Fractionation in the thermal ionization source

International Journal of Mass Spectrometry and Ion Physics, 5 1 (1983) 165-189 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

FRACTIONATION

IN THE THERMAL

IONIZATION

165

SOURCE

K. HABFAST Finn&an

MA T GmbH,

(Received

P. 0. Box 1440 6.2, D - 2800 Bremen

(W. Germany}

28 May 1982)

ABSTRACT In this paper a model for isotopic fractionation in the thermal ionization source is presented The samples, normaBy loaded as salts, are assumed to evaporate, in general, as binary vapors of two chemically different forms. The molecular species in the vapor might either dissociate before ionization and the metal might then be ionized; or, alternatively, molecular ions might be generated which then dissociate into metal ions. Whereas isotopic effects during ionization are negligible, such effects have to be considered for the dissociation process. The dependence of the observed isotope ratio on the chemical form of the loaded sample and on the temperature of the ionization can be explained with this model, whereas the time dependence and the effects of reverse or enhanced fractionation of the observed isotope ratio are readily explained by a generalized Rayleigh distillation equation. The application of the fractionation model to the normalization of observed isotope ratios to an internal standard ratio shows the principal limits for the accuracy of normalization. The commonly used normalization techniques and their inherent errors are considered in the light of the fractionation model and an improved normalization formula is presented which uses the concept of the “apparent mass”. Finally, the model is used to propose experimental methods for the accurate determination of non-normalizable isotopic ratios.

INTRODUCTION

It is well known that the observed isotope ratios in a thermal ionization mass spectrometer normally do not represent the true isotopic composition of the sample. An observed isotope ratio depends not only on the time when it was measured during the run, but also on the chemical form of the sample, as well as on the temperature and the material of the ionization and evaporation filaments [l-5]. Moreover, the specific procedure chosen to load the sample onto the filaments and the individual experimental strategy to measure the isotopic composition have a significant influence on the observed isotope ratios [lo]. All these observations are summarized under the term “fractionation effects”, because it can be assumed that they are generated, in general, by the individual evaporation and ionization behavior QO20-7381/83/$03.00

0

1983 Elsevier Science Publishers B.V.

166

of the sample, which, in some way, depends on the mass of the isotopic species under investigation. Under such circumstances it becomes extremely difficult to compare and qualify experimental results which have been obtained by different laboratories, especially if each detail of the experimental procedure is not described accurately. To overcome these difficulties it is common practice either to use artificially produced isotopic mixtures as external standards in order to calibrate each specific experimental procedure or to use internal standard ratios to which unknown ratios can be related (normalized). In any case, certain assumptions are to be made as to the mass dependence of the fractionation if external or internal calibrations are applied [6]. However, because such assumptions are more or less of empirical nature, the individual experimental procedures applied to measure the isotope ratio are to be performed within strict empirically set limits which often seem to be based on “black magic” rather than a reasonable understanding of the sample’s behavior. Under such circumstances it becomes questionable to transfer a specific experimental procedure from one isotopic system to another and/or to apply suitable diagnostic means to check the identity and/or validity of different runs. During the last 15 years very useful data towards an understanding of the processes happening in the ion source were reported. Eberhardt et al. [4] showed that fractionation in the single-filament source can be explained as a Rayleigh distillation process [ 131 and Kanno [7] gave a theoretical fractionation model based on molecular evaporation which can predict fractionation under certain simplified experimental conditions. Moore et al. [8] expanded Kanno’s model and provided experimental evidence of its validity. Heumann et al. [9] published useful experimental results on the dependence of the observed isotope ratio on the chemical form and on the temperature, and Russel et al. [ 101 showed that even the internal normahzation techniques which traditionally have been considered as much less sensitive to fractionation effects, are to be considered with due scepticism. It must be added that the same holds true for single and multiple spike techniques if they are used with the aim of obtaining the “true” isotopic ratio. Even for straightforward isotope dilution experiments the measured concentration significantly depends on the proper fractionation correction, as was shown, for example, by Moore et al. [ll]. For a special spiking method (on the metal counterpart of the evaporating species) Kanno [ 121 clearly emphasized the need for and the validity of his fractionation model and gave an assessment of the possible errors inherent in these techniques.

167

In the last decade the internal reproducibility of isotope ratio determinations has been improved drastically by the development of fully automatic computer-controlled mass spectrometers. Such spectrometers are no longer the domain of a small number of highly specialized research laboratories which traditionally exchange and cross-check their experience, they are now available to a broad range of users. Therefore, there is an even greater need for a common understanding of fractionation. Otherwise the different automatic instruments may produce precise but, nevertheless, inaccurate and incomparable results. The major difficulty on the way towards a general quantitative fractionation model is the lack of reliable or appropriate thermodynamic data on the evaporation and ionization of isotopic species under the (nonequilibrium) conditions in the thermal ionization source and the incompleteness of the published mass spectrometric data. It might be expected, however, that such data are generated as soon as it is evident which data are needed. So far each fractionation model has to make plausible assumptions, deduced from the -available knowledge of similar or analogous evaporation and ionization processes, in order to quantitatively explain the observed fractionation effects. THE

