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Homogeneity of solids: A proposal for quantitative definition
0039-9140/82/070643-03$03.00/O
Julunru. Vol. 29. pp. 643 to 645. 1982 Printed in Great Britain
Pergamon Press Ltd
HOMOGENEITY OF SOLIDS: A PROPOSAL FOR QUANTITATIVE DEFINITION J.
INCZ~DY
Institute of Analytical Chemistry, University of Veszprkm, VeszprCm, Hungary (Received 9 October 1981. Accepted 29 January 1982) Summary-A
simple formula has been deduced to predict the detectability of chemical homogeneity of
solid substances by use of an analytical method with known spatial resolution. The formula includes the ratio of the spatial resolution of the method to the spatial distribution of the component to be determined. and the standard deviation of the method.
Investigations of the chemical uniformity of solid substances are becoming more and more important. Solid raw materials or products used in electronics or in metallurgy must be, in many cases, rigorously analysed, since their chemical homogeneity decisively influences their use and the physical properties of the product. Danzer et al.’ recently published a paper in which, using statistical criteria, they deduced a formula by which the homogeneity of a solid sample could be evaluated. The resolution and the standard deviation of the analytical procedure were included in the formula. Another approach is described here, in which the effect of the spatial resolution of the analytical method on the detectability of inhomogeneity is also included. Principles of homogeneity
The concentration distribution of an element in the surface layer of a solid sample can be readily followed in the direction of the two co-ordinates by a scanning analytical method, assuming that the concentration change in the direction of the co-ordinates is relatively high and the distances between areas of different compositions are easily measurable. Thus, the local concentration of the components in a very small area on the surface can be determined precisely. If, however, the diameter of the spot analysed is comparable with the distance between the areas of high and low concentrations, or the standard deviation of the method is comparable with the difference between the high and low concentrations, reliable information on the homogeneity or the concentration distribution of an element cannot be obtained. For surface-spot analysis many new high-efficiency scanning methods are now available, e.g., laser-generated atomic-emission spectroscopy, electron-beam microanalysis, secondary ion mass spectrometry, etc. Their spatial resolution varies between 1 and 100 pm. It is clear that the extent of information on homogeneity depends on the following factors. 643
(1) The amount or volume of the subsample which is directly analysed. The smaller this is, the higher will be the geometric resolution of the scanning process, and concentration changes over very short distances can be recognized. (2) The statistical error of spot analysis. Concentration differences can be detected only if they are significantly greater than the statistical error of the analysis. (3) The distance between the points of high and low concentrations of the components in the direction of the space co-ordinates. (4) The mean value of the concentration differences of the elements to be determined. in the direction of the three co-ordinates. It is easy to see that four factors are not independent. The higher the geometric resolution and precision of the analytical method, and the greater the distance and concentration difference between the points of low and high concentrations, the more reliable will be the results of the scanning process, or the information on the inhomogeneity of the solid sample. If, however, the geometric spatial resolution is poor, and the standard deviation is high, inhomogeneity of the sample cannot be detected. Deduction of the formula proposed
For calculation of the approximate limits of detection, or determination of inhomogeneity. the following assumptions are made. (1) For simplification the solid sample is considered to be analysed only in the direction of the z co-ordinate. (2) The change in signal caused by a change in concentration of the component of interest, in the direction Z, can be approximated by a sine curve. which can be characterized by an average amplitude and wavelength. The amplitude is denoted by Ax/Z. and the wavelength by &=. (3) The “window” for spot analysis is characterized
644
ANNOTATIONS
I
I
Fig. 1. Concentration distribution of an element in a solid. The resolution of the instrument is high compared to the concentration change. The error is also very low. (AZ ti 6.~; Ax $ s). The vertical lines indicate the uncertainty of the measurement; the standard deviation s can be taken as a fifth of the range of the signals for a given level.*
by a diameter, or, in the direction of z, a linear width which is denoted by 62. (4) The standard deviation of the analytical measurement is denoted by s. This is used in connection with the determination
of signal difference Ax.
