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Mutual inductance bridge for magnetic relaxation measurements at low temperatures
Journal of Magnetism and Magnetic Materials 28 (1982) 341-347 North-Holland Publishing Company
341
MUTUAL INDUCTANCE BRIDGE FOR MAGNETIC RELAXATION MEASUREMENTS AT LOW TEMPERATURES P.H. MULLER, M. SCHIENLE and A. KASTEN Physikalisches Institut, UniversitiJt Karlsruhe, Fed. Rep. Germany Received 14 April 1982
An apparatus using a mutual inductance bridge has been designed to measure real and imaginary parts of the magnetic ac susceptibility simultaneously in the frequency range from 1 Hz to 230 kHz at helium temperatures. From 230 kHz up to 3 MHz the real part can be determined. The susceptibility as a function of temperature or external magnetic field at constant frequency may be measured. In addition, frequency sweeps can be performed with temperature and magnetic field held constant. The experiment is controlled by an on-line computer.
1. Introduction
2. Bridge design
Mutual inductance bridges are commonly used for magnetic thermometry [1,2] and measurements of the ac susceptibility [3-8]. Most of them are optimised for high sensitivity in a limited frequency range or even at fixed frequency. For the measurement of magnetic relaxation phenomena by the dispersion-absorption method it is important to have a wide frequency range available. Moreover, it is favourable to perform frequency sweeps while the thermodynamically relevant parameters, as temperature and magnetisation, are held constant. These conditions are realised with the help of an on-line computer controlling temperature and magnetic field as well as frequency and amplitude of the ac field. The unavoidable apparative offsets are not compensated by sophisticated electronic circuitry but accounted for by the computer program. The following section gives a brief description of the bridge design, in section 3 the main features of the cryostat are discussed. In the next section it is shown how to compute the complex susceptibility from the measured signals. The expansion of the frequency range up to 3 MHz is discussed in section 5.
A schematic drawing of the experimental set up is given in fig. 1. A Philips synthesizer PM 5190 is used as a signal generator. The primary part of the coil system is driven directly, without the use of any transformers. The output of the secondary coils is detected by a two-phase lock-in amplifier (Ithaco dynatrac 393). The outputs of the detector are monitored with two panelmeters (41 digits), which are connected to the on-fine computer sys-
Room fomperatuPe
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Fig. 1. Scheme of the experimental set up (the labels are the same as in fig. 2).
0304-8853/82/0000-0000/$02.75 © 1982 North-Holland
342
P.H. Miiller et al. / A mutual inductance bridge
Table 1 Coil data
Frequency range Primary turns Secondary turns O of copper wire Resonance frequency Absolute reproducibility
Smallest detectable signal change
Coil set 1 (low frequency)
Coil set 2 (high frequency)
Coil set 3
1 Hz to 10 kHz 1 X 1200 1 X 720; 2 X 360 0.08 m m 50 kHz 0.2%-+ 1 X 10 -5 emu ( 10-300 Hz) 0.2%-+ 1X 10 - 4 emu ( 1 H z - 1 0 kHz) 10 7 e m u (at 200 Hz)
100 Hz to 230 kHz 2 X 75 2 X 75 0.2 m m 1.2 M H z 0.2%--+2X 10 - s emu (200 H z - 10 kHz) 1 % ÷ 2 x l0 4 emu (100 H z - 2 3 0 kHz) 2 x 10 - 7 emu (at 2 kHz)
50 kHz to 3 M H z 1 × 25 1 X 50 0.2 m m 4 MHz
tem (LSI 11). The computer also controls frequency and amplitude of the signal generator. The frequency range of a bridge circuit is limited towards high frequencies by the eigenresonance of the coil system, which is caused by the coil inductance and by the winding capacitance. As the eigenfrequency is approached, the apparative offset increases dramatically. On the other hand, the sensitivity is reduced by partial capacitive shortening of the coils. For a given coil system, the low frequency limit is determined by the sensitivity of the amplifier. A coil system can only be optimised for a limited frequency range. More windings of the secondary coils yield a better sensitivity at low frequencies, but also reduces the resonance frequency of the assembly. In our experiments, we used two different coil systems for the frequency ranges 1 Hz to 10 kHz and 100 Hz to 230 kHz. Typical values of the coil data are given in table 1. The coil systems are wound onto glass tubes (inner diameter 5 mm, outer diameter 7 mm). Each layer is fixed with rubber cement (Marabu, Fixogum). The low frequency set consists of a continuously wound primary coil (length 50 mm; 4 layers with 300 turns each). The secondary system on top of the primary is composed of three coils, a central one (20 mm; 4 X 180 turns) containing the sample, compensated by two others (100 mm; 4 X 90 turns) above and below the first.
10 -5 emu (at 1 MHz) 10 - 3 e m u (at 3 MHz)
The high frequency set is made of two identical pairs of coils. The secondaries (8 mm; 3 × 25 turns) are wound on top of the corresponding primaries (8 mm; 3 X 25 turns). The centres of the two pairs are separated by 16 mm. After winding the coils, balance of the bridge is achieved by adding or removing some windings from one of the secondary coils. In the next step, the coil set is mounted inside the cryostat and cooled down to liquid nitrogen temperature. The drift of the output signal is measured and corrected for after warming up the assembly again. By this procedure, most of the temperature drifts and the influence of the materials surrounding the coils can be compensated. As can be seen in fig. 1, no attempt is made to null the bridge by an additional compensation network. The signal change induced by the sample is monitored directly. There are mainly three reasons for this: i) the additional components would cause stability problems and increase the noise level within the bridge circuit, ii) the compensation is time-consuming and difficult to automate, iii) the compensation circuits would be strongly frequency dependent and limit the usable frequency range. The details of the procedure of calculating the complex susceptibility from the output signals are given in section 4.
