The_Cambridge_Handbook_of_Physics_Formulas

4.5 Angular momentum
101
Quantum paramagnetism
−10
−5
2J +1
coth
2J
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
[ (2J +1)x
B ∞ (x)=L(x)
B 4 (x)
B 1 (x)
B 1/2 (x)=tanhx
5
x
10
]
− 1
2J coth x
2J
B J (x)= (4.147)
2J
⎧
B J (x) Brillouin function
Brillouin
⎨J +1
J total angular momentum
x (x ≪ 1)
function B J (x) ≃ 3J (4.148) quantum number
⎩
L(x) (J ≫ 1)
L(x) Langevin function
=cothx−1/x (see page 144)
B 1/2 (x)=tanhx (4.149)
〈M〉 mean magnetisation
n number density of atoms
( )
g
Mean
µ B B
J Landé g-factor
magnetisation a 〈M〉 = nµ B Jg J B J Jg J (4.150) µ B Bohr magneton
kT
B magnetic flux density
( )
k Boltzmann constant
〈M〉 for isolated
µB B
T temperature
〈M〉
spins (J =1/2)
1/2 = nµ B tanh
(4.151)
kT
〈M〉 1/2 mean magnetisation for
J =1/2 (and g J =2)
a Of an ensemble of atoms in thermal equilibrium at temperature T , each with total angular momentum quantum
number J.
4