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Calibration of Faraday cup efficiency in a multicollector mass spectrometer
105
Chemical Geology (Isotope Geoscience Section), 94 ( 199 1) 1OS- 110 Elsevier Science Publishers B.V.. Amsterdam
Calibration of Faraday cup efficiency in a multicollector mass spectrometer Akio Makishima
and Eizo Nakamura
Institute for Study of the Earth’s Interior, Okayama University, Misasa, Tottori-ken 682-01, Japan (Received March 20, 199 1; accepted for publication
August 5, 199 1)
ABSTRACT Makishima, A. and Nakamura, E., 1991. Calibration Chem. Geol. (Isot. Geosci. Sect.), 94: 105-I 10.
of Faraday cup efficiency in a multicollector
mass spectrometer.
Faraday cup efficiencies (FCE’s) in the multicollector mass spectrometer were examined with a static multicollection technique. The relative Faraday cup efficiency (RFCE) was estimated by measurement of Nd-isotope ratios with five different cup configurations. The application of RFCE correction to the static multicollection technique will increase the reliability of analytical results by eliminating the errors caused by differences in FCE’s, and will also enable us to distinguish which Faraday cups are damaged and should be renewed.
1. Introduction The static multicollection technique for mass spectrometry (e.g., Wagner et al., 1984), in which all isotopes concerned are simultaneously measured using multiple collectors, reduces data acquisition time considerably compared with the conventional single collector peak jumping technique. However, some problems with multicollector mass spectrometers have been identified and the reliability of the static multicollection technique has been questioned. ( 1) It has been observed that Faraday cup efficiencies (FCE’s) are not always the same. Fiedler and Donohue ( 1988 ) reported relative Faraday cup efficiencies (RFCE’s) for their multicollector mass spectrometer to be 0.99971.0047. A variation in FCE of ? 0.5% could lead to systematic errors in the determination of isotopic ratios. However, in the usual static multicollection technique, correction for the difference in the FCE’s is not made and the 0 168-9622/9
l/$03.50
0 199 1 Elsevier Science Publishers
FCE’s are assumed to be equal to each other. If a large difference in FCE’s exists, the analytical reliability of the static multicollection technique falls for Nd- and Sr-isotope analyses, in which analytical precision of < &0.005% is generally required. (2) Although Faraday cup collectors are adjusted to be equal within analytical error in FCE when the multicollector mass spectrometer is assembled, it is becoming clear that Faraday cup collectors can be “damaged” and this damage causes change of the FCE and makes the FCE unstable (H.I. Kuhn and B.M. Jahn, 1990). The resultant change pers. commun., and instability in the FCE may enhance analytical errors. Multicollector mass spectrometers with such a problem could no longer produce precise isotope ratios and consequently the damaged Faraday cup collectors must be ‘renewed. However, this phenomenon is not generally known. Actually we have had a cup damage problem in our mass spectrometer and had to renew all B.V. All rights reserved.
106
Faraday cups. This mass spectrometer was installed in June 1988 and had been used for isotope analysesof Sr, Nd and Ce. The isotope ratio ‘43Nd/144Nd of the La Jolla Nd standard determined by the static multicollection was 0.5 11825at that time. However, in November 1989, there appearedextreme shifts in the isotope analyses of standard samples. The Ndisotope ratio of the La Jolla Nd standard became 0.5 11670 which is 3 t-units ( 1 e-unit= 0.01%) lower than the ratio at installation. Then all Faraday cups were replaced by new ones in August 1990. The dynamic multicollection technique (e.g., Walker et al., 1989) can overcome the systematic errors caused by the difference in FCE’s. In this method, the same massis measuredwith several cups in order to cancel the difference in FCE’s. However, data acquisition time in the dynamic multicollection is much longer (two times or more) than that in the static multicollection and time drift correction is necessary for the dynamic collection. If calibration of RFCE’s and its correction are made for the static multicollection mass spectrometry, the analytical reliability will increase and the utility will be extended, especially for small sample