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Two capacitance dilatometers
Two different dilatometric devices designed for magneto-strictive strain measurements at low temperatures are described. Their sensitivity is o f the order of 1G 1o for relative length changes. They are provided with an adjustable gap system which makes it possible to change the gap between the capacitor plates at helium temperature.
Two capacitance dilatometers G. BrSndli and R. Griessen
Since 1962 when White 1 described the first capacitance cell for length change measurements several capacitance dilatometers have been used by different workers. 2-4 Except for one device 5 the samples must be machined to a well defined shape for mounting in the cell. For example, White 1,6,7 used a cylindrical sample with parallel fiat ends, one extremity being glued or screwed to the bottom of the cell while the other acts directly as a capacitor plate. Fawcett 2,8 also used another type of cell where a hollow cylindrical sample is needed. With these type of dilatometers it is difficult to make length change measurements on very small samples or on soft materials especially if they are in the form of single crystals. When length change measurements are made at low temperatures, materials with a large thermal expansion coefficient will contract during the cooling down and the capacitor gap will be too large at low temperatures so reducing the sensitivity of the device. The purpose of this paper is to describe two different capacitance dilatometers provided with an adjustable gap system which makes it possible to change the gap between the capacitor plates at any temperature. The dilatometers had to fit in the lhnited volume of strong magnets and provide a means of measuring magnetostrictive strains for different sample orientations. The sample holders accept samples of different shapes and dimensions. The two dilatometers to be described have been used successfully for various length change measurements including measurements of the induced magnetostrictive strains on an EuSe 9 single crystal (1 x 1 x 3 mm 3) and the oscillatory magnetostriction of a 6 x 6 x 35 mm 3 indium single crystal, 10 a material which is extremely sensitive to elastic stresses. Length changes at the transition from normal to superconducting state were also measured with these devices. 11 The adjustable gap facility has proved to be extremely useful especially for materials with very large thermal expansion coefficient like zinc along the hexagonal axis. For example, it would have been very difficult to measure the stress GB is with Laboratory of Applied Physics and Measuring Techniques, Brown Boveri, 5401 Baden, Switzerland and RG is with Laboratorium for Festk~perphysilk, Eidgen6ssische Technische Hochschule 8049 Zurich, Switzerland. Received 26 January 1973.
CRYOGENICS
. MAY
1973
dependence of the suscel~tibility of zinc between 4.2 K and room temperature 1~ without an adjustable gap. Using a three terminal method 1,5 including a capacitance bridge (General Radio type 1615 A), these two dilatometers make it possible to detect relative length changes of the order of 10"10. Fig.1 shows a typical recorder tracing obtained with the dilatometer I (see below). This curve represents the oscillatory magnetostrictive strains of a zinc single crystal as a function of the applied field. 13 Curves of similar quality have been obtained with the dilatometer II. Choice of the construction materials The two dilatometers described have been designed primarily for magnetostrictive strain measurements. Thus beside appropriate mechanical properties the material used should have a low electrical conductivity in order to minimize eddy current effects due to magnetic field sweeps. The material should also be non-magnetic. Further because these devices were intended for use at low temperatures the thermal conductivity should not be too small since a uniform temperature over the whole sample holder is needed. To fulfil all these requirements all the parts, including the screw, of the dilatometers were machined from a beryllium-copper alloy called Berylco 25 (obtained from Kawecki Berylco Industries, Reading, Pa,
5 × 1 0 .9
2 kO~
3k0¢
Fig.1
Oscillatory magnetostrictive strains in a single crystal The applied magnetic field is parallel to the hexagonal axis while the length changesare measured in the basal plane. The bar gives the scale for the relative length changes ( 1 0 e = 79.5 A m "1 )
of zinc.
299
USA). Values of the room temperature magnetic susceptibility of different Berylco alloys are given elsewhere. 14 The springs used in our design was also made of Berylco foil because of its extremely low elastic hysteresis.