OBSERVED

ISOTOPE

RATIO

In the multiple filament thermal ionization source [14] the samples are normally loaded as salts of the isotopic metals under investigation. They are evaporated at relatively high temperatures into the vacuum. Once evaporated, the particles will not return to the evaporating surface. Under such nonequilibrium conditions (Langmuir-type evaporation) the vapor pressure of two isotopic species can be assumed to be only negligibly different [7]. Hence, according to Langmuir’s equation, the ratio r of the number N1 or ZV,,of the light or heavy isotopic particles (i.e. the “isotope ratio in the vapor”) evaporating per unit time from the solid phase is given by r=N,/N,=&,/M,-R=y-R where M,, and M, are the molecular weights of the evaporati is the isotope ratio of the species in the solid phase. y =

0) andR is the

fractionation factor. It is well known from Knudsen cell effusion studies [8] that metal salts .merely evaporate as metals or in a single species. Instead, binary vapors of molecular species are observed at such evaporation temperatures as they are commonly used in the thermal ionization source (300-14OOOC). A similar behavior can be assumed for a Langmuir-type evaporation_

168

The ionization of the vaporized molecular species on the hot surface of the ionization filament produces atomic (metal) ions and molecular ions as well. For the production of metal ions it must be taken into consideration, therefore, that the vaporized neutral molecular species either dissociate before the metal ion is generated by surface ionization or that molecular ions form which then dissociate to metal ions. Both processes may even be co-existent. In all cases where incomplete dissociation of the species is involved, isotopic effects during dissociation have to be taken into account, as the energy states of isotopic species are different. As compared to the size of isotopic dissociation effects any isotopic effects of ionization can be neglected, although they would be “amplified” considerably due to the exponential nature of Langmuir’s equation for surface ionization. A fractionation model (Fig. 1) which describes the possible processes during the evaporation and ionization of a salt (MZ), in a multiple filament ionization source should reflect all the facts mentioned above. If two chemical species A [(MX) 9] and B [(MY) 9] are evaporated and ionized, the ion current of the light, metal ions i, which will finally be observed is given by dA h =

nahdt

+

nbycbl

dB - dt + %.V,i(l

- %)Z

dA

+ %.W,+,(l

- et&j-dB

(2)

where n, and n,, are the molar fractions of the isotopically light particles of

Fig. 1. A fractionation model assuming the simultaneous evaporation different species A and B.

of two chemically

169

molecular species A and B; d A and dB are the number of moles of molecular species A and B evaporating per unit time; y = [ N]+/[N,] is the coefficient of ionization to produce metal ions [IV]+ from neutral metal vapor [N,]; ;v, and Y,, are the coefficients of ionization to produce molecular ions from neutral molecular species A and B in the vapor; c,, and cbl are the degrees of dissociation of the isotopically light molecular species A and B into neutral metal vapor; and ca+land cb+lare the degrees of dissociation of the isotopically light molecular ion species A and B into metal ions. A corresponding expression is valid for the heavy metal ions (molar fraction: 1 - n,, 1 - n,; degree of dissociation: cab, ebb, ~a+, c,‘,). The obseped isotope ratio rbbsis given by the ratio of the ion currents of , the light an& of the heavy metal ions. For s&ci&& A and B we have, according to eqn. (1)

and n, = r,/(n

+ ra)

@a)

rb)

nb=rb/(n+

w

Hence, (Ya+habYb)+YaYb(’

Yobs =

(q,+kl]bq,b)+

+k%tb)R

sR

haYb+Yi$~b%~b)~

(5)

and 71a= [%h +_k$W qb= qab

%h>l/[%l +kw - c,,>l

-

[Ebh+jbE~~(l-Ebh)]/[Cb;l+jb~b+l(l-Cbl)] =

kb,

+aibcitdl

-

',,>]/+a1

+.kd&

(6) (7)

-'d]

(8)

where y, and yb are the fractionation coefficients (/m) of evaporating species A and B; k = dB/dA is the ratio of evaporation rates of species A and B; and j, and j, are the ratios of ionization coefficients (u,/u), (Yb/Y) of m&cular and atomic particles of species A and B. If no isotopic effects of dissociation are present (i.e. cal = cab and Ebl= fbh) we have va = r]b = 1. Assuming this and, in addition, the special (rather rare) case that species A is a metal (gal = cab = 1) and species B a molecular species which completely dissociates after ionization ( cbl = 0, l b+l= l), then eqn. (5) reduces t0 the eXpreSSiOnfor robs given by Kanno [7].

170

Equation rObS

=

Yab

(5) can then be written as -R

(9)

yab = (ya+k”9ab’yb)/(rJa+k”~lb.‘)7ab) k’ = [(I

+y;&‘(I

+yb-R)]

(10)

-k=k

W)

