Gyromagnetic Ratio Importance of Magnetic Resonance History of Magnetic Resonance

Lecture 1: Introduction

Lecture aims to explain:

1. History and importance of magnetic resonance

3. Simple resonance theory

4. Gyromagnetic ratio Importance of magnetic resonance History of magnetic resonance

First nuclear magnetic resonance (NMR) experiments followed a long line of research in related fields, and were described and carried out on LiCl molecular beams by Isidor Rabi in USA in 1938. In order for the spin of an atomic nucleus to be determined via Rabi’s technique, a sample needed to be vaporized and then exposed to a magnetic field.

See details at www.magnet.fsu.edu/education/tutorials/pioneers/

Felix Bloch and Edward Purcell, working independently of each other in USA, expanded the technique for use on liquids and solids.

Rabi was awarded the Nobel Prize in Physics for his work in 1944. Bloch and Purcell shared the Nobel Prize in Physics in 1952. Applications of magnetic resonance

Determination of atomic and molecular structure, chemical composition Use in analytical chemistry, biochemistry, oil industry

See examples at

http://www.bruker.com/index.php?id=4816

http://www.process-nmr.com/fox-app/crd_blndng.htm Applications of magnetic resonance

Magnetic resonance imaging (MRI) Primary use in medicine

Available in many hospitals, requires strong non-uniform magnetic field

MRI image of the brain can now be used to understand brain activity

2003 Nobel Prize for Medicine awarded to Peter Mansfield and Paul Lauterbur Applications of magnetic resonance Advanced experiments in nanostructures Non-invasive structural probing, quantum computation

Electron spin resonance in electrostatically defined

quantum dots: essential spin qubit operations

Bz~5.3 T InGaAs/GaAs 15 115 In 69 75As Ga

eV) 71 µ 10 Ga Structural studies

using NMR in σ- σ NMR Signal ( NMR 5 + semiconductor quantum dots

0 36 40 44 48 52 5668 72 rf frequency, ν (MHz) Magnetic moment and angular momentum Relation between magnetic moment and angular momentum μ = γJ

µ - magnetic moment of a nucleus or electron, J - its angular momentum, γ - gyromagnetic ratio

In the quantum theory, instead of vectors we introduce operators. Let’s introduce a dimensionless angular momentum operator Î : Jˆ = Iˆ

Components of Î (for example Îz along Oz direction) have the following eigenvalues: I, I −1, I − 2,...,−I Examples for different I

I=1/2 (spin 1/2): electron, proton, neutron, 31P, 13C, 29Si

I=3/2 (spin 3/2): Na, Cu, Ga, As

I=5/2 (spin 5/2): Al, Mn, 121Sb

I=7/2 (spin 7/2): 123Sb

I=9/2 (spin 9/2), In Simple resonance theory Interaction of magnetic moment with magnetic field

Application of magnetic field H produces an interaction energy of amount ̶ µ⋅H (known from electro-magnetism)

Taking the field H0 along Oz gives a simple Zeeman Hamiltonian: H Zeeman = −γH0I z

These gives energy eigenvalues (to be H0≠0 also referred to as spin levels): -3/2 I=3/2 ∆E E = −γH0m -1/2 m = I, I −1,...,−I H0 =0 ∆E Absorption of energy ∆E possible: +1/2 ∆E ∆E = γH0 +3/2 Realisation of coupling between spin levels (1)

H0 Strong magnetic field The coupling between spin levels (magnetic resonance) is realised by along Oz (~few Tesla) alternating magnetic field applied perpendicular to the static field. 0 H x cosωt

Weak in-plane field (~1 mT)

This will add a perturbing term in our Hamiltonian responsible for coupling:

H = H Zeeman + H pert Realisation of coupling between spin levels (2)

Hamiltonian including effect of the alternating magnetic field will read: 0 H = −γH0I z − γH x I x cosωt H0≠0 -3/2 ∆E Resonance is observed (coupling is the -1/2 strongest) for H0 =0 ∆E ω = ∆ or ω = γ E H0 +1/2 ∆ I=3/2 E +3/2 Gyromagnetic ratio Rough estimation of the order of magnitude

Example 1.1

Compute the magnetic moment and angular momentum of a particle of mass m and charge e moving in a circular path of radius r with period T. Use this result to estimate the gyromagnetic ratio of such particle.

Example 1.2

Estimate the order of magnitude of the gyromagnetic ratio for electron and proton using expression obtained in Example 1.1. Using this result, estimate the frequency at which the magnetic resonance can be observed for the two systems in a magnetic field of 1 Tesla. SUMMARY

Importance of magnetic resonance: wide ranging applications, including chemistry, medicine, and many industrial applications. Predominantly non-invasive techniques (especially NMR). In magnetic field the energies of spin levels are defined as: = −γ E H0m This makes absorption ∆ = γ of energy ∆E possible: E H0 m = I, I −1,...,−I e γ - gyromagnetic ratio, can be estimated as γ = 2m Resonance is observed for The coupling between spin levels (magnetic resonance) is realised by γ alternating magnetic field applied ν = H0 perpendicular to the static field. 2π