A rate model of inductively coupled argon plasma analyte spectra

05M-IS~7182tl KI969-12503.00/0 Q 19X2 Per/mm Press Ltd.

A rate model of inductively

coupled argon plasma analyte spectra R. J.

LOVE-FT

Department of Chemistry, North Dakota State University,

Fargo, ND 58105, U.S.A.

(Received 24 February, 1982; in revised form 7 May, 1982)

Abstract-A model which explicitly considers the various rates of excitation, deexcitation, ionization and recombination for analyte species in an ICAP of defined electron density and temperature is presented. The mode1 reveals that radiative decay, radiative recombination and radiative absorption affect the level populations of a fun~mentaliy collis~naily domi~ted pfasma. fn addition, Penning ionization is shown to have a negligible effect on spectraliy derived temperatures except for elements of high second ionization potential.

EMISSION spectrum of analyte species from an inductively coupled argon plasma (ICAP) has often been noted for its lack of conformity to local thermodynamic equilibrium (LTE) [l-3]. LTE is a situation where all of the equilibrating processes in the system are due to col~sional processes and all radiative processes contribute to such a small degree to the total energy of the system that they can be neglected. In an LTE system, the relative populations of the energy levels within any ionization stage can be calculated by the Boltzmann equation: THE

E=Eexp-(&-ElIIT. and the relative ion level populations calculated using the Saha equation: n3 -= nl

of two adjacent

(1) ionization

stages

can be

6.037 x 102’g3 a 1;“‘”exp (- (EJ - E$ T;) &

where levels 1 and 2 are levels from some ionization stage (densities in cmm3), and 3 is a level in some higher adjacent stage, g is the statisticai weight, T, is the electron temperature in eV, E is the energy of a level (vs the ground atomic state value of zero) in eV, and n, is the electron density (cm-‘). The Boltzmann distribution is maintained from the principle of detailed balance of the collisional excitation rates and decay (superelastic) rates for atom-electron collisions in the plasma. The equation depends on a Maxwellian distribution of electron velocities. The Saha equation likewise is maintained from a detailed balance argument, but between electron-atom collisionat ionization rates and three body recombination rates. Again, a MaxweUian electron dist~bution is assumed. Utilizing the Bolqmann and Saha equations and measuring the intensity (irradiante) of spectral lines originating from levels 1, 2, and 3 (where level 1 and 3 are not, of course, the ground states of the atom or ion), a unique spectroscopic temperature can be derived for any two spectral lines since the irradiance is related to the level population via I = EnA, [I] G. R. KORNBLUM and L. DE GALAN, Spectrochim. Acta 328.71 (1977). 121 P. W. J. M. BOUHANS and F..J. DE BOER, Spectrochim. Ada 32B, 365 (1977). [3] J. F. ALDER, R. M. BGMBELKA and G. F. KIRKBRIGHT, Spectrocitim. Acfa 35B. 163 (1980). $I\‘“) 37-11- c

969

(3)

970

R. J. t.OVFT’l

where Aradis the Einstein spontaneous emission transition probability in s- ‘, and I is the irradiance (here in units of eV cm-j S-‘). Measured relative intensities, yield relative level populations, giving a unique T, from Eqns (1) or (2). Of course one should choose lines for which Arad is well known and for Eqn (2), an independent means of measuring n, must be found (measuring the Z-I, width for example) [3]. The temperatures derived from the above methods are known as the excitation temperature (Boltzmann) and ionization temperature (Saha). Only two reasonably comprehensive studies of these temperatures in the ICAP have been published, one by BOUMANSand DE BOER [2] (hereafter BD) and one by ALDER, BOMBELKA,and KIRKBRIGHT[3] (ABK). Both, in different ways, extract excitation and/or ionization temperatures by spectral line irradiance ratio methods. Both note inconsistencies from element to element in the derived temperatures and ABK notes inconsistencies between excitation temperatures depending on which spectral lines were measured in atomic iron. BD established an excitation temperature using two zinc spectral lines (5850 K) and using that temperature and an assumed electron density of lOI cme3, they calculated what the ion to atom analytical line intensity ratios should have been. Their subsequently measured intensity ratios are larger than expected intensities calculated by the Saha equation. They conclude the ionization temperature is larger than the excitation temperature which requires an overpopulation of the ion line excited state (vs the LTE value at 5850 K). To account for this overpopulation, they postulate that metastable argon atoms (the 4s[l l/2]! (3Pz) and 4s[1/2]! (3P0) configurations) increase the ion level population through collisional excitation or Penning ionization; X+Ar”

e X++Ar+

e-

(4a)

or X+Ar”

$X+*+Ar+e-.

(4b)

This mechanism originates in a similar proposal for direct current argon plasmas [4,5]. No quantitative analysis of the effect is offered in their paper. Keep in mind that BD used an electron temperature of 5850 K. If a higher temperature were used, the degree of departure from LTE in their results changes. Their hypothesis requires a number density of argon metastables in excess of the LTE value, which has not yet been proven. Their discussion of the effect of argon metastables is qualitative and is presented as a perturbation on the Boltzmann-Saha equilibrium backbone. ABK, using iron, derive an excitation temperature for high energy atomic iron levels of 8510 K at a 10 mm viewing height and a measured 3.34 x 10” cme3 electron density. Lower energy iron levels yield a lower temperature. Ionization temperatures calculated for six elements are found to be generally lower than 8510K. In general, their six elements show a range of ionization temperatures from 7720 K (Zn) to 8440 K (Ba). The fact that the low energy atomic iron levels show a lower excitation temperature implies that these levels are more highly populated than expected and ABK attributed this to population through radiative decay processes. They propose the major energy exchange mechanism to be electron collisional processes. Thus, the question of why the ICAP analyte spectra is not in LTE remains open, since no quantitative model exists and qualitative mechanistic explanations are not totally convincing. With this in mind, a quantitative model of ICAP excitation is presented below utilizing the individual rates for all processes which excite, deexcite, ionize, or recombine the analyte atom or ion. The model is used to assess the effect of various processes on the level populations of analyte atoms and to relate the effects to possible changes in temperature or electron density. Certain qualitative observations are addressed, such as elemental variations in ionization temperature, but no attempt is made at this time to completely reproduce the data of BD or ABK. In general a [41 A. CZERNICHOWSKI,J. CHAPELLE and F. CABANNES, C.R. Acod. Sri., Ser. [S] P. RANSON and J. CHAPELLIT. 1. Phys. (Put-is) Colloq.32, CSb-39(1971).

