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Neutron irradiation of gray tin
J. Phys. Chem. &lids
Pergamon Press 1960. Vol. 16. pp. 46-52.
NEUTRON
Printed in Great Britain.
IRRADIATION A. N.
Brookhaven (Received
GOLAND?
National Laboratory, 4 January
OF GRAY TIN*
1960; revised
Upton, New York 18 May
1960)
Abstract-Gray tin was irradiated at 0°C in a reactor in an effort to produce white tin as a result of localized heating caused by thermal spikes. The irradiated powder was compared to an unirradiated specimen by means of X-ray studies at low temperatures. No significant change was found in the amount of white tin present as a result of irradiation. The implications of this result for the thermalspike concept are discussed and various explanations for the experimental results are offered.
INTRODUCTION IN STUDIES of radiation
damage in solids, the concept of the thermal spike has been employed frequently to explain a variety of observed effects. In fact, there is now strong evidence that spikes are produced during the neutron irradiation of materials containing fissionable elements.(l* 29 3) However, the energy released during the fission process is large compared to the energy of the primary knock-on atoms which are produced during similar irradiation of nonfissionable materials. Therefore, it has proved to be more difficult to demonstrate conclusively that the thermal spike phenomenon occurs during the irradiation of the latter materials. A radiation-induced phase change provides a sensitive mechanism for testing the spike hypothesis, and several investigators have performed experiments relying upon this fact.(495,@ It is the purpose of the present note to report on another experiment of this kind, this one involving the transformation of gray (a-) tin to white (p-) tin (the gray phase is stable below 13,2”(Z). In performing an experiment of this kind the question of what the irradiation temperature ought to be arises
immediately.
perature
is
temperature a reactor: nitrogen
Since
13.2”C ranges, low
boiling
the transformation
one
might
all of which
temperatures point),
consider
temthree
are attainable
(around
intermediate
the
in
liquid
temperatures
* Work performed under the auspices of the U.S. Atomic Energy Commission. t Guest Scientist from Ordnance Materials Research Office, Watertown 72, Massachusetts.
(- 30°C to -60”(Z), and temperatures near the phase transition. The low temperature irradiation is a poor choice because the thermal spike may not heat the sample above the transition temperature even if this temperature is depressed momentarily because of its pressure dependence. In order to choose between the two remaining temperature ranges one must consider not only the transformation temperature, but the temperature dependence of the white-to-gray tin growth rate as well. It has been well established for this phase transition,(T, 8) that the growth rate decreases rapidly as the transformation temperature is approached. This is a consequence of the decrease in the free-energy difference between the two phases, this difference being the driving force for the transformation. Therefore, as the temperature is reduced below 13.2”C, the driving force increases. At the same time, however, the temperature becomes low compared to the heat of activation, thereby tending to decrease the growth rate. These two competing processes lead to a bell-shaped curve of growth rate vs. temperature which has a maximum occurring at approximately -30°C. Thus, if the sample were irradiated in the intermediate temperature range, then, upon cooling, the temperature of the transformed region would be near that at which the white-to-gray transformation rate is a maximum and reconversion would be favored. Finally, let US consider the temperature range around 0°C. This temperature is close enough to the transition temperature that a considerable volume of the sample will be heated above 13’2°C by a thermal spike. 46
NEUTRON
IRRADIATION
Furthermore, the transformation rate at 0°C is so low that a great deal of white tin should be retained for a long period of time following its formation. These arguments suggest that an irradiation of gray tin near the transition temperature should offer the best opportunity for successfully observing thermal-spike production in this material. Accordingly, the experiment was carried out at 0°C. The details will be described in the discussion which follows.
OF
GRAY
TIN
47
(3) a spherical spike with TO = 273°K. The curves in Fig. 3 suggest that if one assumes a sphere radius of 200i%then in a flux of 1011 nV about 1 per cent of the volume of a gray tin sample will transform to white tin during a one day irradiation. It
1
I CYLINDRICAL 5 i: Y
600
y
500
THERMAL
SPIKE
IN GRAY
TIN
E. :I Me” To =BO”K
w
THEORY (1) Thermal spike Descriptions of the thermal-spike process have appeared in works by BROOKS(~) SEITZ and KOEHLER@) and DIENES and VINEYARD( among otheis. Following the latter authors, one can calculate the volume fraction of a gray tin sample whose temperature would be raised momentarily above the a++/3 transition temperature by thermal spikes during neutron irradiation. Such a calculation necessarily is based upon a number of assumptions, perhaps the primary one being that the lifetime of a thermal spike is long enough to permit a phase change to take place. As an approximation to reactor irradiation, it is assumed that the neutrons incident upon the sample possess an effective energy of 1 MeV. It is postulated further that a spherical spike is produced by each primary knock-on tin atom; the energy in each spike is taken to be 16 keV, the mean kinetic energy transferred to a tin atom by a 1 MeV neutron. As suggested by the aforementioned authors only the lattice thermal conductivity of the u-tin is considered to be significant when one is investigating the thermal-spike process. Although no data are available for a-tin, one may estimate its thermal conductivity from the data for germanium and silicon;(lz) doing so, one finds that a value of 0.1 cal see-lcm-1 “C-1 is probably a good approximation in the temperature range of interest. This leads to a thermal diffusion constant much greater than that of a metal such as copper, for example. This in turn results in a thermal spike which spreads more rapidly than in copper to a larger region than it would have occupied in that metal. Figures 1, 2 and 3 illustrate these points for three cases: (1) a cylindrical spike in a sample whose ambient temperature, TO, is SOoK, (2) a spherical spike with TO = 80”K, and
FIG. 1. Temperature distributions in a cylindrical thermal spike in gray tin at various times for an initial temperature of 80°K.The thermal diffusion coefficient was taken to be 0.46 cm2/sec., and the energy released per unit path length was assumed to be 0.17 keV/A.
