Quantitative Analysis with an Ion Microanalyzer Tohru Ishitani, Hifumi Tamura, and Toshio Kondo Central Research Laboratory, Hifachi Ltd., Kokubunji, Tokyo, Japan
A preliminary investigation of quantitative analysis with an ion microanalyzer is made by utilizing the dlssoclallon of molecules, based on a local equilibrium plasma model proposed by Andersen. The calculatlon procedures are checked by applying them to the analyses of standard samples of stainless steel. The agreement between the calculated and chemical concentrations is generally good. This approach may develop Into a useful and powerful technique for quantitative analysis since no standard samples are required.
Strictly speaking, the partition functions for molecules demand the consideration of electronic, vibrational, and rotational energy levels. By employing the general approximation that only the lowest electronic level contributes substantially to the partition function of the molecule, the constant D is computed from the practical logarithmic formula:
log
DXy
= 20.432
log(ZxZy)
+
+
1.5 log(MxMy/Mxy) +
log B - log g
+
0 . 5 log T
+
log(1 - 1O-O~625W/T) - ( 5 0 4 o / n v d
Recently, the ion microanalyzer (IMA) has attracted considerable attention as an instrument capable of threedimensional microanalysis with high sensitivity. Though several approaches ( I ) for quantitative analysis have been proposed, they have not been sufficiently complete. The main cause is the difficulty in estimating the degree of ionization of sputtered particles, which is highly sensitive to the surface condition. In 1969, Andersen (2, 3) originally proposed a thermodynamic approach based on an assumption that the sputtered region resembles a local plasma in thermal equilibrium state, and reported hopeful results. Assuming the same model, one of the authors (T.I.) has calculated fundamental data for the degree of ionization ( 4 ) . The work denotes procedures for quantitative calculation where only singly-charged atomic ions are considered as secondary ions. The calculations require that two or more internal standard elements of known concentration be in the analyzed sample to estimate the two unknown parameters (Le., plasma temperature and electron density), which represent the plasma state and determine the degree of ionization. However, by considering both singly-charged atomic ions and molecular ions, internal standard elements are no longer required in the sample. This idea was originated by Andersen and Hinthorne (3). In this paper, the latter quantitative approach is checked by analyzing standard stainless steel samples. The agreement between the calculated concentrations and the chemically determined ones is generally good.
CALCULATION PROCEDURES Following reference 3, let us consider a thermal dissociation in the plasma. For the dissociation equilibrium (XY * X Y) of a diatomic molecule XY, the constant of dissociation D is defined as ( 5 ) :
+
D,
= nxny/nxy.
( 1)
where n is concentration expressed as number per unit volume. The constant D is a function of plasma temperature (5):
where m is the mass, T the absolute temperature, Z the partition function, ed the energy of dissociation, h Planck's constant, and k Boltzman's constant. 1294
ANALYTICAL CHEMISTRY, VOL. 47, NO. 8, JULY 1975
(3) where M is the atomic or molecular weight in atomic mass units, B the rotational constant in cm-l, g the statistical weight of the electronic state, w the vibrational constant in cm-l, and v d the dissociation energy in eV. Here, D is expressed as the number per 1 cm3. Although individual terms in Equation 3 may differ considerably among molecules, their joint effect is such that the term of -(5040/T)Vd is the only essential contribution. Therefore, it generally allows estimates of D to be based on the dissociation energy only ( 5 ) .That is: 5
4tY
1024, 10-(5040/T)Q
(4)
where the coefficient must be assigned an uncertainty factor of f50%. This equation has been applied instead of Equation 3 in the present calculation when the values of B and w of molecules of interest are not found in the literature of Boumans (51, Herzberg (61,Gaydon (7), and Rosen (8).
As sputtered ions, singly-charged atomic and molecular ions are employed in present quantitative correction. Under the assumption of thermal equilibrium, the following ionization processes are also considered: X + X+ e and XY XY+ e. Similarly, the dissociation constants for these equilibrium states are defined as:
+
+
Dx' = %*ne/% Dxy'
0
= nXy'ne/nxyo
( 5)
where nj+, njo, and n, represent the concentration per unit volume of the singly-charged ions, neutral particles and electrons, respectively. Here, the label j (i.e., X or XY) specifies the element or the molecule. In this case, the practical logarithmic formula for these dissociations is given by the Saha-Eggert equation (5):
log(n;n,/n,@) = 15.684
+
log(Zj'/Zjo)
+
1 . 5 log T - 5040(1, - A G / T
(6)
where Zj+ and Zjo are the internal partition functions of these charged states, I , is the ionization potential in eV, and hE the ionization potential depression in eV. The multi-charged ions may generally be neglected since their concentrations are somewhat lower than those of singlycharged ions a t temperatures below several thousand K ( 4 ) . Therefore, the degree of ionization of species j is easily calculated by Equation 6.
