居里定律# 对于顺磁物质，有居里定律：
< M > = N g S μ B B S ( S X ) , X = g μ B μ 0 H k B T \left< M \right>=NgS\mu_BB_S(SX), X=\frac{g\mu_B\mu_0H}{k_BT} ⟨ M ⟩ = N g S μ B B S ( SX ) , X = k B T g μ B μ 0 H 其中，B S B_S B S
B J ( y ) = 2 J + 1 2 J coth ( 2 J + 1 2 J y ) − 1 2 J coth y 2 J B_J(y)=\frac{2J+1}{2J}\coth\left(\frac{2J+1}{2J}y\right)-\frac{1}{2J}\coth\frac{y}{2J} B J ( y ) = 2 J 2 J + 1 coth ( 2 J 2 J + 1 y ) − 2 J 1 coth 2 J y 其高温极限为：
< M > = N g 2 μ B 2 S ( S + 1 ) μ 0 H 3 k B T \left<M\right>=\frac{Ng^2\mu_B^2S(S+1)\mu_0H}{3k_BT} ⟨ M ⟩ = 3 k B T N g 2 μ B 2 S ( S + 1 ) μ 0 H 其中，g g g g g g S S S N N N M M M
TIP 对于铁磁物质的Curie-Weiss 定律，只需将T T T T − Θ T-\Theta T − Θ Θ \Theta Θ
单位问题# emu# VSM测量磁化，给出的物理量的单位是emu/g \text{emu/g} emu/g
1 emu = 1 0 − 3 A ⋅ m 2 1\text{emu}=10^{-3}\text{A}\cdot\text{m}^2 1 emu = 1 0 − 3 A ⋅ m 2 磁化强度# 在电磁学中，磁化强度是对体积归一的磁矩，而此处给出的单位emu/g \text{emu/g} emu/g
磁场的单位为Oe \text{Oe} Oe
1 T = 10000 Oe 1\text{T}=10000\text{Oe} 1 T = 10000 Oe 实际上，在CGS下，可以视为μ 0 \mu_0 μ 0 H H H
H ( CGS ) = μ 0 H ( SI ) 1 Oe = 4 π × 1 0 − 7 N/A 2 × 1 0 3 / 4 π A/m = 1 0 − 4 N / ( A ⋅ m ) = 1 0 − 4 T H(\text{CGS})=\mu_0H(\text{SI})\\ \begin{align} 1\text{Oe}&=4\pi\times10^{-7}\text{N/A}^2\times 10^3/4\pi\; \text{A/m}\\ &=10^{-4}\text{N}/(\text{A}\cdot \text{m})=10^{-4}\text{T} \end{align} H ( CGS ) = μ 0 H ( SI ) 1 Oe = 4 π × 1 0 − 7 N/A 2 × 1 0 3 /4 π A/m = 1 0 − 4 N / ( A ⋅ m ) = 1 0 − 4 T 另外，注意到：
1 T = 1 N / ( A ⋅ m ) = 1 N ⋅ m / ( A ⋅ m 2 ) = 1 0 − 3 J / emu 1\text{T} = 1\text{N}/(\text{A}\cdot \text{m})=1\text{N}\cdot \text{m}/(\text{A}\cdot\text{m}^2)=10^{-3}\text{J}/\text{emu} 1 T = 1 N / ( A ⋅ m ) = 1 N ⋅ m / ( A ⋅ m 2 ) = 1 0 − 3 J / emu 即：
1 Oe = 1 0 − 7 J/emu 1\text{Oe}=10^{-7}\text{J/emu} 1 Oe = 1 0 − 7 J/emu 或：
1 J = 1 0 7 Oe ⋅ emu 1\text{J} = 10^{7}\text{Oe}\cdot\text{emu} 1 J = 1 0 7 Oe ⋅ emu 居里定律：应用# 若使用Oe作为磁场单位，应该将μ 0 \mu_0 μ 0 H H H
χ = g 2 S ( S + 1 ) N μ B 2 3 k B T , χ ≡ M μ 0 H \chi=g^2S(S+1)\frac{N\mu_B^2}{3k_BT},\;\chi\equiv\frac{M}{\mu_0 H} χ = g 2 S ( S + 1 ) 3 k B T N μ B 2 , χ ≡ μ 0 H M 此时，磁化率单位为emu ⋅ Oe − 1 ⋅ g − 1 \text{emu}\cdot\text{Oe}^{-1}\cdot \text{g}^{-1} emu ⋅ Oe − 1 ⋅ g − 1 n s 1 ns^1 n s 1 S = 1 / 2 , g = 2 S=1/2,g=2 S = 1/2 , g = 2
χ = N μ B 2 k B T \chi=\frac{N\mu_B^2}{k_BT} χ = k B T N μ B 2 分析磁化率曲线，可取倒数：
1 χ = k B N μ B 2 T \frac{1}{\chi}=\frac{k_B}{N\mu_B^2}T χ 1 = N μ B 2 k B T 斜率的单位应该为K − 1 emu − 1 ⋅ Oe ⋅ g \text{K}^{-1}\text{emu}^{-1}\cdot\text{Oe}\cdot\text{g} K − 1 emu − 1 ⋅ Oe ⋅ g
