- Email: [email protected]
Grain boundary self-diffusion in zinc
GRAIN BOUNDARY SELF-DIFFUSION
IN ZINC*
EDWARD S. WAJDAt The grain boundary self-diffusion in zinc has been measured in the temperature range from 75°C to 299°C with the use of Znfi as a tracer and the usual lathe sectioning technique. Two different high-purity grades of zinc gave an activation energy of 14.5 kcal/mol to within experimental error. It is concluded that the grain boundary self-diffusion activation energy is 37 per cent less than the lattice self-diffusion energy of activation. L’AUTODIFFUSION
DANS
LES
JOINTS
INTERCRISTALLINS
DU ZINC
L’autodiffusion dans les joints intrecristallins du zinc a et6 mesuree dans l’intervalle tures allant de 75°C B 2CWC, en se servant de Zn@ comme indicateur, et en employant usuelle des sections au tour. Deux qualites differentes de zinc de haute purete ont don@ d’activation de 14,5 kcal/mol, .?I l’erreur experimentale prb. On en conclu_t que l’energte pour l’autodiffusion dans les jomts intercristallins est de 37 pour cent mferreur ?I l’energre pour l’autodiffusion dans le r&au. KORNGRENZEN-SELBSTDIFFUSION
de temperala methode une energie d’act!vation d’actrvatron
IN ZINK
Die Korngrenzen-Selbstdiffusion in Zink wurde mittels Zn” als Indikator und der iiblichen mechan&hen Schnittmethode im Temperaturgebiet zwischen 75°C und 2WC gemessen. Zwei verschiedene Grade Reinstzink ergaben innerhalb der experimentellen Fehlergrenze die gleiche Aktivierungsenergie von 14,5 kcal/mol. Es wird daraus geschlossen, dass die Aktivierungsenergie der Korngrenzen-Selbstdiffusion 37y0 geringer ist als die Aktivierungsenergie der Kristallgitter-Selbstdiffusion.
Introduction In a previous paper [l] the rate of self-diffusion in single crystals of zinc was discussed. The rate was found to depend markedly on the direction in which it was measured. In conjunction with those measurements, the grain-boundary self-diffusion has also been investigated. Information on grainboundary diffusion rates promises to be extremely valuable because it appears that the rates of many important physical and metallurgical processes, such as grain growth, recrystallization, plastic deformation and crystal strength, are more directly related to grain boundary than to latticediffusion coefficients. Also, a quantitative relation for Da and D, which are the grain-boundary and lattice diffusion coefficients, respectively, should be valuable in the development and testing of the different proposed models for grain boundaries. Recent work by Hoffman and Turnbull [2] and Slifkin, Lazarus, and Tomizuka [3] has shown quantitatively that the grain-boundary self-diffusion in silver was much larger than the lattice diffusivity in the same metal at lower temperatures. Numerous other workers have also given qualitative verification that Da > D,. It is the purpose of this work to measure the rate of grain-boundary diffusion in an anisotropic crystal such as zinc, and compare it with the rate of self-diffusion through the bulk crystal lattice of *Received September 8, 1953. tRensselaer Polytechnic Institute, Troy, at Union College, Schenectady, New York. ACTA
METALLURGICA,
VOL.
New York;
2, MAR.
1954
now
the same metal. This work also shows the application of Fisher’s theory [4] for diffusion along uniform grain boundaries, to a material where the diffusion is anisotropic.
Experimental
Procedure
Two grades of zinc as obtained from the New Jersey Zinc Company were used. One grade was the chemically pure (C-P) zinc (99.999%) and the other was a commercially prepared zinc known as Horsehead (H-H) which was 99.9901, pure. The (C-P) specimens were cut directly from the cast rods as received from the supplier. The crystallites in these rods displayed a marked radial orientation. The size of the crystallites used was about 0.1 mm X 3 mm. Back reflection X-ray photographs showed that the crystallites had a preferred orientation with respect to the cylinder specimen axis, and were positioned with their hexagonal axes in the radial direction. The (H-H) specimens were made by melting stock (H-H) zinc and casting it into a slug 1 inch in diameter X 4 inches long. This slug was then alternately cold-rolled and annealed until a small grain size was obtained (2 X 1W2 cm mean linear dimension). Small cylindrical specimens were then cut from the cold-worked slug in a manner such that the cylindrical specimen axis was perpendicular to the direction of rolling of the slug. The specimens were then polished and annealed at the highest temperature used in the investigation to minimize grain-boundary migration and crystal growth during the diffusion anneal. The
WAJDA:
GRAIN
BOUNDARY
SELF-DIFFUSION
185
average grain sizes of the two series of specimens were measured on the part of the specimen that remained after sectioning, and in no case was there an appreciable change in grain size during the diffusion process. After radioactive Zna5 was electrodeposited on the specimen surfaces, the samples were diffused at different temperatures and then sectioned in a lathe, and the activities measured in the manner described in reference [l].
