. Adam Hilger, London (1968). [26] H. KAISER,Z. Anal. Chem. 260, 252 (1972). [27] H. KAISER,Pure Appl. Chem. 34, 35 (1973). [28] S. MANDELSHTAM and V. NEDLER,Spectrochim. Acta 17, 885 (1961). [29] S. L. MANDELSHTAM, Zh. Prikl. Spektrosk. 1, 5 (1964). [30] A. N. ZAIDEL,G. M. MALYSHEVand E. YA. SHREDER,Opt. Spektrosk. 17, 129 (1964). [31] V. A. SLAVNYI,Zh. Prikl. Spektrosk. 6,695 (1967); 7, 123 (1967). [32] K. LAQUA,W. B. HACENAUand H. WAECHTER,Z. Anal. Chem. 225, 143 (1967). [33] U. HAISCH,Spectrochim. Acta 258, 597 (1970). [34] R. KLOCKENKAMPER and K. LAQUA,Spectrochim. Acta 32B, 207 (1977). [35] Spekrrtrlnyi Annliz Chisrykh Veshchestc~, (Edited by Kh. I. ZILBERSTEIN).Khimia, Leningrad (1971); Spectrochemical Anulysis of PureSubstances. Adam Hilger, Bristol (1977). [36] G. KIRCHHOFFund R. BUNSEN,Pogyendo$Ann. 110, 161 (1860).

S. L. MANDELSHTAM

586

these fluctuations. Denoting by o(TLfB) and ~$7”) the standard deviation of these uncorrelated, i.e. statistically independent parts of the fluctuations of the two signals, we obtain 4z.+B

- G) = @(TL+B)

+ 02(Ts)]

(25)

For the evaluation of the detection limit and the precision of the spectrochemical procedure it is reasonable to introduce a quantity II = signal-to-noise ratio [5] :

T<+B- TB

I-I=

(26)

02(r,+,) + aZ(Ts)’

The value of II depends on the concentration of the analyte in the sample, on the spectrum excitation characteristics and on the parameters of the spectral instrument and recording system. The higher the value of II due to an advantageous choice of these parameters, the lower the detection limit of the element in the sample. The criterion for the presence of the analyte in the sample is II 2 3. Practically, working near the detection limit of the element, we have TL 6 TB and ~(TL+B) zz c(G), so 02(%+~)

+

I”

z

202(Te)

and ==

K.+B-

TB

(27) Jz

~TB)

As stated before, the fluctuations of the signals 3, and TB arise in the excitation source and recording system and, in principle, can be partly eliminated or considerably diminished by means of special techniques. It is fundamentally impossible to eliminate the fluctuations in the radiative flux due to its discrete photon character and fluctuations in the recording system resulting from the discrete character of the signal produced by the detector (granularity of the developed photographical emulsion in the case of photographic recording and photocurrent consisting of discrete electrons in the case of a photoelectric detector). It is these fluctuations which determine the theoretically achievable limit of detection. Let us now calculate fI for some methods of spectrum recording. In the case of photoelectric recording by means of a multiplier working in the photocurrent mode, the line and background signals are represented by the corresponding photocurrents, viz.

measuring

Let Qi denote the number of photons from the spectral line per second at the output of the spectral instrument, which are incident upon the detector, and Qk the number of photons from the background in the spectra) interval which corresponds to the width of the spectral line. Let 7 denote the time constant of the readout system. where ‘1 is the quantum efficiency of the photocathode (l/r/ = the Then I = Q’qpz,, and a(l) = (2Beopsl)““, number of photons required for the liberation of one photoelectron), I( is the amplification factor, B is a constant z 1, eO is the electron charge in Coulombs. For this case [28] : (29a)

When a photomultiplier is used in the photon counting mode, the signals from line and background are defined in terms of the numbers of pulses recorded within the time 5 for the line and the background

(28b) Apparently, it = Q’qr and taking into account the Poisson bution we have o(n) = 111/Z. From this it follows [37] that [37]

1. S. ABRAMSON and S. L. MANDELSHTAM, Ix.

Akud

character

of the pulses distri-

Nuuk SSSR. Ser. Fi;. 18,653 (1954)

Some problems

of the emission

spectral

analysis

theory

“y?-& 2

B

587

(29’4

Finally, when photographic recording of the spectra is applied, the signals from line and background are the corresponding densities on the photographic plate [29] and

(284 As usually D = lg lo/Z, where I and I0 are the microphotometer readings for the image area A’ at the spectrum region under study and an unexposed region of the photoplate. Let JV denote the number of developed grains of the photoemulsion in an area A’, and a the average area of the grain projection. Then I = Z,e-“-‘, D = 0.430. bP, and o(D) = 0.43 aJN. For low densities, to a first approximation, it can be assumed that the number of developed grains equals JV = kE’sA’, where E’ is the illuminance on the emulsion E’ = Q’/A’, k the quantum efficiency of the photoemulsion (l/k = the number of photons required for the development of one grain), and r the exposure time. Then +?J& 2 Thus, we obtain a rather general expression