P.H. Mailer et al. / A mutual inductance bridge
3. Cryostat The measurements are performed inside a 4He exchange gas cryostat. Fig. 2 gives a schematic drawing of the inner part. Primary (1) and secondary coils (2) as mentioned above are wound onto a glass tube (3). The sample (4) can be moved vertically inside the tube with the help of the stainless steel sample rod (5). To avoid eddy currents, the end of the rod is made of plexiglass. An Allen-Bradley carbon resistor (6) is used as a thermometer and thermally coupled to the sample by two thin copper wires (0.2 mm). The cryostat incorporates three different helium spaces: the liquid 4He reservoir (7), the inner 4He zone (8), and the sample space (9). The inner 4He zone can be filled with liquid helium via a needle
343
valve (10), and is separated from the helium reservoir by vacuum (11). Usually, the sample chamber is filled with 4He exchange gas at a pressure of about 0.1 mbar and cooled down to 1.15 K by pumping the inner 4He zone. The temperature is regulated by the on-line computer with the help of the heater (12) surrounding coil system and sample. Thus, a temperature stability of 2 m K is achieved below 5 K. The temperature resolution is better than 1 mK, the absolute accuracy of the calibration is about 0.03 K. A 3He insert allowing temperatures down to 0.5 K is under construction. An external static field up to 5 T can be produced by a superconducting magnet (13) inside the 4He reservoir with a homogeneity of 0.03% over 10 mm DSV. The field is measured by the magnet current via a 1 mfl shunt and is also controlled by the computer. To switch from the high frequency to the low frequency design and vice versa, the whole sardple chamber, including coil system and heater, is replaced.
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-_i Fig. 2. Inner part of the helium cryostat (not to scale!). (1) primary coil, (2) secondarycoil, (3) glass tube, (4) sample, (5) sample rod, (6) carbon resistor, (7) liquid 4He reservoir, (8) inner 4He zone, (9) sample space, (10) needle valve, (il) isolatiort vacuum, (12) heater, (13) 5 T magnet.
The ac susceptibility can be measured as a function of either temperature, magnetic field, or frequency of the driving field. Usually, two of these parameters are held constant during a measurement, and the third one is varied. For each frequency, prior to the actual measurement, two calibrations of the coil system are necessary: i) the (complex) offset of the arrangement without sample, but with the sample rod installed, is measured, ii) phase and gain of the coil system is derived inserting a "laboratory standard" crystal (GdVO4; 7.2 × 10 -3 emu). Finally, the complex susceptibility of the sample is measured. The amplitude of the driving field as well as sensitivity and phase settings of the lock-in amplifier have to be identical for the calibration process and the actual measurement. GdVO 4 was chosen as laboratory standard, because it is a Heisenberg antiferromagnet and shows an almost constant susceptibility X' below its ordering temperature TN = 2.49 K [9]. No absorption is found over the entire frequency range
344
P.H. Miiller et aL / A mutual inductance bridge
X " COS
solder points of the temperature sensor which are present in measurements at Bext < 0.1 T and T < 7 K. If experiments in zero external field are required, an additional measurement of the offset at Bext = 0, which has to be subtracted from the signal, is sufficient.
(X+Olcos~o
(X÷O]
sin~O
4.1. Temperature sweeps
"
Fig. 3. Illustration of the relations between complex susceptibility of the sample (x'(S) and X"(S)), complex offset of the apparatus (O) and the signals induced by the laboratory standard crystal (LS; defining the real axis) and the sample (S; complex). All these values are frequency dependent.
covered by our experiments. The crystal we use is a cylinder with a diameter of 2.5 mm, a length of 4.9 mm and a mass of 120.5 mg and with its axis along the crystallographic a-axis of GdVO 4 (perpendicular to the ordering direction). The cylinder is calibrated against a nickel sphere inside a commercial Foner magnetometer (PAR FM- 159). Fig. 3 illustrates, how the complex susceptibility is derived from the three measurements. At first, the complex offset (O) is subtracted from the signals of the laboratory standard (LS) and the sample (S), i.e. the coordinate system is shifted. Secondly, the real part of the sample susceptibility x'(S) is evaluated taking the projection (or scalar product) of the sample signal onto that of the laboratory standard: S LS , x'(S) = LS ~ X (LS).
(1)
The imaginary part of the sample susceptibility is then given by the formula:
X"(S) = ( ( x ' ( L S ) • S / L S ) 2 - x'(S) 2) 1/2.