The reader may wonder why a sine wave is chosen rather than, for example, square waves of random phase and amplitude. We believe that the wave function selected to describe the concentration profiles does not significantly influence the formula deduced, but the use of the sine wave makes the whole deduction much simpler. In Fig. 1 the concentration change of a component in the direction of the z co-ordinate in a solid sample, and the sine wave used for approximation of the changing signal are shown. The frequency of the change is relatively low compared to the “window” or spatial resolution of the profiling methods, and the amplitude is quite high compared to the standard deviation of the analytical measurement (AZ $ 6z; Ax >>s). This is the most favourable case, where the change of concentration can be determined quite rigorously. Figure 2 presents the case of a solid sample with an inhomogeneity occurring with higher frequency, analysed by a method of low spatial resolution (i.e., with large window width), and higher stan-
dard deviation; it can immediately be seen that the presence of inhomogeneity cannot be verified (62 > ) AZ; s > Ax’). This occurs because the large window acts as a filter (moving window average3) giving the average composition of the substance, integrated over the distance 6z. The signal difference observed will be lower than in the case of high resolution, i.e., the observed amplitude will be lower than the true value (Ax’ < Ax). For a concentration change in the direction of the z co-ordinate to be detected with acceptable confidence, the following condition must be fulfilled: Ax’ > 3s,
i.e., the signal difference observed between the points of low and high concentration should be at least three times the statistical error (or noise) of the measurement. This signal difference is connected with the “window” of the single analytical measurement and with the distance between the points of low and high concentration. The relationship can be expressed easily, if we consider spot analysis as giving an integrated value of the composition of the substance over the distance 6z. Using the transfer function of a boxcar window with a width of 6z, we obtain a’ _=a
sin 7rf6z lrfSZ
(2)
where a is the original amplitude, a’ is the amplitude of the smoothed sine wave, and f is its frequency. Since in this case f = I/AZ and 2a = Ax, 2a’ = Ax’, we can write: Ax’ 2
Ax sin (&Z/AZ) 2 n6zlAz
(3)
from equations (1) and (3) we obtain the formula Ax>
3sxSz/ AZ . sm(Ir6z/Az)
For the presence of inhomogeneity to be detectable equation (4) must be satisfied. Taking into consideration the concentration difference, AC, corresponding to Ax, and the standard deviation in concentration units (s,), the following condition must hold for the verification of homogeneity: Ac(Az, 6z, s,) <
Fig. 2. Concentration distribution of an element in a solid. The spatial resolution of the instrument is low compared to the concentration changes. The error is also considerable. Because of the integrating effect, the signal difference observed will be Ax’ rather than Ax. See caption to Fig. I for note on range = 5s. and on uncertainty of measurement.
(1)
3s,xSz/Az sin (n6z/Az)
(5)
The solid sample appears homogeneous if the concentration differences ACat distances AZ/~ are lower than the quantity on the right-hand side, and the measurements are made with a window 6z and standard deviation s,. A similar formula can be obtained if the Fisher significance test is used instead of the 3s criterion.
AF’NOTATIONS
Thus the system appears homogeneous if $
< F(95)
(6)
where F(95) is the critical value for the F-test, u* is the apparent variance of the changing concentration of the component, and s2 is the square of the standard deviation of the measurements. The variance of the sin or cos function is a2/2, where a is the amplitude. From equation (3):
1
645
the mixture be detected by laser emission spectroscopy? The chromium concentration difference between the particles (between chromium oxide and manganese oxide) is AC = 68 - 0 = 68P;; s, = 68 x 0.05 = 3.4”;; Az = 50pm: 6z = 1OOpm: (A: is the average distance between centres of chromium oxide particles). Using formula (5) we obtain 2x 68 < 3 x 3.4 x sin 2n
2
(7)
From (6) and (7) we get the condition
for homogeneity. The critical value of F(95) for n, = 11~= 20 degrees of freedom (easily obtainable by means of the usual methods) is 2.12.
