e. Energy Spectrum of Uniaxial Spin Clusters and Kramer's Theorem

2

B0

3

D= 2E=B²₂

In the following, if not specified otherwise, we will restrict ourselves to the given k = 2 and k

= 4 terms, as well as assume an isotropic g factor. Hence, the generic effective spin Hamiltonian we will use reads

( ) ( )

B

(

x x y y z z

)

V.3.e. Energy Spectrum of Uniaxial Spin Clusters and Kramer's Theorem

The effective spin Hamiltonian for an uniaxial cluster up to second-order (k = 2) shall be considered.

Energy Spectrum in Zero Magnetic Field The Hamiltonian to solve reads

(

^{Sˆ S(S 1)}

)

and the general solution becomes eigen functions n = SM

eigen values ^{E D}

(

^M 3^{S(S 1)}

)

^{DM2 const}

2 1

n = − + = +

For the four examples of a spin with S = 1/2, 1, 3/2, and 2 one finds the energy spectra

S = 1/2 S = 1 S = 3/2 S = 2

- 3 - Vorl. #9 (26 Nov. 2010) Comment

A spin with S = 1/2 does not exhibit any zero-field splitting. This can also be understood as a result form the restriction of the k index, namely that its maximal value is restricted to 2S, which for a spin 1/2 is k = 1, which is smaller than k = 2.

Kramer's Theorem

The above compilation of energy spectra may indicate that for half-integer spins, i.e., for S = 1/2, 3/2, 5/2,... a two-fold degeneracy remains, while for integer spins S = 1, 2, 3,... the ligand-field may remove any degeneracies. This is in fact a general finding, and a result of the Kramer's theorem, which follows directly from time-invariance:

The states in zero field of systems with an ODD number of electrons exhibit an EVEN degeneracy.

Comment

For a system with an odd number of electrons the spin has to be half-integer, while for a system with even number of electrons, the spin gas to be even.

odd ⇔ S half integer

even ⇔ S integer

Consequence

1) Systems with half-integer spin: The states in zero-field are always at least two-fold degenerate.

2) Systems with integer spin: The Kramer's theorem does not apply, hence, all degeneracies may be removed in zero magnetic field.

Representation of the Energies as Function of M

Plotted as function of the magnetic quantum number M, the energies fall onto a parabola, which curvature however depends on the sign of D:

0

D> (hard-axis): parabola resembles a "potential well"

0

D< (easy-axis): parabola resembles a "potential barrier"

( )

Example

-10 -8 -6 -4 -2 0 2 4 6 8 10

E

M

S= 10

M= -10, -9, ..., +9, +10

„down“

„up“

S = 10 with easy-axis

Energy Spectrum in an Applied Magnetic Field The Hamiltonian reads

(

3

)

B xy x x B z z z

2 1

z S(S 1) g Sˆ B g Sˆ B

Sˆ D

Hˆ = − + +μ +μ

Comment

Because of the uniaxial symmetry it doesn't matter in which direction the field is applied in the xy plane, one may hence without restrictions choose B_y = 0. The advantage is then that the Hamiltonian matrix is purely real.

Solution for B || z

In this case the zero-field splitting term and the Zeeman term commute, and hence may be chosen both to be diagonal in the same basis. The general solution is hence simply

eigen functions n = SM

eigen values ^{E D}

(

^M 3^{S(S 1)}

)

^gz B^BM 2 1

n = − + + μ

For the three examples of a spin with S = 1/2, 1, and 3/2 the energy spectra as function of applied magnetic field B look as follows:

S = 1/2 S = 1 S = 3/2

E

B E

B D

E

B 2D

S = 1/2 S = 1 S = 3/2

E

B E

B E

B D

E

B D

E

B 2D

E

B 2D

Representation of the Energies as Function of M

The E-vs-M representation, for the example of a spin with S = 10 and easy-axis anisotropy, is shown together with the E-vs-B representation for three examples of applied magnetic field.

Of most significance are the level-crossings in the E-vs-B representation, which corresponds to resonances in the E-vs-M representation.

-10 -8 -6 -4 -2 0 2 4 6 8 10 E

M

-1.0 -0.5 0.0 0.5 1.0

-100 -80 -60 -40 -20 0 20 40

E/|D|

B (T)

-10 -8 -6 -4 -2 0 2 4 6 8 10 E

M -12-10 -8 -6 -4 -2 0 2 4 6 8 10 12

E

M

M = -10 M = -9 M = -8 M = +10 M = -7

- 5 - Vorl. #9 (26 Nov. 2010) Solution for B ⊥ z

In this case the zero-field splitting term and the Zeeman term doe NOT commute. One hence faces the typical situation of the interplay or competition of two energies which do not commute.

The case of S = 1, however, can be solved analytically, with the result and energy spectrum shown below.