B

270,54 (1970).

Inductively coupled Ar plasma analyte spectra

971

backbone of an LTE system with radiative decay at 8200 K and 3.34 X lOI electrons per cm3 is assumed, and the ability of and extent to which other processes disrupt the LTE backbone is evaluated. The method presented will solve the level population equations but the actual results depend heavily on the quality of the rate equations. Every attempt has been made to use the best available rate equations, but most rates are still approximations. The method depends on the following assumptions: (1) the plasma is in a steady state; (2) all excitation, etc. processes are fast compared to diffusion or analyte transport; (3) the instantaneous electron density can be accurately measured; (4) the line intensity ratios (where experimentally determined) are not affected by self absorption; and (5) the electron-velocity distribution is Maxwellian. Assumption (1) is presumably valid since the source is observed to yield a consistent signal with time (hence its analytical utility), although conceivably the 27.12 MHz frequency of the rf may interact with some rates. Assumption (2) is probably valid for all processes faster than 104s-’ since the analyte stream moves at 1O’cm s-’ [6]; diffusion may change the spatial n, and T, conditions for some analyte atoms. Electron densities have been measured [3] and the degree of self absorption is not severe [7]. No data suggest that there is reason at this time to u priori assume anything other than a Maxwellian velocity distribution for the electrons. This study does not consider argon; no calculations of the level populations of the support gas are made. A set of conditions likely to be found in a particular cm3 of a plasma is postulated and the response of the analyte to those conditions is calculated. The alkaline earth elements are examined because of the large amount of spectroscopic data for them, they have been heavily used in experimental studies, and only three levels of ionization need be considered (e.g. CaI, CaII, CaIII), which simplifies the calculations. Relevant data on these elements is presented in Table 1. 2. THEORY 2.1.

The matrix An atom possessing n energy levels can undergo n(n - 1) transitions, some of which are populating and others depopulating between the various atomic and ionic energy levels. An electron in any particular level N,, is able (conceptually) to be moved into any other unoccupied energy level. Thus level nl is depopulated at some overall rate, which is the sum of the individual rate processes to all other levels, and Table 1. Data on elements used

-.

Elements Energy

(e.v)

Ca

Sr

Ba

7.646

6.113

5.695

5.212

1.P.2

10.004

18.211

15.035

11.871

11.030

lPl Level

5.277

4.346

2.933

2.690

2.239

2P 3,2 Level

3.960

4.434

3.151

3.040

2.122

2P 1,2 Level

3.959

4.422

3.123

2.940

2.512

'so-lPl

234.861

285.213

422.673

460.733

553.5

2s 1,2-2p3/2

313.042

279.553

393.366

407.771

455.403

280.270

396.847

421.552

493.409

2s

M.

M9

9.322

Transition

I61 R.

Be

1.P.l

1/2-

2P

l/2

(nm)

313.107

BARNESand R. G. SCHLEICHER, Spectrochim. Acta 3OB. 109 (1975). J. D. BRADSHAW. M. S. EPSTEIN,R. D. REEVESand J. D. WINEFORDNER. Anal.

[71 N. OMENE~TO,S. NIKDEL, Chem. 51, 1521(1979).

912

R. J. Lovrm

the level is populated from all other levels at various rates. In a the population of level n, will not change with time, thus the must equal the total depopulating rate. Each rate is composed of the level population density of the depopulating level; consider

level system. A set of four equations denote depopulating processes:

describes

the system,

steady state system, total populating rate a rate constant times

for example a four where negative rates

- R,,n, + RIzn2 + R13n3+ R14n4 = 0 RZIn, - RT2n2+ Rz3n3 + Rz4n4 = 0 (5)

R,,n, + Rjznz - R3,n3 + R3.,n4 = 0 R,,n, + R4g2 + Ron3 - R,,.,n4 = 0.

Each rate constant subscript denotes first the level being populated, then the level being depopulated, thus Rz3 means the rate constant from level 3 to level 2. In the first line of Eqn (5), level nl is depopulated at some overall rate, - R,,n,, and is populated from levels n2, n3 and n4 by the other terms. The depopulating rate constant is equal to the sum of Rz,, R3, and R4, which are the rate constants by which nl populates n2, n3, and n4. The equations can be written in matrix form: - RII R2,

-

RIZ

RIG

RM

R22

R23

R24

n2

-

R33

R34

n3

-

&4

n4

R3,

R32

R41

R42

-

R43

-

ni

(6)

which can be solved for all values of n (see below) if the rate constant terms can be evaluated. 2.2. Rates 2.2.1. Electron coNsion processes. The evaluation of the rates requires the calculation of various rate constants, (K), and partial rates equal to the Rab values in Eqn (6). A number of processes are considered. The electron process rate constants (or partial rates) for use in the matrix are: (a) collisional

excitation:

X, + e- + X, + e- [8]

RCE= 1.58 x 10-5[fPq(~,,q)/xT~‘21exp (-x/The

= K,(q,ph,

(7)

where x = transition energy (eV) f,,, = oscillator strength (GP4) = thermally averaged Gaunt factor assumed for simplicity to equal 1 [8]. (b) Collisional decay: X, + e- + X, + e- [81 RCD= 1.58 x 10~5(g,/g,)[f,,(~,,)IXT,1'23n, = &(P, sh. (c) Collisional ionization: Rc, = I .24

x

X, + e-+X,’

(8)

+ 2e- [9,101

10-L,$T;3n[exp( - 4)/4’)][0.915( 1 + 0.64)/4)-’ + 0.42( 1 + 0.5/4)-*]ne = Ke,(c, p)n,

181 H. YASUDA and T. SEKIGUCHI, Jnp. J. Appl. Phys. 18.2245 (1979). 191 R. K. LANDSHOFF and J. D. PEREZ, Phys. Rev. A13, 1619 (1976). (IO] D. SALZMANN and A. KRUMBEIN, 1. Appl. Phys. 49. 3229 (1978).