(2)
The measurement of the complex offset and the laboratory standard are performed at T = 1.5 K and B~xt = 0.2 T and may be used for the temperature range from 1.15 K up to about 15 K and for external fields up to 1 T. At higher temperatures or fields additional calibrations may be necessary because of apparative drifts. The field of 0.2 T is applied during the calibration procedure in order to eliminate the signals caused by superconducting
Measurements of the ac susceptibility as a function of temperature in fixed external magnetic field and frequency are performed as follows: i) Apparative offset and gain for the desired frequency are determined as described above, ii) During the actual measurement the on-line computer controls the temperature such that a constant temperature drift is achieved. This is realised by a PID algorithm with a set point drifting linearly with time. In each measuring cycle the temperature is calculated from the carbon resistor with a polynomial of the form: 1
6
y = E A, (logR)'. i=0
For the final recording of the data, the temperature axis is divided into consecutive windows (typical widths: 2 to 50 mK). All data in a temperature window are added up. When the window is left, the mean values of temperatures and signals are taken and the complex susceptibility is computed on-line using eqs. (1) and (2). All data are recorded on floppy disk. The signal voltmeters are read at a rate of 12Is. The time constant of the lock-in amplifier is adapted to the fast reading rate (typically 40 ms) and integration of the signals is taken over by the computer. This is advantageous for two reasons: i) A large time constant of the lock-in amplifier would delay the signals and cause an apparative hysteresis, ii) There is a practicable possibility to eliminate perturbating pulses: If the difference between the actual value of the data and the preceding values exceeds a certain limit, the data are rejected. The limit is determined on-line by the computer from the signal noise and readjusted continuously.
P.H. Mi~ller et al. / A mutual inductance bridge
Fig. 4. Temperature dependence of the real part (X') of the ac susceptibility of a DyVO 4 sphere (Boxt =0). (a) ~ = 109 Hz; (b) p = 1.10 kHz; (c) ~ =3.03 kHz; (d) p = 10.9 kHz; (e) ~=30.4 kHz; (f) ~=102 kHz; (g) ~=227 kHz; (h) u=350 kHz; (i) u = 6 2 0 kHz; (k) ~ = 9 0 0 kHz; (1) p = l . 2 MHz; (m) p = l . 7 MHz. Curves a - g are obtained with coil design 2 (see table 1) curves h - m with coil design 3 and the procedure described in section 5.
Curves a - g of fig. 4 show typical temperature sweeps at different frequencies on DyVO4 in a temperature and frequency range where there is a relaxation step due to an Orbach spin-lattice process (for details see ref. [10]; curves h - m are discussed in section 5).
4. 2. Sweeps of the external field In the case of field dependent measurements, the external field is not swept continuously, but changed step by step. After each field step the system is given an appropriate time to settle down (1-3 s). The signals are integrated for a few seconds depending on the signal-to-noise ratio and recorded as above. The temperature is stabilised by the PID algorithm. Below 5 K, the temperature stability is better than 2 mK and may be maintained over some hours.
4.3. Frequency sweeps The determination of relaxation times of magnetic materials with the help of ac susceptibility requires the measurement of the dispersion and absorption curves at fixed temperature and external field over a suitable frequency range. In our
345
experiments this is done in a step by step method starting at low frequencies. If the current through the primary coil would be held constant, the induced voltage would constantly increase with increasing frequency, and the sensitivity setting of the lock-in detector had to be changed during the measurement. To avoid this, we reduce the amplitude of the driving field with increasing frequency such, that all settings of the amplifier can be held constant during the sweep and its full range can be used. The procedure is as follows: At first, the laboratory standard crystal is inserted, the lowest frequency applied, and the amplitude of the driving field is regulated that the detector shows the desired value. Frequency, amplitude and the output signals of the lock-in amplifier are recorded. In the next step, the frequency is raised, the same procedure takes place, and so on up to the highest frequency. All this, including the procedure of finding the appropriate amplitudes, is fully automatic. The next frequency' run consists of the measurement of the offset of the arrangement using exactly the values of frequency and amplitude recorded above, just with an empty sample rod installed. Finally, the sample is inserted and measured, applying again the same frequency and amplitude values. In this final run, the complex susceptibility of the sample is computed on-line using eqs. (1) and (2) of the previous section. The absolute error of the frequency setting is less than 10 -6 , the reproducibility of the amplitude is better than 0.2%. Hence, nearly identical conditions can be maintained during these three runs. In this way, frequency dependent phase shifts, including the influence of capacitive losses and the phase response of the lock-in detector are eliminated. The amplitude of the driving field can be kept below 0.16 mT even at the lowest frequencies, which is sufficiently small for our purposes (this value could be reduced, if necessary). Experiments using the coil designs and measuring routines outlined above have up to now been carried out on DyVO 4 [10], DyPO 4, Dy202SO 4 [11] and TbAsO 4 [12]. Fig. 5 shows a typical frequency run on a DyVO4-sphere around a relaxation step (Orbach spin-lattice process; for details see ref.
P . H . Miiller et al. / A m u t u a l i n d u c t a n c e bridge
346
I
[
t
2
Fig. 5. Real (O) and imaginary (©) part of the susceptibilityof a DyVO4 sphere at T=3.0 K and Bext =0.2 T parallel to the crystallographic a-axis as a function of frequency. Drawn curves: X' calculated from X" by the Kramers-Kronig-relation and vice versa.
