- Email: [email protected]
A technique for growing helium crystals in preferred orientations
An apparatus has been developed in which hcp He4 single crystals o f high quafity are grown with a 0.3 probability of obtaining c-axis orientations o f 0 or 90 ° with respect to the direction o f growth. Methods of influencing the relative distribution of crystals between these two angles have been observed and are believed to depend on the anisotropy in the thermal conductivity of hcp He 4. A simple computer simulation of the nucleation process supports our identification.of the orientation angles and our explanation of the observed orientation preferences Some possible applications of similar techniques have been surveyed.
A technique for growing helium crystals in preferred orientations D. T. Lawson
Helium solidifies at very low temperatures and only under high pressure. Measurements of the properties of solid helium therefore must be made with the sample remaining in the container in which it was formed. If large single crystals of a certain orientation are required the problem is further complicated. Some experiments (for example, x-ray or neutron scattering) allow reorientation of the sample container relative to the measurement direction. Others, however, (for example, thermal conductivity, first and second sound propagation, light scattering) are limited by the orientation of the crystal with respect to its container. We wished to undertake a series of thermal conductivit 7 measurements of high quality single crystals of hcp He'~ containing various small amounts of isotopic impurity. The thermal conductivity of hcp He4 is highly anisotropic in the temperature and density region of interest for this experiment. 1 Thus it was necessary to grow from the several solutions crystals which shared a common c-axis orientation with respect to the axis of our sample chamber. Without a sample chamber which produced crystals preferentially in certain orientations the growth of a prohibitively large number of crystals would have been required.
The apparatus It was decided to attempt to nucleate the crystals on a surface which had a periodicity comparable to the lattice dimensions of the helium, in the hope that microscopic
The author is with Department ot Physics, Duke University. Durham, North Carolina, USA. Received 6 November 1972.
276
matching would tend to favour a relatively small number of discrete orientations in the macroscopic samples. We are grateful to Prof H. Meyer for bringing to out attention the discovery, by Barrett, L. Meyer, and Wasserman, 2 that the presence of a specially prepared gold foil favoured the formation of solid hydrogen and deuterium in the fcc rather than the normal hcp phase. Prof Meyer suggested that such foils might be useful in the growth of oriented helium crystals and kindly provided us with a piece of gold foil prepared by L. Meyer and coworkers. The surface of this foil closely approximated the [100] face of a single crystal of fcc gold. Its preparation j had included a 98%reduction in thickness by rolling in one direction followed immediately by an overnight annealing at approximately 5 K below the melting temperature. It was the nucleation of helium crystallites on the surface of this foil which allowed us to influence the orientation of our large single crystals. Our sample chamber, shown in Fig. 1, was a stainless steel tube with an inside diameter of 2.72 ram. At its upper end were a heater and the pressure capillary inlet. The three 0.254 mm thick copper fins labelled B, C, and D deFmed the portion of the sample used in the thermal conductivity measurements. At the lower end of the chamber was an ofhc copper piece, including a nucleation post, which was soldered to the refrigerator with Wood's metal. The gold foilwas soft soldered to the top of the nucleation post, which was .1.9 mm in diameter and extended 7.6 mm into the sample volume to a point 5.1 mm below fin B. Doped germanium resistance thermometers were located on the top and bottom pieces of the sample chamber and on each of the fins. A small He4 pot was used as a growth refrigerator and was pumped through a micrometer needle valve.
CRYOGENICS. MAY 1973
There is a large anisotropy in the thermal conductivity o f h c p helium, first reported by Guyer and Hogan. 1 For temperatures at which umklapp processes are dominant
Frail c a p i l l a r y
K"1 (T, 0) = tc"1 (T, 0) cos20 + K"1 (T, 90 °) sin20
(1)
where 0 is the angle between the hcp c-axis and the direction of heat flow. 4 Thus, once the extrema K(T, 0) and K(T, 90 °) are known for a given density, 0 can be determined from a measurement of umklapp-dominated thermal conductivity. 5
D(
Fig.2 shows the incidence of various values of thermal conductivity, measured at T "1 = 0.65 K "l , among 67 samples grown early in our experiments. (We have tabulated conductivities measured over the B - C , C - D , and B - D regions of each sample and smoothed the histogram by averaging with a A In r = 0.03 window). The large anistropy is evident in the range of values of K represented, while the peaks indicate that certain orientations are strongly favoured by our combination of apparatus and growth technique. The fact that these peaks are due to the influence of the nucleation foil was shown several times experimentally when the preferred orientation behaviour disappeared but was readily restored by measures designed to remove chemical impurities from the foil surface.
C(
- -
Nucleation
post
.3
Fig.1 The sample chamber. B, C, D, E, and 3 are d o p e d germanium resistance thermometers. The b o t t o m of the chamber is thermally anchored to the refrigerator, while the t o p is thermally isolated f r o m other points in the cryostat by a 125 cm length of 0.1524 mm id copper--nickel capillary
We identify the outermost peaks (at In K = - 3 . 2 8 and - 2 . 0 5 for T "1 = 0.65) as the 0 = 0 and 90 ° orientations. Although such an assignment can never be certain in the absence of an independent measure of 0, three separate considerations support our values as the true extrema. 1. Our extrema lie very near, but slightly outside those observed by Hogan, Guyer, and Fairbank 6 at the same density. Those authors observed no strong orientation preference in a sample chamber essentially identical to ours except that their nucleation post was solid copper and terminated in a sharp point. 2. Our extreme data fit the expression
Crystal growth All of the crystals discussed in this paper were grown at a constant pressure of 85.1 atm (1 arm = 101.3 kNm'2), the corresponding melting temperature being 3.125 K. At first the bottom of the sample chamber was cooled to about 3.3 K. An hour later growth began, always with the same setting of the growth refrigerator's regulating valve (I .20 turns). As soon as the bottom of the sample chamber had passed through a predetermined temperature a heater was used to gently return it to that temperature and regulate it there for 2 . 5 - 3 hours. The regulation temperature was chosen so that during this annealing period the solid-liquid interface would advance slowly to a position just below fin B. Growth was resumed at one of two rates which caused the interface to pass through the 5.08 cm region between fins B and D in about 100 or about 200 minutes, respectively. Ala input of 5 x 10 -6 watts was always maintained to the heater at the top of the sample chamber to ensure that the pressure capillary would not be blocked until the chamber was filled with solid. The passage of a liquid-solid interface could be easily detected at each fin as an abrupt change in dR/dt for the thermometric resistor.
