Eur. J. Phys. 18 (1997) 448–452. Printed in the UK PII: S0143-0807(97)84690-9 Curie point of ferromagnets Yaakov Kraftmakher Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Received 4 June 1997, in final form 18 August 1997 Abstract. A student experiment is described concerning Zusammenfassung. Ein Studentenexperiment in Beziehung properties of ferromagnets near the Curie point. Gadolinium der Eigenschaften eines Ferromagnetikums nahe des provides a good opportunity to determine the magnetic Curie-Punktes ist beschrieben. Gadolinium bietet eine gute susceptibility above the transition point and to evaluate the Gelegenheit, die magnetische Suszeptibilitat¨ uber¨ der critical exponent of the susceptibility. The temperature Ubergangstemperatur¨ auszuwerten und den kritischen dependence of the electrical resistance of nickel at enhanced Exponent der Suszeptibilitat¨ berechnen. Die frequencies shows the influence of the magnetic properties of Temperaturabhangigkeit¨ des elektrischen Widerstandes des the sample on the skin depth. From the measurements, the Nickels in hoheren¨ Frequenzen demonstriert den Einfluß der phase transition is clearly seen and the temperature magnetischen Eigenschaften der Probe auf die Skinschicht. dependence of the magnetic permeability below the Curie Von den Messungen, der Phasenubergang¨ ist klar und die point is available. Data on the resistance at an enhanced Temperaturabhandigkeit¨ der magnetischen Permeabilitat¨ unter frequency reveal the phase transition in a nickel-based alloy. der Curie-Temperatur kann berechnet werden. Die Daten des These items may be added to related student experiments Widerstandes in hoheren¨ Frequenzen enthullen¨ den described previously. Phasenubergang¨ in einer Nickellegierung. Vorher beschriebene Studentenexperimente konnen¨ mit obigen Punkten erganzt¨ werden. 1. Introduction One of the most important objectives of the theory and experiment is to determine the critical exponents. At temperatures above the Curie temperature, TC,a The mean-field theory predicted values of the critical ferromagnet loses its intrinsic magnetization. The exponents as follows: α 0,β 0.5,γ 1. transition from the magnetic to the nonmagnetic state However, observed dependences= appeared= to be= quite is a second-order phase transition. Many properties of different. The modern theory of critical phenomena a ferromagnet manifest singularities near the transition can be found elsewhere (Ma 1976, Patashinskii and point, usually with a power-like dependence (e.g. Fisher Pokrovskii 1979, Stanley 1983, Domb 1996). In 1965, 1967, Heller 1967, Kadanoff et al 1967). The 1982, Kenneth G Wilson was awarded the Nobel Prize exponent in such a dependence is referred to as the in Physics ‘for his theory for critical phenomena in critical exponent or critical index. For instance, the connection with phase transitions’ (see Wilson 1993). temperature dependence of the specific heat, C, follows The theory is very complicated but Maris and Kadanoff the equation (1978) have shown how Wilson’s theory could be α incorporated into an undergraduate course of statistical C A B T T − (1) = + | − C| physics. At present, the accepted theoretical values where A and B are constants, generally different below are as follows: α 0.115 0.009,β 0.3645 and above the Curie point, and α is termed the critical 0.0025,γ 1.386 =−0.004 (Domb± 1996).= Generally,± exponent of the specific heat. experimentally= determined± critical exponents are, to Below the transition point, the spontaneous magneti- within experimental errors, in agreement with these zation of a ferromagnet, M, obeys the relation theoretical values. The theory also predicts some relations between the critical exponents, e.g. a relation M A(T T)β (2) = C − fitting the above critical exponents: where β denotes the critical exponent of the spontaneous α 2β γ 2. (4) magnetization. + + ≥ Close to the Curie point, the magnetic susceptibility, Both sets of critical exponents given above fulfil χ, is given by this relation even as an equality. The parameters γ and critical exponents describing the phase transition χ A T T − (3) = | − C| in a ferromagnet are similar to those for the critical where γ is the critical exponent of the susceptibility. point of the liquid–vapour coexistence curve. The 0143-0807/97/060448+05$19.50 c 1997 IOP Publishing Ltd & The European Physical Society 448 Curie point of ferromagnets 449 magnetization is analogous to the difference between the D C density of the fluid and the critical density, whereas the source magnetic susceptibility is analogous to the isothermal compressibility of the fluid. Several student experiments related to the Curie point L F Keithley oscillator thermistor have already been described. Two of these employ sample 199 DMM the mutual inductance technique for determining the Curie point of ferromagnets (Edgar and Quilty 1993, selective Scanner Fisher and Franz 1995). Other experiments include heater amplifier measurements of the electrical resistivity of nickel (Kamal et al 1983, Fox et al 1986, Sullivan et al 1987) differential and observations of changes in the resonant frequency transformer of an LC circuit when the sample placed inside the inductor undergoes the transition (Fox et al 1986). Figure 1. Set-up to measure the magnetic susceptibility Two items described below may be added to the of gadolinium. student experiments reported earlier, namely: (i) the magnetic susceptibility of a ferromagnet above the Curie AC current creates a magnetic flux in the magnetic core. point; (ii) the influence of the magnetic properties of Two secondary windings (n 400) are connected in a ferromagnet on the skin depth when an AC current 2 opposition; thus without a sample= no voltage appears passes through the sample. at the transformer’s output. A small additional coil In the first part, the magnetic susceptibility of connected in series with the secondary windings is used gadolinium is measured above the Curie point. When to finely balance the transformer. This coil is placed approaching the Curie point, the magnetic susceptibility near the core to obtain the necessary compensation depends on the quality of the sample and on the voltage. With a magnetic sample in the gap of the approach to the transition point. In magnetic core, the magnetic flux through this part of the core measurements, this approach was limited by the value 4 3 increases. This causes an increase in the voltage induced of T TC /TC in the range 10− –10− . With such a proximity,| − | the magnetic susceptibility above the in the corresponding secondary winding. To avoid the Curie point may be much smaller than that in the influence of eddy currents in the sample, the operating ferromagnetic phase (e.g. Heller 1967, Herzum et al frequency is reduced to 30 Hz. 1974). Gadolinium is rather an exception that possesses A spherical gadolinium sample (99.9%), 5 mm in a high magnetic susceptibility above the Curie point. diameter, is placed into a small glass container filled It allows one to perform the measurements over a with oil to reduce temperature gradients and to prevent wide temperature interval in the nonmagnetic phase oxidation of the sample. A small thermistor (Fenwal and to evaluate the critical exponent of the magnetic Electronics, model UUA 35J3, 5000 at 25 ◦C) is susceptibility. attached to the sample. Its resistance, R, relates to the absolute temperature as T 3895/ ln(R/0.0106).A In the second part, the temperature dependence of = the electrical resistance of nickel is measured using a bifilar electrical heater is also inserted in the oil. The DC current and an AC current of enhanced frequency. heater and the thermistor contain no magnetic parts. The transition to the nonmagnetic state becomes evident The internal magnetic field in the sample, Hi, should owing to a change in the skin depth. Moreover, the be calculated taking into account the demagnetizing temperature dependence of the magnetic permeability factor, α. Thus, the magnetization is M = of the sample can be evaluated from the ratio of the χHi χ(He αM), where He is the external = − electrical impedance of the sample to its DC resistance. field. Hence, M He/(α 1/χ). The output = + Similar measurements reveal the phase transition in a voltage of the differential transformer is proportional nickel-based alloy. to the magnetization of the sample: V KM = = KHe/(α 1/χ). Below the Curie point, 1/χ α. This assumption+ allows one to determine the external 2. Magnetic susceptibility of gadolinium magnetic field: He αV0/K, where V0 is the output voltage below T .= For a sphere, α 1 (SI units). C = 3 The aim of this experiment is to determine the magnetic Hence, χ 3V /(V0 V). The absolute value of the = − susceptibility of gadolinium above the Curie point. magnetic susceptibility is thus available (Heller 1967). Gadolinium provides a very convenient temperature of The output voltage of the differential transformer is the phase transition and a relatively large interval where fed to an amplifier, PAR model 124A. The amplifier the magnetic susceptibility can be measured. operates in the selective mode and is tuned to the signal For the measurements, an E-shaped transformer is frequency. The amplified voltage is monitored by an employed. The core and the coils have been purchased oscilloscope. For initial balancing of the transformer, from PASCO, catalogue numbers SF-8615, SF-8610 it is more convenient to observe the Lissajous pattern and SF-8611. The primary winding (n1 800) is on the oscilloscope’s screen. For this purpose, the connected to a low-frequency oscillator (figure= 1). The voltage drop across a resistor connected in series with 450 Y Kraftmakher 100 6 290.5 K 4 χ 291 K 290 K 50 ln 2 0 Magnetic susceptibility Magnetic 0 290 295 300 305 K Figure 2. Temperature dependence of the magnetic -2024 susceptibility of gadolinium. − ln (T Tc) Figure 3. Plot of ln χ versus ln(T TC). Determination the primary winding of the transformer is fed to the of γ includes the choice of the transition− temperature.