LS/O
I
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I 2
I 3
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i
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IG
5
0•
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[reAl
i
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I log ( v / H z ) i
b
15
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0
[lO]). The relaxation is not exactly of the Debye type. The relation between X' and X" may, however, be tested by calculating X" from X' and vice versa using the K r a m e r s - K r o n i g relation (drawn curves of fig. 5). Fig. 6a shows the frequency dependent ratio of the moduli of standard signal and offset over the entire frequency range of the apparatus. The ratio being large at low frequencies, decreases as the eigenresonance of the set up is approached. Thus in a frequency run an increasingly larger portion of the panelmeter output is due to the offset rather than the sample susceptibility. Once the ratio gets too small, the reproducibility of the experiment suffers, because noise and apparative drifts are mainly proportional to the overall signal. On the other hand, at low frequencies, where the offset is no problem, the signal decreases since the induced voltage is proportional to the frequency and the integration times need to be larger. This may partially be compensated by increasing the excitation a n d / o r the amplifier sensitivity. There are, however, practical limits because of the problem of either heating the sample by the excitation current or increasing the electronic noise. The use of the two designs described above seems a good compromise for the frequency range from 1 Hz to 230 kHz. The frequency dependence of the excitation current as applied in typical experiments is given in fig. 6b. The limits 8 and 20 mA for the low and high frequency coils are to prevent heating of the sample. In general, the exciting field is kept below 0.16 mT, i.e. below 1300 Hz the low frequency design is used. The magnetic field values (for the coil centres) indicated by arrows are calculated from the excitation current and the geometric dimensions of the coils (although the coils are quite different, the conversion factors are nearly identical).
5. High frequency measurements I
1
2
3
I.
log ( v / H z )
Fig. 6. (a) R a t i o o f the m o d u l i o f the signal i n d u c e d by the l a b o r a t o r y s t a n d a r d c r y s t a l ( L S ) and the o f f s e t ( O ) as a f u n c t i o n of frequency for the two coil designs; (b) excitation current
as applied in a frequency run.
In the course of our experiments it turned out to be desirable to expand the available frequency range to higher frequencies. We describe a simple method to do this with only small modifications of the experimental set up. It is based on the shift of
P.H. Miiller et al. / A mutual inductance bridge
the resonance frequency of an LC-circuit by the real part of the sample susceptibility. The assembly consists of a driven harmonic oscillator, the amplitude of which is monitored by an hf amplifier and rectifier. In a linear approximation the change of the amplitude is given by AA = (dA/dp)A1, cc X'-
(4)
To achieve maximum changes of the oscillator amplitude, the capacitance of the circuit is chosen such, that the amplitude is about half of that at the resonance frequency ( d A / d p = max). The coil of the LC-circuit may be one of the pick-up coils of the high frequency design (usable up to 1 MHz). Alternatively, a different arrangement consisting of only one pair of coils (8 mm length; 1 × 20 and 1 × 50 turns, coil system 3 of table 1) is available. This entire set up, including leads and with generator and amplifier connected, has an eigenfrequency of about 4 MHz, which is shifted proportional to X' of the sample. The resonance frequency of the assembly may be decreased by an external capacitor. In this way, measurements from 50 kHz up to 3 MHz can be performed. From 50 kHz to 230 kHz we get curves of X' identical to those measured with the bridge. The reproducibility over larger periods of time is, however, rather poor, although the short term stability (up to 1 h) is satisfactory. No attempt has been made to calibrate the set up. Thus, up to now, only temperature and field sweeps of X' (not frequency sweeps) in arbitrary units have been performed above 230 kHz. Curves h - m of fig. 4 are derived with this
347
method. Zero is determined by the low temperature value (T < 2 K) of X' and the scaling factor by assuming that all curves meet asymptotically at high temperatures ( T > 10 K).
Acknowledgement We are indebted to Prof. H.G. Kahle for valuable discussions.
References [1] A.C. Anderson, R.E. Peterson and J.E. Robichaux, Rev. Sci. Instr. 41 (1970) 528. [2] S.C. Whitmore, S.R. Ryan and T.M. Sanders, Rev. Sci. Instr. 49 (1978) 1579. [3] H.B.G. Casimir, D. Bijl and F.K. Du Pr6, Physica 8 (1941) 449. [4] W.L. Pillinger, P.S. Jastram and J.G. Daunt, Rev. Sci. Instr. 29 (1958) 159. [5] C.M. Brodbeck, R.R. Barkrey and J.T. Hoeksema, Rev. Sci. Instr. 49 (1978) 1279. [6] J.M. Dundon and M.E. Kretschmar, Rev. Sci. Instr. 49 (1978) 406. [7] M. Kumano and Y. Ikegami, Rev. Sci. Instr. 50 (1979) 921. [8] H.A. Groenendijk, A.J. van Duyneveldt and R.D. Willett, Physica 101B (1980) 320. [9] J.D. Cashion, A.H. Cooke, L.A. Hod, D.M. Martin and M.R. Wells, Colloq. du CNRS No. 180 II (1970) 417. [10] A. Kasten, P.H. Miiller and M. Schienle, Physica ll4B (1982) 77. [11] A. Kasten, P.H. MOiler and M. Sehienle, Physica ll4B (3) (1982) in press. [12] A. Kasten, Solid State Commun. 32 (1979) 973. P.H. Miiller, A. Kasten and M. Schienle, to be published.
