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A method for the preparation of He3 single crystals at ultralow temperatures
A method for the preparation of He 3 single crystals at ultralow temperatures is proposed. The method is based on the shift of the melting curve when a magnetic field is applied. With the use of the proposed technique, i t is thought possible to grow single crystals by the 8ridgman or the zone-melting method. A trial of this method is being planned and the results will hopefully be published soon.
A method for the preparation of He 3 single crystals at ultralow temperatures A. Ikushima Much interest has recently arisen in the properties of solid He 3 at ultralow temperatures. The interest was stimulated by the discovery in 1972 of the spin ordering in solid He3, ~ which occurs at around 1.1 mK if the system is on the melting curve, and therefore, recent work has mainly been devoted to the investigation of its magnetic properties. 2 Except in certain cases such as neutron scattering experiments in which neutrons can penetrate only 100/am or so into He 3, various sorts of experiments would require single crystals of rather large sizes and of desired orientations. However, this problem has not been overcome yet, and so far thin crystals usually were grown around heater wires by heating from the liquid to the solid phases. This method is, however, not ideal since the addition of heat results in a temperature rise and moreover the crystal is often defective or may even be polycrystalline. In this paper, another method for growing single crystals of He a in the mK temperature range is proposed. This method is based on the control of the melting curve by the application of a magnetic field. If the method described here can be successfully applied, the above-mentioned disadvantages will at least be partly avoided, and the research on solid He a should be greatly promoted. A trial of the method is being planned and the results will be published soon.
Basic idea and procedures The method proposed here is based on the downward shift of the melting curve caused by a magnetic field; that is, the solid phase is expanded downward to the lower pressure by a magnetic field. As will be explained below, He a in a crucible is solidified from the bottom upwards to the top, or a molten zone is moved along the length of the specimen.
Procedures Liquid He 3 in a crucible is cooled to the required temperature under an appropriate pressure. The cooling should be done all the way through the liquid phase, otherwise the process is time consuming. The temperature can not be lower than roughly 2 mK, however, since the shift of the melting curve becomes negligible at lower temperatures. The pressure should also be close to the melting pressure, and therefore the pressure should actually be controlled by squeezing the sample chamber. The author is at the Institute for Solid State Physics, The University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan. Paper received
25 July 1979
He 3 is then solidified by applying a magnetic field. If He a is moved downward into the magnetic field, one employs a method very similar to the so-called Bridgman method of crystal growth [Fig. 1 (a)]. If one wants to apply zone-melting or the zone-refining, the specimen is first completely solidified in a 'main' magnetic field, and then, a small coil giving a small magnetic field directed opposite to the main field is moved from the bottom to the top, thus producing a molten zone in the specimen and moving it upwards [Fig. 1 (b)]. Another similar way is to use two magnetic fields of opposite direction with a range of zero field between them, and to move the two coils from the bottom to the top of the specimen [Fig. 1 (c)]. In either case, one can move the magnetic field relative to the specimen not only by actually moving the magnet but by accordingly operating currents in a stack of thin magnets placed along the growth direction. If a satisfactory single crystal can be grown by one of the above methods, the pressure should be increased in order not to re-melt the specimen after the magnetic field is removed. The main magnetic field is finally reduced to zero. One has at first to estimate the amount of the pressure change APm required to compensate for the magnetic field Ho. The Clapeyron-Clausius equation is, ( V L - VS) • z~P m = ( S L - S S ) • A T + ( M L -
Ms) • Mar (1)
where V, S andM are the volume, the entropy and the magnetization of one mole of He 3, respectively, and the subscripts L and S specify liquid and solid. If the temperature is kept constant at T, the pressure change is APm
VS)-1 f H° (ML - M s ) dH
= (VL -
(2)
0
where Vs - V L is given b~P
=
1.2 cma mol a. The magnetizationML
(3)
M L = XLH
XL =
1 . 3 6 x 10 "a
T*
• 37.0 (cma mole q )
(3a)
0011-2275/80/080448-03/:$02.00 © 1980 IPC Business Press
448
CRYOGENICS. AUGUST 1980
ISs (T, 0) - Ss(T, H)] [R = (/3/J/-/)2 (--4 Jxzz + . . -) (6c)
Ss(T, O)/R = ~n2 - (3/2)/~2fix x + . . . . .