sizesbecauseof short data acquisition time without time drift correction. The simplest way of RFCE calibration is the peak jumping method using a very stable ion beam with time drift correction made by interpolation between scans. However, the peak jumping method for five cups takes at least 60 s for one scanwhile data acquisition and idling times are 8 and 4 s for each cup, respectively. During such a long acquisition time, it is difficult to maintain a stable ion beam to calculate the precise RFCE and the time drift correction does not perfectly lit to the actual pattern of beam decay. Fiedler and Donohue ( 1988) examined the RFCE calibration in the multicollector mass spectrometer for U- and Pu-isotope analyses using a fixed-9-collector mass spectrometer. In their method, the RFCE’s were obtained with
A. MAKISHIMA
AND E. NAKAMURA
different Faraday cup configurations by using “absolute” standard samples measured separately by the single-collector jumping technique. However, the obtained RFCE’s are largely affected by the ratios of the absolute standards, so that this method cannot give the unique solution of RFCE in the multicollector mass spectrometer. In this paper, we show a calibration method of the RFCE for a variable-5-collector mass spectrometer using measurementsof Nd. This method does not require knowledge of the absolute ratio of the standard. We also show the presenceof cup damage inferred from the analytical results obtained with the cups before the renewal. 2. Experiment
A Finnigan-MAT @ model-261 mass spectrometer was employed in this study. This mass spectrometer is equipped with 4 movable Faraday cup collectors and a fixed central one. Its resolving power is normally - 500. Accelerating voltage was set to 10 kV. The gains of each amplifier attached to individual collectors (Again) were measured by supplying a uniform current to the amplifiers. La Jolla Nd standard was used in this experiment. The recently reported isotopic compositions from several laboratories are presented in Table I. The Re double-filament technique was used with 400 ng of Nd loaded onto the evaporation filament. The ionbeam current of 144Ndwas kept between l-10-” and 2~10~” TABLE I “‘3Nd/‘44Nd ANU Carnegie Max Planck Rennes Tokyo Toronto Normalized
ratios of La Jolla Nd standard Woodhead and McCulloch ( 1989) Walker et al. (1989) Chauvel et al. (1985) Gruau et al. (1987) Shimizu et al. (1989) Noble et al. ( 1989) to ‘46Nd/‘44Nd=0.7219.
0.5 11869 + 0.000002 0.511857~0.000021 0.5 11847 f 0.000021 0.511875+_0.000009 0.511823~0.000016 0.511858~0.000018
FARADAY
CUP EFFICIENCY
IN A MULTICOLLECTOR
107
MASS SPECTROMETER
TABLE II
0.51 190 : Ri
Cup configurations
Configuration 1 COL COL COL COL COL
5 4 3 2
143 144 146
0.51180
3
4
143 144
143 144 145 146
143 144 146
R3
R4
146
1
0.51185
No. 2
’ i A’ f l 4
for the Nd isotope static analysis
5
143 144 146
0.51190
-
J
4
0.511860r0.000005
II
T
0.511859~0.000006
Isotope ratio R,
&
RS
0.511856'0.000004
1- TL- L 0.511849'0.000003
A, and ion beam intensities for 143Nd,‘44Nd and 146Ndwere determined in the static multicollection mode. Counting time of 8 s was used for each isotope ratio calculation. Since the Sm concentration in the Nd standard is below detection limits, no correction was made for Sm interferences. Atomic ratios were calculated using ‘46Nd/‘44Nd=0.7219 and the power law. One hundred isotope ratios of ‘43Nd/‘44Nd were collected for a single run which took 40 min.; analytical errors are presentedas standard error (2 am). 143Nd/144Ndratios were measured with five different cup configurations (Table II). The cups were reconfigured manually prior to different runs satisfying the configurations presented in Table II. Results with different configurations are denoted as Ri ( i= 1, .... 5 ) . Theoretically, 10 cup configurations ( r0C2) are possible for these three mass peaks in a 5-collector mass spectrometer. However, in practice only five different cup conligurations are available, becauseof the mechanical difficulty of configuring three cups in one mass difference. In addition, the Nd ion beam at COL 3 was used for focusing and peak centering. Six separateruns were carried out using each cup configuration.
Fig. 1. Analytical results of Nd ferent cup configurations. Error with analytical error (20) of cup configuration are given in
isotope ratios with five difbars represent 2q,,. Means six measurements in each the figure.