(

B

)

²

2 2

/

1 D g B

2 D 1 6

E 1 ⎟ + μ

⎠

⎜ ⎞

⎝

± ⎛

−

= 3D E₃ =1

S = 1

0.0 0.5 1.0 1.5 2.0

-2 -1 0 1

2 B⊥z

B||z

E/D

gμ

BB/|D|

M = +1

M = 0 M = -1

ˆ =M′=0 S_x

ˆ =M′=+1 S_x

ˆ =M′=−1 S_x

The low-magnetic-field and large-magnetic-field cases, separated by a crossover regime in which the energies mix as signaled by the curvatures of the levels, can be clearly identified.

At low fields, the zero-field-splitting dominates, and the energy spectrum exhibits only a weak (quadratic) field dependence. At high fields, the Zeeman term dominates, and the spectrum approaches the linear field dependence expected for a free isotropic spin.

VI. Single-Molecule Magnets I: Basic Properties and Giant-Spin Model VI.1. Basic Properties of the SMM Mn12ac

Molecular Structure

Mn O C H Mn Mn O O C C H H

Mn⁴⁺

Mn³⁺

Mn⁴⁺

Mn³⁺

OAc

Weinland, Fischer 1921 T. Lis 1980

[Mn₁₂O₁₂(CH3COO)₁₆(H₂O)₄]·2CH₃COOH·4H₂O

The chemical formula and the structure of the molecule, as determined by X-ray crystallography, are shown to the right.

- 12 Mn ion are bridged by acetate ligands - eight Mn(III) ions with S = 2 form an outer ring

- four Mn(IV) ions with S = 3/2 form an inner core

Magnetic Susceptibility

The magnetic susceptibility, plotted as Tχ vs T, is shown to the right. For the interpretation, is useful to first calculate the Curie constant as expected for uncoupled Manganese ions, i.e., the Curie value as it can be expected to be observed at very high temperatures (see high-temperature expansion).

C = 31.5 emu K/mol C₁₀= 55 emu K/mol

For a spin with S, the Curie constant is given as mol

/ K 8 emu

) 1 S ( S C g

2 S

= +

Hence, for the Mn ions in Mn₁₂ac we find

Mn(IV): S = 3/2 → C_3/2 = 1.875 emu K/mol Mn(III): S = 2 → C₂ = 3.00 emu K/mol

and as a result we obtain the Curie constant for the full Mn12ac molecule (if the spins would be uncoupled) as:

C = 8 × 3.000 + 4 × 1.875 emu K/mol = 31.5 emu K/mol

However, at 300 K the product is not temperature independent and well below the calculated Curie constant. Both observations we do not understand at the moment, but will be explained later to result from the magnetic interactions between the Mn ions in the Mn₁₂ac molecule.

χT

Interestingly, we note that at low temperature the Tχ product seems to approach a constant behavior assuming a value which corresponds to that of a spin with S = 10

S = 10 → C10 = 55 emu K/mol

- 7 - Vorl. #9 (26 Nov. 2010) Magnetization Curves

20μ_B

Observed magnetization curves are shown to the right.

The magnetization curves shall be compared to the behavior expected for a single, isolated spin, which may be recalled to be characterized by 1) Brillouin function

2) a saturation magnetic moment of gμ_BS at high fields

3) scaling if plotted as function of B/T

As seen, the magnetization curves do not follow a Brillouin behavior. In fact, at low temperatures, the magnetization curves show hysteresis as well as funny "steps".

Furthermore, the magnetization curves do not scale.

However, the magnetization shows a clear saturation behavior, with a saturation moment of ca 20μB, which would be consistent with a single spin behavior with S = 10.

Giant-Spin Model for SMMs

For Mn12ac, both the magnetic susceptibility and magnetization curves show a behavior which is substantially more complex than that of a single isolated spin. However, the data also suggest a S = 10 ground state at low temperatures. We in fact will see in the following, that all these effects result form a single spin behavior, but of a spin with a large S and a large easy-axis anisotropy.

At low temperatures, SMMs behave like single spins with large S and large easy-axis anisotropy.

The corresponding model is called the giant-spin model: which is exactly the effective spin Hamiltonian

( ) ( )

B

(

x x y y z z

)

4 4 4 4 4 0 4 0 2 y 2 3 x

2 1

z S(S 1) ESˆ Sˆ B Oˆ B Oˆ gSˆ B Sˆ B Sˆ B

Sˆ D

Hˆ = − + + − + + +μ + + ,

but with a large S and large D<0.

Molecular Nanomagnets ©Oliver Waldmann

VI.2. Giant-Spin Model and Resonant Quantum Tunneling of Magnetization in Mn₁₂ac

Im Dokument Molekulare Nanomagnete: Quantenphysik zum Anfassen Molecular Nanomagnets (Seite 51-57)