341
MUTUAL INDUCTANCE BRIDGE FOR MAGNETIC RELAXATION MEASUREMENTS AT LOW TEMPERATURES P.H. MULLER, M. SCHIENLE and A. KASTEN Physikalisches Institut, UniversitiJt Karlsruhe, Fed. Rep. Germany Received 14 April 1982
An apparatus using a mutual inductance bridge has been designed to measure real and imaginary parts of the magnetic ac susceptibility simultaneously in the frequency range from 1 Hz to 230 kHz at helium temperatures. From 230 kHz up to 3 MHz the real part can be determined. The susceptibility as a function of temperature or external magnetic field at constant frequency may be measured. In addition, frequency sweeps can be performed with temperature and magnetic field held constant. The experiment is controlled by an on-line computer.
1. Introduction
2. Bridge design
Mutual inductance bridges are commonly used for magnetic thermometry [1,2] and measurements of the ac susceptibility [3-8]. Most of them are optimised for high sensitivity in a limited frequency range or even at fixed frequency. For the measurement of magnetic relaxation phenomena by the dispersion-absorption method it is important to have a wide frequency range available. Moreover, it is favourable to perform frequency sweeps while the thermodynamically relevant parameters, as temperature and magnetisation, are held constant. These conditions are realised with the help of an on-line computer controlling temperature and magnetic field as well as frequency and amplitude of the ac field. The unavoidable apparative offsets are not compensated by sophisticated electronic circuitry but accounted for by the computer program. The following section gives a brief description of the bridge design, in section 3 the main features of the cryostat are discussed. In the next section it is shown how to compute the complex susceptibility from the measured signals. The expansion of the frequency range up to 3 MHz is discussed in section 5.
A schematic drawing of the experimental set up is given in fig. 1. A Philips synthesizer PM 5190 is used as a signal generator. The primary part of the coil system is driven directly, without the use of any transformers. The output of the secondary coils is detected by a two-phase lock-in amplifier (Ithaco dynatrac 393). The outputs of the detector are monitored with two panelmeters (41 digits), which are connected to the on-fine computer sys-
Room fomperatuPe
oA c f
'1
t,
Helium tomperofure
" ~
I
Fig. 1. Scheme of the experimental set up (the labels are the same as in fig. 2).
0304-8853/82/0000-0000/$02.75 © 1982 North-Holland
342
P.H. Miiller et al. / A mutual inductance bridge
Table 1 Coil data
Frequency range Primary turns Secondary turns O of copper wire Resonance frequency Absolute reproducibility
Smallest detectable signal change
Coil set 1 (low frequency)
Coil set 2 (high frequency)
Coil set 3
1 Hz to 10 kHz 1 X 1200 1 X 720; 2 X 360 0.08 m m 50 kHz 0.2%-+ 1 X 10 -5 emu ( 10-300 Hz) 0.2%-+ 1X 10 - 4 emu ( 1 H z - 1 0 kHz) 10 7 e m u (at 200 Hz)
100 Hz to 230 kHz 2 X 75 2 X 75 0.2 m m 1.2 M H z 0.2%--+2X 10 - s emu (200 H z - 10 kHz) 1 % ÷ 2 x l0 4 emu (100 H z - 2 3 0 kHz) 2 x 10 - 7 emu (at 2 kHz)
50 kHz to 3 M H z 1 × 25 1 X 50 0.2 m m 4 MHz
tem (LSI 11). The computer also controls frequency and amplitude of the signal generator. The frequency range of a bridge circuit is limited towards high frequencies by the eigenresonance of the coil system, which is caused by the coil inductance and by the winding capacitance. As the eigenfrequency is approached, the apparative offset increases dramatically. On the other hand, the sensitivity is reduced by partial capacitive shortening of the coils. For a given coil system, the low frequency limit is determined by the sensitivity of the amplifier. A coil system can only be optimised for a limited frequency range. More windings of the secondary coils yield a better sensitivity at low frequencies, but also reduces the resonance frequency of the assembly. In our experiments, we used two different coil systems for the frequency ranges 1 Hz to 10 kHz and 100 Hz to 230 kHz. Typical values of the coil data are given in table 1. The coil systems are wound onto glass tubes (inner diameter 5 mm, outer diameter 7 mm). Each layer is fixed with rubber cement (Marabu, Fixogum). The low frequency set consists of a continuously wound primary coil (length 50 mm; 4 layers with 300 turns each). The secondary system on top of the primary is composed of three coils, a central one (20 mm; 4 X 180 turns) containing the sample, compensated by two others (100 mm; 4 X 90 turns) above and below the first.
10 -5 emu (at 1 MHz) 10 - 3 e m u (at 3 MHz)
The high frequency set is made of two identical pairs of coils. The secondaries (8 mm; 3 × 25 turns) are wound on top of the corresponding primaries (8 mm; 3 X 25 turns). The centres of the two pairs are separated by 16 mm. After winding the coils, balance of the bridge is achieved by adding or removing some windings from one of the secondary coils. In the next step, the coil set is mounted inside the cryostat and cooled down to liquid nitrogen temperature. The drift of the output signal is measured and corrected for after warming up the assembly again. By this procedure, most of the temperature drifts and the influence of the materials surrounding the coils can be compensated. As can be seen in fig. 1, no attempt is made to null the bridge by an additional compensation network. The signal change induced by the sample is monitored directly. There are mainly three reasons for this: i) the additional components would cause stability problems and increase the noise level within the bridge circuit, ii) the compensation is time-consuming and difficult to automate, iii) the compensation circuits would be strongly frequency dependent and limit the usable frequency range. The details of the procedure of calculating the complex susceptibility from the output signals are given in section 4.
