Introduction to Magnetic Resonance David J. Keeble

Magnetic Resonance Magnetic Magnetic moments? What matters is matter with moments Matter: Leptons and quarks The simplest fundamental particle is the lepton the electron Classical Physics: What is the ratio of the magnetic moment, m, of a spinning sphere of mass M carrying charge Q, where the charge and mass are identically distributed, to the angular momentum L? The ratio of the magnetic moment, m, to the angular momentum L is called the gyromagnetic ratio, g (or magnetomechanical ratio).

A thin uniform donut, carrying charge Q and mass M, rotates about its axis as shown below: (a) Find the ratio of its magnetic dipole moment to its angular momentum. This is called the gyromagnetic ratio (or magnetomechanical ratio). (b) What is the gyromagnetic ratio for a uniform spinning sphere? [This requires no new calculation; simply decompose the sphere into infinitesimal rings, and apply the result of part (a).] (c) According to quantum mechanics, the angular momentum of a spinning electron is . What then is the electron’s magnetic dipole moment in Am2? z

Magnetic Resonance Magnetic Magnetic moments? The electron Classical Physics: For an electron Quantum Mechanics tells us there is an intrinsic angular momentum of . The Bohr magneton Dirac’s Relativistic Quantum Mechanics Magnetic moment Spin angular momentum define a ‘g-factor’ where now let i.e. we’re here calling S the intrinsic angular momentum of the electron but the units, , are now assigned to the quantity we call the Bohr magneton

Magnetic Resonance Magnetic Magnetic moments? The electron where here we let Dirac’s Relativistic Quantum Mechanics: Feynman, Schwinger, and Tomonaga applied quantum electrodynamics : The most precisely known quantity NB: change of units – try using dimensional analysis to check this

Magnetic Resonance Magnetic Magnetic moments? The proton Quark Charge Spin Up, u +2/3 1/2 Down, d -1/2 The proton is composed of three quarks (uud) Magnetic moment Spin angular momentum The intrinsic angular momentum of the proton The neutron The neutron is composed of three quarks (udd) Magnetic moment Spin angular momentum The intrinsic angular momentum of the neutron

Magnetic Resonance Magnetic Magnetic moments? The proton The neutron Experimental values The neutron The Bohr magneton We define a similar quantity, the nuclear magneton where we substitute the mass of the proton, rather than the electron. The nuclear magneton Comparing the measured magnetic moment values for the proton and neutron with the nuclear magnetron we see they are roughly of the same order.

Magnetic Resonance Magnetic Magnetic moments? The electron The proton Spin angular momentum remember above we define S as a dimensionless number above The proton Magnetic moment Spin angular momentum Define a proton g –factor

Magnetic Resonance Magnetic The proton Nuclear Isotopes Magnetic moment Spin angular momentum Define a proton g –factor Nuclear Isotopes We will be potentially interested in, normally stable, nuclear isotopes that possess a nuclear moment. Most isotope tables list nuclear spin and moment values, the nuclear g-value, defined in the same way as above may be given, or the simple ratio of the moment with the nuclear magneton, and/or the gyromagnetic ratio.

Magnetic Magnetic Resonance Magnetic moments? Nuclear moments Isotope Atomic mass (ma/u) Natural abundance (atom %) Nuclear spin (I) Magnetic moment (μ/μN) 14N 14.003 074 005 2(9) 99.632 (7) 1 0.4037607 15N 15.000 108 898 4(9) 0.368 (7) 1/2 -0.2831892 Isotope Atomic mass (ma/u) Natural abundance (atom %) Nuclear spin (I) Magnetic moment (μ/μN) 46Ti 45.9526294 (14) 8.25 (3) 47Ti 46.9517640 (11) 7.44 (2) 5/2 -0.78848 48Ti 47.9479473 (11) 73.72 (3) 49Ti 48.9478711 (11) 5.41 (2) 7/2 -1.10417 50Ti 49.9447921 (12) 5.18 (2) Isotope Atomic mass (ma/u) Natural abundance (atom %) Nuclear spin (I) Magnetic moment (μ/μN) 63Cu 62.9295989 (17) 69.17 (3) 3/2 2.2233 65Cu 64.9277929 (20) 30.83 (3) 2.3817

Magnetic Resonance Magnetic Magnetic moments? The electron The proton Quantum Mechanics? ‘Observe’ magnetic moments magnetic moment OPERATOR If we assume the non-interacting ‘particles’ each have a total angular momentum (A special case of the Wigner – Eckhart theorem) Here you can choose to pull the h-bar into the angular momentum operator definition. Here you can’t since h-bar is included in the magneton.

Magnetic Resonance Magnetic Magnetic moments in a bulk sample? What we measure is the resulting macroscopic moment per unit volume V, due to the assemble of N magnetic moments in that volume – the Magnetization.

Magnetic Resonance Resonance? So with an assemble of electron spins, or protons…….. Let’s put our magnetic moments into an external magnetic field , B Magnetic moment Spin angular momentum What effect does this have on the energy, E, of our particles carrying magnetic moments?