(9)

Inductively

coupled

Ar plasma analyte

973

spectra

where C#J = EJT, ,$ = number of ionizable electrons. (d) three-body recombination: XC’ + 2e- + X, + eR,,, =

(10)

Rcr11J6.037 x IO*’ET,?” exp (- x/T,) = K,,,(p, c)n,-’

(a detailed balance of process c). (e) Radiative recombination: X,’ + e- + X,, + hv [81 R, = 5.2 x 10-‘4(2 + 1)~#~“*(0.43 + 0.5 In 4 + 0_47&“‘)n, = K,(p, c)n, where 2 is the charge on the recombined (f) Autoionization:

(11)

species.

X$ + XC’+ eR autcl = Adc,

where A,.,, = transition probability (g) Autoionization

recombination:

4)

for autoionization. X,’ + e- --*X: [I 11

R,, = 1.65 x lo-** f

T-‘I* ew (&)A,,&

cb,

(13)

Rdr = T-“*R(Z) x A(Z, i) exp (- 4)~ = K&I, c)n,

(14)

c

= L(q,

(detailed balance of process f) (h) Dielectronic

recombination:

X,’ + e- + X*,,,,, X:,,,

+ X: + hv [ 121

where X,,,,* is an excited atom level above the ionization limit.

i

R(Z) = 6.5 x IO-

,o

z”*(z + 1)2

(z2

(15)

+ 13.4)~/2

fzi(XP2 A(Z) = 1+ O.lOSy + 0.015y2’ where Z is the charge on the recombining

(16)

ion and

y = XNZ + 1)&f;

(17)

IH is the ionization

potential of hydrogen (13.6 eV). 2.2.2. Atom Collision processes. Processes involving atoms are limited to: (i) Atom collisional ionization: X, + A + X,’ + A + e- [ 131 R,i = 5.21

X

IO-” (;>III&.f{&~

M~))W~(b)n*=KJc.

Rev. A 1%.697 (1978). [I21 C. BRETON, C. DE MKHELIS and M. MAITIOLI, J. Quont. Spectrosc.

p)nA

(18)

[I I] V. L. JACOBS and J. DAVIS, Phys.

Rodiot.

Transfer

19,

367

(1978).

974

R. J. Lovt~r

where

In the above T and MA are the atom kinetic temperature (K), and atom mass (8); M, is the electron mass (g), MH is the hydrogen atom mass (g), k is the Boltzmann constant (erg K-‘) and 4 = EJkTA

(20)

where Ei is the ionization energy of the level in question (here in ergs). (j) Atom recombination:

X,’ + A + e---,X + A [ 131

R,,, = R%/6.037 x lo*’ g TA3’*exp (- 4) = K&J,

c)nAnr

Ideally, Eqn 21 should consider both the gas (TJ and electron temperature different) but the error due to the above approximation is small if T, - T.. (k) Penning ionization: Arm + X, 3 X,’ + Ar + e-

(21) (T,) (if

R met = L, FuVnl\, = Kme,(c, p)n&m

(22)

Q_-- 8kT, “* ( rlJ >

(23)

where

(24) and F = (EjEArm)“3.

(25)

The cross section (u) is taken to be in the range of 25-40 x lo-l6 cm* in accordance with line broadening cross sections [14], but the value of most importance is gxn&rn which is varied in the calculations to depict the nature of the effect with increasing n,m values. The variable F reflects the premise that due to the huge amounts of kinetic energy that must be transferred in ionizing highly excited states, these reactions are less favored than reactions with the lower energy atom states. Cross section values are assumed to be constant for all levels considered, which is unrealistic but greatly simplifies the calculations. The Penning process resulting in excited state analyte ion formation, Eqn (4b), was not considered, since the electron collisional rates between ground and first excited states are so much larger in an ICAP. This process is much less important than in a low pressure source. 2.2.3. Radiative processes. Finally radiative processes are considered: (1) Radiative decay: X*, --) X,, + hv

(26) (m) Absorption:

X, + hv + X: Rabs= 6.79 X l@A,d/ &“)I, = R,&q, p)l,

[13] H. W. DRAWIN. 2. Physik 225. 483 (1%9). [14] K. J. LOVEIT and M. L. PARSONS, Appl. Spcctrosc. 31,424 (1977).

(27)

inductively

coupled

Ar plasma

analyte spectra

97s

where u =

$ (s-l)

h is Planck’s constant in eV s I, is in units of W cm-* HZ-’ (the irradiance of the source upon the cm’ under considerations, Einstein absorption probability.

and B,,

is the

(n) Stimulated emission: X: + hv + X,, + 2hu

(2% where Bstim.is the Einstein stimulated emission probability. 3.

CALCULATIONS

3.1. setting up the matrix The rate constants for the transitions from any level to all other levels are calculated. Once some electron temperature, atom temperature, and electron density are assumed, the appropriate processes from those denoted in Section 2 are evaluated. For example, an arbitrary atom level radiates and is collisionally coupled to all lower energy levels with non-zero oscillator strengths. The total rate to any level is the sum of the radiative and collisional rates. The level is also coupled to higher energy levels through collisional excitation. The level ionizes through atom, electron, or metastable argon collisions; the total ionization rate comprises the sum of these processes. The sum of all of the above processes (excitation, deexcitation and ionization) is the rate at which the level depopulated (the negative rate in Eqn (5)). Once all of the rate constants are caculated, the matrix is created. For the alkaline earths, 80 levels are used for beryllium {an 80 x80 matrix), 112 for magnesiuim, 170 for calcium, 141 for strontium and 148 for barium. The input data for the elements consists of the J quantum numbers, the energy of each level [15-173, and all available transition probabilities and oscillator strengths [l&21] 3.2. Solution of the matrix A modification of a Gauss elimination technique (22) is used to solve the matrix. Gauss elimination will not solve a matrix in which the resultant vector is a zero vector (the trivial solution all n, = 0 arises). Since the desired results in this study are normalized level populations, the exact solution of the matrix is not required, that, is, it does not matter if the actual solved population of some level R is 10-‘“cm-3 or 10d cmm3, as long as the ratio of level n to level m is consistent. The solution is to set the dnldt term of the ground state ion (or any other highly populated level) to some small value (typically 10v3) instead of zero. For all practical purposes lo-” is equal to zero, since the total rate constant of depopulation (R times n) of the ground state ion is about 10’ s-’ (R is approximately 108 and n approximately 10”). Thus in adding the rate constants of depopulation (-10’s_‘) and the sum of the rates of population (+lO’) a result of lo-‘, or one part in lo’, is found. This very small resultant is tolerable. [I51 C. E. ?&ORE, Atomic