P.H. Mailer et al. / A mutual inductance bridge
3. Cryostat The measurements are performed inside a 4He exchange gas cryostat. Fig. 2 gives a schematic drawing of the inner part. Primary (1) and secondary coils (2) as mentioned above are wound onto a glass tube (3). The sample (4) can be moved vertically inside the tube with the help of the stainless steel sample rod (5). To avoid eddy currents, the end of the rod is made of plexiglass. An Allen-Bradley carbon resistor (6) is used as a thermometer and thermally coupled to the sample by two thin copper wires (0.2 mm). The cryostat incorporates three different helium spaces: the liquid 4He reservoir (7), the inner 4He zone (8), and the sample space (9). The inner 4He zone can be filled with liquid helium via a needle
343
valve (10), and is separated from the helium reservoir by vacuum (11). Usually, the sample chamber is filled with 4He exchange gas at a pressure of about 0.1 mbar and cooled down to 1.15 K by pumping the inner 4He zone. The temperature is regulated by the on-line computer with the help of the heater (12) surrounding coil system and sample. Thus, a temperature stability of 2 m K is achieved below 5 K. The temperature resolution is better than 1 mK, the absolute accuracy of the calibration is about 0.03 K. A 3He insert allowing temperatures down to 0.5 K is under construction. An external static field up to 5 T can be produced by a superconducting magnet (13) inside the 4He reservoir with a homogeneity of 0.03% over 10 mm DSV. The field is measured by the magnet current via a 1 mfl shunt and is also controlled by the computer. To switch from the high frequency to the low frequency design and vice versa, the whole sardple chamber, including coil system and heater, is replaced.
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-_i Fig. 2. Inner part of the helium cryostat (not to scale!). (1) primary coil, (2) secondarycoil, (3) glass tube, (4) sample, (5) sample rod, (6) carbon resistor, (7) liquid 4He reservoir, (8) inner 4He zone, (9) sample space, (10) needle valve, (il) isolatiort vacuum, (12) heater, (13) 5 T magnet.
The ac susceptibility can be measured as a function of either temperature, magnetic field, or frequency of the driving field. Usually, two of these parameters are held constant during a measurement, and the third one is varied. For each frequency, prior to the actual measurement, two calibrations of the coil system are necessary: i) the (complex) offset of the arrangement without sample, but with the sample rod installed, is measured, ii) phase and gain of the coil system is derived inserting a "laboratory standard" crystal (GdVO4; 7.2 × 10 -3 emu). Finally, the complex susceptibility of the sample is measured. The amplitude of the driving field as well as sensitivity and phase settings of the lock-in amplifier have to be identical for the calibration process and the actual measurement. GdVO 4 was chosen as laboratory standard, because it is a Heisenberg antiferromagnet and shows an almost constant susceptibility X' below its ordering temperature TN = 2.49 K [9]. No absorption is found over the entire frequency range
344
P.H. Miiller et aL / A mutual inductance bridge
X " COS
solder points of the temperature sensor which are present in measurements at Bext < 0.1 T and T < 7 K. If experiments in zero external field are required, an additional measurement of the offset at Bext = 0, which has to be subtracted from the signal, is sufficient.
(X+Olcos~o
(X÷O]
sin~O
4.1. Temperature sweeps
"
Fig. 3. Illustration of the relations between complex susceptibility of the sample (x'(S) and X"(S)), complex offset of the apparatus (O) and the signals induced by the laboratory standard crystal (LS; defining the real axis) and the sample (S; complex). All these values are frequency dependent.
covered by our experiments. The crystal we use is a cylinder with a diameter of 2.5 mm, a length of 4.9 mm and a mass of 120.5 mg and with its axis along the crystallographic a-axis of GdVO 4 (perpendicular to the ordering direction). The cylinder is calibrated against a nickel sphere inside a commercial Foner magnetometer (PAR FM- 159). Fig. 3 illustrates, how the complex susceptibility is derived from the three measurements. At first, the complex offset (O) is subtracted from the signals of the laboratory standard (LS) and the sample (S), i.e. the coordinate system is shifted. Secondly, the real part of the sample susceptibility x'(S) is evaluated taking the projection (or scalar product) of the sample signal onto that of the laboratory standard: S LS , x'(S) = LS ~ X (LS).
(1)
The imaginary part of the sample susceptibility is then given by the formula:
X"(S) = ( ( x ' ( L S ) • S / L S ) 2 - x'(S) 2) 1/2.