Dilatometer
I
This dilatometer is basically similar to that used by Br/indli 5 for length change measurements associated with the normal-superconducting transition in I n - P b alloys. Our new design, however, ensures a better parallelism of the capacitor plates and includes an adjustable gap system. The four principal parts of the dilatometer I shown in Fig.2 are the hollow cylindrical body 1, the moveable part 2 ftxed to 1 with two ring springs 2d to produce a parallel displacement 15,16 of the upper capacitor plate 2c, the adjustable gap system 3, and the lower capacitor
3o 3b 3c 3d 3e
plate system 4. The gap between the capacitor plates 2c and 4a can be adjusted with the screw 3c which moves the plunger 3e, the sample 5, and therefore the capacitor plate 2c. With the connexion 3b and the rod 3a, it is possible to turn the screw 3c from the top of the cryostat. For cooling it is necessary to evacuate the vessel containing the dilatometer (or to have helium as exchange gas in it) because any traces of solid air would hinder the free movement of the screw and the plunger at low temperatures. For the same reason the whole adjustable gap system 3 must be clean and grease-free. This dilatometer was designed with the best possible parallelism of the capacitor plates. This was achieved by lapping simultaneously the capacitor plate 2c, the guard ring head 2b, and the lower edge of the body 1. For this operation we replaced 3e and the sample by a longer plunger and by means of the screw 3c we brought the unlapped capacitor plate to the level of the lower edge of 1, the ring springs being loaded. After lapping the equilibrium position of part 2 is about 1 mm higher than in the lapping position. The capacitor plate 4a isolated from 4b by an Araldite epoxy resin film, and the holder 4b were also lapped together. In this way a very good parallelism of the capacitor plates could be achieved so that small gaps of the order of 0.01 mm could be used for the measurements, thus increasing the sensitivity of the device. The parallelism of the capacitor plates was checked in two different ways. First we measured the capacitance as a function of the number of turns of the screw 3c. This way the value of the capacitance could be changed continuously between 5 pF and about 150 pF before some sort of electric breakthrough occured between the plates. The experimental values so determined are in good agreem ent with the theoretical results obtained with 1
C= - - + d d + 0.22w
2d 2a 2d 2b 2c
1+
(1)
where C is the value of the capacitance for a gap d, r represents the radius of the smaller capacitor plate 2c, and w is the gap between this plate and the guard ring 2b. For our sample holder r = 0.9 cm and w = 0.01 cm. The second test consisted of measuring the same effect (for example the amplitude of the oscillatory magnetostrictive strains of an aluminium single crystal) using different gaps. For the conversion of the observed capacitance changes, Ac, into length changes AI, we used the following relation which is derived directly from (1)
4a
4b
AI=--Ad=AC
+ (d+0.22w) 2
1+
(2)
4c Fig.2
Dilatometer I 1 - f i x e d p a r t ; 2a - - free p a r t ; 2 b - - g u a r d head; 2c -- u p p e r c a p a c i t o r p l a t e ; 2 d -- ring springs; 3a -- string f r o m t h e t o p o f the c r y o s t a t ; 310 - - c o u p l i n g ; 3 c - - screw; 3 d - - b o d y o f the adjustable gap system; 3e - - p l u n g e r ; 4 a - - l o w e r c a p a c i t o r plate; 4 b - - l o w e r c a p a c i t o r p l a t e h o l d e r ; 4 c - - b o d y o f t h e l o w e r plate system; 5 - - sample
300
It was found that the value of the gap did not affect the magnitude of the measured strains. For example, the length change observed at 10 pF was equal within the limits of experimental error to the length change measured at 100 pF. The analysis of these two test runs showed that the capacitor plates are parallel to within 10-4 rad.