The mixture appears completely homogeneous. (2) From the same chromium oxide and manganese oxide another perfectly mixed sample is prepared, but with a 1:40 composition. The mixed sample is carefully spread on a glass plate, and chromium is determined in the surface monolayer. Does the mixture appear homogeneous or not? The dilution is to be taken into consideration when obtaining the AC value:
With V’i ,.!2.12 = 4.12, AC instead of Ax, and s, instead of s, condition (8) becomes: Ac(Az, 6z, s,J < 4.12 s, [ si;~;z;;z)]
(9)
If we treat the spatial resolution problem more rigorously, i.e., the real composition change along the z co-ordinate is decomposed into cosine-wave-components (by the Fourier method), we come to the conclusion that the concentration difference is detectable only when the amplitude (AX/~) and frequency (~/AZ) of the components satisfy the conditions given in equation (5) or (9). All other components with higher frequency or lower amplitude contributing to the real spatial distribution will be filtered out and not be observable. Remarks. For the sake of simplicity only onedimensional changes were considered in the deduction of the formula. However, the pattern in the direction of the other co-ordinates can be treated in a similar manner. Examples. (1) A perfectly mixed 1: 1 (particle ratio) mixture of chromium oxide and manganese oxide is examined by laser-generated emission spectroscopy. The average size of the particles of both components is the same, 25 pm. A circular spot with a diameter of ca. 1OOpm can be analysed by the laser-generation system. The relative standard deviation of the determination for chromium is 5%. Can heterogeneity of
AC = ;
- 0 = 1.77;
Since the concentration is much lower than before, the relative standard deviation of the determination will be taken as 10%: s.L = I7. x 0 .10 = O.l7O,’ 10 6; = 1OOpm as before, but AZ is now z 150 pm. Hence 1.7 > 3 x 0.17 x m = 1.2 an 71~3~
The mixture will behave as a heterogeneous one. Conclusions
By taking into consideration the spatial resolution of the analytical method and the spatial distribution of the elements in the surface of solid materials, and using the formula for filtering by moving window average, it is possible to deduce a simple formula for the prediction of homogeneity. REFERENCES
1. K. Danzer. K. Doerffel, H. Ehrhardt. M. Geissler. G. Ehrlich and P. Gadow. Anal. Chim. Acra. 1979. 105, 1. 2. G. H. Hieftje. And Chum., 1972. 44, No. 6. 81A. 3. D. Brinkley and R. Dessy. J. Chm. EdtIc.. 1979. 56, 148.
Julunru. Vol. 29. pp. 643 to 645. 1982 Printed in Great Britain
Pergamon Press Ltd
HOMOGENEITY OF SOLIDS: A PROPOSAL FOR QUANTITATIVE DEFINITION J.
INCZ~DY
Institute of Analytical Chemistry, University of Veszprkm, VeszprCm, Hungary (Received 9 October 1981. Accepted 29 January 1982) Summary-A
simple formula has been deduced to predict the detectability of chemical homogeneity of
solid substances by use of an analytical method with known spatial resolution. The formula includes the ratio of the spatial resolution of the method to the spatial distribution of the component to be determined. and the standard deviation of the method.
Investigations of the chemical uniformity of solid substances are becoming more and more important. Solid raw materials or products used in electronics or in metallurgy must be, in many cases, rigorously analysed, since their chemical homogeneity decisively influences their use and the physical properties of the product. Danzer et al.’ recently published a paper in which, using statistical criteria, they deduced a formula by which the homogeneity of a solid sample could be evaluated. The resolution and the standard deviation of the analytical procedure were included in the formula. Another approach is described here, in which the effect of the spatial resolution of the analytical method on the detectability of inhomogeneity is also included. Principles of homogeneity
The concentration distribution of an element in the surface layer of a solid sample can be readily followed in the direction of the two co-ordinates by a scanning analytical method, assuming that the concentration change in the direction of the co-ordinates is relatively high and the distances between areas of different compositions are easily measurable. Thus, the local concentration of the components in a very small area on the surface can be determined precisely. If, however, the diameter of the spot analysed is comparable with the distance between the areas of high and low concentrations, or the standard deviation of the method is comparable with the difference between the high and low concentrations, reliable information on the homogeneity or the concentration distribution of an element cannot be obtained. For surface-spot analysis many new high-efficiency scanning methods are now available, e.g., laser-generated atomic-emission spectroscopy, electron-beam microanalysis, secondary ion mass spectrometry, etc. Their spatial resolution varies between 1 and 100 pm. It is clear that the extent of information on homogeneity depends on the following factors. 643