(2)
The measurement of the complex offset and the laboratory standard are performed at T = 1.5 K and B~xt = 0.2 T and may be used for the temperature range from 1.15 K up to about 15 K and for external fields up to 1 T. At higher temperatures or fields additional calibrations may be necessary because of apparative drifts. The field of 0.2 T is applied during the calibration procedure in order to eliminate the signals caused by superconducting
Measurements of the ac susceptibility as a function of temperature in fixed external magnetic field and frequency are performed as follows: i) Apparative offset and gain for the desired frequency are determined as described above, ii) During the actual measurement the on-line computer controls the temperature such that a constant temperature drift is achieved. This is realised by a PID algorithm with a set point drifting linearly with time. In each measuring cycle the temperature is calculated from the carbon resistor with a polynomial of the form: 1
6
y = E A, (logR)'. i=0
For the final recording of the data, the temperature axis is divided into consecutive windows (typical widths: 2 to 50 mK). All data in a temperature window are added up. When the window is left, the mean values of temperatures and signals are taken and the complex susceptibility is computed on-line using eqs. (1) and (2). All data are recorded on floppy disk. The signal voltmeters are read at a rate of 12Is. The time constant of the lock-in amplifier is adapted to the fast reading rate (typically 40 ms) and integration of the signals is taken over by the computer. This is advantageous for two reasons: i) A large time constant of the lock-in amplifier would delay the signals and cause an apparative hysteresis, ii) There is a practicable possibility to eliminate perturbating pulses: If the difference between the actual value of the data and the preceding values exceeds a certain limit, the data are rejected. The limit is determined on-line by the computer from the signal noise and readjusted continuously.
P.H. Mi~ller et al. / A mutual inductance bridge
Fig. 4. Temperature dependence of the real part (X') of the ac susceptibility of a DyVO 4 sphere (Boxt =0). (a) ~ = 109 Hz; (b) p = 1.10 kHz; (c) ~ =3.03 kHz; (d) p = 10.9 kHz; (e) ~=30.4 kHz; (f) ~=102 kHz; (g) ~=227 kHz; (h) u=350 kHz; (i) u = 6 2 0 kHz; (k) ~ = 9 0 0 kHz; (1) p = l . 2 MHz; (m) p = l . 7 MHz. Curves a - g are obtained with coil design 2 (see table 1) curves h - m with coil design 3 and the procedure described in section 5.
Curves a - g of fig. 4 show typical temperature sweeps at different frequencies on DyVO4 in a temperature and frequency range where there is a relaxation step due to an Orbach spin-lattice process (for details see ref. [10]; curves h - m are discussed in section 5).
4. 2. Sweeps of the external field In the case of field dependent measurements, the external field is not swept continuously, but changed step by step. After each field step the system is given an appropriate time to settle down (1-3 s). The signals are integrated for a few seconds depending on the signal-to-noise ratio and recorded as above. The temperature is stabilised by the PID algorithm. Below 5 K, the temperature stability is better than 2 mK and may be maintained over some hours.
4.3. Frequency sweeps The determination of relaxation times of magnetic materials with the help of ac susceptibility requires the measurement of the dispersion and absorption curves at fixed temperature and external field over a suitable frequency range. In our
345
experiments this is done in a step by step method starting at low frequencies. If the current through the primary coil would be held constant, the induced voltage would constantly increase with increasing frequency, and the sensitivity setting of the lock-in detector had to be changed during the measurement. To avoid this, we reduce the amplitude of the driving field with increasing frequency such, that all settings of the amplifier can be held constant during the sweep and its full range can be used. The procedure is as follows: At first, the laboratory standard crystal is inserted, the lowest frequency applied, and the amplitude of the driving field is regulated that the detector shows the desired value. Frequency, amplitude and the output signals of the lock-in amplifier are recorded. In the next step, the frequency is raised, the same procedure takes place, and so on up to the highest frequency. All this, including the procedure of finding the appropriate amplitudes, is fully automatic. The next frequency' run consists of the measurement of the offset of the arrangement using exactly the values of frequency and amplitude recorded above, just with an empty sample rod installed. Finally, the sample is inserted and measured, applying again the same frequency and amplitude values. In this final run, the complex susceptibility of the sample is computed on-line using eqs. (1) and (2) of the previous section. The absolute error of the frequency setting is less than 10 -6 , the reproducibility of the amplitude is better than 0.2%. Hence, nearly identical conditions can be maintained during these three runs. In this way, frequency dependent phase shifts, including the influence of capacitive losses and the phase response of the lock-in detector are eliminated. The amplitude of the driving field can be kept below 0.16 mT even at the lowest frequencies, which is sufficiently small for our purposes (this value could be reduced, if necessary). Experiments using the coil designs and measuring routines outlined above have up to now been carried out on DyVO 4 [10], DyPO 4, Dy202SO 4 [11] and TbAsO 4 [12]. Fig. 5 shows a typical frequency run on a DyVO4-sphere around a relaxation step (Orbach spin-lattice process; for details see ref.
P . H . Miiller et al. / A m u t u a l i n d u c t a n c e bridge
346
I
[
t
2
Fig. 5. Real (O) and imaginary (©) part of the susceptibilityof a DyVO4 sphere at T=3.0 K and Bext =0.2 T parallel to the crystallographic a-axis as a function of frequency. Drawn curves: X' calculated from X" by the Kramers-Kronig-relation and vice versa.