CRYOGENICS
. MAY
1973
K(T) = Ae°/bT quite well and, in particular, form curves which are quite straight when In K is plotted against T "l , as would be expected for the true extrema. 7 3. A simple computer simulation of the nucleation process at the surface of our foil indicated strong preferences for the 0 = 0 and 90 ° orientations (Fig.3c). This computer model will be discussed in more detail in another section.
~- 3 . 2 8
-2.05
1
I-4
-3.3 -3,2 -3.1 - 3 0 - 2 9 - 2 . 8 -2.7 -2.6 -2.5 -24 -2.3 -2.2-2.1 -2.0 In'M,(r'l=0.651,
Wcm "mK-a
Fig.2 The incidence of 67 early samples as a f u n c t i o n of 1 n to(T) w i t h T ] = 0.65. The data have been sorted i n t o A In K = 0.01 bins and smoothed w i t h a A In K = 0.03 w i n d o w
277
Fig.3a shows the data of Fig.2 replotted as a function of 0, using equation 1 and our identification of the extrema. (A more restrictive subset of the same data - including only those samples for which the B - C and C - D region conductivities agree to the precision of the measurements exhibits peaks at the same angles. By the time study of these first 67 samples had been completed our growth techniques had been developed to the point that almost all our samples were single crystals f'filing the entire B - D region and thus fulfdling this more stringent test). After these first 67 samples, measurements were made only on 0 = 0 and 0 = 90 ° crystals, which occurred with a combined probability of about 0.3. In all, over 140 crystals were grown in the course of these experiments. Near the end of the preliminary studies which produced these data it was discovered that slight variations in the growth technique, such as a more abrupt than normal initiation of the annealing period, could substantially increase the relative likelyhood of obtaining a 0 TM 90 °
,~,-
o o
o
o
o
o
o
o
•
~oe
•
o
•
o
2 o
d=2-88 e.4'O7 f=5'76 9 =6"43
I$1 = 3 . 5 0 I~1 = 5 . 7 2
a
b
•
•
oo ~e
o
•
•
•
e 0 oe
o
c
Fig.4 Structural matching a the structure of 85 atmosphere hcp He4. The solid-line vectors generate the hexagonal space lattice with the two-atom basis indicated by the broken-line vector b the [100] plane structure of fcc gold c superposition of the [100] plane o f fcc gold and the hcp basal plane of 85 atmosphere He 4 (open circles represent He4 lattice sites, filled circles represent fcc potential minima -- see text)
crystal vis ~ vis 0 ~ O. As an indication of this effect we show, in Fig.3b, the data of Fig.3a minus those samples grown using the '90 ° optimization' techniques, which will be discussed in more detail below.
4
Evaluation O
Fig.4a indicates the structure of hcp He4. The c/a ratio is known to remain quite close to the ideal value 1.633. 8 The structure of the [100] plane of fcc gold is drawn in Fig.4b. Our preferred orientation growth is believed to occur in the following way. Many crystallites are nucleated on the surface of the gold foil. Because of microscopic matching between the structures of the solid helium and the foil surface, a few specific orientations will occur with high relative probabilities among these crystallites. In the course of growth and annealing a single orientation predominates and a single crystal interface then proceeds along the sample chamber for the duration of growth. Those orientations favoured microscopically in the nucleation process will be correspondingly more likely to appear in the macroscopic single crystals. Thus some of the structure of the microscopic orientation distribution at nucleation can be inferred from the orientation distribution of the observed macroscopic crystals. However other factors, such as the anistropy of heat transport in the bulk solid, also influence the observed statistics.
a
18 o
5"
16
"O
D c
12
In order to survey the structural matchings between the solid helium and the foil surface a computer programme, using a hard sphere model, was designed to simulate the nucleation of hcp crystallites on an fcc substrate. The operation of this programme is outlined in an appendix below. A histogram showing the incidence of 1994 simulated crystallites as a function of 0 appears in Fig.3c. O
IO C
20
30
40
50
60
70
80
90
e
Fig.3 Incidence as a function of c-axis orientation. All data sorted into A O = 1° bins and smoothed with a ,%8 = 5 ° w i n d o w a same data shown in Fig.2 b same data as in a, but with samples grown with '90 ° optimization' techniques not included c results of a 1994-crystallite computer simulation o f the (microscopic) nucleation process
278
The most striking feature of the simulation data is the prediction that many 0 = 0 and 0 = 90 ° crystallites will be nucleated, in agreement with our interpretation of the observed macroscopic incidence. The peak at about 0 = 54 ° in the simulation data also corresponds to a peak in the experimental histograms. Other correlations, if any, are much less obvious. In passing, however, we may note that any component of the experimental incidence which was truly random in orientation would exhibit a sin20
CRYOGENICS
. MAY
1973
6.0
dependence. The over-all effect of such a contribution would be roughly to raise the incidence baseline for angles near the 90 ° end of our experimental plots.