700
: d Y
HERICAL
TtiERMAL
SPIKE
IN
GRAY
/
I
100
150
200
TIN
600
I
I
50
RADIUS.
I
r, IN ANGSTROMS
FIG. 2. Temperature distributions in a spherical thermal spike in gray tin at various times for an initial temperature of 80°K. The thermal diffusion coefficient was taken to be 0.46 cm2/sec, and the energy released at the origin at time t = 0 was assumed to be 16 keV. is worth emphasizing
at this point, that the value of the thermal conductivity which has been used in these calculations is by no means incontestable. BRooKs(13),
for instance,
that
the thermal
might
be closer
to that
of a liquid,
a spike
of longer
duration
conductivity leading estimated the case,
to
in the present then
suggests
calculations.
the temperature
than
that
If this were
of the spike
region
A. N. GOLAND
48
would be close to the transition temperature for a longer period of time than was originally estimated. Consequently, the probability of reversion to gray tin during the final stages of the spike would be even smaller than it is at 0°C. SEITZ and
KOEHLER(~@,on the other hand, have suggested that it may not be proper to equate the spike region with that of a true liquid. At any rate, a longerlived spike than that predicted here would be all to the good for the purposes of this experiment.
from that of p-tin, cr-tin is most readily obtainable in powder form. If white tin were to be produced during neutron irradiation of gray tin powder, one method of detecting the fact would be to compare X-ray diffraction patterns from unirradiated and irradiated powder samples. The theory of quantitative analysis of powder mixtures by means of X-ray diffraction analysis has been developed and verified by a number of investigators,(14) and it may be applied directly to the present problem. We are dealing with the very special case of a binary mixture of allotropic forms of tin. Consequently, the mass absorption coefficients of the separate components of the mixture are equal. This leads to a simplified expression for the diffracted intensity of a reflection from the ith component of the mixture, namely,
(1) 50
100 150 RADIUS. r. IN L\NGSTROMS
200
FIG. 3. Temperature distributions in a spherical thermal spike in gray tin at various times for an initial temperature of 273°K. The thermal diffusion coefficient was taken to be 0.46 cm2/sec, and the energy released per unit path length was assumed to be 0.17 keV/A.
The volume change which accompanies the phase change from gray to white tin is of the order of 27 per cent; consequently, the transition temperature possesses a marked pressure dependence. The normal transformation temperature is 13.2”C, and the Clausius-Clapeyron equation indicates that the pressure dependence of the temperature is -0+06”C atm-1. This means that when a rapidly heated region of gray tin is prevented from expanding by a surrounding rigid gray tin matrix, it stands a good chance of undergoing a transformation to white tin, provided that the initial temperature of the region is not too low. Thus, the heating caused by the thermal spike accompanied by the reduction in transition temperature combine to make the tin system a favorable one in which to search for evidence of thermal spikes. (2) Quantitative
analysis by means of X-ray
diffrac-
tion techniques
Because the density
of cc-tin differs so greatly
In equation (l), Gi depends upon the geometry of the sample, the wavelength of the X-rays and some physical constants; j is the multiplicity of the reflection under consideration; M equals the number
of unit cells per unit volume of the it”
component; F is the crystal structure factor; U’(8) is the Lorentz-polarization factor; Xi is the weight fraction of the ith component and pa its density, and p is the mass absorption coefficient. Two important pieces of information may be derived from the application of this equation to the diffraction pattern of such a binary powder mixture. First, if one forms the ratio of intensities of two reflections from the two components in the same sample, one obtains the expression,
After one obtains the experimental data which appear in equation (2), the only unknown quantity in this ratio is Xi which then may be calculated readily. Second, if the ratio of corresponding reflections from two different patterns (say, 1 and 2) is formed, the result is, (Ii) 1
(X,)1
(I$)2
(X,)2
---=-
(3)
This ratio enables one to detect changes in Xa,
NEUTRON
IRRADIATION
while equation (2) provides the absolute value of &. Therefore, equations (2) and (3) may be used jointly as a check on the internal consistency of the X-ray results. EXPERIMENT
In essence, the experiment consisted of determining by means of X-rays the relative amounts of white and gray tin in a powder sample before and after reactor irradiation. The tin used in this experiment was obtained from the Varlacoid Chemical Company, New York. It had the designation “ultra pure”, and its purity was reputed to be greater than 99.999 per cent, the two detectable impurities being lead (0.00002 per cent) and iron (0.00001 per cent). In order to produce a sample suitable for X-ray powder work an ingot of white tin was placed in contact with a gray tin single crystal, and stored in a deep-freeze at -45°C. Such “seeding” was sufficient to nucleate the transformation from white to gray tin in a matter of hours. The single crystal was removed after the transformation had been initiated leaving the pure tin in the low-temperature phase without contamination. Following several months of storage at -45”C, the powder was transferred to a cold room maintained at 1 f 1°C where it remained throughout the experimental work. Gray tin produced in this manner was sifted through a 325 mesh sieve, and the powder that passed through the sieve was used to make the X-ray samples. At the time the experiment was initiated, there was still approximately 10 per cent white tin in the powder. The irradiation was carried out in hole W-l 3 of the Brookhaven graphite reactor for 82 hr. The thermal flux in this hole at the time was approximately 7.2 x 1012 neutrons cm%ec-1 as determined by activation of a foil composed of an aluminummanganese alloy; the. cadmium ratio was found to be 31.4. From these numbers, one can calculate the resonance flux in any energy range El to Ez for which ln(Ez/El) = l.(u) Since the aforementioned thermal-spike calculation was based upon an effective neutron energy of 1 MeV, it is pertinent to state that the flux in the energy range from 0.55 MeV to 1.5 MeV was of the order of 4.8~ 1011 neutrons cm2 set-1 during the irradiation. An 82 hr exposure in this flux should have caused roughly D
OF
GRAY
TIN
49
16 per cent of the sample volume to transform from gray to white tin by virtue of the thermalspike mechanism, exclusive of the effect of pressure upon the transformation temperature. During the irradiation the sample temperature was maintained at 0 _t 2°C by allowing dry carbon dioxide gas under high pressure to expand into the sample chamber through a fine capillary. Immediately upon removal from the reactor, the powder was placed in a bath of ice and water, and kept there until the radioactivity had decreased to a safe level for handling. At this time, the powder was returned to the cold storage room previously mentioned where the X-ray powder samples were prepared. The basic X-ray equipment consisted of a standard Norelco powder-diffraction unit, and a scintillation counter for detection of the copper Kcc radiation. To this was added a cryostat attachment especially designed to maintain the sample temperature close to that of liquid nitrogen while a diffraction pattern was being recorded. Each sample was prepared and positioned in the cryostat while in the cold room, all equipment and materials being in equilibrium with the room temperature. The cryostat then was packed in dry ice and transferred to the spectrometer. Finally, liquid nitrogen was transferred into the cryostat, and a diffraction pattern obtained. Afterwards, the procedure was reversed in returning the sample to storage. Two samples were employed in the experiment, the powder for each originating in the same batch. One sample was irradiated in the manner just described, while the other was kept as a control in the cold room. Consequently, throughout the study the two sample temperatures never differed from one another by more than 2”, except for the period during which the diffraction patterns were recorded.