Energy Levels, Vol. I. NBS Circular 467, Washington, D.C. (1949). [16] C. E. MOOR&Atomic Energy Levels. Vol. II, NSDRS-NBS-35, Washington, D. C. (1971). [i7] C. E. &ORE, Atomic Energy Levels, Vol. III, NSDRS-NBS-35, Washington, D.C. (1971). [18] C. H. CORLISSand W. R. BOZMAN,~~e~ment~ Trunsition P~babi~ities for Spectrai tines of Seuenty Ekments, NBS Mon~ph 53, W~hi~ton, D. C. (1%2). [I91 W. L. WIIZSB,M. W. SM~H and B. M. GI.ENU?N, Atomic Transition Probabilities, Hydrogen Through Neon. NSDRS-NBS-4. Vol. I, Washington, D. C. (1%9). [20] W. L. WILE, M. W. SMITHand B. M. MILES, Atomic Transition Probabilities, Sodium Through Calcium, NSDRS-NBS-22. Vol. II. Washington, D. C. (1%9). [21] A. LINDGARDand S. E. NIELSEN, Atomic Data and M&ear Data Tables, 19,534 (1977).

976

R. J.LOVE-I-T

Once the level populations are determined, they are substituted into Eqn (5), the matrix, and resultants are calculated for each row. While few resultant values equal zero, all are very low (less than IO-‘). Changing the magnitude of the resultant does not change the relative populations of the levels, but changes the actual calculated value for the levels. The best test for the suitability of the modi~cation is to calculate

the level populations at high electron densities, in which case the method solves exactly for a Boltzman-Saha level population distribution, The approximation will break down for situations in which the total rates are small (i.e. low temperatures and electron densities) but performs well in the areas of interest for the ICAP. 3.3, output The calculated level populations are normalized to a sum of one and are expressed as relative (fractional) level populations. In addition the normalized Boltzmann and Saha populations values are calculated via Eqns (I) and (2). The results are also presented in the form of BN values, the ratio of the rate model populations to the Boltzmann-Saha populations. Thus if BN were less than one, the levels are less populated than the Boltzmann-Saha dis~bution would predict. The ratio of the level populations of the ionic levels responsible for the main analytical spectral ion line (the *p$,~z levels) and the level responsible for the major analytical atom spectral line (‘~3 is also evaluated in order to compare these calculated ratios to the experimental studies [2,3]. It should be noted that the ratio of the intensity ratios,

(30)

the means by which BD [2] express their results is exactly equivalent

to the ratio

BN(ion line) B,(atom line)’ 3.4. Facilities All computer calculations were conducted on an IBM 3601158 computer Fortran program developed in our laboratories.

(31)

using a

4. EVALUATIONOFTHEIMPORTANCE OF PROCESSES WHICH AFFECTICAP LEVEL POPULATIONS 4.1. Electron collisions and radiative decay 4.1.1. Leoel ~a~a~utju~s ~~t~j~ one e~e~e#t. The processes

of electron collisional excitation (Eqn (7)), cotlisional decay (Eqn (8)), collisional ionization (Eqn (9)), three-body recombination (Eqn (IO)), and radiative decay (Eqn (26)) are the only processes considered by ABK, and comprise the backbone of the plasma description. These processes will always be present and any influence of other mechanisms must be viewed as perturbations on this system. When evaluating the level populations of magnesium (Fig. 1) the general tendency is for low energy atomic levels to be more highly over populated (a large BN) than higher energy atomic levels. In general the higher the energy of the level, the lower is its BN value. Figure 1 is plotted as energy level number vs BN, thus the abscissa bears no linear relationship to energy but higher energy levels lie to the right. The fluctuations in Fig. 1 are due to the lack of radiative decay channels for certain levels. The population of these levels is small enough that even providing a radiative decay channel will not increase the level population of the spectroscopic levels. Evaluation of the actual normalized populations of the ground state atom (IS,) and ion (‘S,,r) and the spectral line generating states (here MgI ‘PI and MgII *Pzn) (Table 2) gives some insight into why the system is not in local thermodynamic equilibrium. The

Inductively coupled Ar plasma analyte spectra

LEVEL

NUMBER

977

(ENERGY+)

1. BN values for 112 levels of Mg at 8200 K and Ne = 3.34 X 10” cm’. The abscissa is plotted as level number and is not linearly related to energy. The points are connected to aid visualization. The arrow indicates the ground state singly charged ion state. Fii.

relative populations of the excited states to the ground states for the two spectral lines reveals a lower relative population of the excited state than the Boltxmann distribution predicts (Iable 3). In addition, the ground state atom level, as noted in Pi. 1 and Table 2, is overpopulated with respect to the LTE value. The depopulation of the ion and atom excited states is by both collisional decay (CD), Eqn (8) and radiative decay (BD), Eqn (26) while the populating mechanism is entirely collisional excitation (CE), Eqn (7). It is important to recognize that the relative populations of any two adjacent levels is, in any ionization stage, due entirely to rates coupling those states. The population ratio of the ground and excited state is: n CE ex= n, CD+BD

(32)

while the LTE value should be

Calculating CE (5.575 x 10’s-’ cme3) and CD (8.715 x lo9 s-’ cmV3)for the ‘S,, and ‘P, atom levels yields the LTE ratio, n,/n, = 6.397 X 10e3; addition of BD (4.95 x --

Table 2. Normalized level populations for Mg’ Normalized

Level Designation

Ngl ‘so

Population

l'odcl

LTE

8.