L_
Here again, Jxx is another moment of the Hamiltonean.2'4 Since the solidification is a transition from liquid to solid, viz, from more ordered to less ordered states of spins, the process should be endothermic, and the parameter effecting the process is of course the second term of (6), T • ~S, and the first term is only several ergs/mol at Ho = 10 k0e. This should be a great advantage for work at ultralow temperatures. Table 2 shows the change of the entropy and T • ~S as calculated (if one does not keep the sample temperature constant while the solidification proceeds).
L
a
b
C Fig. 1 Various possible methods for preparation of He 3 single crystals. L and S denote liquid and solid: a -- A He3 sample is moved downwards into a magnetic field and the crystal is grown from the b o t t o m by the Bridgman method; b -- zone melting o f He 3 by moving a small magnetic field inversely directed t o the main field from the b o t t o m to the top o f the specimen, c -- zone melting of He3by zero field at the middle point of two magnets of the fields oppositely directed t o each other.
and M s in the high temperature expansion is given by2
M s = XSH
(4)
1 Np 2 (1 + 4g/xzz + . . . . ) XS- T k B
(4a)
The shift of the melting pressure a function of rio, and is
ZMDm
The crystal grown by the Bridgman method should have a temperature gradient with the bottom part warmer than the top part. Another thing which we should be cautious about is whether the crystal be damaged when the additional pressure AP is applied at the third stage of the abovementioned procedure. The elastic limit, which is usually much less than the fracture stress, of He 4 in the liquid helium temperature range is known to be about l0 s dynes cm -2,s If He 3has a value of the same order of magnitude, this is much more than the change of the melting pressure already calculated. An increase in elastic energy is also associated with this procedure and is given by VK(AP)2/2 where K is the volume compressibility. The amount of this energy increase is negligible, and moreover, it would not convert to heat. Finally, the heat Q2 is associated with the fourth stage of the procedure, that is, the stage in which the field is turned to zero. Q2 will be given by 0
where T* is the magnetic temperature, ~ = (kBT) "1, and Jxzz is a moment of the Hamiltonian defined by Guyer.
_ 1 [ z~Pm - 2.---4- [ 2.84 x 10 "6
(6d)
is then calculated as
5.04 x 10 -7 T
Q2 = f
(7)
Ms(P ) dH /4o
This takes a negative value as is also expected. That is, the process is again endothermic, although the value of Q2 is rather small; Q2 = -2.1x103 ergs mol "1 at 2 mK for H0 = 10 kOe. Discussion
(1
1"66T 10"3)]Ho2
(5)
The method proposed in this paper is greatly advantageous in the sense that the crystal growth does not require any heat input, and instead the method involves endothermic
Zkem/Ito 2 is tabulated in Table 1.
Table 1. One should then check the heat QI associated with the solidification of He 3 by the magnetic field. The expression for Q1 at a fixed temperature is, p*
T, mK 2 5 z~Om/H02 -1.7x10 s -2.7X10 5
10 -1.6X10 "s
100 -8.8X10 "7
(dynes/cm20e)
Ho
QI = _J
Change of melting pressure due to magnetic field
MLdH + T • AS
(6)
0
Table 2. Change of entropy and heat absorbed during solidification under 10 kOe
where ~kS = S L ( T , H ) -
a s ( T , H)
~- S L ( T , O) - S s ( T , 1-I)
(6a)
and
SL(T, 0) = 4.85 R T
CRYOGENICS. AUGUST 1980
7", mK
2
5
10
100
~S/R (per mol) LIS.T (ergs/mol)
-0.39
-0.63
-0.64
-0.21
-5.3x105
-1.7x10 e
-6.5x104 -2.6x105
(6b)
449
processes, and one can expect to get a sizable single crystal in the mK temperature range. A number of problems should also be pointed out here. In the above-mentioned method, the pressure change due to the magnetic field is essential, but the change is actually not appreciable even at H = 10 kOe say. Thus, this method will require a fme pressure control. The next question is whether or not any appreciable heat is generated due to the mutual friction between the solid helium and the crucible wall when the pressure is added at the third stage of the procedure described in the previous Section. This question must be answered by experience. The third point is the cooling of the grown crystal to 1 mK or below to make measurements. The present method, of course, has an advantage in this respect, since the crystal is grown in mK temperature range, but we still should be concerned about the problems. One cannot introduce wire bundles or certain other aids to save the cooling time, if sizable and defect-free crystals are needed.