RESULTS Diffraction data were recorded for scattering angles in the interval from 22” to 90”. Over this range three white tin and eight gray tin reflections were found to be useful in the analysis. Straightforward application of the theory of quantitative analysis by means of X-ray diffraction measurements as described earlier, lead to the results listed in Tables 1 and 2.
50
A.
N.
GOLAND
Table 1. A comparison of corresponding lines in the irradiated and u&radiated change in weight fraction of either phase. .--____.-._. -__ .__~___._____
samples to determine the
Reflection -
,--
---
Xt (unirradiated) Ratio: Xi (irradiated)
__ -_________~ ~__The mean value of the ratio X, (unirr.)/Xar(irrad.) The mean value of Xe (unirr.)/Xs(irrad.)
.-~~
Table 2. A determination of the absolute value of X, ___-___ Reflection Ratio Unirradiated
Ratio Irradiated
1 ! fl(020)/a(lll) I
i
I I
j3(020)/a(lll)
in each sample. ~____
I /
1 fi(Oll):(220) -__
0.101
43
_I
-
-.
was 0.991, and the standard deviation was 5.61 per cent. was 0.988 and the standard deviation was 8.1 per cent.
0.108 ,P
0.102
____
~(Oll)/c((220) ___-
/?(2ll):‘a(311)
;
,B(211)‘(311)z
0.0917 ( 0.109 0.105 XP I .___--. __ __._ __-____ -____~. The mean value of X,s before irradiation was 0.104 with a standard deviation of 0.38 per cent; the corresponding numbers for the irradiated sample were 0.102 and 0.90 per cent.
DISCUSSION
On the basis of the thermal spike calculation alone, one would have expected at least a twofold increase in X, following the irradiation given the powder in this experiment. However, the results in Table 1 indicate that no significant change occurred in this quantity. This is substantiated by the lack of change in X,, as is also shown in Table 1. The results listed in Table 2 are further evidence that no change in the relative amounts of gray and white tin occurred; they provide an excellent check on the inner consistency of the data. As previously stated, the experiment was carried out at approximately O’Cfor two important reasons. The first reason was that the closer the initial temperature of the sample was to the CM$ transition temperature, the greater was the likelihood that an appreciable fraction of the sample would have been heated above the transition temperature by a spike. The second reason was that the linear rate of transformation from @a tin is lower at 0°C than at the temperature where its value is a maximum, by
almost three orders of magnitude.(i) Therefore, any white tin which was produced during the irradiation would have had little chance of reconverting to the gray phase. With regard to this point, although extensive investigations have not been carried out by the author to verify the fact, other investigators@) indicate that the isothermal transformation rate decreases as a function of time at any given temperature. In the work presented here, X-ray studies of powder which had been stored at - 30°C showed that the amount of white tin in the powder approached a constant value even though the P-FCC tin transformation rate is a maximum at that temperature. Other investigators also have expressed the opinion that some white tin always will be present in the gray matrix;(*) the exact amount, however, undoubtedly depends upon the purity of the tin. By way of a more quantitative discussion of the reversion of the white tin formed by the spikes, one can rnake an estimate of the time required at 0°C for 50 per cent completion of this transformation as follows.