(1) The amount or volume of the subsample which is directly analysed. The smaller this is, the higher will be the geometric resolution of the scanning process, and concentration changes over very short distances can be recognized. (2) The statistical error of spot analysis. Concentration differences can be detected only if they are significantly greater than the statistical error of the analysis. (3) The distance between the points of high and low concentrations of the components in the direction of the space co-ordinates. (4) The mean value of the concentration differences of the elements to be determined. in the direction of the three co-ordinates. It is easy to see that four factors are not independent. The higher the geometric resolution and precision of the analytical method, and the greater the distance and concentration difference between the points of low and high concentrations, the more reliable will be the results of the scanning process, or the information on the inhomogeneity of the solid sample. If, however, the geometric spatial resolution is poor, and the standard deviation is high, inhomogeneity of the sample cannot be detected. Deduction of the formula proposed
For calculation of the approximate limits of detection, or determination of inhomogeneity. the following assumptions are made. (1) For simplification the solid sample is considered to be analysed only in the direction of the z co-ordinate. (2) The change in signal caused by a change in concentration of the component of interest, in the direction Z, can be approximated by a sine curve. which can be characterized by an average amplitude and wavelength. The amplitude is denoted by Ax/Z. and the wavelength by &=. (3) The “window” for spot analysis is characterized
644
ANNOTATIONS
I
I
Fig. 1. Concentration distribution of an element in a solid. The resolution of the instrument is high compared to the concentration change. The error is also very low. (AZ ti 6.~; Ax $ s). The vertical lines indicate the uncertainty of the measurement; the standard deviation s can be taken as a fifth of the range of the signals for a given level.*
by a diameter, or, in the direction of z, a linear width which is denoted by 62. (4) The standard deviation of the analytical measurement is denoted by s. This is used in connection with the determination
of signal difference Ax.
The reader may wonder why a sine wave is chosen rather than, for example, square waves of random phase and amplitude. We believe that the wave function selected to describe the concentration profiles does not significantly influence the formula deduced, but the use of the sine wave makes the whole deduction much simpler. In Fig. 1 the concentration change of a component in the direction of the z co-ordinate in a solid sample, and the sine wave used for approximation of the changing signal are shown. The frequency of the change is relatively low compared to the “window” or spatial resolution of the profiling methods, and the amplitude is quite high compared to the standard deviation of the analytical measurement (AZ $ 6z; Ax >>s). This is the most favourable case, where the change of concentration can be determined quite rigorously. Figure 2 presents the case of a solid sample with an inhomogeneity occurring with higher frequency, analysed by a method of low spatial resolution (i.e., with large window width), and higher stan-
dard deviation; it can immediately be seen that the presence of inhomogeneity cannot be verified (62 > ) AZ; s > Ax’). This occurs because the large window acts as a filter (moving window average3) giving the average composition of the substance, integrated over the distance 6z. The signal difference observed will be lower than in the case of high resolution, i.e., the observed amplitude will be lower than the true value (Ax’ < Ax). For a concentration change in the direction of the z co-ordinate to be detected with acceptable confidence, the following condition must be fulfilled: Ax’ > 3s,
i.e., the signal difference observed between the points of low and high concentration should be at least three times the statistical error (or noise) of the measurement. This signal difference is connected with the “window” of the single analytical measurement and with the distance between the points of low and high concentration. The relationship can be expressed easily, if we consider spot analysis as giving an integrated value of the composition of the substance over the distance 6z. Using the transfer function of a boxcar window with a width of 6z, we obtain a’ _=a
sin 7rf6z lrfSZ
(2)
where a is the original amplitude, a’ is the amplitude of the smoothed sine wave, and f is its frequency. Since in this case f = I/AZ and 2a = Ax, 2a’ = Ax’, we can write: Ax’ 2
Ax sin (&Z/AZ) 2 n6zlAz
(3)
from equations (1) and (3) we obtain the formula Ax>
3sxSz/ AZ . sm(Ir6z/Az)
For the presence of inhomogeneity to be detectable equation (4) must be satisfied. Taking into consideration the concentration difference, AC, corresponding to Ax, and the standard deviation in concentration units (s,), the following condition must hold for the verification of homogeneity: Ac(Az, 6z, s,) <
Fig. 2. Concentration distribution of an element in a solid. The spatial resolution of the instrument is low compared to the concentration changes. The error is also considerable. Because of the integrating effect, the signal difference observed will be Ax’ rather than Ax. See caption to Fig. I for note on range = 5s. and on uncertainty of measurement.