In analogy to eqn. (1) we can write

from the fractionation factor yab it appears as if only two isotopic species having the masses ,uI and p2 = p, + AM would evaporate. These masses are therefore called the “apparent masses”. So far, complete spatial separation of the ionization and evaporation processes has been assumed. The multiple filament source [ 141 fulfills this requirement. However, a similar model could be developed for the single filament source, as it can be safely assumed that ionization takes place at the very upper layer of the surface at an instant, when the mass-dependent evaporation processes have already been completed. Although the coefficients of ionization y, y, and Y,, and the rates of evaporation d A and dB will take a slightly different meaning, the concept of the “apparent mass” can be applied as well. In Table 1 we have summarized experimental data which were reported by Heumann et al. [93 to demonstrate the dependence of the observed isotope ratio on the chemical form and even on the crystal size of the loaded salt. In Fig. 2 these data are compared with a plot of eqn. (9) (R = 45.575, AA4 = 4), assuming that the apparent masses of the evaporating species are identical with the molecular weight of the loaded chemical species (i.e. k = 0 of the isotopically lighter species which and qa= 1). The apparent mass r_~~ follows from the observed fractionation factor is given in Table 1, Ekcep t for Ca(NO,), and CaSO,, the salts seem to evaporate in a form that is different from the loaded form. For the small crystal CaCO, and for Ca(OH), it can be concluded that these species disintegrate to CaO before evaporation and that CaO (M, = 56) is the species actually evaporating (i.e. y = 1,0351). Apparently, the large size crystals of CaCO, have a different behavior. Very probably, they evaporate as (CaCO,), . If this is assumed, the majority of the evaporating particles will have unit masses of 200 (40CaC0, 40CaC0,)and 204 (44CaC0, s40CaC03) due to the low abundance of 44Ca; hence y = 1.00995. Near the hot surface of the ionization filament, these particles will nevertheless disintegrate to CaO and will then follow the same ionization and dissociation mechanisms as the CaO particles from the small-crystal-size carbonate. i.e.

2

74 100 111 78 136 100 164

M

47.41f 0.09 47.32f 0.09 46.66~0.15 46.40*0.06 46.28f 0.04 46.13f0.08 46.10f0.12

robs

’ Crystalsize < 5 p. b Perfect rhombic crystal size’> 10 p.

CaCO, a CaCl 1 CaF, CaSO, CaCO, b CaWW 2

wm

Sample 1.0404~0.002 1.0384f0.002 1.0240f 0.003 1.0182f0.001 1.0153*0.001 1.0121+o.O01 1.0116~0.003

Tab

48.6f 2.5 51 f 2.7 83 f12 109 f 8 129 f 8 165 f 6 171 f42

111

[Ca(W),l

W21+G@41 ~~a~~4l IKJa~W21

[CaCl]t[CaCl,]

I~a~l+W~l

[CaOl+[H+Jl

Vapors [A]+[B]

0 0 0.4 1.3 0 0 0

k

0.9966 0.9966 1.0 1.0 1.0 0.9966 1.0

tla

Summary of experimental data for the dependence of the observed isotope ratio on the chemical form of the sample according to Heumann et al. [9]

TABLE 1

c 172

'ohs

no

46.5

Ca(NOi

44.0

Fig. 2. The dependence of the observed isotope ratio rObSon the chemical form of the sample: measured data; 0, proposed chemical species which are evaporating. *3

These mechanisms offer an explanation for the still too large difference between the calculated and the observed fractionation factors. The observed isotope ratio was reported by Heumann to decrease with increasing temperature of the ionization filament (see Table 2). If CaO is the only species which evaporates (k’ = 0), we have, from eqns. (9) and (10) Iohs=

[(d/h)]

-R

(13)

If c, and cf < 1, the assumption can be reasonably made that the lighter molecular species are preferentially dissociated, hence qla< 1 and lobs> y, - R. With increasing temperature, ea, l a* and qa will gradually increase until ~~ and E,+= 1, and lobs will, therefore, decrease with increasing temperature. From the data in Table 2 the overall dissociation factor qa is estimated to be 0.9966 (i.e. 0.8%0per mass unit) at 2000°C with a temperature coefficient of - +20 p.p.m. OC-I. The r a ti o of the vibrational energies E, and E, of the two is0 topic oscillators 4oCa160 and 44Ca160 in their ground state would be &l/E, = 0.987 (or - 3% per mass). To explain the CaF, data we have to assume dimer evaporation (for MgF,

173 TABLE

2

Dependence filament

of the observed isotope ratio 40Ca/44Ca

Temperature of the ionization filament (“C)

40/44 r.bs

1825 2025 2250

47.53 f 0.05 47.39 f 0.13 47.15*0.11

on the temperature

of the ionization

the evaporation of the dimer form from a Knudsen cell has been reported 151). With a ratio k = 1.3 of the evaporation rates of the dimer and the monomer forms of CaF, the apparent mass of p1 = 109 in Table 1 can be explained. In CaCl,, both isotopes of chlorine have comparable abundances (37C1 : 3sC1 = 1 : 3). Therefore, a total of 6 isotopic species with the following masses m and molar fractions n have to be considered. [

44Ca

40Ca m

n (%I

m

n @,)

110 112 114

56.2 35.9 5.75

114 116 118

1.23 0.79 0.13

Hence, the fractionation factor for CaCl, should be y = 1.01786 (if dissociation effects are neglected). However, the corresponding observed value is much higher and can only be explained by the coexistence of CaCl, and CaCl in the vapor phase with a ratio of k = 0.4. The “true” isotope ratio R = 45.575 of these data is unusually low as compared to other published values [lo] for R, ( = 47.15). This can be explained by an inherent mass discrimination of the mass spectrometer (ATLAS CH4) itself for the lighter ions. The above explanations may show the usefulness of the fractionation model, but some at least plausible assumptions are needed. The model is still far from a satisfactory quantitative prediction as long as the necessary thermodynamic data or, at least, complete and precise experimental data from the isotope ratio measurements (i.e. atomic/molecular ion ratios) are not available.