F Ii° '/2 f I I
Although our data do not include anything approaching a polycrystalline sample, 5 we note that a polycrystalline average 1 over 0 would yield a value for K equal to that for a single 0 = 53 ° crystal.
k =
V
V7
0 0 0 121 o
j I
Ii s.o
The microscopic matching which favours the 0 = 90 ° orientation may be seen easily by comparing Figs 4a and 4b. The hcp c-axis constant is a very close match to twice the fcc nearest-neighbour distance. The matching which favours nucleation of 0 = 0 crystallites is not so obvious. Consider the comparison, in Fig.4c, of the [100] plane fcc gold structure with the hcp basal plane structure of He 4 at the density of our experiments. The first helium atom to become bound on a vacant area of substrate is quite likely to do so at one of the fcc potential minima. When a second helium atom becomes bound to the first, an fcc nearest-neighbour pair provides a potential well for the two-atom system. When a third atom becomes bound to the first two, the foil provides a stable potential minimum for the system, with atoms at positions marked 1, 2, and 3 in Fig.4c. At least four more atoms may become bound to these three in the same hcp basal plane without serious opposition to the substrate potential. The effect of the '90 ° optimization' techniques can be explained in terms of the relative numbers of 0 and 90 ° crystallites at nucleation and the anisotropy in the thermal conductivity near the melting temperature. Although 0 = 90 ° crystals have a much higher conductivity at temperatures near the thermal conductivity peak, near the melting temperature they have a conductivity which is only about half that of 0 = 0 crystals (see Fig.5). During periods of sustained growth, then, a 0 = 0 crystallite will have a competitive advantage over an otherwise identical crystallite oriented at 90 °.
IOO
o
V
L
O0
V
o
V
g
o
ro
E u
V
121
I =
L
4.0
0 0 0
Q
V v o
[043
.J
,,
3.o .oL L
0
@i
& &
He 3 - _ _ He 4 ~
2.0,_._,, ,, , fcc hcp bcc
0
O0
i
i
t
,
,
,
,
,
,
1
,
,
i
i
i
NiCuPtAIAuAcjFeCrNbLi BeCoZnCdMg fcc bcc hop Moteriol
Fig.6 Some structural matches between metal lattices and solid helium. Ranges are indicated at the left for various characteristic dimensions in solid He 3 and He 4. Symbols f o r the characteristic dimensions in the metals are • a or 2a, ~ a~¢/2,~ a ~ , o aV/3/2 or a~/3, and t= c
The simulation data indicate that more 90 ° than 0 ° crystallites (874/432, see Fig.3c) are formed at nucleation. The anisotropy effect, favouring the predomination of 0 = 0 crystals during sustained growth, tends to balance this advantage. Our '90 ° optimization' techniques, by minimizing the period of sustained growth preceeding the predomination of a single 0, largely remove the balancing influence, resulting in a macroscopic incidence more favourable to 0 = 90 ° crystals (Fig.3a, b).
IO Other applications The technique we have outlined consists of the use of partial, imperfect matchings between the substrate and the sample lattice to produce an ensemble of microscopic seed crystallites whose number density is strongly peaked for certain orientation angles. An appropriate growth technique then produces single macroscopic crystals from this ensemble. The technique was crucial to the success of a study requiring the reproduction of a highly anisotropic thermal conductivity. It may be of use in other experiments as well.
TE u
io.i
~
E
~
=
O
~
O.OI O.OOI ' ' 0.3 05
. .0.7 . . . 0.9
I1'. ' I.'3 ' I.'5 ' T-a K-i
I17
Fig.5 Thermal c o n d u c t i v i t y anisotropy in hcp He 4 grown at a constant pressure of 8 5 a t m . The high temperature region i s a p p r o x i mate, being based on an extrapolation of equation 2 using the parameters determined f r o m lower temperature data 6
CRYOGENICS . MAY 1973
As an indication of this potential usefulness, we have plotted, in Fig.6, some matchings between the lattices of a few metals and various phases and densities of He 4 and He 3. As examples, we note the near match between fcc silver and the bcc phase of He 4, and the exact matches between the fcc metals and fcc helium (corresponding to helium growth pressures/> about 2 500 kg cm'2).
279
This survey is i n t e n d e d to be suggestive o f potential applications for the technique. It is certainly not exhaustive, even for solid helium, as is obvious from the complicated incidence s p e c t r u m p r e d u c t e d by our c o m p u t e r simulation. I wish to thank P r o f H. A. Fairbank for his guidance and support o f this work. Thanks are also due to Profs R. A. G u y e r and H. Meyer and Dr E. M. Hogan for their help in planning these experiments and Mr T. N. Roberts for assistance in taking the data. This w o r k was supported in part by the National Science F o u n d a t i o n and the Office o f Naval Research.