NEUTRON
IRRADIATION
The data of FLEEMANand DIENES(~@indicate that at -30°C it takes approximately 500 min for 50 per cent completion of the transformation. It then follows from the temperature dependence of the rate of transformation,t7) that at 0°C the equivalent time would be approximately 4 x 10s min or about 10 months. Allowing for the time during which the sample was stored at 0°C while its radioactivity diminished, one can estimate that at most ten per cent of the white tin reconverted. The only other opportunity for the white tin to reconvert would have occurred while the X-ray sample was being cooled to liquid nitrogen temperature in the cryostat. That reconversion was negligible during this short period was verified by repetition of the X-ray measurements on the same sample at liquid nitrogen temperature after the sample temperature had been allowed to reach 0°C between runs. During this process, the sample temperature passed through the maximum in the transformation rate three times. Nevertheless, the new mean value of the ratio shown in Table 1 as calculated from the second set of data was well within the indicated standard deviation, showing that the cooling-down process did not contribute significantly to reconversion. These considerations lead to the conclusion that most of the white tin produced during the experiment would have been detected readily. CONCLUSIONS Several explanations may be offered for the apparent failure to observe the effects of thermal spikes in this experiment, and at present one would find it difficult to lay emphasis upon any one in particular. To begin with, examination of the assumptions which are implicit in a thermal spike calculation raises some questions about the validity of such a calculation. One must realize that it is based upon thermodynamic considerations which, therefore, imply the existence of equilibrium conditions. However, the times involved are so short that one wonders if such conditions can prevail, and if the concept of temperature has any meaning during a thermal spike process. Even if thermodynamics may be applied legitimately to this problem, the times involved might have been too short to permit the transformation fram gray to white tin to occur. Alternatively, the spike temperature may have been too low to induce the
OF
GRAY
TIN
51
transition; the temperatures derived in the calculations, after all, are only approximations to the true state of affairs. Of course, one cannot rule out the possibility that new regions of white tin are produced by a spike mechanism but that they immediately reconvert to gray tin. By envisioning a lattice in which both crystal structures coexist, one can see that distortions will be present;(l7) perhaps the new white tin finds itself in a strain field which it cannot support, and so it reconverts. One would have difficulty in verifying this fact experimentally. It is extremely doubtful that the production of lattice defects during the irradiation, and their subsequent annealing could have enabled reconversion to occur rapidly. If this were the case one would be led to expect some evidence of enhanced diffusion as well. Quite to the contrary, however, the difference in the amount of white tin present before and after the irradiation was insignificant. In summarizing, perhaps it would be instructive to mention specifically the previous experiments whose aim also has been the detection of spikes by means of radiation induced phase changes. Some of these have involved fissionable materials, and rightfully should be placed in a separate category. Of those in which non-fissionable matter was employed, four deserve mention, those of DENNEY@)(Fe, --f Fr) and similar experiments by BOLTAX( GONSERand OKKERSE(@ (GaSb solid-+ liquid), and WITTELS and SHEnmLL(rg)(ZrOs monoclinic-tcubic). Positive results were reported in the first three experiments while the fourth reported in the negative. One has difficulty in comparing these experiments with the present one because of differences in the materials investigated. Iron being metallic and zirconia, ionic, one might expect some differences between them and gray tin. There should be a greater similarity between GaSb and a-Sn, but here the conclusions are in opposition. The disordering of ordered alloys by radiation has been reported frequently, and this effect often has been attributed to the production of thermal spikes. However, it seems more likely to the author that replacement collisions(ss~ 21) may account for the entire process. The threshold for a replacement collision is much lower than that for a displacement at least in certain crystallographic directions, and this might account for the sensitivity of ordered
A.
52
N.
GOLAND
alloys to radiation. The only incontestable statement one can make is that more evidence must be accumulated before the question of thermal spike production in nonfissionable matter is settled to the satisfaction of all interested investigators.
Acknowledgements-The author would like to express his gratitude to D. T. KEATINGfor giving so generously of his time and X-ray equipment so that this study might be pursued. He would also like to acknowledge helpful discussions with G. J. DIENESand G. H. VINEYARD.The irrradiations could not have been performed without the assistance of J. J. FLOYD, F. P. REEVE,W. J. FEDRICH and G. J. HUMMER.
REFERENCES 1. BLEIBERGM. L., JONESL. J. and LUSTMAN B., /. Appl. Phys. 27,127O (1956). 2. KONOBEEVSKY S. T.,J. Nuclear Energy 1,356 (1956). 3. WITTELS M. C. and SHERRILLF. _4., J. Appl. Phys. 27, 643 (1956). 4. TUCKER C. W. and SENIO P., J. Appl. Phys. 27,207 (1956). 5. DENNEY J. M., Phys. Rev. 94,1417 (1954). 6. GONSER A. and OKKER~EB., Phys. Rev. 105, 757 (1957).
BECKERJ. H., J. Appl. Pkys. 29, 1110-21 (1958).
ii: BURGERSW. G. and GROEN L. J., Disc. Faraday Sot. No. 23, 183 (1957). 9. BROOKSH., Ann. Rev. Nuclear Sci. 6,215-76 (1956). 10. SEITZ F. and KOEHLER J. S., Solid State Physics (edited by SEITZ F. and TURNBULL D.) Vol. II, p. 351. Academic Press, New York (1956). 11. DIENESG. J. and VINEYARDG. H., Radiation Eflects in Solids, p. 32. Interscience, New York (1957). 12. WHITE G. K. and WOODS S. B., Phys. Rev. 103, 569-71 (1956). 13. BROOKSH.‘, private communication. 14. KLUC H. P. and ALEXANDERL. E.. X-rav Diffraction Procedures, p. 410. John Wiley,‘New kork(1954). 15. See, for example, HUGHESD. J., Pile Neutron Research, pp. 62-64. Addison-Wesley Publishing Co., Cambridge, Mass. (1953). 16. FLEEMANJ. and DIENES G. J., J. AppZ. Phys. 26, 652-54 (1955). 17. PRASAD S. C. and WOOSTER W. A., Acta Cryst., Cati. 9,35 (1956). 18. BOLTAX A., Bull. Amer. Phys. Sot. (ser. 2) 4, 136 (1959). 19. WITTELS M. C. and SHERRILL F. A., Phys. Rev. Letters 3,176-7 (1959). 20. KINCHIN G. H. and PEASER. S., Reports on the Progress of Physics Vol. XVIII p. 1. (1955). 21. VINEYARDG. H., GIBSON J. B., GOLAND A. N. and MILGRAM M., J. Appl. Pkys. 30, 1322 (1959); Bull. Amer. Ph$js. Sot. (ser. 2) 5, 26 (1960).