2.394-Zb

2.253-2

1.06

MgI 'P,

1.449-4

1.442-4

1.00

WI

9.653-1

9.665-l

0.999

25,,2

blq:1 2?3,2

3.428-3

3.640-3

0.942

II9111 's,

2.594-4

2.970-4

0.871

‘T, = 8200 K. n, = 3.34 x 10” cm-’ 9.394-2 means 2.394 X lo-*

R. J. I.nvwr Table

3. Relative

populations Relative Model

Levels Mql

'P, T

of energy

levels

Populations Boltzman

6.050-3

6.398-3

3.552-3

3.766-3

0

M911 2P3,2 e51/2

108s-’ cm-‘) [20] and use of Eqn 31 gives nex/ng = 6.050 x 10V3, which corresponds exactly to the model calculated ratio (Table 3). Thus, deviations from LTE can be expected at the normally accepted ICAP temperatures due solely to the contribution of radiative decay. Note also that the BN value for the ground atomic state is 1.06 (Table 2). The ratio of the RD rate to the CD rate is 0.057, thus one would expect the ground state to be about 6% more highly populated than if collisional decay were totally dominant. All of the lower levels in Fig. 1 that have I&., values greater than 1 reflect the contribution of radiative decay from higher energy levels on their population. This will occur only when the radiative rates are high compared to the CD mechanism. The ground state ion, for example, has a BN value of about 1 since that level is mostly populated through ionization, but it is noteworthy that a small variation in a level with a relative population of about 1.00 can have a significant effect on levels with lower relative populations while not changing the BN value of the highly populated level. The change in Table 2 for the ion ground state is relatively insignificant but large in its absolute magnitude (~~s,,+TE)- w,,~(M~~~I)= 0.(3012) The ratio of populations for the ion ground and excited state can also be exactly equaled by consideration of radiative decay, but note that the 2P3,2 level has a BN value below 1.00, indicating that when considering equilibrium the most highly populated state seems to control the deviation from LTE at all other levels. The ion ground state, being highly buffered to population changes due to its high initial population, is established via collisional ionization and three body recombination. The highly excited levels of the atom are coupled by these mechanisms to the ion ground state and maintain a BN value of about 1.00. The lower energy atom levels are less significantly coupled to the ion ground state, thus can be overpopulated via their excitation and decay coupling to the more highly excited states. The ion ground state will not change much in population, since even with a large relative overpopulation in the low energy atom states, their absolute populations are small. The doubly charged ion state for Mg at LTE has a low population. Any radiative decay (after recombination) into the low energy ion levels will be able to deplete the doubly charged ion population. Comparison of the populations of the zP312ion state and the double charged ion ‘S,, state reveals the excited state to be more highly populated. The excited ion state is thus populated mostly by collisional excitation from the ground state ion and very little contribution can be expected from the doubly charged state or high energy ion levels. The fact that the excited ion state BN value is below 1.00 is due to the rate of radiative decay to *SllZexceeding radiative populating rates from higher energy levels. Usually, the experimental determination of the ionization temperature depends on the measurement of the relative intensities of the atomic and ionic analytical lines, here from the ‘P, atom and ‘P3,* ion levels, and use of Eqn (2). The level populations can be derived, if necessary from Eqn (3). Calculation of n2P31Jn’P, (which is related through Eqn (3) to the intensity ratio), from Table 2 gives a ratio of 23.67 for the kinetic model and 25.25 for LTE. Since a spectral measurements will provide a value of 23.67 presuming adherence to the kinetic model, Saha based analysis of the ratio yields a temperature near 8150 K, not 8200 K. Even in this simplistic model, the

Inductively coupled Ar plasma analyte

SPeCIra

979

measured analytical line intensity ratios underestimate the temperature if a Saha based calculation is used. The high energy atom lines all possess BN values near 1.00, thus a comparison of two lines from these levels (analogous to ABK’s Fe(I) analysis) yields a Saha-derived temperature of 8200 K. Temperatures derived from low lying atom (or ion) levels tend to yield low temperature values since the lower energy level is more highly populated (larger BN) than the higher energy level. In the case of Mg the population of the ‘PI atom level is higher than its LTE value (Table 2), and the population of the *Pj12ion level is lower than its LTE value. Previous discussions of non-equilibrium in the ICP discuss the overpopulated atom level [3], but no mention of an underpopulated ion level has been advanced. The net effect is to lower Saha derived temperatures more than if only an increased atom level population were considered. When ABK [3] determined their temperatures in an ICP, the conditions which yielded a 3.34 x 10” cm’ electron density lead to an iron atom line excitation temperature (high energy levels) of 8510 K. Their Mg temperature using the levels discussed here is found to be 7950 K. The large extent of the decrease of the derived ionization temperature from 8510 K cannot be easily reconciled to a simple electron collision-radiative decay model unless the electron collisional rate constants are shown to be much lower than in Eqns (7) and (8). Evaluation of Eqns (7), (8) and (26) reveals that any increase in electron temperature or density changes the collisional excitation and decay rates and does not affect the radiative decay rate. An increase in electron density ‘favors a closer adherence to LTE, thus at electron densities above about 1 x lOI cm3, LTE (particularly for the analytical line levels) is achieved. The temperature dependence is less obvious, but since the collisional decay is dependent on T;“*, an increase in T, fosters an increasing departure from LTE. The collisional decay rate is about 10% lower at 10000 K than at 8200 K, thus the relative contribution from radiative decay is larger. Apparently the more critical parameter when experimentally determining the temperature is the electron density. 4.1.2. Interelement comparisons. The normalized populations of comparable levels in two additional alkaline earth elements, Be and Ba, illustrates the specific differences which can occur due to the unique properties of various elements (Table 4). This diversity is important when one considers the variations in temperature ABK and BD note from element to element. For beryllium, the Be11 *SIcr state is not much more populated than the Be1 ‘So state. Thus any losses in population from the z&12state due to radiative decay tend to change (lower) the BN value. The doubly charged ion state (Be111 IS,,) is very poorly populated and would not affect the singly charged ion levels at all. In most respects, beryllium is like magnesium, except due to the high Be ionization potential, a decrease in the ‘Sl12 level can be seen. For barium, with two low ionization potentials, the doubly charged state is reasonaly well populated. Radiative decay from highly excited (and reasonably highly populated) BaII levels will contribute enough to the BaII 2Pu2 population to raise its BN value above 1.0. This contribution from BaIII will also affect the BaII 2S,n ground Table 4. Normalized level populations of Be and Ba’ se state