References 1 Guyer, R. A., Hogan, E. M. Solid State Comm 5 (1967) 909; Hogan, E. M., Guyer, R. A., Fairbank, H. A. PhysRev 185 (1969) 356 2 Barrett, C. S., Meyer, L., Wasserman, J. J Chem Phys 45 (1966) 834 3 Meyer, L. private communication to Meyer, H. (1967). The techniques by which such foils are prepared have been thoroughly investigated for a number of metals and alloys. See, for example, Barrett, C. S. Structure of Metals (McGraw Hill, NY, 1943) 4 The decision as to which extreme in conductivity corresponds to 0 = 0 is not trivial. The assignment was first made by Guyer and Hogan (ref 1 above) on the basis of a comparison of polycrystalline average over O with results from samples believed to be good approximations of polycrystalline specimens. The simultaneous x-ray scattering and thermal conductivity measurements of Fain, S. C., Jr, Lazarus, D. Phys Rev A 1 (1970) 1460, while limited by difficulties in obtaining large single crystals, supported the assignment of Guyer and Hogan and Hogan, Guyer, and Fairbank. 5 It is important to note that our thermal conductivity measurements not only provided a sensitive indication of c-axis orientation, but also were extremely sensitive indicators of the size and quality of our crystals. Comparison of conductivities for the B-C and C - D portions of each sample allowed a determination of crystal size. Common umldapp and boundary-scatteringdominated regions for the two halves, with the boundary-limited mean free path longer than the sample diameter, ensured that a single crystal l-filed the entire B - D volume. In addition, our entire thermal conductivity curves - umklapp, Poiseuille, and boundary regions - were reproducible within a few percent. The sensitivity to point defects is illustrated by the observed decrease in conductivity by a factor of 1.9 in 0 = 90 ° crystals due to an isotopic impurity concentration of 10 "s The reproducibility of the Poiseuille region with changes in growth rate of a factor of 2 and the scaling of phonon scattering with absolute isotopic concentration ensured that any unintentional point defects caused less phonon scattering than an isotopic concentration of l0 "6. Non-point defects were observable with the same sensitivity and were recognizable as such through their lower order dependences on phonon frequency. Because of the great sensitivity of our measurements to any source of phonon scattering, our growth techniques were required not only to fill the B - D region with a large single crystal, but also to minimize strain and point defects. These techniques are discussed in more detail in Lawson, D. T., Fairbank, H. A. J Low Ternp Phys I 1 (1973) 363, and Lawson, D. T., thesis, Duke University, 1971 (unpublished) 6 Hogan, E. M., Guyer, R. A., Fairbank, t L A . (reference 1), foundA (0) = 2.31 x 10 "4 Wcm "1 K ' I , A (90 °) = 1.12 x 10 "s, b(0) = 4.62, b(90 °) = 2.58 for the umldapp thermal conductivity ~(T, 0) = A (0) exp [®/b(O)T] ; our values are A (0) = 2.21 x 10-4 W cm-lK "1, A(90 °) = 8.79 x 10 -6, b(0) = 4.67. b(90 °) = 2.49; those authors reported 85.0 -+0.1 atmospheres as their growth pressure. Subsequent disassembly of their high pressure system indicated that 85.1 atms is a more accurate value. We have matched our growth pressure accordingly 7 Hogan, E. M., Guyer, R. A., Fairbank, H. A. (see reference 1) 8 Franck, J. P., Wanner, R. PhysRev Lett 25 (1970) 345 9 The solid helium ranges are derived from the ranges in molar volume V [see, for example, Wilks, L The Properties of Liquid and Solid Helium (Clarendon, Oxford, 1967)]. Assuming perfect structure, we have a 3 = 2.347 Vand c = 1.633 a for hop,
280
6.639 V for fcc, and a 3 = 3.320 V for bcc. The metal lattice dimensions are from Kittel, C. Introduction to Solid Sate Physics (Wiley, NY, 1966) and have not been corrected for low temperatures. Fabrication difficulties and oxide layer problems might, in fact, render some of these metals impractical as substrates a 3 =
APPENDIX The nucleation simulation programme In an a t t e m p t to understand the role o f nucleation in our preferred orientation technique, a c o m p u t e r e x p e r i m e n t was devised. A simple m o d e l simulates the r a n d o m deposition o f hard spheres on a [100] fcc substrate. CrystaUites are identified and their orientations determined and tallied. They are then removed f r o m the substrate, allowing the continuous collection o f orientation statistics for a m o d e l substrate of manageable size. The substrate lies in the z = 0 plane. Its texture is determined by specifYing the length o f the fcc cube edge, RS. The location o f that substrate potential m i n i m u m closest to a given (x, y, 0) point then is simply
+
RC, ~ -
+
RC, 0
where [c] is the largest integer ~<[cl. R C = RS/x/2. The particles are ' d r o p p e d ' from positions on a 100 E x 100 E x-y grid. The grid position of each o f the N M A X drops is determined by four integers extracted f r o m a c o m p u t e r generated r a n d o m number. The choice of R S and the grid spacing E determine the relative fineness o f the randomization and the area of substrate to be utilized by the experiment. For convenience the grid corner m a y be offset from the z axis by (A, A, 0). Other input parameters include the hard sphere nearestneighbour distance ( d i a m e t e r ) R and interaction length RO. Particles whose centres c o m e within R O of each other are attracted, while the substrate potential minima attract the particles more weakly and affect particle positions only within the constraints imposed by p a r t i c l e - p a r t i c l e bonds. A parameter EP determines the accuracy required for orientation pattern-matching. The coordinates of each particle are stored along with a serial n u m b e r which identifies b o t h the individual and the crystallite (if any) o f which it is a part. The orientation o f a crystallite is determined by a pattern recognition technique based on the hcp basal plane triangles and the c-axis lattice constant. In order to be tallied and r e m o v e d a crystallite must fulf'd two criteria. 1. It must have an unambiguous and well-defined c-axis orientation. 2. It must exceed a specified m i n i m u m size. Provisions are made for removing crystallites which achieve a seconcL m u c h larger, specified size and are of ambiguous orientation or of fcc structure. Our serial n u m b e r i n g technique allows either t h e consolidation o f crystallites as growth proceeds or the possibility o f non-interactive interpenetration, decreasing the n u m b e r o f ' a m b i g u o u s ' discards at the cost o f an unphysical situation. We find that the two approaches yield the same statistical results for our model.
CRYOGENICS. MAY 1973
The positioning of a new particle is accomplished in a series of logical steps, each of which begins with a search for other particles appearing within the interaction distance, proceeds to choose an appropriate geometric model, and ends by repositioning the particle(s) and updating serial numbers as necessary. A hierarchy of influences governs these movements. 1. Particle-particle interactions 2. Particle- substrate potential interactions 3. 'Gravity' (assumed acting in the - z direction) When alternatives still remain in repositioning a particle, a second hierarchy is applied. 1. Transport the particle the shortest distance 2. Choose a direction of motion most consistent with the last movement of the same particle 3. Randomize the choice Fig.3c represents a 14 500-particle computer experiment with the following parameters.
CRYOGENICS. M A Y 1973
E
= 0.407 A
RS
= 4.07 A
A
= 4.00 A
R
= 3.503 A
RO
= 4.10 A
EP
= 0.01 A
The crystallites were sorted to the nearest angular degree but have been averaged with a 5 ° wide window for comparison with the incidence of various orientations in the macroscopic crystals. We have determined that these results are not qualitatively altered by small changes in the programme parameters (that is by variations which do not destroy the ~general correspondence of our model to the 85 atm H e ' a n d [100] gold matching). However, no systematic study of this model has been undertaken. Some thought has been given to modifying the programme for the evaluation of phonon coupling between solid helium and various substrates.