(29~)

(Vl’ II II, viz. II = ___ J2 O where II, = QL/(Q;p)“’ = signal/noise, i.e. the quantity which reflects the discrete character of the photon emission by the excitation source, coefficient q is the quantum efficiency of the detector and reflects the discrete character of the signal produced by the detector. For photocathodes v z 0.34.4, for photoemulsions y z 0.01 [38]. It should be noted that when deriving the expressions (29), we neglected the photoemulsion fog, the dark current of the photomultiplier and the background of the photon counter. However, it is not difficult to take into account also these additional sources of “noise” of the detectors [28,29]. Consider now the dependence of II on the parameters of the spectral instrument. Denote by qL the integral brightness of the spectral line in the source radiation, expressed in photons, and by qe the brightness of the background in photons per unit wavelength band. Then the photon flux from the line and the background emerging from the spectral instrument is Qi = q,shR

and

for the quantity

Q’s = qsshQAA’

where s and h are the width and the height of the entrance slit respectively, R = (d/“2, where d is the diameter of the aperture of the instrument, f the focal distance of the collimator, AE,’ is the width of the spectral line image in the focal plane of the spectral instrument (in case of photoelectric recording the width of the exit slit is chosen larger than line image). Designating by s’ the geometric width of the line image and by L’ = ds’/dX the linear dispersion of the spectral instrument, we can write AX = s’JL’. For s’ we can use the approximation [39,29] : s’ = ($2 + si2 + s;2y

where the first term is the geometric width of the entrance slit image s; = s/I’, the second term is the width of the diffraction image of the infinitely narrow entrance slit s; = s,p’, where s, = iv/d) is the so-called normal width of the slit, and the third term is the width of the spectral image of the interval which corresponds to the spectral line width in the source radiation s; = s~pl, where sL = 6X Here /I’ is the magnification of the spectral instrument /I’ = f’/f(l/sin CY), where c(is the angle between the central ray in the beam of rays which form the line image and the plane of the plate. Expressing E in terms of the angular dispersion L = Of’(l/sin IX),where 0 = d4/dl and introducing the maximum resolution [38] [39]

R. SHAW, J. Photogr. Sci. 11, 199 (1963). H. KAISER and G. HANSEN, Spectrochinl. Acrcc 3,433

(1949).

S. L. MANDELSHTAM

588 Table 1. Main parameters Spectrograph ISP-28 DFS-8 grating DFS-13 grating

type

(quartz prism) (diffraction 600 lines/mm) (diffraction 1200 lines/mm)

d/f

since

1:15

0.67

1:35 1:42

for three types of spectrographs

l/L’A/mm

(1 = 3000 A)

Of’W6

Ro

Pd

s. (cm)

sL (cm)

13.5

4.5 x 1om4

1.2 X lo-4

0.4

20000

28

1.00

6.0

1 X 1om3

6.3 x 1O-4

2.1

60000

32

1.00

2.0

1.5 X 10-3

5.0

120000

42

1.3 x 10-j

of the spectral instrument dictated by the diffraction at the aperture, R0 = Od, and also the maximum slit height ho (for the given spectral instrument), we obtain I-I = (Ypp2 KP(j

-!?L $2 4Y

(30)

where (31)

The quantity Pd can be considered as a characteristic of the spectral instrument detection power, and the coefficient K shows how this power is used in the given experiment. Similar to the case of informing power, the detection power of the spectral instrument is determined primarily by the value of its resolving power and the collimator speed. The detection power of the instrument is little affected by the slit width. Table 1 gives typical values of the parameters for three instruments: one with a prism and the other two with diffraction grating (the slit height ho is assumed to be 5 x 10-l cm, the spectral line width in the source radiation 6il= 0.03 A). As follows from (31), it is advantageous to increase the width of the entrance slit s. It seems to contradict the fact that the widening of the entrance slit results in the decrease of the line-to-background ratio. In practice, however, the increase in the slit width in excess of s z (2-3)s” does not lead to a further increase of K. The maximum value of K is, evidently, determined by the distance to the nearest interfering line in the sample spectrum. If we denote by AI. its distance from the analytical line, then, evidently, there must be AA/2 5 A>.. and

(32) where R, and RI, are resolving

powers corresponding

to A& and SE.