Level

Population

Ba 6N

Population

6,

I

'5,

2.091-l

1.10

2.566-4

1.18

I

'Pl

3.250-4

0.99

3.147-5

1.15

II

251/2

7.439-l

0.97

3.020-l

1.03

II

2P 312

7.a66-3b

0.93

1.249-2

1.01

'5"

1.609-6

0.61

9.25&2

0.83

111

‘8200 K, 3.34 x IO” electrons per cm3 b2;,2 + *PI,2

R. J. LOVES

980

state whose BN is larger than that of BaII *PwZby the amount contributed by radiative decay from 2 PS12.Because BN for the *S,,Z state is greater than 1.0, the populations of atom levels will tend to be larger than for Be and Mg. The temperatures derived from spectral measurements will tend to vary depending on the relative extent by which BN values for the *Pvz level and the ‘P, level deviates from 1.00. 4.2. Radiative recombination Inclusion of radiative recombination Eqn (11) into the model presented in Section 4.1 changes the results. A comparison of the level populations of magnesium and barium with and without radiative recombination (Table 5), reveals that the general tendency is to increase the populations of the atom levels and greatly decrease the doubly charged ion level. This further increase of the low energy atom levels over the collisional-radiative decay model tends to lower even more the temperature calculated from measured spectra since the population of the atom excited Ievel is increased more than the excited ion level population, which changes little and may even decrease (Mg). Using the Saha equation for Mg, the relative population ratio (n2Prlz/n’PJ when radiative recombination is considered is 21.76 (vs 23.67 for the model without radiative recombination). The temperature derived is 8090 K (vs 8150 K). The electron temperature for both systems is 8200 K. Evidently at the electron densities in an ICAP, radiative recombination competes effectively with three body recombination as a depopulating mechanism from both ionic ground states. The populating effect is greatest for the ground atomic state, implying that the effect at least partially involves recombination to that state, whereupon the atomic excited levels are mostly populated according to the collisional-radiative decay model presented previously. The large decline in the doubly charged ion state can be related to the dependence of radiative recombination on charge in addition to the larger second ionization potential (see Eqn (11)). The larger decrease in the MgIII state vs the BaIII state is largely artificial. The current data bank for Ba contains many more radiative transition probabilities than Mg in the singly ionized state and the population of the doubly charged level is very dependent on the number of radiative transitions supplied. The effect of these added transitions on the populations of the ‘So, ‘Pi, *Sm, and *Prj2 states is insignificant. The ground state ion BN value decreases for Mg while the value increases for Ba. For Mg, the population of the MgIII ‘S,, state is very small, as noted before, thus its greatly increased recombination rate effects the MgII *Pin state very little (the buffer state), yet loss from the buffer state to the MgI ‘S,, state will diminish *Sl12somewhat. For Ba, the BaIII state is highly populated, thus increased recombination from ‘S, will be capable of increasing the *Sin population. Since the rate of radiative recombination for ‘S0--j2Sm will exceed that of *Sm + ‘So (atom), the ionic ground state population will show a net increase. With increasing electron density, three-body recombination begins to dominate due

Table 5. 8,

w state

Ba with

RR

Level

Model

I

‘%I

1.06

1.15

1.18

1.30

1

'Pl

1.00

1.09

1.15

1.26

0.999

0.997

1.03

1.05

0.942

0.940

1.01

1.02

0.871

0.637

0.830

0.707

II

2s

II

2P

111

1%

l/2 3/2

%200 K . n, = 3.34

x

IO” cm-

’

Model

with

RR

Inductively

coupled

Ar plasma anaiyte

spectra

981

of radiative recombination with densities radiative recombination will predominate (microwave plasmas). Increasing the temperature decreases the radiative recombination rate (by roughly jre-“*)and the three body rate (by about ‘RI”?-‘), resulting in a net increase in the relative contribution from radiative recombination. The model becomes essentially a collisional-radiative model [23] with the inclusion of radiative recombination (this notation refers to ionization and recombination, not excitation and deexcitation as used in Section 4. I). The ICAP apparently occupies the interfacial regime between the pure LTE approximations and the corona model [241 and neither model appears to be sulliciently applicable to describe the analyte environment in an ICAP. In evaluating experimentally measured ion to atom line intensity ratios, radiative to

its

n,*

dependence

vs the linear dependence

electron density. Alternatively, at lower electron

recombination will lower the apparent temperature. The magnitude of the effect is a function of the first and second ionization potentials of the element studied. As noted previously, for Mg ABK found a temperature of 7950K (vs 8510 for FeI) while mciusion of radiative recombination only lowers an 8200 K temperature to an apparent 8090 K. Nonetheless, assuming the rate equations proper trend is demonstrated.