Results
and Discussion
As yet no general solution of the diffusion equation for diffusion along uniform grain boundaries has been obtained. However, Fisher [4] has derived an approximate solution appropriate to the region beyond the influence of pure bulk diffusion, which is easily applicable to the interpretation of the experimental data. Roe [5] and Whipple [6] have also considered the same problem in more detail using more rigorous mathematical analysis, but their results are rather complicated and extremely difficult to correlate with experimental data. With the same assumptions outlined in Fisher’s paper [4], one obtains a simple analytical expression for the ratio of the lattice diffusivity to the grain boundary diffusivity as follows:
where a is the isotope activity per unit volume at a distance g measured in a direction normal to the free surface on which the original isotope was deposited and 6 is the thickness of the uniform grain boundary. Using the experimentally measured value of the slope of the activity-penetration curve, together with the known value of the bulk diffusion coefficient obtained from the work of Shirn, Wajda, and Huntington [l] at the same temperature, and assuming a value of 5 X lo-* cm for 6, one can find the absolute value of the grain boundary diffusivity (Db). Typical activity penetration curves are shown in Figures 1, 2, and 3. The vertical lines through some of the points give an indication of the counting errors involved. In Figure 1 is shown penetration data on a polycrystalline sample which was diffused at a high temperature where lattice diffusion is more pronounced than grain-boundary diffusion. When the data are plotted against (g)2, the average diffusion distance squared, a straight line results as is required by the theory of homogenous diffusion, whereas a plot against $ gives a
I 2 3 Distance Squared
FIGURE 1. 1 = 169 hours.
High-temperature
4 5 (to“ cm*)
diffusion:
T = 268.1T;
pronounced curvature. This indicates that the major portion of the diffusion took place through the lattice. Figure 2 shows similar data on a polycrystalline sample which was diffused at a very low temperature, where one would expect the grainboundary diffusion to be more effective than the lattice diffusion. This expectation is verified very nicely because a linear relationship appears when log activity is plotted against Q as predicted by Fisher’s theory, whereas a marked curvature is observed for the log activity versus ($)2. Further-
I FIGURE 2. 501.5 hours.
Distance
Low-temperature
IIF’
cm)
diffusion:
T = 90°C;
t =
186
ACTA
METALLURGICA,
more, one would expect, in a temperature range where both types of diffusion are important, that the experimental data would not agree with either the lattice or grain-boundary diffusion theories. This is seen in Figure 3. Here a plot of log activity
2 20 40 60 Distance Squared
FIGWRE 3. hours,
Apparent
ki IO \ 80 i30 (IO-~ cm’)
diffusion:
T = 157°C;
t = 312.5
TABLE
=
VOL.