~L
The last problem would be how to control the growth direction of the crystal. We have not mentioned anything about the problem in this paper. Nothing has been established yet with regard to this question for solid helium even at high temperatures, and it is important that this problem is solved in the near future. The author would like to thank Professor S. Nakajima, Professor S. Ushioda and Dr. A. Tominaga for valuable discussions.
References 1
Halperin, W.P. Arehie, C.N. Rasmussen, F.B. Buhrman, R.A.
2 3
Richardson, R.C.Phys Rev Lett 32 (1974) 927 ibid 34 (1975) 718 Guyer, R.A. J Low Temp Phys 30 (1978) 1 Wheatley,J.C. The Helium Liquids J.G.M. Armitage and J.E. Farquhar (ed) (Academic Press, 1975) 241 Suzuki,H. JPhys Soc Japan 35 (1973) 1472
4
®
Manufacturing Engineering Proceedings of the 2nd Joint Polytechnics Symposium, '1'1-'i3 June 1979, Lanchester Polylechn~c, UK The aim of this symposium was to bring together industrialists, researchers and students in a common forum where their problems and achievements could be examined and discussed, Topics covered include: Machine tools; Bearings and tribology; Computers/Microprocessors; Forming and fabrication; Electrochemical machining; Material removal and processes.
Further information, available on reauest from:
The Sales Manager IPC Science and Technology PressLimited, PO Box 63, WestburyHouse, BuryStreet, Guildford, Surrey, England GU2 5BH
July 1979/cloth/260 pages/O 86103 013 3 £15.00 net [$39.00]
450
C R Y O G E N I C S . AUGUST 1980
A method for the preparation of He 3 single crystals at ultralow temperatures A. Ikushima Much interest has recently arisen in the properties of solid He 3 at ultralow temperatures. The interest was stimulated by the discovery in 1972 of the spin ordering in solid He3, ~ which occurs at around 1.1 mK if the system is on the melting curve, and therefore, recent work has mainly been devoted to the investigation of its magnetic properties. 2 Except in certain cases such as neutron scattering experiments in which neutrons can penetrate only 100/am or so into He 3, various sorts of experiments would require single crystals of rather large sizes and of desired orientations. However, this problem has not been overcome yet, and so far thin crystals usually were grown around heater wires by heating from the liquid to the solid phases. This method is, however, not ideal since the addition of heat results in a temperature rise and moreover the crystal is often defective or may even be polycrystalline. In this paper, another method for growing single crystals of He a in the mK temperature range is proposed. This method is based on the control of the melting curve by the application of a magnetic field. If the method described here can be successfully applied, the above-mentioned disadvantages will at least be partly avoided, and the research on solid He a should be greatly promoted. A trial of the method is being planned and the results will be published soon.
Basic idea and procedures The method proposed here is based on the downward shift of the melting curve caused by a magnetic field; that is, the solid phase is expanded downward to the lower pressure by a magnetic field. As will be explained below, He a in a crucible is solidified from the bottom upwards to the top, or a molten zone is moved along the length of the specimen.