LS/O
I
I
l
I
I 1
I 2
I 3
I ~
i
i
i
15
IG
5
0•
o
[reAl
i
/~
i
0 2t~mT
I log ( v / H z ) i
b
15
_g 10 5
0
[lO]). The relaxation is not exactly of the Debye type. The relation between X' and X" may, however, be tested by calculating X" from X' and vice versa using the K r a m e r s - K r o n i g relation (drawn curves of fig. 5). Fig. 6a shows the frequency dependent ratio of the moduli of standard signal and offset over the entire frequency range of the apparatus. The ratio being large at low frequencies, decreases as the eigenresonance of the set up is approached. Thus in a frequency run an increasingly larger portion of the panelmeter output is due to the offset rather than the sample susceptibility. Once the ratio gets too small, the reproducibility of the experiment suffers, because noise and apparative drifts are mainly proportional to the overall signal. On the other hand, at low frequencies, where the offset is no problem, the signal decreases since the induced voltage is proportional to the frequency and the integration times need to be larger. This may partially be compensated by increasing the excitation a n d / o r the amplifier sensitivity. There are, however, practical limits because of the problem of either heating the sample by the excitation current or increasing the electronic noise. The use of the two designs described above seems a good compromise for the frequency range from 1 Hz to 230 kHz. The frequency dependence of the excitation current as applied in typical experiments is given in fig. 6b. The limits 8 and 20 mA for the low and high frequency coils are to prevent heating of the sample. In general, the exciting field is kept below 0.16 mT, i.e. below 1300 Hz the low frequency design is used. The magnetic field values (for the coil centres) indicated by arrows are calculated from the excitation current and the geometric dimensions of the coils (although the coils are quite different, the conversion factors are nearly identical).
5. High frequency measurements I
1
2
3
I.
log ( v / H z )
Fig. 6. (a) R a t i o o f the m o d u l i o f the signal i n d u c e d by the l a b o r a t o r y s t a n d a r d c r y s t a l ( L S ) and the o f f s e t ( O ) as a f u n c t i o n of frequency for the two coil designs; (b) excitation current
as applied in a frequency run.
In the course of our experiments it turned out to be desirable to expand the available frequency range to higher frequencies. We describe a simple method to do this with only small modifications of the experimental set up. It is based on the shift of
P.H. Miiller et al. / A mutual inductance bridge
the resonance frequency of an LC-circuit by the real part of the sample susceptibility. The assembly consists of a driven harmonic oscillator, the amplitude of which is monitored by an hf amplifier and rectifier. In a linear approximation the change of the amplitude is given by AA = (dA/dp)A1, cc X'-
(4)
To achieve maximum changes of the oscillator amplitude, the capacitance of the circuit is chosen such, that the amplitude is about half of that at the resonance frequency ( d A / d p = max). The coil of the LC-circuit may be one of the pick-up coils of the high frequency design (usable up to 1 MHz). Alternatively, a different arrangement consisting of only one pair of coils (8 mm length; 1 × 20 and 1 × 50 turns, coil system 3 of table 1) is available. This entire set up, including leads and with generator and amplifier connected, has an eigenfrequency of about 4 MHz, which is shifted proportional to X' of the sample. The resonance frequency of the assembly may be decreased by an external capacitor. In this way, measurements from 50 kHz up to 3 MHz can be performed. From 50 kHz to 230 kHz we get curves of X' identical to those measured with the bridge. The reproducibility over larger periods of time is, however, rather poor, although the short term stability (up to 1 h) is satisfactory. No attempt has been made to calibrate the set up. Thus, up to now, only temperature and field sweeps of X' (not frequency sweeps) in arbitrary units have been performed above 230 kHz. Curves h - m of fig. 4 are derived with this
347
method. Zero is determined by the low temperature value (T < 2 K) of X' and the scaling factor by assuming that all curves meet asymptotically at high temperatures ( T > 10 K).
Acknowledgement We are indebted to Prof. H.G. Kahle for valuable discussions.
References [1] A.C. Anderson, R.E. Peterson and J.E. Robichaux, Rev. Sci. Instr. 41 (1970) 528. [2] S.C. Whitmore, S.R. Ryan and T.M. Sanders, Rev. Sci. Instr. 49 (1978) 1579. [3] H.B.G. Casimir, D. Bijl and F.K. Du Pr6, Physica 8 (1941) 449. [4] W.L. Pillinger, P.S. Jastram and J.G. Daunt, Rev. Sci. Instr. 29 (1958) 159. [5] C.M. Brodbeck, R.R. Barkrey and J.T. Hoeksema, Rev. Sci. Instr. 49 (1978) 1279. [6] J.M. Dundon and M.E. Kretschmar, Rev. Sci. Instr. 49 (1978) 406. [7] M. Kumano and Y. Ikegami, Rev. Sci. Instr. 50 (1979) 921. [8] H.A. Groenendijk, A.J. van Duyneveldt and R.D. Willett, Physica 101B (1980) 320. [9] J.D. Cashion, A.H. Cooke, L.A. Hod, D.M. Martin and M.R. Wells, Colloq. du CNRS No. 180 II (1970) 417. [10] A. Kasten, P.H. Miiller and M. Schienle, Physica ll4B (1982) 77. [11] A. Kasten, P.H. MOiler and M. Sehienle, Physica ll4B (3) (1982) in press. [12] A. Kasten, Solid State Commun. 32 (1979) 973. P.H. Miiller, A. Kasten and M. Schienle, to be published.