Pergamon Press 1960. Vol. 16. pp. 46-52.
NEUTRON
Printed in Great Britain.
IRRADIATION A. N.
Brookhaven (Received
GOLAND?
National Laboratory, 4 January
OF GRAY TIN*
1960; revised
Upton, New York 18 May
1960)
Abstract-Gray tin was irradiated at 0°C in a reactor in an effort to produce white tin as a result of localized heating caused by thermal spikes. The irradiated powder was compared to an unirradiated specimen by means of X-ray studies at low temperatures. No significant change was found in the amount of white tin present as a result of irradiation. The implications of this result for the thermalspike concept are discussed and various explanations for the experimental results are offered.
INTRODUCTION IN STUDIES of radiation
damage in solids, the concept of the thermal spike has been employed frequently to explain a variety of observed effects. In fact, there is now strong evidence that spikes are produced during the neutron irradiation of materials containing fissionable elements.(l* 29 3) However, the energy released during the fission process is large compared to the energy of the primary knock-on atoms which are produced during similar irradiation of nonfissionable materials. Therefore, it has proved to be more difficult to demonstrate conclusively that the thermal spike phenomenon occurs during the irradiation of the latter materials. A radiation-induced phase change provides a sensitive mechanism for testing the spike hypothesis, and several investigators have performed experiments relying upon this fact.(495,@ It is the purpose of the present note to report on another experiment of this kind, this one involving the transformation of gray (a-) tin to white (p-) tin (the gray phase is stable below 13,2”(Z). In performing an experiment of this kind the question of what the irradiation temperature ought to be arises
immediately.
perature
is
temperature a reactor: nitrogen
Since
13.2”C ranges, low
boiling
the transformation
one
might
all of which
temperatures point),
consider
temthree
are attainable
(around
intermediate
the
in
liquid
temperatures
* Work performed under the auspices of the U.S. Atomic Energy Commission. t Guest Scientist from Ordnance Materials Research Office, Watertown 72, Massachusetts.
(- 30°C to -60”(Z), and temperatures near the phase transition. The low temperature irradiation is a poor choice because the thermal spike may not heat the sample above the transition temperature even if this temperature is depressed momentarily because of its pressure dependence. In order to choose between the two remaining temperature ranges one must consider not only the transformation temperature, but the temperature dependence of the white-to-gray tin growth rate as well. It has been well established for this phase transition,(T, 8) that the growth rate decreases rapidly as the transformation temperature is approached. This is a consequence of the decrease in the free-energy difference between the two phases, this difference being the driving force for the transformation. Therefore, as the temperature is reduced below 13.2”C, the driving force increases. At the same time, however, the temperature becomes low compared to the heat of activation, thereby tending to decrease the growth rate. These two competing processes lead to a bell-shaped curve of growth rate vs. temperature which has a maximum occurring at approximately -30°C. Thus, if the sample were irradiated in the intermediate temperature range, then, upon cooling, the temperature of the transformed region would be near that at which the white-to-gray transformation rate is a maximum and reconversion would be favored. Finally, let US consider the temperature range around 0°C. This temperature is close enough to the transition temperature that a considerable volume of the sample will be heated above 13’2°C by a thermal spike. 46
NEUTRON
IRRADIATION
Furthermore, the transformation rate at 0°C is so low that a great deal of white tin should be retained for a long period of time following its formation. These arguments suggest that an irradiation of gray tin near the transition temperature should offer the best opportunity for successfully observing thermal-spike production in this material. Accordingly, the experiment was carried out at 0°C. The details will be described in the discussion which follows.
OF
GRAY
TIN
47
(3) a spherical spike with TO = 273°K. The curves in Fig. 3 suggest that if one assumes a sphere radius of 200i%then in a flux of 1011 nV about 1 per cent of the volume of a gray tin sample will transform to white tin during a one day irradiation. It
1
I CYLINDRICAL 5 i: Y
600
y
500
THERMAL
SPIKE
IN GRAY
TIN
E. :I Me” To =BO”K
w
THEORY (1) Thermal spike Descriptions of the thermal-spike process have appeared in works by BROOKS(~) SEITZ and KOEHLER@) and DIENES and VINEYARD( among otheis. Following the latter authors, one can calculate the volume fraction of a gray tin sample whose temperature would be raised momentarily above the a++/3 transition temperature by thermal spikes during neutron irradiation. Such a calculation necessarily is based upon a number of assumptions, perhaps the primary one being that the lifetime of a thermal spike is long enough to permit a phase change to take place. As an approximation to reactor irradiation, it is assumed that the neutrons incident upon the sample possess an effective energy of 1 MeV. It is postulated further that a spherical spike is produced by each primary knock-on tin atom; the energy in each spike is taken to be 16 keV, the mean kinetic energy transferred to a tin atom by a 1 MeV neutron. As suggested by the aforementioned authors only the lattice thermal conductivity of the u-tin is considered to be significant when one is investigating the thermal-spike process. Although no data are available for a-tin, one may estimate its thermal conductivity from the data for germanium and silicon;(lz) doing so, one finds that a value of 0.1 cal see-lcm-1 “C-1 is probably a good approximation in the temperature range of interest. This leads to a thermal diffusion constant much greater than that of a metal such as copper, for example. This in turn results in a thermal spike which spreads more rapidly than in copper to a larger region than it would have occupied in that metal. Figures 1, 2 and 3 illustrate these points for three cases: (1) a cylindrical spike in a sample whose ambient temperature, TO, is SOoK, (2) a spherical spike with TO = 80”K, and
FIG. 1. Temperature distributions in a cylindrical thermal spike in gray tin at various times for an initial temperature of 80°K.The thermal diffusion coefficient was taken to be 0.46 cm2/sec., and the energy released per unit path length was assumed to be 0.17 keV/A.