Procedures Liquid He 3 in a crucible is cooled to the required temperature under an appropriate pressure. The cooling should be done all the way through the liquid phase, otherwise the process is time consuming. The temperature can not be lower than roughly 2 mK, however, since the shift of the melting curve becomes negligible at lower temperatures. The pressure should also be close to the melting pressure, and therefore the pressure should actually be controlled by squeezing the sample chamber. The author is at the Institute for Solid State Physics, The University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan. Paper received
25 July 1979
He 3 is then solidified by applying a magnetic field. If He a is moved downward into the magnetic field, one employs a method very similar to the so-called Bridgman method of crystal growth [Fig. 1 (a)]. If one wants to apply zone-melting or the zone-refining, the specimen is first completely solidified in a 'main' magnetic field, and then, a small coil giving a small magnetic field directed opposite to the main field is moved from the bottom to the top, thus producing a molten zone in the specimen and moving it upwards [Fig. 1 (b)]. Another similar way is to use two magnetic fields of opposite direction with a range of zero field between them, and to move the two coils from the bottom to the top of the specimen [Fig. 1 (c)]. In either case, one can move the magnetic field relative to the specimen not only by actually moving the magnet but by accordingly operating currents in a stack of thin magnets placed along the growth direction. If a satisfactory single crystal can be grown by one of the above methods, the pressure should be increased in order not to re-melt the specimen after the magnetic field is removed. The main magnetic field is finally reduced to zero. One has at first to estimate the amount of the pressure change APm required to compensate for the magnetic field Ho. The Clapeyron-Clausius equation is, ( V L - VS) • z~P m = ( S L - S S ) • A T + ( M L -
Ms) • Mar (1)
where V, S andM are the volume, the entropy and the magnetization of one mole of He 3, respectively, and the subscripts L and S specify liquid and solid. If the temperature is kept constant at T, the pressure change is APm
VS)-1 f H° (ML - M s ) dH
= (VL -
(2)
0
where Vs - V L is given b~P
=
1.2 cma mol a. The magnetizationML
(3)
M L = XLH
XL =
1 . 3 6 x 10 "a
T*
• 37.0 (cma mole q )
(3a)
0011-2275/80/080448-03/:$02.00 © 1980 IPC Business Press
448
CRYOGENICS. AUGUST 1980
ISs (T, 0) - Ss(T, H)] [R = (/3/J/-/)2 (--4 Jxzz + . . -) (6c)
Ss(T, O)/R = ~n2 - (3/2)/~2fix x + . . . . .
L_
Here again, Jxx is another moment of the Hamiltonean.2'4 Since the solidification is a transition from liquid to solid, viz, from more ordered to less ordered states of spins, the process should be endothermic, and the parameter effecting the process is of course the second term of (6), T • ~S, and the first term is only several ergs/mol at Ho = 10 k0e. This should be a great advantage for work at ultralow temperatures. Table 2 shows the change of the entropy and T • ~S as calculated (if one does not keep the sample temperature constant while the solidification proceeds).
L
a
b
C Fig. 1 Various possible methods for preparation of He 3 single crystals. L and S denote liquid and solid: a -- A He3 sample is moved downwards into a magnetic field and the crystal is grown from the b o t t o m by the Bridgman method; b -- zone melting o f He 3 by moving a small magnetic field inversely directed t o the main field from the b o t t o m to the top o f the specimen, c -- zone melting of He3by zero field at the middle point of two magnets of the fields oppositely directed t o each other.
and M s in the high temperature expansion is given by2
M s = XSH
(4)
1 Np 2 (1 + 4g/xzz + . . . . ) XS- T k B
(4a)
The shift of the melting pressure a function of rio, and is
ZMDm
The crystal grown by the Bridgman method should have a temperature gradient with the bottom part warmer than the top part. Another thing which we should be cautious about is whether the crystal be damaged when the additional pressure AP is applied at the third stage of the abovementioned procedure. The elastic limit, which is usually much less than the fracture stress, of He 4 in the liquid helium temperature range is known to be about l0 s dynes cm -2,s If He 3has a value of the same order of magnitude, this is much more than the change of the melting pressure already calculated. An increase in elastic energy is also associated with this procedure and is given by VK(AP)2/2 where K is the volume compressibility. The amount of this energy increase is negligible, and moreover, it would not convert to heat. Finally, the heat Q2 is associated with the fourth stage of the procedure, that is, the stage in which the field is turned to zero. Q2 will be given by 0
where T* is the magnetic temperature, ~ = (kBT) "1, and Jxzz is a moment of the Hamiltonian defined by Guyer.