3. Result and discussion
3.1. Nd isotope ratios with Jive different cup configurations Results of Nd isotope analysis with live different cup configurations are shown in Fig. 1. The averagesof RI, R2 and R3 are identical to each other when the analytical errors are considered. However, R4 and R5 are apparently lower than RI-R3 by up to 0.6 t-units, substantially exceedinganalytical errors.Since only the A-gain calibration was used in the determination of these isotope ratios, it is suspectedthat this diversity is causedby differencesin FCE’s. .3.2. Calculation ofRFCE The true gain for eachcollector contains both the A-gains and FCE’s, and the true gain of a collector is given by the equation:
108
A. MAKISHIMA
G =J;g;
(1)
where g, andJ; are the A-gain and FCE, respectively, of the ith collector. In the usual static multicollection technique, only A-gains ( gi) are measured by introducing a constant electrical current to the amplifiers of each collector, and FCE Cr;) correction is not performed. We cannot obtain absolute FCE’s becauseit is difficult to introduce exactly equal amounts of ion current into the individual Faraday cups. Therefore, we have determined the relative FCE (RFCE) , rather than the FCE. RFCE’s can be obtained from the Nd-isotope measurements of a standard sample with different cup configurations (Table II), by assuming that the true gain-corrected ratios determined with the different cup configurations are equal to each other. An ion beam intensity correctedwith the true gain ( IFoRR) in the ith Faraday cup is given by the equation: ]C=‘RR= 1;G; =KLtY Z IQ”SJ;
(2)
where 1’I and IyBS refer to the measured ion current before application of any gain corrections and to the ion current after the A-gain correction, respectively. The true gain-corrected ratio, C for the No. 1 cup configuration in Table II, is given by the equation using the power law: C1=(14CoRR/ GORR) [ (I$oRR/I$oRR)/0.7219]1’2 = (IYBS/IYBS)
AND
E. NAKAMURA
ratio obtained in the usual isotope analysiswith the static multicollection. Similarly, C for the other configurations are given by the following equations: C2
=R2Wh)
(41
UZ>“*
G =&WhKti/h)“*
(5)
C4 =&CfS/f4) Wf4>1’2
(6)
C5
(7)
=Rsti/S2)
ti/f2)1’2
Since: c=c,=c*=c3=c4=c5
(8)
we obtain: fi/jZ,=R:Ry’R:‘R;Rr’
(9)
.fi/.fi=R&R:Rs’ j&=R2Ry3R$R$
(10)
fs/f4=R,R2’.5R:.5R44.5R:.5
(12)
(11)
In order to calculate RFCE’s for N (number) Faraday cups, more than (N- 1) equations are necessaryif C is known. When C is unknown, N equations are required. We have five Faraday cup collectors with our mass spectrometer and can obtain five equations (eqs. 3-7 ) from five different cup configurations. Therefore, both the RFCE and C can be calculated. Errors in the RFCE calculation are estimated by the method shown in the Appendix. 3.3. Cup damage and implication of RFCE calibration,for multicollection technique
[ (1$‘BS/I$)BS)/0.7219]1’2 x tilh)
w.w*
=R1 Wh) ti/h> l’* (3) where RI is the fractionation corrected isotope ratio with A-gain correction; this is the isotope
The calculated RFCE’s relative to COL 4 are presented in Table III. All RFCE’s are nearly unity and identical within the analytical error, so that the correction with RFCE does not make a noticeable difference in the isotope ra-
FARADAY
CUP EFFICIENCY
IN A MULTICOLLECTOR
109
MASS SPECTROMETER
TABLE III
TABLE V
Calculated RFCE’s against COL 4 with the new Faraday cups
Calculated RFCE’s against COL 4 before the renewal of the Faraday cups
RFCE
Error ( 1 a)
0.999990 0.999986 1.000014 0.999987
0.000027 0.000026 0.000017 0.000023
RFCE fi /f4 fi/jl, h/f4 h/f4
fi/f4 fi/f4 fi/f4 fsljI$
1.00115 1.00111 1.00085 0.99948
TABLE IV Analytical results of Nd isotope ratios with five different cup configurations by the mass spectrometer before the renewal of the Faraday cups
R, R2 R, R4 R5
Run I
Run 2
Average
0.512034+0.000012 0.51202210.000018 0.511671~0.000017 0.511745i0.000015 0.511813~0.000010
0.51207610.000011 0.512067~0.000010 0.511669+0.000011 0.511729~0.000010
0.512055 0.512045 0.511670 0.511737
