Magnetic Properties in the Ordered Phase
In our theoretical framework, the magnetic isotherm is determined by solving a differential equation of the inverse of magnetic susceptibility \(y(\sigma ,t)\) as a function of \(\sigma \) . Its initial condition is equivalent to determine the temperature
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Magnetic Properties in the Ordered Phase
4.1 Initial Value Problem Even in the magnetically ordered phase, the magnetic isotherm is evaluated by solving the same ordinary differential equation as (3.25) given by Φ(σ 2 , y, yz , t) = 0,
yz = y + σ
∂y , ∂σ
(4.1)
Φ(σ 2 , y, yz , t) ≡ 2 A(y, t) + A(yz , t) − c(2y + yz ) +
TA 2 σ − 3A(0, tc ). 3T0
As a solution, inverse magnetic susceptibility y(σ, t) is obtained as a function of σ . In reference to the magnetic isotherms in the ground state and in the paramagnetic phase, i.e., (3.7) and (3.48), respectively, it is reasonable to introduce the σ dependence of the inverses of magnetic susceptibilities in the ordered phase. They are given by y(σ, t) = y0 (t) + y1 (t)σ 2 + · · · = y1 (t)[σ 2 − σ02 (t)] + · · · , yz (σ, t) =
2y1 (t)σ02 (t) + 3y1 (t)[σ 2
− σ02 (t)]
(4.2)
= yz0 (t) + 3y(σ, t),
where y(σ, t) and yz (σ, t), corresponding to H/M and ∂ H/∂ M, are the perpendicular and the parallel components, respectively, to the uniform spontaneous magnetization. The value of yz (σ, t) in the absence of the external field for σ = σ0 (t) is denoted by yz0 (t) ≡ 2y1 (t)σ02 (t). In the ordered phase, y0 (t) in the first line becomes negative, and the spontaneous moment squared, σ02 (t) = −y0 (t)/y1 (t), appears. Both of these values, σ0 (t) and y1 (0), are therefore expected to be in agreement with σ0 and y1 in (3.10) and (3.11) in the ground state at t = 0. All the parameters, y0 (t), σ0 (t), and σ0 (t), are regarded as functions of the reduced temperature t. To find the initial conditions of (4.1), let us first consider the case, H = 0, in the absence of the magnetic field. Since y(σ, t) = 0 is then satisfied, σ = σ0 (t) has to be satisfied in the ordered phase. Initial values of y(σ, t) and yz (σ, t) at H = 0, after putting these values into (4.1), are shown in Table 4.1. Values in Y. Takahashi, Spin Fluctuation Theory of Itinerant Electron Magnetism, Springer Tracts in Modern Physics 253, DOI: 10.1007/978-3-642-36666-6_4, © Springer-Verlag Berlin Heidelberg 2013
79
80
4 Magnetic Properties in the Ordered Phase
Table 4.1 Initial values of σ and inverse of magnetic susceptibilities, y and yz , at H = 0 in the ordered phase Ordered phase Paramagnetic phase
σ
y(σ, t)
yz (σ, t)
σ0 (t) 0
0 y0 (t)
2y1 (t)σ02 (t) y0 (t)
Same corresponding values in the paramagnetic phase are also shown
the paramagnetic phase are also shown in the same table for reference. Number of independent parameters is determined by putting these values into (4.1). In the ordered phase, two independent parameters, σ0 (t) and y1 (t), are involved in the initial values, whereas in the paramagnetic phase, only the single parameter y0 (t) is present. An extra condition seems to be necessary to determine these parameters simultaneously, other than the single equation of (4.1). To solve the problem, note that variables σ 2 and yz (σ, t) in the weak field limit are expanded with respect to the small parameter y(σ, t) up to the linear term by σ 2 = σ02 (t) +
1 y(σ, t), y1 (t)
yz (σ, t) = 2y1 (t)σ
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