2,
against (g)z and g gives a straight line over a limited portion only. This temperature region is termed lattice diffusion region.” It is the “apparent extremely difficult to separate the contributions to diffusion from the lattice and grain-boundary effects in this region. The rate of polycrystalline self-diffusion was measured in the temperature range from 75°C to 200°C and the results are summarized in Table I. The diffusion coefficients calculated in the apparent lattice diffusion region were termed DA when calculated on the basis of the lattice diffusion theory and Db when calculated from Fisher’s grainboundary theory. The average root-mean-square probable error for the diffusion coefficient was 11 per cent. This error arose from many sources, such as weighing of the sections, time of diffusion, sample activity counting, and extrapolated values for lattice diffusion coefficients at low temperatures. Assuming that the temperature dependence of the diffusivity is described by D = Doexp (-Q/RT), the log of the diffusivity I
(OK) X 10-a 1.6420 1.8484 2.075 2.174 2.217 2.217 2.268 2.299 2.326 2.335 2.387 2.392 2.439 2.445 2.481 2.493 2.532 2.597 2.653 2.667 2.674 2.703 2.710 2.755 2.824 2.857
’
Crystal
t (hrs)
1 2 3 20 7 10 21 11 23 6 33 9 32 25 24 18 29 17 27 16 28 13 30 26 31 15
16 169 261 92 314 287.5 92 96 312.5 361.5 69.5 576 166 189 312.5 170.5 337.5 241 355.5 93.5 528 167 404 501.5 959 118
(C-P)
(II-II)
Da (cmz/sec)
x
IO-‘0
19.18 2.00 0.146 0.0916 0.059 0.0614 0.0548 0.0247 0.0176
803 1180 1286 401 183.6 182.8
can be plotted against l/T
=
D, (cm*/sec) X lo-r0
l/T
1954
I
297 70.0 171.6
0.016 64.1
56.1 32.6 33.2 28.3 22.9 10.1 9.02 12.21 7.23 6.69 5.75 3.50 2.53
Dv (cm*/sec) x 10-u
570 170 100 100 56 37 27 43.3 12 12.6 6.4 4.2 3.6 4.4 1.0 0.50 0.43 0.83 0.29 0.55 0.13 0.15 0.043
WAJDA:
GRAIN
BOUNDARY
and the slope and intercept will give, respectively, the activation energy (Q) and the temperatureindependent part of the diffusivity (Do). The plot of the data is shown in Figure 4. A least-square fit was applied to the points and the data can be described by: Do (cmz/sec) Q (kcal/mol)
SELF-DIFFUSION
187
In summary, it can be concluded that the grainboundary activation energy is about 37 per cent less than the lattice activation energy, giving a larger self-diffusivity for the grain boundary than for pure bulk seIf-diffusion in zinc. CEO 5ccJ
(C-P) (99.999%) (H-H) (99.99%)
0.22 0.38
14.3 * 0.2 14.6 f 0.2
$ (OK-' IO-? ‘1 \,\, I6
I8,
2,O
FIGURE 5.
FIGURE 4.
2.4
2.5
LogD vs l/T.
2.6 I .
2.7
= (C-P),
2.6 I +
= (H-H).
The difference between the (C-P) and (H-H) is very small and it is questionable whether any special significance can be attached to this difference beyond experimental error. However, a part of this difference might be attributed to the anisotropy for motion in the grain boundaries, as was found by Couling and Smoluchowski [7]. Figure 5 shows the over-all picture of selfdiffusion in zinc as determined from the experiments in this Laboratory. On the left are the single-crystal data which were reported on in reference [l]. On the right are the grain-boundary data described here. In the reciprocal temperature range (l/T = 2.07 to 2.36) is the apparent lattice diffusion region where both types of diffusion phenomena are prominent, and a marked deviation from both straight-line plots is observed. The diffusivity in this region can be described by: DA = (2 X lO_*) exp (-
$I
2,.6
2,8
Acknowledgement
IO-‘)
f(‘K
23
‘3,;’
Self-diffusion in zinc.
17,50O/RT)
(cm*/sec).
This apparent diffusion coefficient for the lattice is not a unique constant like D, or Da but is a mixed coefficient made up from some average of intragranular and intergranular diffusion.
The author wishes to express his appreciation to Professor Hillard B. Huntington for his enthusiastic direction and encouragement in this project. The preparation of the radioactive plating solution was made with the help of Professor Herbert Clark. A special note of thanks is also extended to Mr. George A. Shirn for his help in making some of the measurements and to Professor experimental Francis T. Worrell for the kind use of the X-ray equipment. The work was performed under the sponsorship of the Atomic Energy Commission, Contract AT-(30-l)-1044.
References 1. SHIRN, G. A., WAJDA, E. S., HUNTINGTON,H. B. Acta Met. 1 (1953) 513. 2. HOFFMAN, R. E., and TURNBULL, D. J. Appl. Phys. 22 (1951) 634. 3. SLIFKIN, L., LAZARUS, D., and TOMIZUKA, T. J. Appl. Phys. 23 (1952) 1032. 4. FISHER, J. C. J. Appl. Phys. 22 (1951) 74. 5. ROE, G. M. Knolls Atomic Laboratory, Schenectady, New York. 6. WHIPPLE, R. T. A.E.R.E. Report T/R 1026 (1952). Ministry of Supply, Harwell, England. 7. COIJLING,L., and SMOLUCHOWSKI,R. Phys. Rev. 91 (1953) 245.