tio measurements. However, there exists substantial difference in the isotope ratio so that the isotope ratio measurement is more sensitive to the difference of RFCE than the calculation of RFCE itself. The corrected ratio of 0.5 1185 is consistent with those obtained in previous studies (0.5 1182-0.5 1188;Table I). The analytical results of Nd-isotope ratios with five different cup configurations before the renewal of Faraday cups are shown in Table IV. Although data setsare limited, the Ndisotope ratios shift up to 4 ~-units, which is unacceptably larger than the analytical errors in the Nd-isotope analysis.The RFCE’s before the cup renewal are calculated using the averages given in Table IV and are shown in Table V. Compared with the RFCE’s of the renewed cups (0.99999-l.OOOOl ), the RFCE’s before the cup renewal largely deviate from unity (0.9995-l .OOl). It is inferred that these large deviations of RFCE’s were caused by cup damage although the mechanism of cup damageis still not clear. In conclusion, the application of RFCE presentedin this study may be used to detect pos-
sible cup damage by continually monitoring RFCE’s and distinguish which cups should be renewed in a multicollector mass spectrometer. Furthermore, the RFCE correction minimizes the effect of FCE’s and will increasethe reliability of analytical results in the static multicollection mass spectrometry. Acknowledgements
We are very thankful to S. Akimoto, H. Honma, H. Kagami and K. Nagao (ISEI) for supporting this work. We owe a great deal to the discussion with the Finnigan-MAT engineer, H.I. Kiihn, who visited Misasa to repair our mass spectrometer. We acknowledge B.M. Jahn for discussionsconcerningthe problem of the cup damage in multicollector mass spectrometer. We are also very grateful to A. Masuda and H. Shimizu (The University of Tokyo) for offering the La Jolla Nd standard. Our thanks are extended to R. Hill (Australian National University) for brushing up our English. This research was supported by Monbusho (Ministry of Education, Science and Culture, Japan) International Scientific ResearchProgram and by a Grant-in-Aid for Scientific Researchfrom Monbusho. Appendix
-
Estimation
of errors of RFCE
Since RFCE is given by the non-linear equations (9)( 12), estimation of RFCE errors caused by the isotopic analysis errors is not simple. This Appendix shows the method used for error propagation. We want to know a variance-covariance matrix (S,)
110
A. MAKISHIMA
of a data vector, Y, while Y is a function X and defined as: Y=f(X).
of a data vector (A-1 1
A variance-covariance matrix and a mean vector of Xare defined as S, and m, respectively. Taylor expansion of Y around m gives the approximation:
Y=f(w) +(gx >xzp(X-p) where
(>( ax, ax, Jf -zz ax
af af --...-
af
ax, >
(A-3)
Sy is given by: Sy= BSxTB
(A-4)
where (A-5) and TB is a transposed matrix of B. In this study, as Y is RFCE and a function of a vector of R=T(R,,R,,R3,R4,R,), eq. A-l is given as:
Y=f(R)=f(R,,R,,...,R,)
(‘4-6)
Means and variances of R are expressed as r= ‘( r,,v, ,... Ye) and s,*, s,‘, . . . . sg2, respectively. Since R, and R, are considered to be independent, covariance of R is 0 and S, becomes a diagonal matrix. Consequently, the variance of Y is given by:
(A-7)
AND E. NAKAMURA
References Chauvel, C., Dupre, B. and Jenner, G.A., 1985. The SmNd age of Kambalda volcanics is 500 Ma too old! Earth Planet. Sci. Lett., 74: 3 15-324. Fiedler, R. and Donohue, D., 1988. Pocket sensitivity calibration of multicollector mass spectrometers. Fresenius Z. Anal. Chem., 33 1: 209-2 13. Gruau, G., Jahn, B.M., Glikson, A.Y. Davy, R., Hickman, A.H. and Chauvel, C., 1987. Age of the Archean Talga-Talga Subgroup, Pilbara Block, Western Australia, and early evolution of the mantle: new Sm-Nd isotopic evidence. Earth Planet. Sci. Lett., 85: 105-l 16. Noble, S.R.,Lightfoot, P.C. and Scharer, U., 1989. A new method for single-filament isotopic analysis of Nd using in situ reduction. Chem. Geol. (Isot. Geosci. Sect.), 79: 15-19. Shimizu, H., Mori, K. and Masuda, A., 1989. REE, Ba, and Sr abundances and Sr, Nd, and Ce isotopic ratios in.Hole 504B basalts, ODP Leg I1 1, Costa Rica Rift. In: K. Becker, H. Sakai, H. et al. (Editors), Proc. Ocean Drill. Prog., Sci. Results, Vol. 1 11. Ocean Drill. Prog., College Station, Texas, pp. 77-83. Wagner, G., Rache, H. and Tuttas, D., 1984. Variable multicollection - Speed, precision and versatility with the Mode1 261 thermal ionization mass spectrometer. Finnigan-MAT G.m.b.H., Bremen, Tech. Rep. No. 416, 10 PP. Walker, R.J., Carlson, R.W., Shirey, S.B. and Boyd, F.R., 1989. OS, Sr, Nd, and Pb isotope systematics of southern African peridotite xenolith: Implication for the chemical evolution of subcontinental mantle. Geochim. Cosmochim. Acta, 53: 1583-l 595. Woodhead, J.D. and McCulloch, M.T., 1989. Ancient sea floor signals in Pitcairn Island lavas and evidence for large amplitude, small length-scale mantle heterogeneities. Earth Planet. Sci. Lett., 94: 257-273.