281
A technique for growing helium crystals in preferred orientations D. T. Lawson
Helium solidifies at very low temperatures and only under high pressure. Measurements of the properties of solid helium therefore must be made with the sample remaining in the container in which it was formed. If large single crystals of a certain orientation are required the problem is further complicated. Some experiments (for example, x-ray or neutron scattering) allow reorientation of the sample container relative to the measurement direction. Others, however, (for example, thermal conductivity, first and second sound propagation, light scattering) are limited by the orientation of the crystal with respect to its container. We wished to undertake a series of thermal conductivit 7 measurements of high quality single crystals of hcp He'~ containing various small amounts of isotopic impurity. The thermal conductivity of hcp He4 is highly anisotropic in the temperature and density region of interest for this experiment. 1 Thus it was necessary to grow from the several solutions crystals which shared a common c-axis orientation with respect to the axis of our sample chamber. Without a sample chamber which produced crystals preferentially in certain orientations the growth of a prohibitively large number of crystals would have been required.
The apparatus It was decided to attempt to nucleate the crystals on a surface which had a periodicity comparable to the lattice dimensions of the helium, in the hope that microscopic
The author is with Department ot Physics, Duke University. Durham, North Carolina, USA. Received 6 November 1972.
276
matching would tend to favour a relatively small number of discrete orientations in the macroscopic samples. We are grateful to Prof H. Meyer for bringing to out attention the discovery, by Barrett, L. Meyer, and Wasserman, 2 that the presence of a specially prepared gold foil favoured the formation of solid hydrogen and deuterium in the fcc rather than the normal hcp phase. Prof Meyer suggested that such foils might be useful in the growth of oriented helium crystals and kindly provided us with a piece of gold foil prepared by L. Meyer and coworkers. The surface of this foil closely approximated the [100] face of a single crystal of fcc gold. Its preparation j had included a 98%reduction in thickness by rolling in one direction followed immediately by an overnight annealing at approximately 5 K below the melting temperature. It was the nucleation of helium crystallites on the surface of this foil which allowed us to influence the orientation of our large single crystals. Our sample chamber, shown in Fig. 1, was a stainless steel tube with an inside diameter of 2.72 ram. At its upper end were a heater and the pressure capillary inlet. The three 0.254 mm thick copper fins labelled B, C, and D deFmed the portion of the sample used in the thermal conductivity measurements. At the lower end of the chamber was an ofhc copper piece, including a nucleation post, which was soldered to the refrigerator with Wood's metal. The gold foilwas soft soldered to the top of the nucleation post, which was .1.9 mm in diameter and extended 7.6 mm into the sample volume to a point 5.1 mm below fin B. Doped germanium resistance thermometers were located on the top and bottom pieces of the sample chamber and on each of the fins. A small He4 pot was used as a growth refrigerator and was pumped through a micrometer needle valve.
CRYOGENICS. MAY 1973
There is a large anisotropy in the thermal conductivity o f h c p helium, first reported by Guyer and Hogan. 1 For temperatures at which umklapp processes are dominant
Frail c a p i l l a r y
K"1 (T, 0) = tc"1 (T, 0) cos20 + K"1 (T, 90 °) sin20
(1)
where 0 is the angle between the hcp c-axis and the direction of heat flow. 4 Thus, once the extrema K(T, 0) and K(T, 90 °) are known for a given density, 0 can be determined from a measurement of umklapp-dominated thermal conductivity. 5
D(
Fig.2 shows the incidence of various values of thermal conductivity, measured at T "1 = 0.65 K "l , among 67 samples grown early in our experiments. (We have tabulated conductivities measured over the B - C , C - D , and B - D regions of each sample and smoothed the histogram by averaging with a A In r = 0.03 window). The large anistropy is evident in the range of values of K represented, while the peaks indicate that certain orientations are strongly favoured by our combination of apparatus and growth technique. The fact that these peaks are due to the influence of the nucleation foil was shown several times experimentally when the preferred orientation behaviour disappeared but was readily restored by measures designed to remove chemical impurities from the foil surface.
C(
- -
Nucleation
post
.3
Fig.1 The sample chamber. B, C, D, E, and 3 are d o p e d germanium resistance thermometers. The b o t t o m of the chamber is thermally anchored to the refrigerator, while the t o p is thermally isolated f r o m other points in the cryostat by a 125 cm length of 0.1524 mm id copper--nickel capillary
We identify the outermost peaks (at In K = - 3 . 2 8 and - 2 . 0 5 for T "1 = 0.65) as the 0 = 0 and 90 ° orientations. Although such an assignment can never be certain in the absence of an independent measure of 0, three separate considerations support our values as the true extrema. 1. Our extrema lie very near, but slightly outside those observed by Hogan, Guyer, and Fairbank 6 at the same density. Those authors observed no strong orientation preference in a sample chamber essentially identical to ours except that their nucleation post was solid copper and terminated in a sharp point. 2. Our extreme data fit the expression
Crystal growth All of the crystals discussed in this paper were grown at a constant pressure of 85.1 atm (1 arm = 101.3 kNm'2), the corresponding melting temperature being 3.125 K. At first the bottom of the sample chamber was cooled to about 3.3 K. An hour later growth began, always with the same setting of the growth refrigerator's regulating valve (I .20 turns). As soon as the bottom of the sample chamber had passed through a predetermined temperature a heater was used to gently return it to that temperature and regulate it there for 2 . 5 - 3 hours. The regulation temperature was chosen so that during this annealing period the solid-liquid interface would advance slowly to a position just below fin B. Growth was resumed at one of two rates which caused the interface to pass through the 5.08 cm region between fins B and D in about 100 or about 200 minutes, respectively. Ala input of 5 x 10 -6 watts was always maintained to the heater at the top of the sample chamber to ensure that the pressure capillary would not be blocked until the chamber was filled with solid. The passage of a liquid-solid interface could be easily detected at each fin as an abrupt change in dR/dt for the thermometric resistor.