IN ZINC*
EDWARD S. WAJDAt The grain boundary self-diffusion in zinc has been measured in the temperature range from 75°C to 299°C with the use of Znfi as a tracer and the usual lathe sectioning technique. Two different high-purity grades of zinc gave an activation energy of 14.5 kcal/mol to within experimental error. It is concluded that the grain boundary self-diffusion activation energy is 37 per cent less than the lattice self-diffusion energy of activation. L’AUTODIFFUSION
DANS
LES
JOINTS
INTERCRISTALLINS
DU ZINC
L’autodiffusion dans les joints intrecristallins du zinc a et6 mesuree dans l’intervalle tures allant de 75°C B 2CWC, en se servant de Zn@ comme indicateur, et en employant usuelle des sections au tour. Deux qualites differentes de zinc de haute purete ont don@ d’activation de 14,5 kcal/mol, .?I l’erreur experimentale prb. On en conclu_t que l’energte pour l’autodiffusion dans les jomts intercristallins est de 37 pour cent mferreur ?I l’energre pour l’autodiffusion dans le r&au. KORNGRENZEN-SELBSTDIFFUSION
de temperala methode une energie d’act!vation d’actrvatron
IN ZINK
Die Korngrenzen-Selbstdiffusion in Zink wurde mittels Zn” als Indikator und der iiblichen mechan&hen Schnittmethode im Temperaturgebiet zwischen 75°C und 2WC gemessen. Zwei verschiedene Grade Reinstzink ergaben innerhalb der experimentellen Fehlergrenze die gleiche Aktivierungsenergie von 14,5 kcal/mol. Es wird daraus geschlossen, dass die Aktivierungsenergie der Korngrenzen-Selbstdiffusion 37y0 geringer ist als die Aktivierungsenergie der Kristallgitter-Selbstdiffusion.
Introduction In a previous paper [l] the rate of self-diffusion in single crystals of zinc was discussed. The rate was found to depend markedly on the direction in which it was measured. In conjunction with those measurements, the grain-boundary self-diffusion has also been investigated. Information on grainboundary diffusion rates promises to be extremely valuable because it appears that the rates of many important physical and metallurgical processes, such as grain growth, recrystallization, plastic deformation and crystal strength, are more directly related to grain boundary than to latticediffusion coefficients. Also, a quantitative relation for Da and D, which are the grain-boundary and lattice diffusion coefficients, respectively, should be valuable in the development and testing of the different proposed models for grain boundaries. Recent work by Hoffman and Turnbull [2] and Slifkin, Lazarus, and Tomizuka [3] has shown quantitatively that the grain-boundary self-diffusion in silver was much larger than the lattice diffusivity in the same metal at lower temperatures. Numerous other workers have also given qualitative verification that Da > D,. It is the purpose of this work to measure the rate of grain-boundary diffusion in an anisotropic crystal such as zinc, and compare it with the rate of self-diffusion through the bulk crystal lattice of *Received September 8, 1953. tRensselaer Polytechnic Institute, Troy, at Union College, Schenectady, New York. ACTA
METALLURGICA,
VOL.
New York;
2, MAR.
1954
now
the same metal. This work also shows the application of Fisher’s theory [4] for diffusion along uniform grain boundaries, to a material where the diffusion is anisotropic.
Experimental
Procedure
Two grades of zinc as obtained from the New Jersey Zinc Company were used. One grade was the chemically pure (C-P) zinc (99.999%) and the other was a commercially prepared zinc known as Horsehead (H-H) which was 99.9901, pure. The (C-P) specimens were cut directly from the cast rods as received from the supplier. The crystallites in these rods displayed a marked radial orientation. The size of the crystallites used was about 0.1 mm X 3 mm. Back reflection X-ray photographs showed that the crystallites had a preferred orientation with respect to the cylinder specimen axis, and were positioned with their hexagonal axes in the radial direction. The (H-H) specimens were made by melting stock (H-H) zinc and casting it into a slug 1 inch in diameter X 4 inches long. This slug was then alternately cold-rolled and annealed until a small grain size was obtained (2 X 1W2 cm mean linear dimension). Small cylindrical specimens were then cut from the cold-worked slug in a manner such that the cylindrical specimen axis was perpendicular to the direction of rolling of the slug. The specimens were then polished and annealed at the highest temperature used in the investigation to minimize grain-boundary migration and crystal growth during the diffusion anneal. The
WAJDA:
GRAIN
BOUNDARY
SELF-DIFFUSION
185
average grain sizes of the two series of specimens were measured on the part of the specimen that remained after sectioning, and in no case was there an appreciable change in grain size during the diffusion process. After radioactive Zna5 was electrodeposited on the specimen surfaces, the samples were diffused at different temperatures and then sectioned in a lathe, and the activities measured in the manner described in reference [l].