C R Y O G E N I C S . M A Y 1973
II
Dilatometer
This is an entirely new type of capacitance dilatometer. The main difference from dilatometer I is that the capacitor plates do not remain parallel but rotate when the length of the sample changes. A schematic picture of the device is shown in Fig.3. Fig.4 shows an exploded view of the dilatometer II which was used to measure transverse and parallel magnetostriction in the gap of a conventional electromagnet. It consists of three different parts: the body la with the fixed capacitor plate lb, the moveable part 2c with the capacitor plate 2b, and the adjustable gap system 3. To carry out measurements the sample is weakly clamped between point A o f part 2c and point B of part 3c. The moveable part 2c is fixed to the body la by two different springs: the stronger spring 2d acts as rotation axis while the weak spring 2a increases the lateral stiffness for movements parallel to the plane of the capacitor plates. The plates which are isolated with an Araldite epoxy resin film have been lapped before mounting and the gap is chosen as zero when the plates are parallel. The adjustable gap system 3 transforms a screw turn (3a) in a small horizontal displacement of the block 3c, and therefore changes the angle between the two plates by a small amount. It was therefore possible to calibrate the dilatometer by measuring the capacitance as a function o f the screw turns, that is, as a function o f the angle between the plates. It was found that the calibration curve is in excellent agreement with the theoretical curve as long as the mean gap is not larger than 0.5 mm.
3a )
3b
2a
2c
la
For larger mean gap values the edge distortions o f the electric field are no longer negligible. The theoretical curve was obtained from the following relation 17 C = - In
q~
1+
2d (3) Fig.4 Dilatometer I I l a -- fixed part; 1 b -- fixed capacitor plate; 2a - spring for lateral stiffness; 2b -- free capacitor plate; 2c -- moveable part; 2d suspension spring; 3a -- screw; 3b -- fork (vertical displacements); 3c -- block (horizontal displacements) For measurement the sample is weakly clamped between A and B
Capacity plates
where C is the capacitance of two rectangular plates of area a, b which enclose an angle ~. The different lengths in this equation and the following two relationships, are explained in Fig.3. For small angles between the plates r TM S I + S 2 because the gap is zero when the plates are parallel. So a very good approximation is
s2
s
i ampie
C~
x In 1- 2e
1+ - S1 + S2
(4)
From (4) it follows immediately that AC A/~
( 1 - 2e)
(5)
C Fig.3 Schematic picture of dilatometer II for the calculation of its capacitance and for the relation between the length changes of the sample and the corresponding capacitance changes
CRYOGENICS.MAY 1973
where Al is the length change corresponding to the capacitance change, AC. Note that for this type of dilatometer
301
A / a n d AC are o f opposite signs. This equation offers the same degree o f accuracy as (1).
References
Conclusions Experimental values obtained with the two dilatometers on the same sample did not show any deviation greater than the experimental error of 4%. A non-parallel plate dilatometer is therefore as reliable and precise as a more conventional parallel plate dilatometer, the sensitivity being the same, 10"10 for relative length changes. Both dilatometers are compact and insensitive to mechanical vibrations. We are greatly indebted to Prof J. L. Olsen for his advice and help with several aspects of the construction of the dilatometers. It is also a pleasure to thank A. Kundig for much help in the design o f the devices and A. Vogelsanger for his high precision machining. This work was supported in part financially by the Schweizerischer Nationalfonds.
302
10 11 12 13 14 15 16 17
White, G. K. Cryogenics 1 (1961) 151 Fawcett, E. Phys Rev B2 (1970) 1604 Green, B. A., Chadrasekhat, B. S. PhysRev Lett 11 (1963) 331 Reitz, L. M., Spatlin, D. M. PhysRevB 5 (1972) 3803 Brandli, G. Phys kondensMaterie 11 (1970) 93 and 111 White, G. K., Collins, J. G. J L o w Temp Phys 7 (1972) 43 White, G. K. JPhys C: Solid StatePhys 5 (1972) 2731 Fawcett, E. private communication Gtiessen, R., Landolt, M., Ott, H. R. Solid State Comm 9 (1971) 2219 Griessen, R., Sorbello, R. S. (forthcoming) Ott, H. R. SolidState Comm 9 (1971) 2225 Griessen, R., Ott, H. R. PhysRev Lett 29 (1972) 1150 Griessen, R., Kiindig, A. Solid State Comm 11 (1972) 295 Griessen, R. Cryogenics (forthcoming) Jones, R. V. JScilnstr 39 (1962) 193 Braddik, H. J. J. The Physics of Experimental Method (Chapman and Hall, 1954) 72 Landau, L. D., Lifshitz, E. M. Elcctrodynamics of Continuous Media (Pergamon Press, 1960) 100