(1)
3s,xSz/Az sin (n6z/Az)
(5)
The solid sample appears homogeneous if the concentration differences ACat distances AZ/~ are lower than the quantity on the right-hand side, and the measurements are made with a window 6z and standard deviation s,. A similar formula can be obtained if the Fisher significance test is used instead of the 3s criterion.
AF’NOTATIONS
Thus the system appears homogeneous if $
< F(95)
(6)
where F(95) is the critical value for the F-test, u* is the apparent variance of the changing concentration of the component, and s2 is the square of the standard deviation of the measurements. The variance of the sin or cos function is a2/2, where a is the amplitude. From equation (3):
1
645
the mixture be detected by laser emission spectroscopy? The chromium concentration difference between the particles (between chromium oxide and manganese oxide) is AC = 68 - 0 = 68P;; s, = 68 x 0.05 = 3.4”;; Az = 50pm: 6z = 1OOpm: (A: is the average distance between centres of chromium oxide particles). Using formula (5) we obtain 2x 68 < 3 x 3.4 x sin 2n
2
(7)
From (6) and (7) we get the condition
for homogeneity. The critical value of F(95) for n, = 11~= 20 degrees of freedom (easily obtainable by means of the usual methods) is 2.12.
The mixture appears completely homogeneous. (2) From the same chromium oxide and manganese oxide another perfectly mixed sample is prepared, but with a 1:40 composition. The mixed sample is carefully spread on a glass plate, and chromium is determined in the surface monolayer. Does the mixture appear homogeneous or not? The dilution is to be taken into consideration when obtaining the AC value:
With V’i ,.!2.12 = 4.12, AC instead of Ax, and s, instead of s, condition (8) becomes: Ac(Az, 6z, s,J < 4.12 s, [ si;~;z;;z)]
(9)
If we treat the spatial resolution problem more rigorously, i.e., the real composition change along the z co-ordinate is decomposed into cosine-wave-components (by the Fourier method), we come to the conclusion that the concentration difference is detectable only when the amplitude (AX/~) and frequency (~/AZ) of the components satisfy the conditions given in equation (5) or (9). All other components with higher frequency or lower amplitude contributing to the real spatial distribution will be filtered out and not be observable. Remarks. For the sake of simplicity only onedimensional changes were considered in the deduction of the formula. However, the pattern in the direction of the other co-ordinates can be treated in a similar manner. Examples. (1) A perfectly mixed 1: 1 (particle ratio) mixture of chromium oxide and manganese oxide is examined by laser-generated emission spectroscopy. The average size of the particles of both components is the same, 25 pm. A circular spot with a diameter of ca. 1OOpm can be analysed by the laser-generation system. The relative standard deviation of the determination for chromium is 5%. Can heterogeneity of
AC = ;
- 0 = 1.77;
Since the concentration is much lower than before, the relative standard deviation of the determination will be taken as 10%: s.L = I7. x 0 .10 = O.l7O,’ 10 6; = 1OOpm as before, but AZ is now z 150 pm. Hence 1.7 > 3 x 0.17 x m = 1.2 an 71~3~
The mixture will behave as a heterogeneous one. Conclusions
By taking into consideration the spatial resolution of the analytical method and the spatial distribution of the elements in the surface of solid materials, and using the formula for filtering by moving window average, it is possible to deduce a simple formula for the prediction of homogeneity. REFERENCES
1. K. Danzer. K. Doerffel, H. Ehrhardt. M. Geissler. G. Ehrlich and P. Gadow. Anal. Chim. Acra. 1979. 105, 1. 2. G. H. Hieftje. And Chum., 1972. 44, No. 6. 81A. 3. D. Brinkley and R. Dessy. J. Chm. EdtIc.. 1979. 56, 148.