174 FRACTIONATION

WITH TIME

Since the lighter isotopic species are preferentially evaporated they are depleted in the solid phase. The isotope ratio in the sample and, hence, the observed isotope ratio will decrease during the evaporation of the sample 1161. Kanno [7] gave a general solution to this time-dependent fractionation effect by solving the Rayleigh distillation problem under the assumption that two different species evaporate simultaneously. If y= and y,, are the fractionation factors of the two isotopic species and if k is the ratio of their evaporation rates, then the isotope ratio R in the solid phase at a time when the quantity Q of the sample has been evaporated, is given by (Y -

lb(Q/Qd

=ln(W%) +b(y-

+ (Y - %[(R

+ %‘(h,

l)ln[(c+d.R)/(c+d-R,)]

+ I)] 04)

where Y = (ya+k-y,,)/(k+

1)

(19

b= [(ya-yd*.k],‘(c*d)

06)

c=(y-

07)

d=(Y,-

l)(k+

1)

l)Y, + Y,(YIJ - 0-k

08)

Q, and R, are the originally loaded amount of sample and the true isotope ratio at the beginning of the evaporation, respectively. If eqn. (14) and eqn. (5) are combined, the total fractionation pattern of a two-isotope system which evaporates with two different chemical forms can be predicted. Apparently, this has to be done by some suitable iterative method using a digital computer. The result of such a computation is shown in Fig. 3. Uranium oxide U,O, has been assumed to evaporate as

where the rate of evaporation of the two oxide species has been assumed to change during the evaporation from k = 3 at 4 = Q/Q, = 0.95 to k = 0.3 ((U,O,), gradually converts to 3U0, + 0,) at q = 0.55 (i.e. when 45% of the sample has evaporated). Similar fractionation patterns having near-zero fractionation are observed very often during the measurement of uranium oxide. If the variation of k with time is different from run to run, then the slope of the fractionation

175

1.006 1.005 +l.ooll 1.003 1.002 I.004

%OOO

Fig. 3. Computer-generated fractionation of a L&O, sample, which partially decomposed UO, during evaporation. The dotted lines indicate how UO, and U,O, would fractionate single compounds.

to as

pattern is also different and a significant reduction of the run-to-run reproducibility is observed (see the following two sections}. For most practical purposes eqn. (14) can be replaced by a straightforward Taylor-series approximation. If R’ is the approximation of R in eqn. (14), then @‘/&I)

= 1+ KY - WYlh

4

(20)

The relative error of R’ is given by CR’ - W/R

= (Y - 1J2 *f@,,

Y, k

4)

(20

For all practically occurring values of y, R, and k the error function f is independent of y and k within sufficiently narrow limits of accuracy and, therefore, can be written as

fo%9 4) = -

[

R

0

1+,+g%7 lnq

(22)

1

and - 2.7
(0.1 < 4-c 1)

q) -= 0.5

Hence, the fractionation pattern can be approximated sion (using eqns. (9)-( 11) and (20)): & ‘ s/%

=

Yobs +8'

bobs

-

l)ln 4

(23) by an explicit expres(24)

176 Yohs =

(Y, + ‘blabYb)/b?a + k%bvb)

(y,+ k aTI~,,Y,,)(Y~ -

g = (y, + ky,)( Ya -

1+

1+ h,,(Y,

(25) -

ktub

-

1)) qb))

(26)

where yobs is called the “observed fractionation factor”. In analogy to eqn. ( 12) the apparent mass is given by fut= AM/(Y&

- 1)

(27)

If only one species evaporates (k = 0), g = 1. If two different species are evaporating, the numeric value of g depends on the possibly different ratios of the ionization coefficients ( j,, j,) of the different molecular and atomic species. Without giving further details on the characteristics of fractionation, which can be derived from eqn. (24), we note that the values of “k” and “qab. k” are separated in g and Y&. This offers a chance to assess their numeric values for an isotopic system in which the true R, and the evaporating species are known. As shown above, fractionation (i.e. fractionation correction) is governed by the amount of sample consumed during evaporation. In most expetimental situations, however, the amount of sample, which has been evaporated up to the time when the ratio is measured, is not known. Hence, it must be measured separately, if a fractionation correction is needed. There are two basic procedures to do this: (a) The sample is mixed with a spike of known isotopic content (or, even more convenient, one ratio of a multiple-ratio system is known). The fractionation of the known ratio can be used to measure sample consumption and, hence, to compute the true isotopic ratio of simultaneously evaporating species with an unknown ratio. This procedure is called “normalization to an internal standard”. (b) The amount of evaporated sample is determined by the time integral of the emitted ion currents, if no internal standard is available. These methods will be discussed in the following two sections. NORMALIZATION

TO AN INTERNAL

STANDARD

If two separate isotope ratios can be measured in a system (i.e. if at least three isotopes are present), then one of these ratios can be used to correct for fractionation. Traditionally, the most simple correction method has been to assume that fractionation will be proportional to the mass difference of the measured isotopic species [8], i.e. instead of eqn. (5) it is assumed that Yohs= R,(l

+ 6 - AM)

(23)

177

If two ratios robs1and robs2 are measured and if one true ratio R,, the other true ratio is obtained as (by elimination of c)

ROl =

[Gb,,l/[Q - 44+ ekbs*/R02~1

is known,

(29)

$a = $I~= AMJAM,

(30) It has been shown by Russel et al. [lo] that this linear approach is insufficient for an accurate normalization. Instead, an “exponential” fractionation law was proposed which requires, in its first approximation, the use of 9=+,

M3 AM, =M,‘aM,

(31)

instead of c#+-, in eqn. (30). M, and (M, + AM,) or M3 and (MS + AM,) are the isotopic masses of the two measured ratios. If the evaporation of the sample is governed by Rayleigh-evaporation, an accurate method for normalization can be given under the plausible assumption that all isotopic species in the system show the same chemical behavior. Equation (14) is applied twice to two ratios R, and R, and thus Q/Q, can be eliminated. This, finally, requires the computation of R,, from the following equation: [ 2]‘“r+).