C R Y O G E N I C S . M A Y 1973
Two capacitance dilatometers G. BrSndli and R. Griessen
Since 1962 when White 1 described the first capacitance cell for length change measurements several capacitance dilatometers have been used by different workers. 2-4 Except for one device 5 the samples must be machined to a well defined shape for mounting in the cell. For example, White 1,6,7 used a cylindrical sample with parallel fiat ends, one extremity being glued or screwed to the bottom of the cell while the other acts directly as a capacitor plate. Fawcett 2,8 also used another type of cell where a hollow cylindrical sample is needed. With these type of dilatometers it is difficult to make length change measurements on very small samples or on soft materials especially if they are in the form of single crystals. When length change measurements are made at low temperatures, materials with a large thermal expansion coefficient will contract during the cooling down and the capacitor gap will be too large at low temperatures so reducing the sensitivity of the device. The purpose of this paper is to describe two different capacitance dilatometers provided with an adjustable gap system which makes it possible to change the gap between the capacitor plates at any temperature. The dilatometers had to fit in the lhnited volume of strong magnets and provide a means of measuring magnetostrictive strains for different sample orientations. The sample holders accept samples of different shapes and dimensions. The two dilatometers to be described have been used successfully for various length change measurements including measurements of the induced magnetostrictive strains on an EuSe 9 single crystal (1 x 1 x 3 mm 3) and the oscillatory magnetostriction of a 6 x 6 x 35 mm 3 indium single crystal, 10 a material which is extremely sensitive to elastic stresses. Length changes at the transition from normal to superconducting state were also measured with these devices. 11 The adjustable gap facility has proved to be extremely useful especially for materials with very large thermal expansion coefficient like zinc along the hexagonal axis. For example, it would have been very difficult to measure the stress GB is with Laboratory of Applied Physics and Measuring Techniques, Brown Boveri, 5401 Baden, Switzerland and RG is with Laboratorium for Festk~perphysilk, Eidgen6ssische Technische Hochschule 8049 Zurich, Switzerland. Received 26 January 1973.
CRYOGENICS
. MAY
1973
dependence of the suscel~tibility of zinc between 4.2 K and room temperature 1~ without an adjustable gap. Using a three terminal method 1,5 including a capacitance bridge (General Radio type 1615 A), these two dilatometers make it possible to detect relative length changes of the order of 10"10. Fig.1 shows a typical recorder tracing obtained with the dilatometer I (see below). This curve represents the oscillatory magnetostrictive strains of a zinc single crystal as a function of the applied field. 13 Curves of similar quality have been obtained with the dilatometer II. Choice of the construction materials The two dilatometers described have been designed primarily for magnetostrictive strain measurements. Thus beside appropriate mechanical properties the material used should have a low electrical conductivity in order to minimize eddy current effects due to magnetic field sweeps. The material should also be non-magnetic. Further because these devices were intended for use at low temperatures the thermal conductivity should not be too small since a uniform temperature over the whole sample holder is needed. To fulfil all these requirements all the parts, including the screw, of the dilatometers were machined from a beryllium-copper alloy called Berylco 25 (obtained from Kawecki Berylco Industries, Reading, Pa,
5 × 1 0 .9
2 kO~
3k0¢
Fig.1
Oscillatory magnetostrictive strains in a single crystal The applied magnetic field is parallel to the hexagonal axis while the length changesare measured in the basal plane. The bar gives the scale for the relative length changes ( 1 0 e = 79.5 A m "1 )
of zinc.
299
USA). Values of the room temperature magnetic susceptibility of different Berylco alloys are given elsewhere. 14 The springs used in our design was also made of Berylco foil because of its extremely low elastic hysteresis.