Chemical Geology (Isotope Geoscience Section), 94 ( 199 1) 1OS- 110 Elsevier Science Publishers B.V.. Amsterdam
Calibration of Faraday cup efficiency in a multicollector mass spectrometer Akio Makishima
and Eizo Nakamura
Institute for Study of the Earth’s Interior, Okayama University, Misasa, Tottori-ken 682-01, Japan (Received March 20, 199 1; accepted for publication
August 5, 199 1)
ABSTRACT Makishima, A. and Nakamura, E., 1991. Calibration Chem. Geol. (Isot. Geosci. Sect.), 94: 105-I 10.
of Faraday cup efficiency in a multicollector
mass spectrometer.
Faraday cup efficiencies (FCE’s) in the multicollector mass spectrometer were examined with a static multicollection technique. The relative Faraday cup efficiency (RFCE) was estimated by measurement of Nd-isotope ratios with five different cup configurations. The application of RFCE correction to the static multicollection technique will increase the reliability of analytical results by eliminating the errors caused by differences in FCE’s, and will also enable us to distinguish which Faraday cups are damaged and should be renewed.
1. Introduction The static multicollection technique for mass spectrometry (e.g., Wagner et al., 1984), in which all isotopes concerned are simultaneously measured using multiple collectors, reduces data acquisition time considerably compared with the conventional single collector peak jumping technique. However, some problems with multicollector mass spectrometers have been identified and the reliability of the static multicollection technique has been questioned. ( 1) It has been observed that Faraday cup efficiencies (FCE’s) are not always the same. Fiedler and Donohue ( 1988 ) reported relative Faraday cup efficiencies (RFCE’s) for their multicollector mass spectrometer to be 0.99971.0047. A variation in FCE of ? 0.5% could lead to systematic errors in the determination of isotopic ratios. However, in the usual static multicollection technique, correction for the difference in the FCE’s is not made and the 0 168-9622/9
l/$03.50
0 199 1 Elsevier Science Publishers
FCE’s are assumed to be equal to each other. If a large difference in FCE’s exists, the analytical reliability of the static multicollection technique falls for Nd- and Sr-isotope analyses, in which analytical precision of < &0.005% is generally required. (2) Although Faraday cup collectors are adjusted to be equal within analytical error in FCE when the multicollector mass spectrometer is assembled, it is becoming clear that Faraday cup collectors can be “damaged” and this damage causes change of the FCE and makes the FCE unstable (H.I. Kuhn and B.M. Jahn, 1990). The resultant change pers. commun., and instability in the FCE may enhance analytical errors. Multicollector mass spectrometers with such a problem could no longer produce precise isotope ratios and consequently the damaged Faraday cup collectors must be ‘renewed. However, this phenomenon is not generally known. Actually we have had a cup damage problem in our mass spectrometer and had to renew all B.V. All rights reserved.