Consider now the quantity qL/q812which is determined by the spectrum excitation source. Using (22), we can write 1 1 ” = G 1 + R/Are assuming the plasma to be transparent. As to the continuous spectrum, we assume that it has a minimum value for the source, i.e. it is due only to the interaction of electrons and ions in the plasma, i.e. the free-bound and free-free transitions. Actually, there are usually some additional sources of continuous spectra due to the glow of graphite particles and of the ends of electrodes in an arc, the scattering by the optics of the spectral instrument, etc. ; however, in principle, they can be eliminated. The spectral structure of the continuous background radiation is rather complicated. In the spectral range of interest for the analyst it can be assumed to a first approximation that qB is frequency independent [16] and 32~~ qB

=

-

3JI

e6Z2 ,,+,Ni Cl c3(27W) 3’2 hv(kT,)“* ‘zi

(34)

Some problems

of the emission

spectral

analysis

589

theory

where ni is the total number of ions of all elements present in the source and z is the average charge of the ions. Since usually the degree of ionization is small, z z 1, and J+; Z &Ve. Substituting the values of the constants, we obtain N*

qB = 0.96 x lo-*’

o”z

1

>.

(35)

With (33) this can be reduced to 1 = 5.8 x lo8 1 + RlNe Jz qA’*

4L ____

Sl~“2~10(~T,)“4 SON,

e_E ,kT J” ’

e

011,2

(36)

and for the quantity FI we finally obtain : I-I = 5.8 x lo8 x (VT)“* x KPd

1 ’ 1 + R/&k

91~“2A10(kT,)“4 e_E ,,kTe &. 11,* 0 . so-J,+/,

(37)

Here the numerical factor is determined by the choice of the constants, the second factor by the recording device, the third one by the parameters of the spectral instrument and the fourth one by the parameters of the spectral line and the excitation source. For the evaluations of the detection limit we assume reasonable values of the various quantities in (37): q=O.l, T=loS, K=l, Pd=40, ;.=30OOA, Alo=lO*s-‘, T, = 6000 K (0.5 eV), E, = 4 eV, I = 0.1 cm, l+R/.l~=l,y~=y0=1.Thenweobtainfor~=3,.~~=.I’~,,~lO~ ~1’e-- lO”cm~ j. Wealsoassume atoms per cm3 (as to order of magnitude). To derive the minimum detectable amount g,,,i”(g) of the analyte, one has to specify the process that dictates the entry of the sample material into the discharge. For a direct current arc, where the sample is evaporated into the discharge and atoms depart from the discharge zone by the diffusion, the calculations yield [28]: hi. z lo- ” y. Recently similar calculations were made for laser microanalysis. The values obtained are lo- ‘5~10-‘3 g [34].

In both cases theoretically calculated detection limits are several orders of magnitude below those achieved actually at present. There are various reasons for this. First of all, it should be stressed once again that the above calculations based on very schematical pictures should not be considered as strictly quantitative. Their only purpose is to demonstrate the general relationships in which the signal-to-noise ratio in the spectral analysis and the limit of detection (which is determined by this ratio) are involved. In spite of these reservations, the difference between the theoretical results and practically achievable values for the analytical detection limits is, evidently, large enough. It is explained primarily by the presence of fluctuations in the radiation of the excitation sources [31]. Further, additional noises are introduced by actual detectors as compared to the idealized values assumed in the calculations. Finally, a considerable loss of sensitivity obviously results from the inefficiency of the link C -+ ~1; i.e. by the loss of atoms in the transition from sample to source. If one could retain these atoms in the excitation zone, for example, by ionization with strong U.V.radiation and retaining the ions by a magnetic field, then we would probably improve the detection limits by one or two orders of magnitude. It should, however, be noted that the theoretically calculated detection limit (x lo6 atoms per cm3 in the given example) is not an absolute limit. Recent advances in laser spectroscopy enable the detection of only few atoms [40]. These methods, however, have not yet reached such a stage of development that they can be used in common analytical practice. 5. SAMPLE

SUBSTANCE

TRANSPORT

All the above results refer to the right-hand side of the analytical procedure chain. The left-hand side of this chain, the link C -+ .a4b, has been relatively little studied as yet. It refers in particular to the processes of the sample transport to the excitation zone. Purely phenomenologically, based on the simplest representation, it can be assumed that the [40]

G. I. BEKOV, V. S. LETOKHOV and V. I. MISHIN, Pis’ma Zh. E.xper. Tear. Fiz. 27,52

(1978).

590

S. L. MANDELSHTAM

number of atoms of the given element which are transported per unit time to the excitation zone, for example, by evaporation, is proportional to their concentration in the sample, i.e. equals EC, and the number of atoms of this element which depart from the source per unit time, mostly by diffusion, convection, etc., is proportional to their concentration in the source cloud, i.e. equals /?A ‘,,. Hence we have for the rate of transport of atoms in the source [16,21] :

and in a stationary state aC = BeA‘0 or ,4 ‘0 = (x/fi)C (or more generally log .A ‘0 = bZ log C + n2). With the use of (29) it gives the so-called Scheibe-Lomakin formula : logZ=blogC+cr.

(39)

This expression holds with a surprising accuracy and forms the basis of all practical methods of analysis. The expression was found empirically about 50 years ago, fortunately without any theory at all.