700
: d Y
HERICAL
TtiERMAL
SPIKE
IN
GRAY
/
I
100
150
200
TIN
600
I
I
50
RADIUS.
I
r, IN ANGSTROMS
FIG. 2. Temperature distributions in a spherical thermal spike in gray tin at various times for an initial temperature of 80°K. The thermal diffusion coefficient was taken to be 0.46 cm2/sec, and the energy released at the origin at time t = 0 was assumed to be 16 keV. is worth emphasizing
at this point, that the value of the thermal conductivity which has been used in these calculations is by no means incontestable. BRooKs(13),
for instance,
that
the thermal
might
be closer
to that
of a liquid,
a spike
of longer
duration
conductivity leading estimated the case,
to
in the present then
suggests
calculations.
the temperature
than
that
If this were
of the spike
region
A. N. GOLAND
48
would be close to the transition temperature for a longer period of time than was originally estimated. Consequently, the probability of reversion to gray tin during the final stages of the spike would be even smaller than it is at 0°C. SEITZ and
KOEHLER(~@,on the other hand, have suggested that it may not be proper to equate the spike region with that of a true liquid. At any rate, a longerlived spike than that predicted here would be all to the good for the purposes of this experiment.
from that of p-tin, cr-tin is most readily obtainable in powder form. If white tin were to be produced during neutron irradiation of gray tin powder, one method of detecting the fact would be to compare X-ray diffraction patterns from unirradiated and irradiated powder samples. The theory of quantitative analysis of powder mixtures by means of X-ray diffraction analysis has been developed and verified by a number of investigators,(14) and it may be applied directly to the present problem. We are dealing with the very special case of a binary mixture of allotropic forms of tin. Consequently, the mass absorption coefficients of the separate components of the mixture are equal. This leads to a simplified expression for the diffracted intensity of a reflection from the ith component of the mixture, namely,
(1) 50
100 150 RADIUS. r. IN L\NGSTROMS
200
FIG. 3. Temperature distributions in a spherical thermal spike in gray tin at various times for an initial temperature of 273°K. The thermal diffusion coefficient was taken to be 0.46 cm2/sec, and the energy released per unit path length was assumed to be 0.17 keV/A.
The volume change which accompanies the phase change from gray to white tin is of the order of 27 per cent; consequently, the transition temperature possesses a marked pressure dependence. The normal transformation temperature is 13.2”C, and the Clausius-Clapeyron equation indicates that the pressure dependence of the temperature is -0+06”C atm-1. This means that when a rapidly heated region of gray tin is prevented from expanding by a surrounding rigid gray tin matrix, it stands a good chance of undergoing a transformation to white tin, provided that the initial temperature of the region is not too low. Thus, the heating caused by the thermal spike accompanied by the reduction in transition temperature combine to make the tin system a favorable one in which to search for evidence of thermal spikes. (2) Quantitative
analysis by means of X-ray
diffrac-
tion techniques
Because the density
of cc-tin differs so greatly
In equation (l), Gi depends upon the geometry of the sample, the wavelength of the X-rays and some physical constants; j is the multiplicity of the reflection under consideration; M equals the number
of unit cells per unit volume of the it”
component; F is the crystal structure factor; U’(8) is the Lorentz-polarization factor; Xi is the weight fraction of the ith component and pa its density, and p is the mass absorption coefficient. Two important pieces of information may be derived from the application of this equation to the diffraction pattern of such a binary powder mixture. First, if one forms the ratio of intensities of two reflections from the two components in the same sample, one obtains the expression,
After one obtains the experimental data which appear in equation (2), the only unknown quantity in this ratio is Xi which then may be calculated readily. Second, if the ratio of corresponding reflections from two different patterns (say, 1 and 2) is formed, the result is, (Ii) 1
(X,)1
(I$)2
(X,)2
---=-
(3)
This ratio enables one to detect changes in Xa,
NEUTRON
IRRADIATION
while equation (2) provides the absolute value of &. Therefore, equations (2) and (3) may be used jointly as a check on the internal consistency of the X-ray results. EXPERIMENT
In essence, the experiment consisted of determining by means of X-rays the relative amounts of white and gray tin in a powder sample before and after reactor irradiation. The tin used in this experiment was obtained from the Varlacoid Chemical Company, New York. It had the designation “ultra pure”, and its purity was reputed to be greater than 99.999 per cent, the two detectable impurities being lead (0.00002 per cent) and iron (0.00001 per cent). In order to produce a sample suitable for X-ray powder work an ingot of white tin was placed in contact with a gray tin single crystal, and stored in a deep-freeze at -45°C. Such “seeding” was sufficient to nucleate the transformation from white to gray tin in a matter of hours. The single crystal was removed after the transformation had been initiated leaving the pure tin in the low-temperature phase without contamination. Following several months of storage at -45”C, the powder was transferred to a cold room maintained at 1 f 1°C where it remained throughout the experimental work. Gray tin produced in this manner was sifted through a 325 mesh sieve, and the powder that passed through the sieve was used to make the X-ray samples. At the time the experiment was initiated, there was still approximately 10 per cent white tin in the powder. The irradiation was carried out in hole W-l 3 of the Brookhaven graphite reactor for 82 hr. The thermal flux in this hole at the time was approximately 7.2 x 1012 neutrons cm%ec-1 as determined by activation of a foil composed of an aluminummanganese alloy; the. cadmium ratio was found to be 31.4. From these numbers, one can calculate the resonance flux in any energy range El to Ez for which ln(Ez/El) = l.(u) Since the aforementioned thermal-spike calculation was based upon an effective neutron energy of 1 MeV, it is pertinent to state that the flux in the energy range from 0.55 MeV to 1.5 MeV was of the order of 4.8~ 1011 neutrons cm2 set-1 during the irradiation. An 82 hr exposure in this flux should have caused roughly D
OF
GRAY
TIN
49
16 per cent of the sample volume to transform from gray to white tin by virtue of the thermalspike mechanism, exclusive of the effect of pressure upon the transformation temperature. During the irradiation the sample temperature was maintained at 0 _t 2°C by allowing dry carbon dioxide gas under high pressure to expand into the sample chamber through a fine capillary. Immediately upon removal from the reactor, the powder was placed in a bath of ice and water, and kept there until the radioactivity had decreased to a safe level for handling. At this time, the powder was returned to the cold storage room previously mentioned where the X-ray powder samples were prepared. The basic X-ray equipment consisted of a standard Norelco powder-diffraction unit, and a scintillation counter for detection of the copper Kcc radiation. To this was added a cryostat attachment especially designed to maintain the sample temperature close to that of liquid nitrogen while a diffraction pattern was being recorded. Each sample was prepared and positioned in the cryostat while in the cold room, all equipment and materials being in equilibrium with the room temperature. The cryostat then was packed in dry ice and transferred to the spectrometer. Finally, liquid nitrogen was transferred into the cryostat, and a diffraction pattern obtained. Afterwards, the procedure was reversed in returning the sample to storage. Two samples were employed in the experiment, the powder for each originating in the same batch. One sample was irradiated in the manner just described, while the other was kept as a control in the cold room. Consequently, throughout the study the two sample temperatures never differed from one another by more than 2”, except for the period during which the diffraction patterns were recorded.