_ 1 [ z~Pm - 2.---4- [ 2.84 x 10 "6
(6d)
is then calculated as
5.04 x 10 -7 T
Q2 = f
(7)
Ms(P ) dH /4o
This takes a negative value as is also expected. That is, the process is again endothermic, although the value of Q2 is rather small; Q2 = -2.1x103 ergs mol "1 at 2 mK for H0 = 10 kOe. Discussion
(1
1"66T 10"3)]Ho2
(5)
The method proposed in this paper is greatly advantageous in the sense that the crystal growth does not require any heat input, and instead the method involves endothermic
Zkem/Ito 2 is tabulated in Table 1.
Table 1. One should then check the heat QI associated with the solidification of He 3 by the magnetic field. The expression for Q1 at a fixed temperature is, p*
T, mK 2 5 z~Om/H02 -1.7x10 s -2.7X10 5
10 -1.6X10 "s
100 -8.8X10 "7
(dynes/cm20e)
Ho
QI = _J
Change of melting pressure due to magnetic field
MLdH + T • AS
(6)
0
Table 2. Change of entropy and heat absorbed during solidification under 10 kOe
where ~kS = S L ( T , H ) -
a s ( T , H)
~- S L ( T , O) - S s ( T , 1-I)
(6a)
and
SL(T, 0) = 4.85 R T
CRYOGENICS. AUGUST 1980
7", mK
2
5
10
100
~S/R (per mol) LIS.T (ergs/mol)
-0.39
-0.63
-0.64
-0.21
-5.3x105
-1.7x10 e
-6.5x104 -2.6x105
(6b)
449
processes, and one can expect to get a sizable single crystal in the mK temperature range. A number of problems should also be pointed out here. In the above-mentioned method, the pressure change due to the magnetic field is essential, but the change is actually not appreciable even at H = 10 kOe say. Thus, this method will require a fme pressure control. The next question is whether or not any appreciable heat is generated due to the mutual friction between the solid helium and the crucible wall when the pressure is added at the third stage of the procedure described in the previous Section. This question must be answered by experience. The third point is the cooling of the grown crystal to 1 mK or below to make measurements. The present method, of course, has an advantage in this respect, since the crystal is grown in mK temperature range, but we still should be concerned about the problems. One cannot introduce wire bundles or certain other aids to save the cooling time, if sizable and defect-free crystals are needed.
~L
The last problem would be how to control the growth direction of the crystal. We have not mentioned anything about the problem in this paper. Nothing has been established yet with regard to this question for solid helium even at high temperatures, and it is important that this problem is solved in the near future. The author would like to thank Professor S. Nakajima, Professor S. Ushioda and Dr. A. Tominaga for valuable discussions.
References 1
Halperin, W.P. Arehie, C.N. Rasmussen, F.B. Buhrman, R.A.
2 3
Richardson, R.C.Phys Rev Lett 32 (1974) 927 ibid 34 (1975) 718 Guyer, R.A. J Low Temp Phys 30 (1978) 1 Wheatley,J.C. The Helium Liquids J.G.M. Armitage and J.E. Farquhar (ed) (Academic Press, 1975) 241 Suzuki,H. JPhys Soc Japan 35 (1973) 1472
4
®
Manufacturing Engineering Proceedings of the 2nd Joint Polytechnics Symposium, '1'1-'i3 June 1979, Lanchester Polylechn~c, UK The aim of this symposium was to bring together industrialists, researchers and students in a common forum where their problems and achievements could be examined and discussed, Topics covered include: Machine tools; Bearings and tribology; Computers/Microprocessors; Forming and fabrication; Electrochemical machining; Material removal and processes.
Further information, available on reauest from:
The Sales Manager IPC Science and Technology PressLimited, PO Box 63, WestburyHouse, BuryStreet, Guildford, Surrey, England GU2 5BH
July 1979/cloth/260 pages/O 86103 013 3 £15.00 net [$39.00]
450
C R Y O G E N I C S . AUGUST 1980