K j + = nI+/nj
( 7)
0 2 ' -+STAINLESS STEEL
where
n j = n j O+ nI+
7-
( 8)
The value of Kj+ is represented as a function of T and ne. It is noted that the value of ne is equal to the sum of the concentrations of all kinds of ions, that is:
(9) The secondary ion intensity the ion concentration n,+:
I]+ is intimately related with
Ij* = cqjnj+
-p -$I----
,
I . . . 1.1 . . , , . : _-_
(10) _i
where a?] is the instrumental parameter which gives the mass spectrometer transmission rate and the ion detector sensitivity. In the present calculation, the former is supposed to be constant. As for the latter, the most commonly used correction for mass discrimination ( 9 ) , which is based on the assumption that ions of equal velocity produce the same number of electrons a t the first electrode of the electron multiplier, is applied. This correction is not required when the pulse count mode is applied instead of the current mode. It is not necessary in the present calculation to measure the constant a . It can be estimated by integrating Equation 9 with Equation 10 as follows:
By using Equations 7 and 10, the value of nj may be rewritten as:
n j = Ij+/((&)VjKj+)
(12)
This is true, of course, only when the numbers of negative ions are small relative to that of positive ions ( 3 ) . Consequently, the experimental value of dissociation constant D'xy, defined by Equation 1,may be expressed as: ~t~~
= (l/(a))(Ix+ry+/Ixy+)x
(VXY /7]xVy) (Kxy*/Kx'Ky')
50
55
' '
I
6or&i O 'A,
' .
Figure 1. A pertinent part of a mass spectrum chart obtained from stainless steel sample bombarded by 10-keV 02+ ions
a
Table I. Comparative Analysis of Stainless Steel (Cr18-Ni9) IhM, ,Yt %
Chemical anal)sis,wt
Fe Cr
".
S e c . ion i n t . , arb units
SO
s-Ma
69.14 18.35 9.11 0.53 0.19 1.19 0.43 0.60 0.34 0.03
117. (68.9)* 69.0 77.0 (18.3Ib 18.2 Ni 6.76 9.32 9.34 Mn 1.70 0.94 0.93 cu 0.14 0.25 0.25 Si 1.23 0.93 0.93 Mo 0.38 0.27 0.27 Nb 1.33 0.72 0.72 Ti 2.71 0.36 0.35 V 0.22 0.04 0.04 FeO: 0.39 CrO: 0.34 02: 0.037 0: 0.31 a S = singly-charged ion; M = molecular ion. Internal standard element.
(13)
On the other hand, the theoretical one D x y may be obtained from Equation 3. In order to acquire the optimum set of ( T , n e ) ,the following minimum seeking method is applied:
A t least two molecules are required to estimate these two parameters (Le., T and n e ) . Finally, the concentration of element 1 in the sample may be calculated from: I
where m
Here, nl and nl, are the concentrations of element I and of molecule Im, respectively, see Equation 8. It should be noted that in the summation in Equation 15, the elements in the primary ion beam are withdrawn.
RESULTS AND DISCUSSION Hitachi's IMA-2 ion microanalyzer has been employed to check the present quantitative approach. The apparatus has been described earlier (IO), and so it is not presented
here. The target chamber was evacuated to a base pressure of less than 1 X Torr with oil diffusion pumps and was Torr during the ion bommaintained a t about 1 X bardment. All the samples were bombarded by 10-keV 0 2 + ions a t a current of 1 pA and about 0.5 mm in diameter. When the sample was bombarded, the secondary ion intensity initially showed a high value owing to the surface-oxidized thin atomic layers. It gradually became a stationary value where the oxide formation and the sputtering were in a state of equilibrium. All the input data for the quantitative calculation related to the secondary ion intensity have been taken from the mass spectrum chart obtained at this equilibrium state. Experiments have been carried out mainly on standard samples of stainless steel (NBS 446, 447,449, and 450). A reproducibility of the weak ion intensities is less than a few percent. It might be due to the micro-homogeneity of the samples. Figure 1 shows a pertinent part of the mass spectrum obtained from a stainless steel sample. For the quantitative analysis, it is desirable that the shape of peaks in the spectrum be trapezoidal, as seen in Figure 1. This minimizes interference to the measurement of peak intensity from adjacent peaks. As a typical example of quantitative correction, the calculated concentrations for an NBS 446 sample are shown in the last column (S-M) of Table I. The chemically ANALYTICALCHEMISTRY, VOL. 47,
NO. 8,
JULY 1975
1295
mi ./
//*
/
/
/.