RESULTS Diffraction data were recorded for scattering angles in the interval from 22” to 90”. Over this range three white tin and eight gray tin reflections were found to be useful in the analysis. Straightforward application of the theory of quantitative analysis by means of X-ray diffraction measurements as described earlier, lead to the results listed in Tables 1 and 2.
50
A.
N.
GOLAND
Table 1. A comparison of corresponding lines in the irradiated and u&radiated change in weight fraction of either phase. .--____.-._. -__ .__~___._____
samples to determine the
Reflection -
,--
---
Xt (unirradiated) Ratio: Xi (irradiated)
__ -_________~ ~__The mean value of the ratio X, (unirr.)/Xar(irrad.) The mean value of Xe (unirr.)/Xs(irrad.)
.-~~
Table 2. A determination of the absolute value of X, ___-___ Reflection Ratio Unirradiated
Ratio Irradiated
1 ! fl(020)/a(lll) I
i
I I
j3(020)/a(lll)
in each sample. ~____
I /
1 fi(Oll):(220) -__
0.101
43
_I
-
-.
was 0.991, and the standard deviation was 5.61 per cent. was 0.988 and the standard deviation was 8.1 per cent.
0.108 ,P
0.102
____
~(Oll)/c((220) ___-
/?(2ll):‘a(311)
;
,B(211)‘(311)z
0.0917 ( 0.109 0.105 XP I .___--. __ __._ __-____ -____~. The mean value of X,s before irradiation was 0.104 with a standard deviation of 0.38 per cent; the corresponding numbers for the irradiated sample were 0.102 and 0.90 per cent.
DISCUSSION
On the basis of the thermal spike calculation alone, one would have expected at least a twofold increase in X, following the irradiation given the powder in this experiment. However, the results in Table 1 indicate that no significant change occurred in this quantity. This is substantiated by the lack of change in X,, as is also shown in Table 1. The results listed in Table 2 are further evidence that no change in the relative amounts of gray and white tin occurred; they provide an excellent check on the inner consistency of the data. As previously stated, the experiment was carried out at approximately O’Cfor two important reasons. The first reason was that the closer the initial temperature of the sample was to the CM$ transition temperature, the greater was the likelihood that an appreciable fraction of the sample would have been heated above the transition temperature by a spike. The second reason was that the linear rate of transformation from @a tin is lower at 0°C than at the temperature where its value is a maximum, by
almost three orders of magnitude.(i) Therefore, any white tin which was produced during the irradiation would have had little chance of reconverting to the gray phase. With regard to this point, although extensive investigations have not been carried out by the author to verify the fact, other investigators@) indicate that the isothermal transformation rate decreases as a function of time at any given temperature. In the work presented here, X-ray studies of powder which had been stored at - 30°C showed that the amount of white tin in the powder approached a constant value even though the P-FCC tin transformation rate is a maximum at that temperature. Other investigators also have expressed the opinion that some white tin always will be present in the gray matrix;(*) the exact amount, however, undoubtedly depends upon the purity of the tin. By way of a more quantitative discussion of the reversion of the white tin formed by the spikes, one can rnake an estimate of the time required at 0°C for 50 per cent completion of this transformation as follows.