106
Faraday cups. This mass spectrometer was installed in June 1988 and had been used for isotope analysesof Sr, Nd and Ce. The isotope ratio ‘43Nd/144Nd of the La Jolla Nd standard determined by the static multicollection was 0.5 11825at that time. However, in November 1989, there appearedextreme shifts in the isotope analyses of standard samples. The Ndisotope ratio of the La Jolla Nd standard became 0.5 11670 which is 3 t-units ( 1 e-unit= 0.01%) lower than the ratio at installation. Then all Faraday cups were replaced by new ones in August 1990. The dynamic multicollection technique (e.g., Walker et al., 1989) can overcome the systematic errors caused by the difference in FCE’s. In this method, the same massis measuredwith several cups in order to cancel the difference in FCE’s. However, data acquisition time in the dynamic multicollection is much longer (two times or more) than that in the static multicollection and time drift correction is necessary for the dynamic collection. If calibration of RFCE’s and its correction are made for the static multicollection mass spectrometry, the analytical reliability will increase and the utility will be extended, especially for small sample sizesbecauseof short data acquisition time without time drift correction. The simplest way of RFCE calibration is the peak jumping method using a very stable ion beam with time drift correction made by interpolation between scans. However, the peak jumping method for five cups takes at least 60 s for one scanwhile data acquisition and idling times are 8 and 4 s for each cup, respectively. During such a long acquisition time, it is difficult to maintain a stable ion beam to calculate the precise RFCE and the time drift correction does not perfectly lit to the actual pattern of beam decay. Fiedler and Donohue ( 1988) examined the RFCE calibration in the multicollector mass spectrometer for U- and Pu-isotope analyses using a fixed-9-collector mass spectrometer. In their method, the RFCE’s were obtained with
A. MAKISHIMA
AND E. NAKAMURA
different Faraday cup configurations by using “absolute” standard samples measured separately by the single-collector jumping technique. However, the obtained RFCE’s are largely affected by the ratios of the absolute standards, so that this method cannot give the unique solution of RFCE in the multicollector mass spectrometer. In this paper, we show a calibration method of the RFCE for a variable-5-collector mass spectrometer using measurementsof Nd. This method does not require knowledge of the absolute ratio of the standard. We also show the presenceof cup damage inferred from the analytical results obtained with the cups before the renewal. 2. Experiment
A Finnigan-MAT @ model-261 mass spectrometer was employed in this study. This mass spectrometer is equipped with 4 movable Faraday cup collectors and a fixed central one. Its resolving power is normally - 500. Accelerating voltage was set to 10 kV. The gains of each amplifier attached to individual collectors (Again) were measured by supplying a uniform current to the amplifiers. La Jolla Nd standard was used in this experiment. The recently reported isotopic compositions from several laboratories are presented in Table I. The Re double-filament technique was used with 400 ng of Nd loaded onto the evaporation filament. The ionbeam current of 144Ndwas kept between l-10-” and 2~10~” TABLE I “‘3Nd/‘44Nd ANU Carnegie Max Planck Rennes Tokyo Toronto Normalized
ratios of La Jolla Nd standard Woodhead and McCulloch ( 1989) Walker et al. (1989) Chauvel et al. (1985) Gruau et al. (1987) Shimizu et al. (1989) Noble et al. ( 1989) to ‘46Nd/‘44Nd=0.7219.
0.5 11869 + 0.000002 0.511857~0.000021 0.5 11847 f 0.000021 0.511875+_0.000009 0.511823~0.000016 0.511858~0.000018
FARADAY
CUP EFFICIENCY
IN A MULTICOLLECTOR
107
MASS SPECTROMETER
TABLE II
0.51 190 : Ri
Cup configurations
Configuration 1 COL COL COL COL COL
5 4 3 2
143 144 146
0.51180
3
4
143 144
143 144 145 146
143 144 146
R3
R4
146
1
0.51185
No. 2
’ i A’ f l 4
for the Nd isotope static analysis
5
143 144 146
0.51190
-
J
4
0.511860r0.000005
II
T
0.511859~0.000006
Isotope ratio R,
&
RS
0.511856'0.000004
1- TL- L 0.511849'0.000003
A, and ion beam intensities for 143Nd,‘44Nd and 146Ndwere determined in the static multicollection mode. Counting time of 8 s was used for each isotope ratio calculation. Since the Sm concentration in the Nd standard is below detection limits, no correction was made for Sm interferences. Atomic ratios were calculated using ‘46Nd/‘44Nd=0.7219 and the power law. One hundred isotope ratios of ‘43Nd/‘44Nd were collected for a single run which took 40 min.; analytical errors are presentedas standard error (2 am). 143Nd/144Ndratios were measured with five different cup configurations (Table II). The cups were reconfigured manually prior to different runs satisfying the configurations presented in Table II. Results with different configurations are denoted as Ri ( i= 1, .... 5 ) . Theoretically, 10 cup configurations ( r0C2) are possible for these three mass peaks in a 5-collector mass spectrometer. However, in practice only five different cup conligurations are available, becauseof the mechanical difficulty of configuring three cups in one mass difference. In addition, the Nd ion beam at COL 3 was used for focusing and peak centering. Six separateruns were carried out using each cup configuration.