CRYOGENICS
. MAY
1973
K(T) = Ae°/bT quite well and, in particular, form curves which are quite straight when In K is plotted against T "l , as would be expected for the true extrema. 7 3. A simple computer simulation of the nucleation process at the surface of our foil indicated strong preferences for the 0 = 0 and 90 ° orientations (Fig.3c). This computer model will be discussed in more detail in another section.
~- 3 . 2 8
-2.05
1
I-4
-3.3 -3,2 -3.1 - 3 0 - 2 9 - 2 . 8 -2.7 -2.6 -2.5 -24 -2.3 -2.2-2.1 -2.0 In'M,(r'l=0.651,
Wcm "mK-a
Fig.2 The incidence of 67 early samples as a f u n c t i o n of 1 n to(T) w i t h T ] = 0.65. The data have been sorted i n t o A In K = 0.01 bins and smoothed w i t h a A In K = 0.03 w i n d o w
277
Fig.3a shows the data of Fig.2 replotted as a function of 0, using equation 1 and our identification of the extrema. (A more restrictive subset of the same data - including only those samples for which the B - C and C - D region conductivities agree to the precision of the measurements exhibits peaks at the same angles. By the time study of these first 67 samples had been completed our growth techniques had been developed to the point that almost all our samples were single crystals f'filing the entire B - D region and thus fulfdling this more stringent test). After these first 67 samples, measurements were made only on 0 = 0 and 0 = 90 ° crystals, which occurred with a combined probability of about 0.3. In all, over 140 crystals were grown in the course of these experiments. Near the end of the preliminary studies which produced these data it was discovered that slight variations in the growth technique, such as a more abrupt than normal initiation of the annealing period, could substantially increase the relative likelyhood of obtaining a 0 TM 90 °
,~,-
o o
o
o
o
o
o
o
•
~oe
•
o
•
o
2 o
d=2-88 e.4'O7 f=5'76 9 =6"43
I$1 = 3 . 5 0 I~1 = 5 . 7 2
a
b
•
•
oo ~e
o
•
•
•
e 0 oe
o
c
Fig.4 Structural matching a the structure of 85 atmosphere hcp He4. The solid-line vectors generate the hexagonal space lattice with the two-atom basis indicated by the broken-line vector b the [100] plane structure of fcc gold c superposition of the [100] plane o f fcc gold and the hcp basal plane of 85 atmosphere He 4 (open circles represent He4 lattice sites, filled circles represent fcc potential minima -- see text)
crystal vis ~ vis 0 ~ O. As an indication of this effect we show, in Fig.3b, the data of Fig.3a minus those samples grown using the '90 ° optimization' techniques, which will be discussed in more detail below.
4
Evaluation O
Fig.4a indicates the structure of hcp He4. The c/a ratio is known to remain quite close to the ideal value 1.633. 8 The structure of the [100] plane of fcc gold is drawn in Fig.4b. Our preferred orientation growth is believed to occur in the following way. Many crystallites are nucleated on the surface of the gold foil. Because of microscopic matching between the structures of the solid helium and the foil surface, a few specific orientations will occur with high relative probabilities among these crystallites. In the course of growth and annealing a single orientation predominates and a single crystal interface then proceeds along the sample chamber for the duration of growth. Those orientations favoured microscopically in the nucleation process will be correspondingly more likely to appear in the macroscopic single crystals. Thus some of the structure of the microscopic orientation distribution at nucleation can be inferred from the orientation distribution of the observed macroscopic crystals. However other factors, such as the anistropy of heat transport in the bulk solid, also influence the observed statistics.
a
18 o
5"
16
"O
D c
12
In order to survey the structural matchings between the solid helium and the foil surface a computer programme, using a hard sphere model, was designed to simulate the nucleation of hcp crystallites on an fcc substrate. The operation of this programme is outlined in an appendix below. A histogram showing the incidence of 1994 simulated crystallites as a function of 0 appears in Fig.3c. O
IO C
20
30
40
50
60
70
80
90
e
Fig.3 Incidence as a function of c-axis orientation. All data sorted into A O = 1° bins and smoothed with a ,%8 = 5 ° w i n d o w a same data shown in Fig.2 b same data as in a, but with samples grown with '90 ° optimization' techniques not included c results of a 1994-crystallite computer simulation o f the (microscopic) nucleation process
278
The most striking feature of the simulation data is the prediction that many 0 = 0 and 0 = 90 ° crystallites will be nucleated, in agreement with our interpretation of the observed macroscopic incidence. The peak at about 0 = 54 ° in the simulation data also corresponds to a peak in the experimental histograms. Other correlations, if any, are much less obvious. In passing, however, we may note that any component of the experimental incidence which was truly random in orientation would exhibit a sin20
CRYOGENICS
. MAY
1973
6.0
dependence. The over-all effect of such a contribution would be roughly to raise the incidence baseline for angles near the 90 ° end of our experimental plots.