NEUTRON
IRRADIATION
The data of FLEEMANand DIENES(~@indicate that at -30°C it takes approximately 500 min for 50 per cent completion of the transformation. It then follows from the temperature dependence of the rate of transformation,t7) that at 0°C the equivalent time would be approximately 4 x 10s min or about 10 months. Allowing for the time during which the sample was stored at 0°C while its radioactivity diminished, one can estimate that at most ten per cent of the white tin reconverted. The only other opportunity for the white tin to reconvert would have occurred while the X-ray sample was being cooled to liquid nitrogen temperature in the cryostat. That reconversion was negligible during this short period was verified by repetition of the X-ray measurements on the same sample at liquid nitrogen temperature after the sample temperature had been allowed to reach 0°C between runs. During this process, the sample temperature passed through the maximum in the transformation rate three times. Nevertheless, the new mean value of the ratio shown in Table 1 as calculated from the second set of data was well within the indicated standard deviation, showing that the cooling-down process did not contribute significantly to reconversion. These considerations lead to the conclusion that most of the white tin produced during the experiment would have been detected readily. CONCLUSIONS Several explanations may be offered for the apparent failure to observe the effects of thermal spikes in this experiment, and at present one would find it difficult to lay emphasis upon any one in particular. To begin with, examination of the assumptions which are implicit in a thermal spike calculation raises some questions about the validity of such a calculation. One must realize that it is based upon thermodynamic considerations which, therefore, imply the existence of equilibrium conditions. However, the times involved are so short that one wonders if such conditions can prevail, and if the concept of temperature has any meaning during a thermal spike process. Even if thermodynamics may be applied legitimately to this problem, the times involved might have been too short to permit the transformation fram gray to white tin to occur. Alternatively, the spike temperature may have been too low to induce the
OF
GRAY
TIN
51
transition; the temperatures derived in the calculations, after all, are only approximations to the true state of affairs. Of course, one cannot rule out the possibility that new regions of white tin are produced by a spike mechanism but that they immediately reconvert to gray tin. By envisioning a lattice in which both crystal structures coexist, one can see that distortions will be present;(l7) perhaps the new white tin finds itself in a strain field which it cannot support, and so it reconverts. One would have difficulty in verifying this fact experimentally. It is extremely doubtful that the production of lattice defects during the irradiation, and their subsequent annealing could have enabled reconversion to occur rapidly. If this were the case one would be led to expect some evidence of enhanced diffusion as well. Quite to the contrary, however, the difference in the amount of white tin present before and after the irradiation was insignificant. In summarizing, perhaps it would be instructive to mention specifically the previous experiments whose aim also has been the detection of spikes by means of radiation induced phase changes. Some of these have involved fissionable materials, and rightfully should be placed in a separate category. Of those in which non-fissionable matter was employed, four deserve mention, those of DENNEY@)(Fe, --f Fr) and similar experiments by BOLTAX( GONSERand OKKERSE(@ (GaSb solid-+ liquid), and WITTELS and SHEnmLL(rg)(ZrOs monoclinic-tcubic). Positive results were reported in the first three experiments while the fourth reported in the negative. One has difficulty in comparing these experiments with the present one because of differences in the materials investigated. Iron being metallic and zirconia, ionic, one might expect some differences between them and gray tin. There should be a greater similarity between GaSb and a-Sn, but here the conclusions are in opposition. The disordering of ordered alloys by radiation has been reported frequently, and this effect often has been attributed to the production of thermal spikes. However, it seems more likely to the author that replacement collisions(ss~ 21) may account for the entire process. The threshold for a replacement collision is much lower than that for a displacement at least in certain crystallographic directions, and this might account for the sensitivity of ordered
A.
52
N.
GOLAND
alloys to radiation. The only incontestable statement one can make is that more evidence must be accumulated before the question of thermal spike production in nonfissionable matter is settled to the satisfaction of all interested investigators.
Acknowledgements-The author would like to express his gratitude to D. T. KEATINGfor giving so generously of his time and X-ray equipment so that this study might be pursued. He would also like to acknowledge helpful discussions with G. J. DIENESand G. H. VINEYARD.The irrradiations could not have been performed without the assistance of J. J. FLOYD, F. P. REEVE,W. J. FEDRICH and G. J. HUMMER.
REFERENCES 1. BLEIBERGM. L., JONESL. J. and LUSTMAN B., /. Appl. Phys. 27,127O (1956). 2. KONOBEEVSKY S. T.,J. Nuclear Energy 1,356 (1956). 3. WITTELS M. C. and SHERRILLF. _4., J. Appl. Phys. 27, 643 (1956). 4. TUCKER C. W. and SENIO P., J. Appl. Phys. 27,207 (1956). 5. DENNEY J. M., Phys. Rev. 94,1417 (1954). 6. GONSER A. and OKKER~EB., Phys. Rev. 105, 757 (1957).
BECKERJ. H., J. Appl. Pkys. 29, 1110-21 (1958).
ii: BURGERSW. G. and GROEN L. J., Disc. Faraday Sot. No. 23, 183 (1957). 9. BROOKSH., Ann. Rev. Nuclear Sci. 6,215-76 (1956). 10. SEITZ F. and KOEHLER J. S., Solid State Physics (edited by SEITZ F. and TURNBULL D.) Vol. II, p. 351. Academic Press, New York (1956). 11. DIENESG. J. and VINEYARDG. H., Radiation Eflects in Solids, p. 32. Interscience, New York (1957). 12. WHITE G. K. and WOODS S. B., Phys. Rev. 103, 569-71 (1956). 13. BROOKSH.‘, private communication. 14. KLUC H. P. and ALEXANDERL. E.. X-rav Diffraction Procedures, p. 410. John Wiley,‘New kork(1954). 15. See, for example, HUGHESD. J., Pile Neutron Research, pp. 62-64. Addison-Wesley Publishing Co., Cambridge, Mass. (1953). 16. FLEEMANJ. and DIENES G. J., J. AppZ. Phys. 26, 652-54 (1955). 17. PRASAD S. C. and WOOSTER W. A., Acta Cryst., Cati. 9,35 (1956). 18. BOLTAX A., Bull. Amer. Phys. Sot. (ser. 2) 4, 136 (1959). 19. WITTELS M. C. and SHERRILL F. A., Phys. Rev. Letters 3,176-7 (1959). 20. KINCHIN G. H. and PEASER. S., Reports on the Progress of Physics Vol. XVIII p. 1. (1955). 21. VINEYARDG. H., GIBSON J. B., GOLAND A. N. and MILGRAM M., J. Appl. Pkys. 30, 1322 (1959); Bull. Amer. Ph$js. Sot. (ser. 2) 5, 26 (1960).