Results
and Discussion
As yet no general solution of the diffusion equation for diffusion along uniform grain boundaries has been obtained. However, Fisher [4] has derived an approximate solution appropriate to the region beyond the influence of pure bulk diffusion, which is easily applicable to the interpretation of the experimental data. Roe [5] and Whipple [6] have also considered the same problem in more detail using more rigorous mathematical analysis, but their results are rather complicated and extremely difficult to correlate with experimental data. With the same assumptions outlined in Fisher’s paper [4], one obtains a simple analytical expression for the ratio of the lattice diffusivity to the grain boundary diffusivity as follows:
where a is the isotope activity per unit volume at a distance g measured in a direction normal to the free surface on which the original isotope was deposited and 6 is the thickness of the uniform grain boundary. Using the experimentally measured value of the slope of the activity-penetration curve, together with the known value of the bulk diffusion coefficient obtained from the work of Shirn, Wajda, and Huntington [l] at the same temperature, and assuming a value of 5 X lo-* cm for 6, one can find the absolute value of the grain boundary diffusivity (Db). Typical activity penetration curves are shown in Figures 1, 2, and 3. The vertical lines through some of the points give an indication of the counting errors involved. In Figure 1 is shown penetration data on a polycrystalline sample which was diffused at a high temperature where lattice diffusion is more pronounced than grain-boundary diffusion. When the data are plotted against (g)2, the average diffusion distance squared, a straight line results as is required by the theory of homogenous diffusion, whereas a plot against $ gives a
I 2 3 Distance Squared
FIGURE 1. 1 = 169 hours.
High-temperature
4 5 (to“ cm*)
diffusion:
T = 268.1T;
pronounced curvature. This indicates that the major portion of the diffusion took place through the lattice. Figure 2 shows similar data on a polycrystalline sample which was diffused at a very low temperature, where one would expect the grainboundary diffusion to be more effective than the lattice diffusion. This expectation is verified very nicely because a linear relationship appears when log activity is plotted against Q as predicted by Fisher’s theory, whereas a marked curvature is observed for the log activity versus ($)2. Further-
I FIGURE 2. 501.5 hours.
Distance
Low-temperature
IIF’
cm)
diffusion:
T = 90°C;
t =
186
ACTA
METALLURGICA,
more, one would expect, in a temperature range where both types of diffusion are important, that the experimental data would not agree with either the lattice or grain-boundary diffusion theories. This is seen in Figure 3. Here a plot of log activity
2 20 40 60 Distance Squared
FIGWRE 3. hours,
Apparent
ki IO \ 80 i30 (IO-~ cm’)
diffusion:
T = 157°C;
t = 312.5
TABLE
=
VOL.