F Ii° '/2 f I I
Although our data do not include anything approaching a polycrystalline sample, 5 we note that a polycrystalline average 1 over 0 would yield a value for K equal to that for a single 0 = 53 ° crystal.
k =
V
V7
0 0 0 121 o
j I
Ii s.o
The microscopic matching which favours the 0 = 90 ° orientation may be seen easily by comparing Figs 4a and 4b. The hcp c-axis constant is a very close match to twice the fcc nearest-neighbour distance. The matching which favours nucleation of 0 = 0 crystallites is not so obvious. Consider the comparison, in Fig.4c, of the [100] plane fcc gold structure with the hcp basal plane structure of He 4 at the density of our experiments. The first helium atom to become bound on a vacant area of substrate is quite likely to do so at one of the fcc potential minima. When a second helium atom becomes bound to the first, an fcc nearest-neighbour pair provides a potential well for the two-atom system. When a third atom becomes bound to the first two, the foil provides a stable potential minimum for the system, with atoms at positions marked 1, 2, and 3 in Fig.4c. At least four more atoms may become bound to these three in the same hcp basal plane without serious opposition to the substrate potential. The effect of the '90 ° optimization' techniques can be explained in terms of the relative numbers of 0 and 90 ° crystallites at nucleation and the anisotropy in the thermal conductivity near the melting temperature. Although 0 = 90 ° crystals have a much higher conductivity at temperatures near the thermal conductivity peak, near the melting temperature they have a conductivity which is only about half that of 0 = 0 crystals (see Fig.5). During periods of sustained growth, then, a 0 = 0 crystallite will have a competitive advantage over an otherwise identical crystallite oriented at 90 °.
IOO
o
V
L
O0
V
o
V
g
o
ro
E u
V
121
I =
L
4.0
0 0 0
Q
V v o
[043
.J
,,
3.o .oL L
0
@i
& &
He 3 - _ _ He 4 ~
2.0,_._,, ,, , fcc hcp bcc
0
O0
i
i
t
,
,
,
,
,
,
1
,
,
i
i
i
NiCuPtAIAuAcjFeCrNbLi BeCoZnCdMg fcc bcc hop Moteriol
Fig.6 Some structural matches between metal lattices and solid helium. Ranges are indicated at the left for various characteristic dimensions in solid He 3 and He 4. Symbols f o r the characteristic dimensions in the metals are • a or 2a, ~ a~¢/2,~ a ~ , o aV/3/2 or a~/3, and t= c
The simulation data indicate that more 90 ° than 0 ° crystallites (874/432, see Fig.3c) are formed at nucleation. The anisotropy effect, favouring the predomination of 0 = 0 crystals during sustained growth, tends to balance this advantage. Our '90 ° optimization' techniques, by minimizing the period of sustained growth preceeding the predomination of a single 0, largely remove the balancing influence, resulting in a macroscopic incidence more favourable to 0 = 90 ° crystals (Fig.3a, b).
IO Other applications The technique we have outlined consists of the use of partial, imperfect matchings between the substrate and the sample lattice to produce an ensemble of microscopic seed crystallites whose number density is strongly peaked for certain orientation angles. An appropriate growth technique then produces single macroscopic crystals from this ensemble. The technique was crucial to the success of a study requiring the reproduction of a highly anisotropic thermal conductivity. It may be of use in other experiments as well.
TE u
io.i
~
E
~
=
O
~
O.OI O.OOI ' ' 0.3 05
. .0.7 . . . 0.9
I1'. ' I.'3 ' I.'5 ' T-a K-i
I17
Fig.5 Thermal c o n d u c t i v i t y anisotropy in hcp He 4 grown at a constant pressure of 8 5 a t m . The high temperature region i s a p p r o x i mate, being based on an extrapolation of equation 2 using the parameters determined f r o m lower temperature data 6
CRYOGENICS . MAY 1973
As an indication of this potential usefulness, we have plotted, in Fig.6, some matchings between the lattices of a few metals and various phases and densities of He 4 and He 3. As examples, we note the near match between fcc silver and the bcc phase of He 4, and the exact matches between the fcc metals and fcc helium (corresponding to helium growth pressures/> about 2 500 kg cm'2).
279
This survey is i n t e n d e d to be suggestive o f potential applications for the technique. It is certainly not exhaustive, even for solid helium, as is obvious from the complicated incidence s p e c t r u m p r e d u c t e d by our c o m p u t e r simulation. I wish to thank P r o f H. A. Fairbank for his guidance and support o f this work. Thanks are also due to Profs R. A. G u y e r and H. Meyer and Dr E. M. Hogan for their help in planning these experiments and Mr T. N. Roberts for assistance in taking the data. This w o r k was supported in part by the National Science F o u n d a t i o n and the Office o f Naval Research.
References 1 Guyer, R. A., Hogan, E. M. Solid State Comm 5 (1967) 909; Hogan, E. M., Guyer, R. A., Fairbank, H. A. PhysRev 185 (1969) 356 2 Barrett, C. S., Meyer, L., Wasserman, J. J Chem Phys 45 (1966) 834 3 Meyer, L. private communication to Meyer, H. (1967). The techniques by which such foils are prepared have been thoroughly investigated for a number of metals and alloys. See, for example, Barrett, C. S. Structure of Metals (McGraw Hill, NY, 1943) 4 The decision as to which extreme in conductivity corresponds to 0 = 0 is not trivial. The assignment was first made by Guyer and Hogan (ref 1 above) on the basis of a comparison of polycrystalline average over O with results from samples believed to be good approximations of polycrystalline specimens. The simultaneous x-ray scattering and thermal conductivity measurements of Fain, S. C., Jr, Lazarus, D. Phys Rev A 1 (1970) 1460, while limited by difficulties in obtaining large single crystals, supported the assignment of Guyer and Hogan and Hogan, Guyer, and Fairbank. 