[ RRql=;][

s,=“;,$j”~

(32) which can be performed by application of Newton’s algorithm for the roots of functions. Doing this, the values of R, and R, are to be computed first from robs1 and %bs2 by application of eqn. (5), which is quadratic in R. yi, bi, Ci, and di are given by eqns. (15)-(18). Apparently, this approach is not a practical one, because the requested input data (e.g. ya, yb, kja,_ib,Ei,etc.) are normally not known. However, it shows the principal limits of an accurate normalization. Accurate normalization is an insoluble problem if only ratio data and isotopic masses are available. This statement is not limited to a Rayleigh-type evaporation,. as all basic parameters of’evaporation and ionization used in this model have also to be used in any other fractionation model. For the Rayleigh-type evaporation, Table 3 and Fig. 4 show the principal limits of the accuracy of normalization for some arbitrarily chosen sets of parameters. The computation of R,, (i.e. a normalization of the ratio 40Ca/44Ca to the ratio 44Ca/48Ca) -was p erformed using eqn. (32) and eqn. (5) under the

178 TABLE

3

Relative error of normalization for 40Ca/44Ca vs. 44Ca/48CaB Evaporating species

j,

actual

assumed

act.

ass. -

act.

ass.

act.

ass.

CaO CaO CaO CaO CaO CaO CaO CaCl 2 + CaCl (k = 1.3)

CaO CaO CaO CaO Ca Ca Ca Ca

10 10 1 0.01 0.01 1 10 0

10 1 1 0.01 1 1 1 0

0.8 0.8 0.4 0.4 0.4 0.4 0.4 1

0.8 0.8 0.4 0.4 1 1 1 1

1.0004 1.0004 1.004 1.004 1.004 1.004 I.004 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

c,

eal/cah

a Arbitrarily chosen sets of parameters for the evaporation, dissociation and ionization of CaO and CaCl, samples. The isotopic effects of dissociation are valid for AM = 4 and are assumed to decrease with increasing degree of dissociation.

Fig. 4. Errors of the normalization of 40Caj44Ca to 44Ca/48Ca, if the evaporation behavior of a CaO (CaCl,) sample is not reflected correctly. Numbers (l)-(8) refer to the line numbers in Table 3, where the set of parameters is given. Isotopic effects of dissociation produce a nearly constant error offset, independent of the time of the measurement. The assumption of a wrong mass for fractionation correction leads to large errors, depending also on the time of the measurement.

179

(at q = 0.9) act. 0.2 0.2 0.4 0.8 0.4 0.4 0.4 0

0.2 0.2 0.4 0.8 0 0 0 0

1.008 1.008 1.004 1.ooo4 1.004 1.004 1.004

1.028939 1.028939 1.028323 1.027371 1.027311 1.028323 1.029649 1.020009

122 122 153 208 -671 - 750 - 855 - 1342

assumption of a certain actual sample behavior for the evaporation of CaO and CaCl,. The normalization was then computed a second time using evaporation and ionization parameters which are different from the actual ones but which are similar to commonly used assumptions for normalization. This results in different values, R&. The relative error of Rb, vs. R,, is given in Table 3 for 4 = 0.9 and is plotted in Fig. 4 as a function of 1 - q_ Nevertheless, using the approximations and the “apparent mass” approach of the previous section, a variety of practical normalization procedures can be deduced which are accurate enough for most practical purposes. If eqn. (24) is applied twice for two ratios robsI{R,,) and robs2(R02), eqn. (33) is obtained, by elimination of 4,

where 9 = ‘#$=

[glhobsl

-

1)l/[g2(Y,bs2-

'11

For all cases where only one chemical species is evaporating (i.e. k = 0)and = 1, g, = g, = 1) where isotopic effects of dissociation are excluded (i.e. q77a this reduces to (35)

180

where + = +, = (Xzr - 1)/k,

- 1)

(36)

With the existence of isotopic effects g, and g, * 1 but g,/g, = 1, hence eqns. (35) and (36) are still valid. It should be noted that a linear approximation of the square roots in ya, and yaz results in the normalization formula ((eqns. (29) and (31)) given by Russel et al. [lo]. In Fig. 5 the results of Russel et al. [lo] are plotted. From eqn. (35) the following was derived: Gt&%

= (1 - G) + 9kJs2/kl2

>

(37)

The measured data fit this relation if two different evaporation processes are assumed to occur successively during the run: (1) An apparent lower mass of - 60 is evaporating first {this may indicate that the majority of the particles which leave the filament are neutral CaO vapor). Equation (37) results in:

ro4bos/44 R40/44 0

= 0.066 + 0.934 -

ro;y48 R44/48 0

(38)

