(4/9/03)Ferromagnetism–The Curie Temperature of GadoliniumAdvanced Laboratory, Physics 407University of WisconsinMadison, Wisconsin 53706AbstractThe Curie temperature of Gadolinium is determined by measuringthe magnetic susceptibility of a Gadolinium sample as a function oftemperature. The data are analyzed using the Curie-Weiss law whichcontains the Curie temperature as a parameter. Gadolinium is unusualin that the Curie temperature is very close to room temperature.11 IntroductionFerromagnetic materials show ferromagnetic behavior only below a criticaltemperature called the Curie temperature, above which the material has nor-mal paramagnetic behavior. The approach to ferromagnetism as a functionof temperature from above is described by the Curie-Weiss Law which givesthe magnetic susceptibility as a function of temperature.χ = µ − 1 =CT − TC(1)where χ and µ are the magnetic susceptibility and and relative magneticpermeability of the material respectively. C is a constant characteristic fora given substance and TCis the Curie temperature. Eqn. 1 is only validabove the Curie temperature.The relative magnetic susceptibility of a material is readily determined byplacing a sample of the material inside a small coil and measuring the induc-tance of the coil with and without the sample. If the inductance is measuredas a function of temperature from above to below the Curie temperature, theCurie-Weiss law, Equ. 1, can be used to determine the Curie temperature.The relative magnetic permeability of the sample can be written asµ =L(T )L0(2)where L(T ) is the inductance of the coil at temperature T and L0is theinductance of the coil without the sample. This is not exactly the relativepermeability since not all the magnetic flux will couple to the sample. FromEqn. 1 we can write an equation linear in T as L(T )L0− 1!−1=T − TCC. (3)The left hand side of this equation is zero when T = TCso a plot of theleft hand side vs temperature extrapolated to zero will intersect the x-axis atT = TC. The lack of a 100% fill factor for the coil will not affect this result.2 TheoryParamagnetism is a property exhibited by substances which, when placed ina magnetic field, are magnetised parallel to the field to an extent prop or-tional to the field (except at very low temperatures or very large magnetic2fields). Paramagnetic materials always have permeabilities greater than 1,but the values are in general not as great as those of ferromagnetic materials.Ferromagnetic materials have the property, that below a certain temperaturecalled the Curie temperature, the atomic magnetic moments tend to line upin a common direction.The classical theory of paramagnetism treats the substance as a collec-tion of magnetic dipoles with no interactions between them. In an externalmagnetic field, each magnetic dipole has a potential energy given by:E = − µ · H = −µH cos θ (4)where µ is the magnetic moment of the dipole, H is the applied magnetic fieldand θ is the angle between the dipole and the direction of H. If there are Ndipoles per unit volume the magnetization would be given by M = Nµ wherethe direction of the magnetization would be that of the applied field. Howeverin the prese nce of thermal agitation it is necessary to use the Boltzmanndistribution to average over the dipole distribution in thermal equlibrium attemperature T. We then have:M = Nµcos θ = NµZe−E/kTcos θdΩ/Ze−E/kTdΩ (5)where dΩ is the element of solid angle and e−E/kTis the Boltzmann dis-tibution of a dipole at angle θ with respect to the applied field at absolutetemperature T. The integration yields a result in terms of the Langevin func-tion L(x) = coth x − 1/x with x = µH/kT giving:M = NµL(x). (6)When x  1 the Langevin function L(x) approaches x/3 so that Equ. 6becomes:M∼=Nµ2H/3kT. (7)This is a very good approximation except at low temperature or extremelyhigh magnetic fields and gives us the Curie law χ ≡ M/H = C/T.The theory outlined above neglects the interaction between the magneticmoments. For c ertain metals this additional interaction is very important andleads to ferromagnetism. Ferromagnetism is chacterized by a crtical temper-ature called the Curie temperature above which the substance behaves asa paramagnet and below the Curie temperature the substance possesses a3spontaneous magnetization in the absense of a magnetic field. Upon applica-tion of a weak magnetic field, the magnetization increases rapidly to a highvalue called the saturation magnetization, which is in general a function oftemperature.The basis of the Weiss molecular theory of ferromagnetism is that belowthe Curie temp erature, a ferromagnet is composed of small, spontaneouslymagnetized regions called domains and the total magnetic moment of the ma-terial is the vector sum of the magnetic moments of the individual domains.Each domain is magnetized due to the strong magnetic interaction withinthe domain which tends to align the individual magnetic moments withinthe domain. The spontaneous magnetization below the Curie temperaturecomes about from an internal magnetic field called the Weiss molecular fieldwhich is proportional to the magnetization of the domain. Thus the effectivefield acting on any magnetic moment within the domain may be written as:H = H0+ λM (8)where H0is an externally applied field and λM is the Weiss molecular fieldwhose order of magnitude in iron is 107oersteds. Above the Curie tempera-ture, the Curie law now becomes:M/(H0+ λM) = C/T. (9)Solving for χ = M/H0we obtain:χ = M/H0= C/(T − Cλ) = C/(T − TC) (10)which defines the Curie temperature as TC= Cλ giving us the Curie-Weisslaw for the paramagnetic behavior of ferromagnetic substances above theCurie temperature. We see that the Curie-Weiss law leads to a nonzeromagnetization when H0= 0 at T = TC.A quantitative description of ferromagnetism requires a quantum the-ory treatment. The Heisenberg theory is based on an effective interactionbetween the electron spins, the exchange interaction energy, given by:ExchangeEnergy = −2JijSi· Sj(11)where Siand Sjare the spin angular momentum vectors of two electronsi and j, and Jijis the exchange integral between the electrons. Dependingon the balance between the Pauli Principle and the electrostatic interaction4energy of the electrons, the exchange integral may be positive or negative cor-responding to parallel spins (ferromagnetism) or antiparallel spins (antiferro-magnetism or ferrimagnetism). The