Fig. 1. Analytical results of Nd ferent cup configurations. Error with analytical error (20) of cup configuration are given in
isotope ratios with five difbars represent 2q,,. Means six measurements in each the figure.
3. Result and discussion
3.1. Nd isotope ratios with Jive different cup configurations Results of Nd isotope analysis with live different cup configurations are shown in Fig. 1. The averagesof RI, R2 and R3 are identical to each other when the analytical errors are considered. However, R4 and R5 are apparently lower than RI-R3 by up to 0.6 t-units, substantially exceedinganalytical errors.Since only the A-gain calibration was used in the determination of these isotope ratios, it is suspectedthat this diversity is causedby differencesin FCE’s. .3.2. Calculation ofRFCE The true gain for eachcollector contains both the A-gains and FCE’s, and the true gain of a collector is given by the equation:
108
A. MAKISHIMA
G =J;g;
(1)
where g, andJ; are the A-gain and FCE, respectively, of the ith collector. In the usual static multicollection technique, only A-gains ( gi) are measured by introducing a constant electrical current to the amplifiers of each collector, and FCE Cr;) correction is not performed. We cannot obtain absolute FCE’s becauseit is difficult to introduce exactly equal amounts of ion current into the individual Faraday cups. Therefore, we have determined the relative FCE (RFCE) , rather than the FCE. RFCE’s can be obtained from the Nd-isotope measurements of a standard sample with different cup configurations (Table II), by assuming that the true gain-corrected ratios determined with the different cup configurations are equal to each other. An ion beam intensity correctedwith the true gain ( IFoRR) in the ith Faraday cup is given by the equation: ]C=‘RR= 1;G; =KLtY Z IQ”SJ;
(2)
where 1’I and IyBS refer to the measured ion current before application of any gain corrections and to the ion current after the A-gain correction, respectively. The true gain-corrected ratio, C for the No. 1 cup configuration in Table II, is given by the equation using the power law: C1=(14CoRR/ GORR) [ (I$oRR/I$oRR)/0.7219]1’2 = (IYBS/IYBS)
AND
E. NAKAMURA
ratio obtained in the usual isotope analysiswith the static multicollection. Similarly, C for the other configurations are given by the following equations: C2
=R2Wh)
(41
UZ>“*
G =&WhKti/h)“*
(5)
C4 =&CfS/f4) Wf4>1’2
(6)
C5
(7)
=Rsti/S2)
ti/f2)1’2
Since: c=c,=c*=c3=c4=c5
(8)
we obtain: fi/jZ,=R:Ry’R:‘R;Rr’
(9)
.fi/.fi=R&R:Rs’ j&=R2Ry3R$R$
(10)
fs/f4=R,R2’.5R:.5R44.5R:.5
(12)
(11)
In order to calculate RFCE’s for N (number) Faraday cups, more than (N- 1) equations are necessaryif C is known. When C is unknown, N equations are required. We have five Faraday cup collectors with our mass spectrometer and can obtain five equations (eqs. 3-7 ) from five different cup configurations. Therefore, both the RFCE and C can be calculated. Errors in the RFCE calculation are estimated by the method shown in the Appendix. 3.3. Cup damage and implication of RFCE calibration,for multicollection technique
[ (1$‘BS/I$)BS)/0.7219]1’2 x tilh)
w.w*
=R1 Wh) ti/h> l’* (3) where RI is the fractionation corrected isotope ratio with A-gain correction; this is the isotope
The calculated RFCE’s relative to COL 4 are presented in Table III. All RFCE’s are nearly unity and identical within the analytical error, so that the correction with RFCE does not make a noticeable difference in the isotope ra-
FARADAY
CUP EFFICIENCY
IN A MULTICOLLECTOR
109
MASS SPECTROMETER
TABLE III
TABLE V
Calculated RFCE’s against COL 4 with the new Faraday cups
Calculated RFCE’s against COL 4 before the renewal of the Faraday cups
RFCE
Error ( 1 a)
0.999990 0.999986 1.000014 0.999987
0.000027 0.000026 0.000017 0.000023
