Magnetic Property Measurement System 4

Nutshells of Magnetic Properties Measurement System

Magnetism where science meets philosophy !

Magnetic Properties of Solids

Magnetic (with unpaired electron)

diamagnetic (no unpaired electron)

Superconductor Gouy method

SQUID (Superconducting QUantum Interference Device)

Solids in Magnetic Field

Magnetic Properties of Solids

Magnetic induction Applied magnetic field (G, Oe)

Magnetization : magnetic moment of the material per unit volume (G/cm3)

Magnetic permeability χv : Magnetic susceptibility per unit volume (dimensionless, emu/cm3)

χg : (emu/g, cm3/g)

χmol : (emu/mol, cm3/mol)

the degree of magnetization of a material in response to a magnetic field = M/H

Magnetic Susceptibility

Magnetic Properties of Solids

Curie Law : χ = C/T

TC : Curie Temperature TN : Neel Temperature

Curie-Weiss Law : χ = C/(T-θ)

Ferromagnetism : T < TC

Antiferromagnetism : T < TN

Paramagnetism : competition between magnetic and thermal motion

Magnetic Susceptibility

Magnetic Properties of Solids Magnetic Susceptibility

Magnetic Susceptibility of Diamagnetic Solids

χdia = χdia atoms + χdia bonds

Pascal’s constants

All molecules have contributions from diamagnetic effects

χdia = χdia atoms + χdia bonds

Pascal’s constants

Magnetic Susceptibility of Diamagnetic Solids

Magnetic Moment of Electron

Two sources of magnetic moment – spin (S) and angular(L) motions of electrons

spin quantum number orbital (angular momentum) quantum number

µS+L = g [J(J+1)]1/2β

Landé g-factor (gyromagnetic ratio) = 1 + J(J+1) + S(S+1) – L(L+1)

µS+L = g [S(S+1) + 0.25L(L+1)]1/2β

total angular momentum quantum number

2J(J+1)

When spin-orbit coupling is negligible,

true for most cases except heavy metals such as Lanthanides

Theoretically, electron Bohr magneton

µS+L = g [S(S+1) + 0.25L(L+1)]1/2β

2

Landé g-factor (gyromagnetic ratio) = 1 + J(J+1) + S(S+1) – L(L+1)

2J(J+1)

µS = g [S(S+1)]1/2β = µeff

In most cases, L is effectively quenched,

J = S, L = 0 g = 2, gfree electron = 2.0023

0 0 1 ½ 0 0 1 ½

1 ½

1 ½

Why is L quenched in crystal field ?

Q) Why do the transition metal ions have so much diversified magnetic moments (spin states)?

Magnetic Moment of Electron

Magnetic Susceptibility of Paramagnetic Solids

S = 1/2

ms = + 1/2

ms = - 1/2

𝜇𝑛 = −𝑚𝑠𝑔𝛽 Magnetic moment of electron spin g = g-factor β = electron Bohr magneton

H

𝐸𝑛 = 𝑚𝑠𝑔𝛽H

Boltzman distribution 𝑃𝑛 =𝑁𝑛𝑁 =

𝑒−𝐸𝑛𝑘𝑘

∑ 𝑒−𝐸𝑛𝑘𝑘

Molar macroscopic magnetic moment 𝑀 = 𝑁𝐴�𝜇𝑛𝑃𝑛 = 𝑁𝐴𝑔2𝛽2𝐻4𝑘𝑘

𝑚𝑠

Molar macroscopic magnetic moment 𝑀 = 𝑁𝐴�𝜇𝑛𝑃𝑛 =𝑁𝐴𝑔2𝛽2𝐻4𝑘𝑘

𝑚𝑠

Molar magnetic susceptibility 𝜒 =

𝑀𝐻

=𝑁𝐴𝑔2𝛽2

4𝑘𝑘 Curie Law : χ = C/T

Curie-Weiss Law : χ = C/(T-θ)

𝜒 =𝑁𝐴𝑔2𝛽2

3𝑘𝑘 𝑆 𝑆 + 1 S: electron spin angular momentum quantum number

𝜒 =𝑁𝐴

3𝑘𝑘 (𝜇𝑒𝑒𝑒)2

generalize for S

Effective magnetic moment of an atom or a molecule

𝜇𝑒𝑒𝑒 = (3𝑘𝑁𝐴

)1/2(𝜒𝑘)1/2 𝜇𝑒𝑒𝑒 = (3𝑘𝑁𝐴𝛽2

)12 𝜒𝑘

12 = 2.8279 𝜒𝑘 1/2 in β unit

Magnetic Susceptibility of Paramagnetic Solids

From Experimental Data to χm 2D sheet array of

Longitudinal magnetic moment

From Experimental Data to χm 2D sheet array of

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 3200.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

MUe

ff (M

Ub)

Temp (K)

0 50 100 150 200 250 3000

20

40

60

80

100

1/Xm

(mol

/em

u)

Temp (K)0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

L M

omen

t (em

u)

Temp (K)

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

Xm (e

mu

/ mol

)

Temp (K)

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

XmT

(em

u K/

mol

)

Temp (K)

Example

TIP = 0.00589, C = 0.42509 and Tc = 0.0369 K

Almost paramagnetic

Ferro- and Antiferromagnetism

𝐸𝑛 = 𝑚𝑠𝑔𝛽H Previous slide (Spin only)

Actually there and spin-orbit coupling and higher order Zeeman interaction

depending on spin-orbit coupling

𝑚𝑠𝑔𝛽H 2nd order Zeeman term 1st order Zeeman term

van Vleck equation When spin-orbit coupling is ignored, van Vleck eq is equal to Curie law.

2 electron spins

There exists some type of magnetic interaction between the two unpaired electrons in the molecule, and the process by which they interact is called magnetic exchange (Heigenberg exchange interaction).

J > 0 : Ferromagnetic interaction J < 0 : Antiferromagnetic interaction

2 electron spins Ferro- and Antiferromagnetism

Total Hamiltonian in magnetic field

2 electron spins Ferro- and Antiferromagnetism

ms

1

0

-1

0

H

0

-1

0

H

Total Hamiltonian in magnetic field

into van Vleck eq

Bleaney-Bowers equation

J. Chem. Soc. 1956, 3837

2J = -334 cm-1 and Tmax = 255 K.

S = S1+ S2

2 electron spins Ferro- and Antiferromagnetism

Bleaney-Bowers equation (ρ: paramagnetic impurities) J = -7.11 cm-1

2 electron spins Ferro- and Antiferromagnetism

More complicated systems Ferro- and Antiferromagnetism

More complicated systems Ferro- and Antiferromagnetism

julX MAGPACK PHI