are approximately

correct, the

4.3 Autoionization, dielectronic recombination, atom collisional processes. The evaluation of the above processes by the model, reveals no significant effects. Consideration of autoionization piaces those levels which autoionize into LTE,

otherwise the levels tend to be tremendously over populated. No ability to increase the electron density due to the addition of autoionization generated electrons is currently written into the model, thus the effect of autionization on n, could not be evaluated. Dielectronic recombination is expected to be important for high electron temperatures and low densities [ 113, and contributes an amount of recombination under ICAP conditions that is about 1% of the total recombination by other processes. Atom colhsion processes do have an effect, although small. The effect is due entirely to the argon support gas, not to introduced hydrogen atoms from the nebulized water. The effect on level populations is much less than that due to the processes considered in sections 4.1 and 4.2. 4.4 Penning ionization Penning ionization has been postulated to be a viable mechanism for analyte excitation in an ICAP [23 by analogy to d.c. plasma studies [4,5]. Utilizing a series of cross section times metastable argon densities (axnJ (the LTE value for metastables is about 10” cm-’ for an 8200 K plasma) the effect of Penning ionization on level populations is easily evaluated. Cross sections of approximately 30 x lO+ cm2 are used. For this study radiative recombination is considered (Section 4.2) and the atom temperature is 6000 K 1251. A comparison of the level populations of all levels for Be (Fig. 2) and Ba (Fig. 3) reveals a noticeable distinction between the elements. For beryllium the consideration of a 10” cma3 metastable argon density lowers the population of the lowest atom levels and raises the population of all other levels, Note that the high energy atom levels are about in equilibium with the ion ground state due to ionization-recombination processes. For barium, at a me&u&able density of 1Or3cmW3the population of all levels except the BaIII state decreases rather uniformly. The effect is summarized for the spectroscopic levels in Table 6. In Figs. 2 and 3, the peaks are due to the lack of radiative transition probabilities to ground states for some high energy levels. f22] ANONYMOUS,System 350 Scienfi& Subroutine Package, Version 3 Programmer’s Manual, 5th Ed. IBM Corp., TechnicalPublications, White Plains, NY, p. 121 (1970). [23] D. R. BATES, A. E. KIN~XTON and R. W. P. NCWHIRTER,Prac. Roy. Sot. A267,297 (1962). [24] R. H. HUDDLESTONI?, and S. L. LEONARD,Plasma Diagnostic Techniques, Academic Press, New York (1965). f251 H. G. C. HUMANand R. H. SCOTT.Spectrochim. Acta 318,459 (19763.

982 It ‘5

,

I ,

B” I!

0.1

LEVEL

NUMBER(ENERGY -3)

Fig. 2 EN for Be (80 levels) with (---) and without (-) Penning ionization (10” cm-‘Arm) at 8200 K and n, = 3.34 x 10” cm-‘. The arrow indicates the ground state singly charged ion state.

1

\ \

.

,-

i

LEVEL

NUMBER (ENERGY ---+I

Fig. 3. BN for Ba (148 levels) with (---) and without (-) Penning ionization (10” cm-‘Arm) at 8200 K and a, = 3.34 x 10” cm-3. The arrow indicates the ground state singly charged ion state.

Table 6. Effect of Penning ionization on level populations

for Be and Ba

Level Populations Ba

Be Ar';c;%;ity

,112

2P

3/2+1/z

7.39-3 7.39-3

‘Pl 3.91-4

2P

3/2+1/Z

"1

2.13-Z

3.46-5

10'3

7.99-3

2.99-4

1.55-Z

2.52-5

10'4

9.93-3

9.79-5

4.17-3

6.75-6

Inductively coupled Ar plasma analyte spectra

983

These results are unanticipated in previous discussions of the metastable argon effect [2]. What is clearly indicated is that a metastable argon atom will ionize a metal as often as capable. The I I .55 and I I .76 eV energies of the metastable states are capable of ionizing all of the alkaline earth atoms, but, Be with a second ionization potential of 18.2 eV cannot be easily ionized from either the Be11 ground state or the ‘PI excited state (5.28 eV). The other alkaline earths can be ionized from one or both of these states. As a result, beryllium largely ionizes by Penning collisions to the Be11 state at the expense of the lowest energy atom states since the high energy atomic states have collisional recombination rates high enough to equilibrate them with ground state ion. Metastable ionization depletes the atom ground state, in the absence of a fast recombination mechanism to repopulate it. For barium (and magnesium, calcium and strontium), the ground and excited states ionize by a metastable argon collision to the BaIII state, which “piles up”. The energy levels below the BaIII state reequilibrate via the modes in section 4.2. The absolute populations of the atom and singly charged ion levels decrease (Table 6), but the relative populations of the levels remain unchanged (Table 7). For beryllium, the ion excited state population increases, while the atom excited state decreases (Table 6). For Ba, Sr, Ca, and Mg, the experimentally derivable ionization temperatures will not change, but for Be, the temperature increases when compared to the collisional radiative system. For the metastable argons to have an effect, the densities have to greatly exceed their LTE values or the collisional cross sections must greatly exceed 30 X lo-l6 cm*. Consideration of the results of BD reveals no large discrepancy for Be vs Mg in their intensity ratio measurements. They find the Be ion to atom line ratio to be 320 times the value expected at 5850 K, while the Mg ratio is 310 times greater. The model predicts that beryllium should be anomalous (Table 7) if met&able argon were important and research to this time notes no such anomaly in an ICAP. Interestingly, WILLIAMS and COLEMAN [26] remark in a study of the d.c. plasma that “the ion emission [for Be] . . . is much more intense than that of the atom which is surprising in view of’beryllium’s high (9.3 eV) ionization potential.” Their observation at least qualitatively fits the scenario for metastable argon participation as predicted by this model.