5 It is important to note that our thermal conductivity measurements not only provided a sensitive indication of c-axis orientation, but also were extremely sensitive indicators of the size and quality of our crystals. Comparison of conductivities for the B-C and C - D portions of each sample allowed a determination of crystal size. Common umldapp and boundary-scatteringdominated regions for the two halves, with the boundary-limited mean free path longer than the sample diameter, ensured that a single crystal l-filed the entire B - D volume. In addition, our entire thermal conductivity curves - umklapp, Poiseuille, and boundary regions - were reproducible within a few percent. The sensitivity to point defects is illustrated by the observed decrease in conductivity by a factor of 1.9 in 0 = 90 ° crystals due to an isotopic impurity concentration of 10 "s The reproducibility of the Poiseuille region with changes in growth rate of a factor of 2 and the scaling of phonon scattering with absolute isotopic concentration ensured that any unintentional point defects caused less phonon scattering than an isotopic concentration of l0 "6. Non-point defects were observable with the same sensitivity and were recognizable as such through their lower order dependences on phonon frequency. Because of the great sensitivity of our measurements to any source of phonon scattering, our growth techniques were required not only to fill the B - D region with a large single crystal, but also to minimize strain and point defects. These techniques are discussed in more detail in Lawson, D. T., Fairbank, H. A. J Low Ternp Phys I 1 (1973) 363, and Lawson, D. T., thesis, Duke University, 1971 (unpublished) 6 Hogan, E. M., Guyer, R. A., Fairbank, t L A . (reference 1), foundA (0) = 2.31 x 10 "4 Wcm "1 K ' I , A (90 °) = 1.12 x 10 "s, b(0) = 4.62, b(90 °) = 2.58 for the umldapp thermal conductivity ~(T, 0) = A (0) exp [®/b(O)T] ; our values are A (0) = 2.21 x 10-4 W cm-lK "1, A(90 °) = 8.79 x 10 -6, b(0) = 4.67. b(90 °) = 2.49; those authors reported 85.0 -+0.1 atmospheres as their growth pressure. Subsequent disassembly of their high pressure system indicated that 85.1 atms is a more accurate value. We have matched our growth pressure accordingly 7 Hogan, E. M., Guyer, R. A., Fairbank, H. A. (see reference 1) 8 Franck, J. P., Wanner, R. PhysRev Lett 25 (1970) 345 9 The solid helium ranges are derived from the ranges in molar volume V [see, for example, Wilks, L The Properties of Liquid and Solid Helium (Clarendon, Oxford, 1967)]. Assuming perfect structure, we have a 3 = 2.347 Vand c = 1.633 a for hop,
280
6.639 V for fcc, and a 3 = 3.320 V for bcc. The metal lattice dimensions are from Kittel, C. Introduction to Solid Sate Physics (Wiley, NY, 1966) and have not been corrected for low temperatures. Fabrication difficulties and oxide layer problems might, in fact, render some of these metals impractical as substrates a 3 =
APPENDIX The nucleation simulation programme In an a t t e m p t to understand the role o f nucleation in our preferred orientation technique, a c o m p u t e r e x p e r i m e n t was devised. A simple m o d e l simulates the r a n d o m deposition o f hard spheres on a [100] fcc substrate. CrystaUites are identified and their orientations determined and tallied. They are then removed f r o m the substrate, allowing the continuous collection o f orientation statistics for a m o d e l substrate of manageable size. The substrate lies in the z = 0 plane. Its texture is determined by specifYing the length o f the fcc cube edge, RS. The location o f that substrate potential m i n i m u m closest to a given (x, y, 0) point then is simply
+
RC, ~ -
+
RC, 0
where [c] is the largest integer ~<[cl. R C = RS/x/2. The particles are ' d r o p p e d ' from positions on a 100 E x 100 E x-y grid. The grid position of each o f the N M A X drops is determined by four integers extracted f r o m a c o m p u t e r generated r a n d o m number. The choice of R S and the grid spacing E determine the relative fineness o f the randomization and the area of substrate to be utilized by the experiment. For convenience the grid corner m a y be offset from the z axis by (A, A, 0). Other input parameters include the hard sphere nearestneighbour distance ( d i a m e t e r ) R and interaction length RO. Particles whose centres c o m e within R O of each other are attracted, while the substrate potential minima attract the particles more weakly and affect particle positions only within the constraints imposed by p a r t i c l e - p a r t i c l e bonds. A parameter EP determines the accuracy required for orientation pattern-matching. The coordinates of each particle are stored along with a serial n u m b e r which identifies b o t h the individual and the crystallite (if any) o f which it is a part. The orientation o f a crystallite is determined by a pattern recognition technique based on the hcp basal plane triangles and the c-axis lattice constant. In order to be tallied and r e m o v e d a crystallite must fulf'd two criteria. 1. It must have an unambiguous and well-defined c-axis orientation. 2. It must exceed a specified m i n i m u m size. Provisions are made for removing crystallites which achieve a seconcL m u c h larger, specified size and are of ambiguous orientation or of fcc structure. Our serial n u m b e r i n g technique allows either t h e consolidation o f crystallites as growth proceeds or the possibility o f non-interactive interpenetration, decreasing the n u m b e r o f ' a m b i g u o u s ' discards at the cost o f an unphysical situation. We find that the two approaches yield the same statistical results for our model.
CRYOGENICS. MAY 1973
The positioning of a new particle is accomplished in a series of logical steps, each of which begins with a search for other particles appearing within the interaction distance, proceeds to choose an appropriate geometric model, and ends by repositioning the particle(s) and updating serial numbers as necessary. A hierarchy of influences governs these movements. 1. Particle-particle interactions 2. Particle- substrate potential interactions 3. 'Gravity' (assumed acting in the - z direction) When alternatives still remain in repositioning a particle, a second hierarchy is applied. 1. Transport the particle the shortest distance 2. Choose a direction of motion most consistent with the last movement of the same particle 3. Randomize the choice Fig.3c represents a 14 500-particle computer experiment with the following parameters.
CRYOGENICS. M A Y 1973
E
= 0.407 A
RS
= 4.07 A
A
= 4.00 A
R
= 3.503 A
RO
= 4.10 A
EP
= 0.01 A
The crystallites were sorted to the nearest angular degree but have been averaged with a 5 ° wide window for comparison with the incidence of various orientations in the macroscopic crystals. We have determined that these results are not qualitatively altered by small changes in the programme parameters (that is by variations which do not destroy the ~general correspondence of our model to the 85 atm H e ' a n d [100] gold matching). However, no systematic study of this model has been undertaken. Some thought has been given to modifying the programme for the evaluation of phonon coupling between solid helium and various substrates.
281