2,
against (g)z and g gives a straight line over a limited portion only. This temperature region is termed lattice diffusion region.” It is the “apparent extremely difficult to separate the contributions to diffusion from the lattice and grain-boundary effects in this region. The rate of polycrystalline self-diffusion was measured in the temperature range from 75°C to 200°C and the results are summarized in Table I. The diffusion coefficients calculated in the apparent lattice diffusion region were termed DA when calculated on the basis of the lattice diffusion theory and Db when calculated from Fisher’s grainboundary theory. The average root-mean-square probable error for the diffusion coefficient was 11 per cent. This error arose from many sources, such as weighing of the sections, time of diffusion, sample activity counting, and extrapolated values for lattice diffusion coefficients at low temperatures. Assuming that the temperature dependence of the diffusivity is described by D = Doexp (-Q/RT), the log of the diffusivity I
(OK) X 10-a 1.6420 1.8484 2.075 2.174 2.217 2.217 2.268 2.299 2.326 2.335 2.387 2.392 2.439 2.445 2.481 2.493 2.532 2.597 2.653 2.667 2.674 2.703 2.710 2.755 2.824 2.857
’
Crystal
t (hrs)
1 2 3 20 7 10 21 11 23 6 33 9 32 25 24 18 29 17 27 16 28 13 30 26 31 15
16 169 261 92 314 287.5 92 96 312.5 361.5 69.5 576 166 189 312.5 170.5 337.5 241 355.5 93.5 528 167 404 501.5 959 118
(C-P)
(II-II)
Da (cmz/sec)
x
IO-‘0
19.18 2.00 0.146 0.0916 0.059 0.0614 0.0548 0.0247 0.0176
803 1180 1286 401 183.6 182.8
can be plotted against l/T
=
D, (cm*/sec) X lo-r0
l/T
1954
I
297 70.0 171.6
0.016 64.1
56.1 32.6 33.2 28.3 22.9 10.1 9.02 12.21 7.23 6.69 5.75 3.50 2.53
Dv (cm*/sec) x 10-u
570 170 100 100 56 37 27 43.3 12 12.6 6.4 4.2 3.6 4.4 1.0 0.50 0.43 0.83 0.29 0.55 0.13 0.15 0.043
WAJDA:
GRAIN
BOUNDARY
and the slope and intercept will give, respectively, the activation energy (Q) and the temperatureindependent part of the diffusivity (Do). The plot of the data is shown in Figure 4. A least-square fit was applied to the points and the data can be described by: Do (cmz/sec) Q (kcal/mol)
SELF-DIFFUSION
187
In summary, it can be concluded that the grainboundary activation energy is about 37 per cent less than the lattice activation energy, giving a larger self-diffusivity for the grain boundary than for pure bulk seIf-diffusion in zinc. CEO 5ccJ
(C-P) (99.999%) (H-H) (99.99%)
0.22 0.38
14.3 * 0.2 14.6 f 0.2
$ (OK-' IO-? ‘1 \,\, I6
I8,
2,O
FIGURE 5.
FIGURE 4.
2.4
2.5
LogD vs l/T.
2.6 I .
2.7
= (C-P),
2.6 I +
= (H-H).
The difference between the (C-P) and (H-H) is very small and it is questionable whether any special significance can be attached to this difference beyond experimental error. However, a part of this difference might be attributed to the anisotropy for motion in the grain boundaries, as was found by Couling and Smoluchowski [7]. Figure 5 shows the over-all picture of selfdiffusion in zinc as determined from the experiments in this Laboratory. On the left are the single-crystal data which were reported on in reference [l]. On the right are the grain-boundary data described here. In the reciprocal temperature range (l/T = 2.07 to 2.36) is the apparent lattice diffusion region where both types of diffusion phenomena are prominent, and a marked deviation from both straight-line plots is observed. The diffusivity in this region can be described by: DA = (2 X lO_*) exp (-
$I
2,.6
2,8
Acknowledgement
IO-‘)
f(‘K
23
‘3,;’
Self-diffusion in zinc.
17,50O/RT)
(cm*/sec).
This apparent diffusion coefficient for the lattice is not a unique constant like D, or Da but is a mixed coefficient made up from some average of intragranular and intergranular diffusion.
The author wishes to express his appreciation to Professor Hillard B. Huntington for his enthusiastic direction and encouragement in this project. The preparation of the radioactive plating solution was made with the help of Professor Herbert Clark. A special note of thanks is also extended to Mr. George A. Shirn for his help in making some of the measurements and to Professor experimental Francis T. Worrell for the kind use of the X-ray equipment. The work was performed under the sponsorship of the Atomic Energy Commission, Contract AT-(30-l)-1044.
References 1. SHIRN, G. A., WAJDA, E. S., HUNTINGTON,H. B. Acta Met. 1 (1953) 513. 2. HOFFMAN, R. E., and TURNBULL, D. J. Appl. Phys. 22 (1951) 634. 3. SLIFKIN, L., LAZARUS, D., and TOMIZUKA, T. J. Appl. Phys. 23 (1952) 1032. 4. FISHER, J. C. J. Appl. Phys. 22 (1951) 74. 5. ROE, G. M. Knolls Atomic Laboratory, Schenectady, New York. 6. WHIPPLE, R. T. A.E.R.E. Report T/R 1026 (1952). Ministry of Supply, Harwell, England. 7. COIJLING,L., and SMOLUCHOWSKI,R. Phys. Rev. 91 (1953) 245.