Excursion I on the Effective Hamiltonian
VII. Magnetic Coupling
VIII.1. c. Magnetic Susceptibility
Calculating the magnetic susceptibility corresponds to calculating the thermodynamic expectation value of the magnetic moment operator for small fields (see chapter III.5). For a general many-spin system the magnetic moment operator reads =−μ
∑
⋅i for the present case, where we assumed isotropic and identical g factors for all spin centers, the operator simplifies to
Sˆ g mˆ
B
r =−μ r.
The thermal average of the magnetic moment may then be calculated from
∑
For a spin-1/2 dimer, the energy eigen values and functions were calculated in the previous chapter. Introducing them and calculating the expectation values of the Sˆz operator yields
( )
The magnetic susceptibility is obtained from the magnetic moment at small magnetic fields. It can be calculated from the above equation by expanding the exponentials up to first order,
( )
which yields the magnetic susceptibility
J
The equation is valid for both positive and negative J values, i.e., ferromagnetic and antiferromagnetic interactions.
Limiting cases
1) high temperatures, T→∞
For very high temperatures we physically expect that the correlations between the two spins introduced by the interaction become negligible, and that hence at high temperatures the two
spins should essentially behave like to independent spins. That is, at high temperatures the magnetic susceptibility is expected to follow the Curie law, with a Curie constant corresponding to that of two spin-1/2 spin, i.e., C = 2CSpin-1/2. Indeed, expanding the exponentials at high temperatures yields the limiting behavior
2 / 1 Spin B
2 2 B
T 2C
T k 2
T →∞ μ g = −
= χ
2) low temperatures, T→0
At low temperatures only the lowest multiplet is expected to be thermally populated, which for antiferromagnetic coupling is a singlet, S = 0, and for a ferromagnetic coupling is the multiplet wit the largest possible total spin Smax = S1 + S2, or S = 1 in the case of the spin-1/2 dimer. Hence
⎩⎨
⎧
>
∀
<
= ∀ χ
→ C − J 0
0 J T 0
1 Spin 0
T
In the following two plots the full temperature dependence of the magnetic susceptibility of the spin-1/2 dimer is shown for the three cases of a ferromagnetic (red), zero (black) and antiferromagnetic (green) coupling is shown in the two representations χ vs T and χT vs T.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0
χ (emu/mol)
T/|J|
J > 0 (F) J = 0 (Curie) J < 0 (AF)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
χT (emu/mol K)
T/|J|
J > 0 (F) J = 0 (Curie) J < 0 (AF) C = 0.75 emu K/mol CS=1= 1.00 emu K/mol
The limiting cases of low and high temperatures can be clearly observed. In the χ vs T plot, the difference between zero and a ferromagnetic coupling is not easily recognized. However, the antiferromagnetic coupling reveals it self clearly by a characteristic maximum at a temperature of ca. kBT≈ J . In the χT vs T plot the correlations between the spins introduced by the interaction are clearly seen. In the case of a ferromagnetic coupling the spins prefer to stay parallel enhancing the magnetism, while for an antiferromagntic coupling the spins prefer to align anti-parallel such as to compensate each other reducing the magnetism.
Comments
1) The magnetic susceptibility exhibits a scaling with T/|J|, i.e., the magnetic susceptibility depends on the scaling variable T/|J| and not T and |J| independently. Physically this expected because the magnetic susceptibility is governed by the competition of to energy scales, i.e., the temperature kBT and the coupling strength |J|.
2) The maximum in χ vs T at kBT ≈ |J| for J < 0 provides a first estimate for the strength of the coupling
3) The magnetic couplings lead to a deviation from the Curie law.
- 3 - Vorl. #15 (13. Jan. 2011) Example
The text book example is that of the Cu-Acetate molecule shown to the right (Bleany
& Bowers 1954). The strong antiferomagntic coupling between the two spin-1/2 Cu(II) centers in the molecule of about J = -300 K is immediately evident from the χ vs T measurement.
At very low temperatures a small upturn,
called "Curie tail", is observed, which is due to a minor amount of paramagnetic impurities in the sample.
[Cu2(CH3COO)4(H2O)2]
VIII.1.d. Magnetization
The magnetization for the spin-1/2 dimer can also be calculated analytically, which however shall not be done here; only limiting cases and the general features will be discussed. The magnetization could be calculated using the same starting equation as in the above, however, it is useful to recall that it may alternatively be calculated from the free energy F(B,T) (see also chapter III.a.):
B ) T , B ( F e
e n S n g
) T , B ( m
n E n
E z
B n
n
∂
−∂
= μ
−
=
∑
∑
β
− β
− F=−kTlnZ
∑
−β= exp( E )
Z n
In particular, at very low temperatures, such that only the ground state is thermally populated, the magnetization is given by
B
m E0
∂
∂ for T = 0,
−
= ) B ( E0
where is the (field dependent) ground state energy. Hence, the low temperature magnetic moment is given as the negative of the field derivative of the ground state energy.
The two cases of antiferromagnetic and ferromagnetic couplings need to be differentiated.
Spin Dimer with AF coupling 1) T = 0
At very low temperatures only the field dependent ground state is thermally populated, and the magnetization given by the field derivative, i.e., m=−∂E0/∂B. Hence, since at each level crossing as function of field the slope of the ground state changes abruptly, a sharp step in teh magnetization occurs. The height of such a step is gμB since the magnetic quantum number M changes by one.
It can be noted that the position of the level crossing fields and hence the magnetization steps on the field axis scale with gμBB/|J|, which can again be physically expected since again two counter playing energies are involved, the antiferromagnetic interaction |J| which favors anti-parallel alignment and the magnetic field B which aims at aligning the spins..
As regards the orders of magnitudes, one finds that gμB /kB ≈ 1.5 K/T. Hence, an exchange energy of |J| = 15 K corresponds to a field of 10 T, and the observation of the steps typically requires large magnetic fields.
2) T << |J|
At non-zero but still small temperatures, i.e., temperatures significantly smaller than the exchange energy |J|, the steps in the magnetization are still seen, but are washed out or broadened, respectively, because of the influence of the thermal excitation of the higher-lying states. The behavior resembles that of the Fermi function. A detailed calculation shows that the magnetization change a level crossing from S to S+1 follows
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ⎟⎟⎠
⎜⎜ ⎞
⎝ + ⎛ − +
μ
= 2k T
B tanh B
2 1 2 S 1 g m
B n B
The derivative of this broadened magnetization step yields a Gauss function centered at the level crossing field with a FWHM of
B / m ∂
∂ 4ln
(
2+1)
kBT = 3.5 kBT.Typically, in order to observe the magnetization steps, the temperature should be less than 0.1|J|, which typically is less than 1 K.
3) T >> |J|
At high temperatures the magnetization steps are completely washed out and disappear, i.e., a smooth field dependent magnetization curve is observed. At such temperatures the disordering effect of temperature overcomes the ordering effect of the exchange interactions, and physically one expect the magnetization steps to disappear.
Beispiel
The above description of the magnetization curves are exemplified for a spin-10 dimer with S1 = S2 = 10. Here the total spin ranges from S = 0 to S = 20, and accordingly twenty level crossings and twenty magnetization steps occur with increasing field.
0 5 10 15 20
0 10 20 30 40
J = -1 K S=10 Dimer
0 K 0.1 K 0.3 K 1.0 K 3.0 K 10 K
m (μ B)
B (T)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 2 4 6
S=10 Dimer
0 K 0.1 K 0.3 K 1.0 K 3.0 K 10 K
m (μB)
B (T)