Paramagnetic Interaction

Investigations into paramagnetic materials present an interesting challenge com- pared to diamagnetic compounds. A material is deemed paramagnetic when the magnetic susceptibility (χ) is positive. Ifχ is negative, then the material is domi- nated by diamagnetism. Positiveχis a result of the presence of a localised unpaired electron which upon interaction with B0 forms an internal induced magnetic field parallel to this applied magnetic field.

The paramagnetic HamiltonianHˆp can generally be described as the sum of

two distinct interactions:

ˆ

Hp =HˆC+HˆD (2.92)

whereHˆC is the through bond or Fermi contact interaction andHˆD is the through space or electron-nuclear dipolar interaction. These interactions contain contribu- tions to the isotropic shift and thus it can be considered that these contributions are added to the diamagnetic shiftδdiamagnetici.e. the shift that would be measured if the material contained no unpaired electrons

δmeasured=δdiamagnetic+δhyperf ine (2.93)

where

δhyperf ine=δcontact+δdipolar (2.94)

withδcontactandδdipolarbeing the contribution from both the Fermi contact and the electron-nuclear dipolar interactions respectively.

To understand how the interaction between the nucleus (I) and the electron (S) forms this paramagnetic Hamiltonian Hˆp, contributions toδhyperf ine from the

magnetic susceptibility needs to be understood. For a given group of spins placed in an external magnetic field the induced magnetic field per unit volume, the mag- netisation per unit volumeM can be described as

M = µind

V =

1 µ0

where the applied magnetic field is proportional to the magnetic susceptibility per unit volumeχV and the inverse of the magnetic constantµ0. Therefore, its possible to calculate the magnetic susceptibility per moleχM by

χM =VMχV =VM

µ0M B0

(2.96)

whereVM is the molar volume. This can be further defined as the magnetic suscep-

tibility per moleculeχ being divided by Avogadro’s constant NA:

χ= χM NA

. (2.97)

From Equation 2.95 and fromµind=NAhµiV /VM,χcan be expressed as:

χ= µ0hµi B0

(2.98)

wherehµi is the average induced magnetic moment per particle.

The relaxation and precession dynamics of the electron spin S are several orders of magnitude faster than the NMR timescale. This causes the electron spin S to be averaged over several Zeeman states giving rise to the Curie spin hSzi

hSzi= P Ms MS n₋ geµBMsB0 kBT o P Ms n −geµBMsB0 kBT o (2.99) hS±i= 0 (2.100)

With the high temperature approximation this simplifies to

hSzi=−

geµBB0 3kBT

S(S+ 1). (2.101)

As the induced magnetic moment per particle is proportional to the curie spin hµi=−µgehSzi[33] therefore the Curie moment can be determined

hµsi=

g2

eµ2BS(S+ 1)

3kBT

B0. (2.102)

isotropic molecular magnetic susceptibility (Equation 2.102)

χiso =µ0

g2_eµ2_BS(S+ 1) 3kBT

, (2.103)

and in terms of the Curie spin (Equation 2.101)

hSi=hSzi=−

χiso

µ0µBge

B0. (2.104)

The Hamiltonian of the through bond coupling between the nucleusI and the elec- tronS is proportional to thesorbital for the atom containing the spin I

ˆ

HC _{=Aχ·Iˆ (2.105)}

where the through bond coupling of the nucleus to the average electron spin is described by the hyperfine Fermi constantA is

A= µ0

3S~geµBγIρs (2.106)

where theγI is the nuclear gyromagnetic ratio. The gyromagnetic ratio and mag-

netic moment of the electron (g-value) is represented byge and the spin density at

the nucleus,ρswhich is described by the sumiof the spin density for theith nuclear

orbitals

ρs=

X

i

[|Ψ_i−(0)|2− |Ψ_i+(0)|2] (2.107) with the positive|Ψ_i−(0)|2 _and|Ψ+

i (0)|2 negative spin densities respectively.