/
/
,/
CHEMICAL ANALYSIS ( w t
/
'/e)
Figure 2. Comparison of the analyses with the ion microanalyzer (iMA) with the chemical analyses for a number of elements in stain-
less steel samples analyzed concentrations and the secondary ion intensities are also given in this table. In the calculation, three molecular ions (FeO+, CrO+, and 0 2 + ) with a fairly strong intensity have been employed to estimate ( T ,n e ) values through Equation 14. The agreement between the chemically analyzed concentrations and the calculated ones is generally good. In order to compare the present approach "without standard elements" with the previous approach "with more than two standard elements" ( 4 ) ,the concentrations calculated from the latter approach are also presented in column S in Table I. Here, Fe and Cr were employed as the standard elements. The estimated values of the T and ne for S and S-M approaches were 6930 K, 5.5 X 10l6 electrons/cm3 electrons/cm3, respectively. The difand 6920 K, 5.5 X ferences between these calculated concentrations were small. In the present calculation, it has been assumed that the value of K M +is used as a substitute for that of KMO+. This is because of little knowledge of partition function ZMOfor the monoxide molecule MO. This assumption may play an important role in the accuracy of the quantitative analysis a t the present stage. Figure 2 illustrates the comparative results for the stainless steel samples. The ion microprobe analyses and the chemical analyses are presented along the vertical axis and the horizontal one, respectively. Both axes are graduated in log-scale. The broken lines on the figure are one to one per-
1296
fect correlation lines. Each line is labeled a t 1 wt % concentration. Generally, the IMA results for Mo and Si appear to be too small by about 30% and those for Mn to be too large by about 80%.This systematic deviation is considered to be related to a number of factors, Le., 1) the errors in the physical constant used in the calculation, 2) the correction for the ion detector response depending on the mass, chemical nature of the impinging ions, etc., 3) the effective secondary ion source viewed from the mass analyzer, which has some energy acceptance, and 4) the accuracy of some approximated equations introduced in calculations. Such a systematic error has already been reported in the paper of Andersen and Hinthorne (3). Some factors causing the error have also been mentioned, but it has not clear which factor is most to blame. The source of error is being investigated further.
ANALYTICAL CHEMISTRY, VOL. 47, NO. 8, JULY 1975
CONCLUSION The quantitative calculation utilizing the dissociation of molecules, based on a thermal equilibrium model, has been applied to standard samples of stainless steel. The calculated results from the present approach have shown a generally good agreement with the chemically analyzed concentrations. This prliminary investigation has supported the belief that the present calculation may develop into a useful and powerful technique for quantitative analysis with IMA since no standard samples are required. However, fundamental studies on sputtering, ionization process, and plasma physics are still lacking.
LITERATURE CITED (1) G. K. Wehner, Proc. 6th Int. Vacuum Congr. 1974, Jpn, J. Appl. Phys., Suppl. 2, Pt. 1, 495 (1974). (2) C. A. Andersen. hf.J. Mass Specfrom. /on Phys., 2, 61 (1969). (3) C. A. Andersen and J. R. Hinthorne, Anal. Chem., 45, 1421 (1973). (4) R. Shimizu, T. lshitani, and Y. Ueshima, Jpn. J. Appl. Phys., 13, 249 (1974). (5)P. W. J. M. Boumans, "Theory of Spectrochemical Excitation", Plenum Press, New York, 1966. (6) G. Herzberg, "Molecular Spectra and Molecular Structure. 1, Spectra of Diatomic Molecules", D. van Nostrand. Princeton, NJ, 1950. (7) A. G. Gaydon, "Dissociation Energies and Spectra of Diatomic Molecules", 3rd ed., Chapman and Hall, London, 1968. (8)B. Rosen, "Spectroscopic Data Relative to Diatomic Molecules", Pergamon Press, New York, NY, 1970. (9) J. Roboz, "Introduction to Mass Spectrometry", Interscience Pub,, A Division of John Wiley & Sons, New York, 1968. p 149. (10) H. Tamura, T. Kondo, and H. Doi, "Proc. 6th Int. Conf. X-ray Optics and Microanalysis". G. Shinoda, K. Kohra, and T. Ichinokawa. Ed., Univ. of Tokyo Press, Tokyo, 1972, p 423.
RECEIVEDfor review January 31, 1975. Accepted March 28, 1975.