4.5. Radiative absorption The absorption of photons can be modeled as a rate process as well (stimulated emission, Eqn (29), must necessarily also be included). The effect of increasing continuous photon h-radiance on the level populations (Table 8) at 8510 K and an electron density of 3.34 x 10” cmm3(metastable argon was not considered) explains the qualitative observation that atomic fluorescence spectroscopy (AFS) in an ICP is basically no more sensitive than emission spectroscopy 17, 27, 281. Table 7. Relative populations of ion excited levels (2P,n+ *Pin) and the atom excited level (‘PI) of the alkaline earths vs metastable argon density’ “(%,* + *P,,*)/“(‘P,) Be

ns

Cl

Sr

Ba

0

18.9

32.9

246

376

616

10’2

18.9

32.9

246

376

616

10’3

26.7

33.0

246

376

616

10’4

96.0

34.6

247

377

616

!‘A’-” an3

1

‘8200 K *
984

R. J. Lov~ri Table

8. Effect

of absorption

Irradiance' 0 X3-14

of 234.9 nm” radiation '%I

'PI

on the level populations 2s

112

%I2

+

2P

1.68-l

3.43-4

7.87-l

1.02-2

1.68-l

3.43-4

7.87-I

1.02-z 1.03-2

10-12

1.62-I

3.45-4

7.93-l

10-10

3.60-2

3.99-4

9.13-l

1.19-2

10-8

5.90-4

4.14-4

9.46-l

1.23-2

~0-6

1.42-4

4.13-4

9.47-I

1.23-2

of Beh

l/2

‘234.9 nm is the wavelength of the ‘Se- ‘P, atom t~~s~tion b8510K. 3.34 x $0” cm+ = n, ‘Units of W cm-* Hz-’

Increasing the irradiance will increase the population of the ‘Pr atom level, but, as shown in Section 4.1, the electron collisional rates in an ICP are so high that once excited, the state is more likely to ionize than emit. The result is that the loss of population from the atom ground state, normally transformed into a large increase in the excited state, shows up as an increase in the ion ground state. The increased ground state population will create an increased ion excited state population through electron collisions. At 10d6W cm-‘Hz-’ the excited atom state is almost saturated with respect to the atom ground state, but the net increase over no irradiation is only 20%. In addition the population of the 2P3,2,tn levels increases by 21% due to the increased population of the *SllZ ion state. This effect should be experimentally observable, but has not yet been noticed. One would expect for Be a 20% improvement in the detectability of the atom under saturation. The important detail is that excitation of the atom ground state results in an increase in population of the ion ground state through ionizing collisions. The effect of irradiation on the analyte spectra will depend on the relative populations of the ground atom and ion states, thus for Mg (Table 2) one expects little increase in the ion emission since the ion ground state is initially so highly populated, whereas an element with a high ionization potential such as beryllium or zinc will show an increase in spectral intensity from the ion lines upon irradiation with atomic resonance radiation. 5. CONCLUSIONS

Development of a specific rate model for the calculation of the level populations of analyte species in the ICAP has allowed the evaluation of a number of processes possibly responsible for the analyte spectra emitted from an analytical ICAP. The model indicates that radiative decay can in fact disrupt the system from local thermodynamic equilibrium, but to a relatively small extent. Obviously the extent of disruption will depend on the specific collisional deexcitation rate and radiative decay rate for the level of interest in the element of interest, thus there will be element to element variations in measured ionization temperatures. The model also indicates that radiative recombination will affect the level populations to an extent dependent upon ionization potential. The overall result is an increase in the ground state atom population for elements whose ground state ion is very heavily populated. Most importantly, the ICAP is in the regime where three-body and radiative recombination are both important. Any attempt to approximate the level populations of an ICAP at an electron density near lOi crnm3 would fail without the consideration of both processes. Some of the discrepancies in ICAP spectral interpretation (VIZ, LTE) may revolve around the actual electron density in specific plasmas. At a 10*6cm-3 electron density, the level population would be much more agreeably approximated by an LTE model. A number of processes, autoionization, dielectronic recombination and atom collisional effects would appear to be unimportant in establishing the level populations of

Inductively the

main

ion

coupled

Ar plasma

analyte spectra

985

and

atom resonance lines, but may be of importance to specific levels (notably autoionizing b3elS). Penning ionization occurs to as large an extent as possible; any element with a low second ionization potential will be greatly overpopulated in the doubly charged ion state. The model predicts that sharp differences should be observable between beryllium and the other alkaline earth elements if Penning ionization were important. No evidence to date suggests this to be the case for an ICAP, thus the Penning ionization mechanism should be reevaluated experimentally. In addition a large excess over the LTE population of metastable argon atoms or a very high collisional cross section would be required to observe the effect at all. In any event, Penning ionization (based on the approximation used here) does not appear to have any effect on the ionization temperature, except for elements such as beryllium.

Radiative absorption disrupts the level populations, but since the collisional rates of the model are high, the major effect is to increase the ion ground state population: only small increases in signal should be seen under conditions of resonance fluoressence. The size of the resonance fluorescence signal will depend on the extent to which the ion ground state can be increased, thus elements with high first ionization potential will show an increase (notably of the ion resonance lines) while for elements of low first ionization potential, the ion ground state is already very heavily populated and one should see little increase. The model itself is only as good as the rate expressions used and improvements in

the rate approximation would improve the results, but the method gives the first useful estimates of the effect of specific processes on the analyte element level populations. Experimental verification of the effects predicted in the analyte spectra are imperative and will be pursued in our laboratory. The high electronic collision rates in the ICAP cause certain processes well studied in low pressure environments to behave much differently. As long as the electron collision rate is larger than the radiative decay rate radiative absorption will not contribute meaningfully to specific states, but have an effect smeared over the entire elemental energy level manifold. Specific “four level” studies are therefore limited in modeling the ICAP due to this collisional coupling. Penning ionization at this point is empirically added to the model. Inclusion of argon in the model will improve the ability to model argon metastable populations. Obviously the high electron collision rate indicates electron-Arm collisions to be the predominant deactivator of Arm, not Arm-atom collisions. Of course the Arm energy is then transferred kinetically to the electron resulting in a possibly skewed electron energy distribution. Penning ionization should only be important in systems of low electron density with mostly atoms present, which would allow an atom to ionize and emit (if possible) before the next collision. This scenario is very unlikely in the ICAP analytical zone, as the calculations indicate. Acknowledgemenrs--Thanks to Jeffrey Kingsley for his contributions. The financial assistance provided by the Society for Analytical Chemists of Pittsburgh Starter Grant and Research Corporationand the donation of computer Philadelphia,

time by NDSU are greatly PA., September. 1981.

appreciated.

This work was presented

in Part at FACSS

VIII.