(2) An apparent lower mass of - 40 is evaporating later, hence r44/48 R 40/44 0

=

0.089 + 0.911

ObS

R44/48 0

(3%

It may be concluded that a single normalization procedure is not appropriate for these data. But the accuracy of normalization can be improved if some plausible assumptions are made on the molecular species which evaporates at the time the ratio is measured. If such dramatic effects can be excluded, i.e. if the evaporation behavior stays in a steady process with no apparent rapid changes of the fractionation pattern, then the accuracy of normalization can be improved even further by the additional use of eqn. (21) and by using experimental data only. This is done in several steps. First, the normalizing ratio r,,,, is plotted against the emitted charge of ions, which is a measure for the consumption of sample as shown in the next section. This plot is extrapolated to zero (i.e. 4 = 1) which, according to eqn. (24), results in the quantity ).(O) -

obs2 = Y&s2 = R 02

J1+ ( AM2/h)

(40)

p3 and p4 = p.3 + AM, are the apparent masses of the normalizing ratio. Equation (40) reflects the overall evaporation and ionization behavior of the

181

e

Fig. 5. Plot of the normalized ratios 44Ca/48Ca vs. 4o Ca/44Ca. For these high precision results of Russel et al. [ 101 it has to be assumed that two. evaporation processes occur successively in order to explain the two different slopes in the plot.

sample. Using eqn. (40) and the apparent masses pI and p2 = pI + AM, of the ratio robs,, we can compute yobsl since the mass differences are known (~1~= CL,+ AM3) and since it can be safely assumed that all isotopic species have the same chemical form. Hence, l/2

AM1Mbr2 - 1)

Yobsl

AM,-AJ43(y,2bs2-

(41) 1) 1

If, in the second step, the true ratio R,, is computed using 9 obs =

hM

-

Who,,,-

1)

(42)

eqn. (43) is obtained Ro1 = [(I - &,,,)/A]

+ 6?‘,

-+obs -

(43) hh~z/~O,)

A and B are correction terms which result from eqns. (21) and (22). These equations require knowledge of 4, R,, and R,,. The quantity 4 can be assessed by application of eqn. (24) to robs2in all cases where k = 0.For Rot, as an appropriate approximation, the measured value rdbo,iat the beginning

182

of the run can be used. Hence, finally:

A

=

1

robs2

_

- R,,

-

Yobs2

(4)

)I(

An alternative procedure would be to use r,(ti, only as a first estimate for R,, and to perform an iterative computation of R,, by inserting this preliminary value R,, obtained from eqn. (43) into eqn. (44) and so on. For all cases where two different chemical species are evaporated (k =+= 0) and where a different ionization and dissociation behavior cannot be excluded, the better the processes in the source are known, the better is the accuracy of normalization. As an example, the observed isotope ratios can be corrected for a possible isotopic effect of dissociation by the assumption of a linear dependence of the degree of dissociation on the mass differences before they are used in eqn. (43). In other cases, the evaporation behavior can be studied by examining standard samples of known composition or by the addition of a spike to the co-evaporating metal counterpart, as was proposed by Kanno [12]. For some seIected examples we computed the normalization errors of different normalization procedures which are to be expected if the actual sample behavior is not reflected correctly. The results are given in Figs. 6(a)-6(d). In all examples given, the normalizing error of eqn. (43), which uses the concept of the apparent mass, is less than 0.1 X 10W6. It should be noted that the “exact” solution (eqn. (32)) produces relatively large errors if the real sample behavior is not reflected correctly in the normalization procedure. The linear (eqn. (30)) and semi-linear (eqn. (31)) approximations are appropriate if the lower masses of both ratios are identical. If the “middle” masses of the ratios are identical, all approximations are equally good or bad, depending on the practical requirements. For relative mass differences AM/M of more than 5% the errors might well reach the 1%0 range. In such cases it appears to be mandatory to apply the more elaborate normalization procedure of eqn. (43). From the facts mentioned above two experimental rules can be deduced to minimize the errors of normalization to an internal standard: (a) the sample should be loaded in such a form that only one chemical species will evaporate; and (b) the lower masses of the two ratios should be identical in such cases where it must be assumed that two chemical species are evaporating.

-

8‘f&

YS. 86188

----

86&

VS. 67188 (b)

A

.

-

~43/‘4W VS. A44/..46

----

n4$WI

Fig. 6. Normalization errors, obtained from different normalization algorithms, if the actual and the assumed sample behavior is different: (E), fractionation correction using the exact Rayleigh distillation formula; (L), fractionation correction if fractionation is assumed to be proportional to the mass difference (eqns. (29) and (30)). (R), fractionation correction according to Russcl et al. [lo] (eqns. (29) and (3 1)); and (Q), fractionation correction using an approximated Rayleigh-type evaporation formula (eqns. (35) and (36)). (a) Normalization ‘errors of *‘Sr/s’Sr vs. s%r/**Sr are practically negligible if the lower masses of both ratios are identical. The sample is assumed to evaporate as SrReO, [8]; (b) normalization errors of s4Sr/86Sr vs. *%r/**Sr () and of ‘%r/“Sr vs. *‘Sr#**Sr (- - - - - -) are larger than the internal and external precision of modem isotope ratio instrumentation; (c) normalization errors of ‘43Nd/‘44Nd vs. ‘44Nd/‘46Nd if Nd,Os has been loaded; and (d) normalization errors 143Nd/144Nd vs. ‘44Nd/‘46Nd ( -) and of *43Nd/144Nd vs. ‘42Nd/*44Nd (- - - - - -) if NdCl, has been loaded.