N N N g − 1 g^{-1} g − 1 n n n
μ B \mu_B μ B × 1 0 − 24 \times10^{-24} × 1 0 − 24 ⋅ \cdot ⋅ 2 ^2 2 9.274 × 1 0 − 21 emu 9.274\times10^{-21}\text {emu} 9.274 × 1 0 − 21 emu
k B k_B k B 1.380649 × 1 0 − 23 1.380649 × 10^{-23} 1.380649 × 1 0 − 23 1.380649 × 1 0 − 16 1.380649 × 10^{-16} 1.380649 × 1 0 − 16 ⋅ \cdot ⋅
综上，斜率为：
k = 16.053 × 1 0 23 / n [ Oe.g/K/emu ] k=16.053\times 10^{23}/n \;[\text{Oe.g/K/emu}] k = 16.053 × 1 0 23 / n [ Oe.g/K/emu ] 其中，n为 1g 物质中的顺磁原子数：
N = 2.66 / k [ mol/g ] N=2.66/k \;\;[\text{mol/g}] N = 2.66/ k [ mol/g ] 非顺磁# 此时，由于未知g , S g,S g , S μ B \mu_B μ B μ S \mu_S μ S
χ = N A n ( g S ( S + 1 ) μ B ) 2 3 k B T = N A μ S 2 3 k B T \chi = \frac{N_An(g\sqrt{S(S+1)}\mu_B)^2}{3k_BT}=\frac{N_A\mu_S^2}{3k_BT} χ = 3 k B T N A n ( g S ( S + 1 ) μ B ) 2 = 3 k B T N A μ S 2 其中，N = n N A N=nN_A N = n N A N A = 6.02 × 1 0 23 N_A=6.02\times 10^{23} N A = 6.02 × 1 0 23 g g g
N N N n n n
通常讨论原子磁矩时，会以μ B \mu_B μ B
χ = n N A ( μ μ B ) 2 3 k B T \chi=\frac{nN_A(\mu\mu_B)^2}{3k_BT} χ = 3 k B T n N A ( μ μ B ) 2 取倒数：
1 χ = 3 k B n N A μ B 2 μ 2 T \frac{1}{\chi}=\frac{3k_B}{nN_A\mu_B^2\mu^2}T χ 1 = n N A μ B 2 μ 2 3 k B T 仍然采用CGS单位制。代入常数：
χ − 1 = 8 n μ 2 T \chi^{-1}=\frac{8}{n\mu^2}T χ − 1 = n μ 2 8 T 要求出每个Fe-Site中的顺磁原子数，需要知道1g物质中的Fe-Site mol数(设为x x x
1 χ / x = 8 n ′ μ 2 T \frac{1}{\chi/x}=\frac{8}{n'\mu^2}T χ / x 1 = n ′ μ 2 8 T 此处，χ / x \chi/x χ / x
emu ⋅ Oe − 1 ⋅ g − 1 ⋅ ( mol/g ) − 1 = emu ⋅ Oe − 1 ⋅ mol − 1 \text{emu}\cdot\text{Oe}^{-1}\cdot g^{-1}\cdot(\text{mol/g})^{-1}=\text{emu}\cdot\text{Oe}^{-1}\cdot \text{mol}^{-1} emu ⋅ Oe − 1 ⋅ g − 1 ⋅ ( mol/g ) − 1 = emu ⋅ Oe − 1 ⋅ mol − 1 Conclusion# 在CGS单位制(磁场-Oe, 质量-g, 磁矩-emu, 温度-K, 真空磁导率-1)下，居里定律在数值上可以表述为：
χ − 1 = 8 n μ 2 T \chi^{-1}=\frac{8}{n\mu^2}T χ − 1 = n μ 2 8 T 其中，χ \chi χ emu ⋅ Oe − 1 ⋅ g − 1 \text{emu}\cdot\text{Oe}^{-1}\cdot \text{g}^{-1} emu ⋅ Oe − 1 ⋅ g − 1 emu ⋅ Oe − 1 ⋅ mol − 1 \text{emu}\cdot\text{Oe}^{-1}\cdot \text{mol}^{-1} emu ⋅ Oe − 1 ⋅ mol − 1 n n n μ \mu μ μ B \mu_B μ B T T T
举个例子# 如，处理FeGa3体系时，若要研究每个Fe-Site上的平均磁矩，则首先取一个Unit为包含单一Fe-Site的Unit，不妨就取为FeGa3。1g的FeGa3包含0.00377mol的Unit，因而：
1 χ = 8 x n ′ μ 2 T \frac{1}{\chi}=\frac{8}{xn'\mu^2}T χ 1 = x n ′ μ 2 8 T 此处，x = 0.00377 x=0.00377 x = 0.00377 n ′ n' n ′
在Magnetic Moments of Transition Metals - Chemistry LibreTexts 上可以查到过渡族的局域磁矩值。比如对Fe(III) lowspin，μ = 1.73 \mu = 1.73 μ = 1.73
记拟合斜率为k k k
n ′ = 8 k x μ 2 n'=\frac{8}{kx\mu^2} n ′ = k x μ 2 8 