Dilatometer
I
This dilatometer is basically similar to that used by Br/indli 5 for length change measurements associated with the normal-superconducting transition in I n - P b alloys. Our new design, however, ensures a better parallelism of the capacitor plates and includes an adjustable gap system. The four principal parts of the dilatometer I shown in Fig.2 are the hollow cylindrical body 1, the moveable part 2 ftxed to 1 with two ring springs 2d to produce a parallel displacement 15,16 of the upper capacitor plate 2c, the adjustable gap system 3, and the lower capacitor
3o 3b 3c 3d 3e
plate system 4. The gap between the capacitor plates 2c and 4a can be adjusted with the screw 3c which moves the plunger 3e, the sample 5, and therefore the capacitor plate 2c. With the connexion 3b and the rod 3a, it is possible to turn the screw 3c from the top of the cryostat. For cooling it is necessary to evacuate the vessel containing the dilatometer (or to have helium as exchange gas in it) because any traces of solid air would hinder the free movement of the screw and the plunger at low temperatures. For the same reason the whole adjustable gap system 3 must be clean and grease-free. This dilatometer was designed with the best possible parallelism of the capacitor plates. This was achieved by lapping simultaneously the capacitor plate 2c, the guard ring head 2b, and the lower edge of the body 1. For this operation we replaced 3e and the sample by a longer plunger and by means of the screw 3c we brought the unlapped capacitor plate to the level of the lower edge of 1, the ring springs being loaded. After lapping the equilibrium position of part 2 is about 1 mm higher than in the lapping position. The capacitor plate 4a isolated from 4b by an Araldite epoxy resin film, and the holder 4b were also lapped together. In this way a very good parallelism of the capacitor plates could be achieved so that small gaps of the order of 0.01 mm could be used for the measurements, thus increasing the sensitivity of the device. The parallelism of the capacitor plates was checked in two different ways. First we measured the capacitance as a function of the number of turns of the screw 3c. This way the value of the capacitance could be changed continuously between 5 pF and about 150 pF before some sort of electric breakthrough occured between the plates. The experimental values so determined are in good agreem ent with the theoretical results obtained with 1
C= - - + d d + 0.22w
2d 2a 2d 2b 2c
1+
(1)
where C is the value of the capacitance for a gap d, r represents the radius of the smaller capacitor plate 2c, and w is the gap between this plate and the guard ring 2b. For our sample holder r = 0.9 cm and w = 0.01 cm. The second test consisted of measuring the same effect (for example the amplitude of the oscillatory magnetostrictive strains of an aluminium single crystal) using different gaps. For the conversion of the observed capacitance changes, Ac, into length changes AI, we used the following relation which is derived directly from (1)
4a
4b
AI=--Ad=AC
+ (d+0.22w) 2
1+
(2)
4c Fig.2
Dilatometer I 1 - f i x e d p a r t ; 2a - - free p a r t ; 2 b - - g u a r d head; 2c -- u p p e r c a p a c i t o r p l a t e ; 2 d -- ring springs; 3a -- string f r o m t h e t o p o f the c r y o s t a t ; 310 - - c o u p l i n g ; 3 c - - screw; 3 d - - b o d y o f the adjustable gap system; 3e - - p l u n g e r ; 4 a - - l o w e r c a p a c i t o r plate; 4 b - - l o w e r c a p a c i t o r p l a t e h o l d e r ; 4 c - - b o d y o f t h e l o w e r plate system; 5 - - sample
300
It was found that the value of the gap did not affect the magnitude of the measured strains. For example, the length change observed at 10 pF was equal within the limits of experimental error to the length change measured at 100 pF. The analysis of these two test runs showed that the capacitor plates are parallel to within 10-4 rad.
C R Y O G E N I C S . M A Y 1973
II
Dilatometer
This is an entirely new type of capacitance dilatometer. The main difference from dilatometer I is that the capacitor plates do not remain parallel but rotate when the length of the sample changes. A schematic picture of the device is shown in Fig.3. Fig.4 shows an exploded view of the dilatometer II which was used to measure transverse and parallel magnetostriction in the gap of a conventional electromagnet. It consists of three different parts: the body la with the fixed capacitor plate lb, the moveable part 2c with the capacitor plate 2b, and the adjustable gap system 3. To carry out measurements the sample is weakly clamped between point A o f part 2c and point B of part 3c. The moveable part 2c is fixed to the body la by two different springs: the stronger spring 2d acts as rotation axis while the weak spring 2a increases the lateral stiffness for movements parallel to the plane of the capacitor plates. The plates which are isolated with an Araldite epoxy resin film have been lapped before mounting and the gap is chosen as zero when the plates are parallel. The adjustable gap system 3 transforms a screw turn (3a) in a small horizontal displacement of the block 3c, and therefore changes the angle between the two plates by a small amount. It was therefore possible to calibrate the dilatometer by measuring the capacitance as a function o f the screw turns, that is, as a function o f the angle between the plates. It was found that the calibration curve is in excellent agreement with the theoretical curve as long as the mean gap is not larger than 0.5 mm.