184 NORMALIZATION

TO AN EXTERNAL

STANDARD

If an internal standard ratio is not available for the assessment of the consumed sample at the time of the measurement of the ratios, the amount of sample consumed must be measured directly. This requires two independent but nevertheless identical runs, one with the unknown sample and one with a standard sample of known composition. The observed ratios of both runs for a given identical sample consumption are then related to each other for the purpose of normalization. It is obvious that this procedure will require not only apparently identical runs but also some means to check the actual identity of the two runs, concerning the samples’ behavior. The amount of sample which has been evaporated can be assessed, in principle, by integrating eqn. (2) and its corresponding expression for the heavier ion current. As d A + dB = dQ, the following (assuming va = qb = 1) for the total ion current i,.is obtained: i,(t)

=Yobs'

Yohs=

(dQ/dt)

(46)

[Yha+k%ib~b)]/(l

+d

(47)

yobs is called the “observed ion yield”. Hence, at time t,, the following relative amount of sample will be left on the evaporating filament q(t,)

= 1 -~*$/fmi,(r)dl 0

obs

0

The yield is measured in the dimensionless quantity “ions per molecule”. MO is the molecular weight of the sample (g . mol- ‘>; F = 96 494 A - s; and Q. is the amount of sample loaded. Using this relation together with eqn. .(24), fractionation with time can be predicted for any given ion current/time profile (see Fig. 7). The yield y&. is measured once by evaporation of the sample to exhaustion:

For the purpose of the normalization of an unknown sample to a known external standard sample the current/time profiles of both runs are integrated over a reasonable time period followed by regression, of the observed ratios VS. the current/time integral, i.e. reconstruction of eqn. (24) from experimental data. The regression is extrapolated to the beginning of the run (i.e. q(t) = 1). For the standard run this results in #O) obsl

=

Yobs2

-02

(50)

Ion

currentx IO-11 ..,.:

7

l

_,/+ FL .:.-?::::;::::l’:::::~::;:l;:::::~::::.’:::::; 1

!

,

j

hours

4

1

,.

.

..I”.‘.

Fig. 7. Computer-generated fractionation patterns of two s5Rb/87Rb runs (2% yield, 50 ng sample), having similar, but different ion current/time profiles, resulting from different evaporation temperatures. Only higher-order regression of robbS vs. time would result in the true isotope ratio from both runs. Two runs having the same evaporation temperature but - 30% different ion yields (or 30% different sample size) would show the same differences in their current/time profile and in the fractionation profile as shown in this figure.

and for the sample run, I r$il = yobsl . R 01 :

(51)

If both the sample and the standard have the same evaporation (and only &en), then Y&l = YObS2,or

This is the general procedure

for a two-isotope

behavior

system with no internal

186

standard. The expression R 0 2/r0t,S2 is often called the “filament bias”. It is independent of the size of the sample only if it is determined as described above and if the evaporation behavior does not change during the run. Hence, a comparison of the normalized slopes of fractionation /dq) of different standard runs, loaded with the same amount (WC&-(d&S, of sample, will be an appropriate means to assess the. reliability of the statistical limits of the filament bias. It is apparent from eqns. (24) and (48) that such methods cannot be as accurate as the internally normalizing ones, because there are several possibilities that parameters vary in such a way that they compensate each other. This may result in two apparently but not actually identical runs (see Fig. 7). However, there are still some other means to check and to limit the possibilities of undetected errors. The most important one is to load and to heat the sample in such a way that the ion current decays exponentially from an initial value of i, during the run. In this case, fractionation is linear with time, as follows from eqns. (24) and (48):

%,s/~O=

Yobs

-

d&b,

-

MO

i,

‘k-FQo Yobs ’ t

(5%

The fractionation slopes of sample and standard are constant. and can be compared directly. The extrapolation to f = 0 is a linear regression which may be used to assess the inherent reproducibility by statistical means. For all cases where only one species evaporates,

robs/RO=

Yobs

-

bobs

- l>(t/te>

t, is the time at which the ion current has decayed to the value i,/e. c = t,, r&,s = R, and the observed ion yield should be MO * i, - t, Yobs =

FQ

0

(54) At

(55)

It should be consistent with the value obtained from eqn. (49). The number dN of particles evaporating from the surface per unit time will be dN/dt=z-Fe L

Qo -z/t, e

(56)

(L = 6.02 x 1O23 mol-‘). This type of evaporation (i.e. an exponentially decaying ion current) will be obtained only under one of the following alternative experimental suppositions: (a) The loaded amount of sample is small enough to cover only one molecular layer on the loaded spot. If one molecule covers the area F,

187

(= 3 x lo-l5 cm2) and if F, ( = 0.01 cm2) is the area of the loaded spot, then an upper limit for the loaded amount Q, (in g) of sample is obtained as

F,

MO -

Q OGF’

(5 5 x lo--l2 MO)

L

m

(57)

and, hence, for the observed initial ion current i,:

‘, ’

F’ F -.-.Fm

L

Yobs

(=

t,

5

x lo-‘yo,,&.)