RFCE fi /f4 fi/jl, h/f4 h/f4
fi/f4 fi/f4 fi/f4 fsljI$
1.00115 1.00111 1.00085 0.99948
TABLE IV Analytical results of Nd isotope ratios with five different cup configurations by the mass spectrometer before the renewal of the Faraday cups
R, R2 R, R4 R5
Run I
Run 2
Average
0.512034+0.000012 0.51202210.000018 0.511671~0.000017 0.511745i0.000015 0.511813~0.000010
0.51207610.000011 0.512067~0.000010 0.511669+0.000011 0.511729~0.000010
0.512055 0.512045 0.511670 0.511737
tio measurements. However, there exists substantial difference in the isotope ratio so that the isotope ratio measurement is more sensitive to the difference of RFCE than the calculation of RFCE itself. The corrected ratio of 0.5 1185 is consistent with those obtained in previous studies (0.5 1182-0.5 1188;Table I). The analytical results of Nd-isotope ratios with five different cup configurations before the renewal of Faraday cups are shown in Table IV. Although data setsare limited, the Ndisotope ratios shift up to 4 ~-units, which is unacceptably larger than the analytical errors in the Nd-isotope analysis.The RFCE’s before the cup renewal are calculated using the averages given in Table IV and are shown in Table V. Compared with the RFCE’s of the renewed cups (0.99999-l.OOOOl ), the RFCE’s before the cup renewal largely deviate from unity (0.9995-l .OOl). It is inferred that these large deviations of RFCE’s were caused by cup damage although the mechanism of cup damageis still not clear. In conclusion, the application of RFCE presentedin this study may be used to detect pos-
sible cup damage by continually monitoring RFCE’s and distinguish which cups should be renewed in a multicollector mass spectrometer. Furthermore, the RFCE correction minimizes the effect of FCE’s and will increasethe reliability of analytical results in the static multicollection mass spectrometry. Acknowledgements
We are very thankful to S. Akimoto, H. Honma, H. Kagami and K. Nagao (ISEI) for supporting this work. We owe a great deal to the discussion with the Finnigan-MAT engineer, H.I. Kiihn, who visited Misasa to repair our mass spectrometer. We acknowledge B.M. Jahn for discussionsconcerningthe problem of the cup damage in multicollector mass spectrometer. We are also very grateful to A. Masuda and H. Shimizu (The University of Tokyo) for offering the La Jolla Nd standard. Our thanks are extended to R. Hill (Australian National University) for brushing up our English. This research was supported by Monbusho (Ministry of Education, Science and Culture, Japan) International Scientific ResearchProgram and by a Grant-in-Aid for Scientific Researchfrom Monbusho. Appendix
-
Estimation
of errors of RFCE
Since RFCE is given by the non-linear equations (9)( 12), estimation of RFCE errors caused by the isotopic analysis errors is not simple. This Appendix shows the method used for error propagation. We want to know a variance-covariance matrix (S,)
110
A. MAKISHIMA
of a data vector, Y, while Y is a function X and defined as: Y=f(X).
of a data vector (A-1 1
A variance-covariance matrix and a mean vector of Xare defined as S, and m, respectively. Taylor expansion of Y around m gives the approximation:
Y=f(w) +(gx >xzp(X-p) where
(>( ax, ax, Jf -zz ax
af af --...-
af
ax, >
(A-3)
Sy is given by: Sy= BSxTB
(A-4)
where (A-5) and TB is a transposed matrix of B. In this study, as Y is RFCE and a function of a vector of R=T(R,,R,,R3,R4,R,), eq. A-l is given as:
Y=f(R)=f(R,,R,,...,R,)
(‘4-6)
Means and variances of R are expressed as r= ‘( r,,v, ,... Ye) and s,*, s,‘, . . . . sg2, respectively. Since R, and R, are considered to be independent, covariance of R is 0 and S, becomes a diagonal matrix. Consequently, the variance of Y is given by:
(A-7)
AND E. NAKAMURA
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