3a )
3b
2a
2c
la
For larger mean gap values the edge distortions o f the electric field are no longer negligible. The theoretical curve was obtained from the following relation 17 C = - In
q~
1+
2d (3) Fig.4 Dilatometer I I l a -- fixed part; 1 b -- fixed capacitor plate; 2a - spring for lateral stiffness; 2b -- free capacitor plate; 2c -- moveable part; 2d suspension spring; 3a -- screw; 3b -- fork (vertical displacements); 3c -- block (horizontal displacements) For measurement the sample is weakly clamped between A and B
Capacity plates
where C is the capacitance of two rectangular plates of area a, b which enclose an angle ~. The different lengths in this equation and the following two relationships, are explained in Fig.3. For small angles between the plates r TM S I + S 2 because the gap is zero when the plates are parallel. So a very good approximation is
s2
s
i ampie
C~
x In 1- 2e
1+ - S1 + S2
(4)
From (4) it follows immediately that AC A/~
( 1 - 2e)
(5)
C Fig.3 Schematic picture of dilatometer II for the calculation of its capacitance and for the relation between the length changes of the sample and the corresponding capacitance changes
CRYOGENICS.MAY 1973
where Al is the length change corresponding to the capacitance change, AC. Note that for this type of dilatometer
301
A / a n d AC are o f opposite signs. This equation offers the same degree o f accuracy as (1).
References
Conclusions Experimental values obtained with the two dilatometers on the same sample did not show any deviation greater than the experimental error of 4%. A non-parallel plate dilatometer is therefore as reliable and precise as a more conventional parallel plate dilatometer, the sensitivity being the same, 10"10 for relative length changes. Both dilatometers are compact and insensitive to mechanical vibrations. We are greatly indebted to Prof J. L. Olsen for his advice and help with several aspects of the construction of the dilatometers. It is also a pleasure to thank A. Kundig for much help in the design o f the devices and A. Vogelsanger for his high precision machining. This work was supported in part financially by the Schweizerischer Nationalfonds.
302
10 11 12 13 14 15 16 17
White, G. K. Cryogenics 1 (1961) 151 Fawcett, E. Phys Rev B2 (1970) 1604 Green, B. A., Chadrasekhat, B. S. PhysRev Lett 11 (1963) 331 Reitz, L. M., Spatlin, D. M. PhysRevB 5 (1972) 3803 Brandli, G. Phys kondensMaterie 11 (1970) 93 and 111 White, G. K., Collins, J. G. J L o w Temp Phys 7 (1972) 43 White, G. K. JPhys C: Solid StatePhys 5 (1972) 2731 Fawcett, E. private communication Gtiessen, R., Landolt, M., Ott, H. R. Solid State Comm 9 (1971) 2219 Griessen, R., Sorbello, R. S. (forthcoming) Ott, H. R. SolidState Comm 9 (1971) 2225 Griessen, R., Ott, H. R. PhysRev Lett 29 (1972) 1150 Griessen, R., Kiindig, A. Solid State Comm 11 (1972) 295 Griessen, R. Cryogenics (forthcoming) Jones, R. V. JScilnstr 39 (1962) 193 Braddik, H. J. J. The Physics of Experimental Method (Chapman and Hall, 1954) 72 Landau, L. D., Lifshitz, E. M. Elcctrodynamics of Continuous Media (Pergamon Press, 1960) 100
C R Y O G E N I C S . M A Y 1973