(58)

(b) The sample is loaded in such a quantity that it will cover several hundreds of molecular layers, i.e.: Q,>5~10~~

F’ .q

(= 2.5 x 1O-9 MO)

(5%

To ensure nearmexponential decay, the rate of diffusion of particles from the bulk of the sample to the evaporating surface must be small as compared to the rate of evaporation from the surface. This condition can be reached over a broad range of sample sizes if a supporting “lattice” like silica gel or graphite (resin bead) is used to absorb the sample on the evaporation filament (or on the single ionization filament). Sandwich loading techniques [17] (the sample is covered by a layer of rhenium) and several often-used high temperature loading techniques [8] (the sample is dissolved in the bulk of the filament material by preheating) also result in similar evaporation behavior. When using such loading techniques the filaments must be heated before measurement in such a way that a diffusion barrier (i.e. a negative concentration gradient and a negative temperature gradient normal to the surface) is built up in the bulk of the sample, just below the emitting surface. Very often, during the heating-up phase, a relatively fast fractionation of the upper layers and a conversion of the rest of the sample to the finally evaporating compound is observed [4]. It should be noted that the consumption of sample during the heating-up phase should be assessed by measuring the emitted charge, in order to be able to assess the apparent mass from the fractionation factor. In a multiple isotope system (like Ca, Sr, Pb, U or Pu) a plot of two normalized ratios should result in a straight line for a given steady sample behavior (see Fig. 5). Furthermore, in a multiple isotope system the fractionation per unit mass of the different isotopic species must be consistent with the true ratios. If in eqn. (54) yobs is linearly approximated, a linear expression for the fractionation per unit mass is obtained: kobs/RO)AM=!

+$+Fyio:; ) obs

0

(60)

Very often, the fractionation per unit mass is given as a function of filament temperature. However, as dN/dr is a fairly complicated (mostly empiric) function of temperature, it makes little sense to give a number for the temperature dependence of the fractionation without giving all experimental conditions such as pt, yobs, i, = f( T). The most appropriate characteristic number to indicate the temperature dependence of fractionation (and to compare different runs) would probably be “the difference in fractionation per mass unit per mole of evaporated sample for a given temperature difference”. SUMMARY

The following major conclusions can be drawn from the results of this paper: (1) The concept of molecular evaporation and of temperature and mass dependent dissociation of molecular species helps to understand most of the observed fractionation effects. (2) For a quantitative prediction of fractionation effects the appropriate thermodynamic data of evaporation and dissociation must be available. (3) The accuracy of normalization to an internal standard is limited, in principle, if the details of the evaporation and ionization processes are unknown, The knowledge of a simple mass dependent “fractionation law” is not sufficient to solve this problem. By application of the concept of the experimentally accessible apparent mass, approximations of sufficient accuracy can be made to compute the true isotope ratio. (4) The accuracy of normalization to an external standard is limited by experimentally inseparable systematic error factors. Exponentially decaying evaporation rates, however, may offer some means to control, at least, the true confidence limits of external reproducibility. REFERENCES A.K. Brewer, J. Chem. Phys., 4 (1936) 350. W.R. Shields, E.L. Gamer, C.F. Hedge and S.S. Goldich, J. Geophys. Res., 68 (1963) 2331. K. Habfast, 2. Naturforschg., Teil A, 15 (1960) 273. A. Eberhardt, R. Delwiche and J. Geiss, 2. Naturforschg., Teil A, 19 (1964) 736. J.M. Ozard and R.D. Russel, Earth Planet. Sci. Lett., 4 (1968) 310. M.H. Dodson, J. Sci. Instrum., 40 (1963) 289.

189 7 H. Kanno, Bull. Chem. Sot. Jpn., 44 (1971) 1808. 8 L.J. Moore, E.F. Heald and J.F. Filliben, in N.R. Daly (Ed.), Advances in Mass Spectrometry,Vol. 7A, The Institute of Petroleum, London, 1978, p, 448. 9 KG. Heumann, K.H. Lieser and H. Elias, in K. Ogata and T. Hayakawa (Eds.), Recent Developments in Mass Spectroscopy, University of Tokyo Press, 1970, p. 457. 10 W.A. Russel, D.A. Papanastassiouand T.A. Tombrello, Geochim. Cosmochim. Acta, 42 (1978) 1075. I 1 L.J. Moore, L.A. Ma&an, W.R. Shields and E.L. Garner, Anal. Chem., 46 (1974) 1082. 12 H. Kanno, Bull. Chem. Sot. Jpn., 52 (1979) 2299. 13 (a) Lord Rayleigh, Philos. Mag., 42 (1896) 493; (b) K. Cohen, The Theory of Isotope Separation, McGraw-Hill, New York, 1951. 14 M.G. I&ram and P. Chupka, Rev. Sci. Instrum., 24 ( 1953) 5 18. 15 J.W. Green, G.D. Blue, T.C. Ehlert and J.L. Margrave, J. Chem. Phys., 41 (1964) 2245. 16 P. de Bicvre, in N.R. Daly (Ed.), Advances in Mass Spectrometry, Vol. 7A, The Institute of Petroleum, London, 1978, p. 395. 17 R.E. Perrin, D.J. Rokop, J-H. Cappis and W.R. Shields, presented at the 29th Annu. Conf. on Mass Spectrometryand Allied Topics, Minneapolis, MN, 198 1.