To determine the shift contribution of the Fermi contactδcontacttowards the hyperfine shift δhyperf ine the energy of the contact interaction needs to be related to the size and strength of the magnetic field

δcontact= Econtact γI~IzB0

, (2.108)

whereEcontact is the energy for this isotropic hyperfine coupling

Econtact=AIzhSzi. (2.109)

Thus, when the Curie spin (Equation 2.104) and the isotropic magnetic susceptibility (Equation 2.103) are substituted in, the Fermi contact interaction is:

δcontact=AgeµBS(S+ 1) 3γI~kBT

From this it is clear where the inverse reliance on temperature for this interaction originates. If the electronic field at the paramagnetic nucleus is anisotropic, the g factor should be replaced by the term 1₃(gk+ 2g⊥) which represents the parallel gk and perpendicular g⊥ components, thus representing an axially symmetric tensor. In this case the magnetic susceptibility has an orientation dependence relative to the external magnetic field. Therefore, the average induced magnetic field represented in Equation 2.98 instead becomes

hµˆi= χˆBˆ0 µ0

(2.111)

The second part of the paramagnetic Hamiltonian, the electron nuclear in- teraction ( ˆHD), can be described as analogous to the dipolar interaction discussed in Section 2.4.2. The determination of the interaction energy is fundamentally the same as Equation 2.35 in Section 2.4.2. However, this change from a nuclear-nuclear to an electron nuclear interaction produces a few key differences. Firstly the dis- tance between the electron and the nucleusr can be approximated by the distance from the nucleus to the metal centre. This approximation allows us to ignore any fractional electron density covered by the Fermi contact shift. By displaying the dipolar Hamiltonian Hˆd from Equation 2.36 in terms of the nucleus and the elec-

tron by the taking magnetic moment of the nucleus as described in Section 2.1.1

ˆ

µ=γIˆ (2.2)

and describing the magnetic moment of the electron ˆµas the averaged induced elec- tron magnetic moment hµˆi in the magnetic field. This Hamiltonian now represents the electron nuclear interaction

ˆ HD ₌₋µ0 4π " 3(~γIIˆ·r)(hµi ·ˆ r) r5 − ~γIIˆhµiˆ r3 # . (2.112)

Therefore, by using Equations 2.97 & 2.111 and describing ˆI = ˆIκκsince the nucleus

is quantized along the direction of the external magnetic fieldκ. The Hamiltonian can be rewritten as ˆ HD ₌₋~γIB0 4πr5 Iˆkκ· 3r(r·χ)−r2χ·κ. (2.113) A second rank magnetic susceptibility tensorσp can then be defined as

which results in the electron nuclear Hamiltonian having the form

ˆ

HD ₌₋ ~γI

4πr5Iˆ·σp·B. (2.115) resulting in a remarkable similarity to the chemical shielding Hamiltonian

ˆ

HCS =γIˆ·σ·B. (2.25)

And as with CSA (Section 2.4.1), the paramagnetic interaction is observed to scale linearly with magnetic field.

To transform this interaction in the LAB frame, it can be easier forHˆD to be expressed as PAS frame in spherical tensor form:

ˆ

HD ₌ X

m=−1,0,1

D2,0d20m(β)T2,m (2.116)

Whered2_0m(β) is the reduced Wigner matrix element, withβbeing the angle between the PAS and the magnetic field. D2,0 is the only non-zero component of the dipolar tensorD:

D2,0 = √

6~γIµBge

4πr3 (2.117)

and the spin operator terms (T2,m), in a large applied magnetic field are:

T2,0 = 2 3IzB0− √ 2 3 χ0,0IzB0 (2.118) T2,±1 =∓ 1 2hS±iIz = 0 (2.119) T2,±2 =0 (2.120)

Where only the terms that commute withIz are retained. The dipolar Hamiltonian

then becomes ˆ HD ₌ ( 2 3D2,0χ2,0− 1 2(D2,1χ2,−1+D2,−1χ2,1)− √ 2 3 D2,0χ0,0 ) IzB0 (2.121)

whereχ2,n are represented by: χ0,0=− √ 3χiso (2.122) χ0,±1= r 2 3∆χax (2.123) χ0,±2= 3 4 ∆χrh ∆χax (2.124)

where the axial and rhombic components of the tensorχ, in the PAS frame, are:

χax=χzz −

χxx+χyy

2 (2.125)

χrh=χxx−χyy (2.126)

Equation 2.121 can then be simplified to: [27]

ˆ HD ₌ ( δdipolar+ r 1 6∆σp(3cos 2_β−1) ) IzB0 (2.127) where the shift anisotropy is expressed as

∆σp =

χiso ~γIr3

(2.128)

and the dipolar shift term is:

δdipolar 1 12πr3 ∆χax(3 cos2θ−1) + 3 2∆χrhsin 2_{θcos 2ϕ} (2.129)

where the anglesθand ϕdescribe the orientation of the nuclear spin relative to the PAS ofχ, as this component solely relies on the spins relative to each other in the molecule.