Magnetic Resonance Resonance? So with an assemble of electron spins, or protons…….. Let’s put our magnetic moments into an external magnetic field , B Magnetic moment Spin angular momentum Energy – to determine the quantum mechanical operator that allows us to predict the results of energy measurements we can start with the classical expression a substitute the appropriate observable operators.

Magnetic Resonance Classical perfect magnetic dipole Let’s first go back to the classical case and consider the forces acting on a loop area ab carrying current I, it’s not too difficult to establish that a torque must act and that it’s given by the expression : The force on an infinitesimal loop, with dipole moment m, in a field B is: Magnetic moment Here we’ve moved the dipole in from infinite and rotated it. Then as long as B is zero at infinity the energy associated with the torque is :

Magnetic Resonance Resonance? Classical E&M Quantum Mechanics Magnetic moment Spin angular momentum Quantum Mechanics For a ‘static’ (it can rotate, but let’s not deal with translation) dipole moment m, in a field B we now have:

Magnetic Resonance Resonance? Magnetic moment Spin angular momentum So let’s remember the fundamental issues regarding J, L, S, and I in quantum mechanics: The algebraic theory of spin is identical to the theory of orbital angular momentum; we call it spin angular momentum. However, physically these are very different : The eigenfunctions of orbital angular momentum are spherical harmonics we get from solving the differential equations that we get from the Schrödinger time-independent equation The eigenfunctions of spin angular momentum are expressed as column matrices. This physics emerges from Dirac equation, but we use it with the Schrödinger time-independent equation

Magnetic Resonance Resonance? No spin stands lone – unfortunately? If they did the simple story we’ve developed would be it, and as we’ll learn we would measure ‘text book’ magnetic resonance spectra. Simple, elegant, understandable – but we’d be out of a job! Magnetic moment Spin angular momentum Spins couple – to eachother, to the orbital motion of the particles, to vibrations, to…………… But before we break out into the ‘real world’ let’s stick with our ideal isolated magnetic moments for a bit longer and look at the basic principles of ‘resonance’.

Magnetic Resonance Resonance? Let’s consider an I = 3/2 nucleus placed in a magnetic field B. Magnetic moment Spin angular momentum

Magnetic Resonance Resonance? Let’s consider an I = 3/2 nucleus placed in a magnetic field B. Magnetic moment Spin angular momentum

Magnetic Resonance Resonance? Let’s consider an I = 3/2 nucleus placed in a magnetic field B. Magnetic moment Spin angular momentum

Magnetic Resonance Resonance? Consider and assembly of particles, each having total angular momentum Let’s assume they are noninteracting – the greatest possible simplification The probability that a dipole within the assembly at temperature T has potential energy Ei is, according to Boltzmann: Here: Why Boltzmann statistics? It is the fact they are non-interacting, and hence distinguishable that’s key The differences in population of the levels means that energy can be absorbed, there can be a net moving of spins ‘up’

Magnetic Resonance Resonance? The probability that a dipole within the assembly at temperature T has potential energy Ei is, according to Boltzmann statistics. So at a finite temperature multiple levels can be populated To get a transition from one level to another - we need to apply an oscillating magnetic field with the correct orientation with respect to the external magnetic field. We can tackle this using time-dependent perturbation theory which can give us Fermi’s golden rule, which for our purposes can take the form for the probability per unit time that a paramagnet initially in state m will be found it state m’: B1 is the magnitude of a magnetic field oscillating at frequency w perpendicular to B0 In this last expression, we’ve let the ‘real world’ butt in again and are assuming the there is a distribution of effective magnetic fields across our assembly giving a lineshape g(w)

Magnetic Resonance Resonance? We can tackle this using time-dependent perturbation theory which can give us Fermi’s golden rule, which for our purposes can take the form for the probability per unit time that a paramagnet initially in state m will be found it state m’: The other important consequence of this expression is that the term in the square brackets defines the ‘selection rules ‘ for these transitions.

Magnetic Resonance Resonance? How about a single electron, S = 1/2, placed in a magnetic field B. Magnetic moment Spin angular momentum Two eigenstates

Magnetic Resonance Resonance? Magnetic moment Spin angular momentum

Quantitative, Sensitivity ~ 1010 spins Electron Paramagnetic Resonance (EPR) Zeeman B (T) S = 1/2 g = 2 94 GHz 3.36 34 GHz 1.22 9.5 GHz 0.34 Quantitative, Sensitivity ~ 1010 spins

Electron Paramagnetic Resonance (EPR) Zeeman No spin stands lone ……. The expression on the right is the first, normally dominant, term in a general ‘spin’ Hamiltonian expression for EPR. The left hand expression is exact for a mythical assembly of non-interacting ‘free’ electrons. In a real sample those normally ‘special’ electrons that are not spin-paired and so are detectable by EPR will be occupying an orbital, an electronic state, that may also have some orbital angular momentum ‘character’ due to say to a spin-orbit interaction. In consequence, the true eigenstates of that electron involve angular momentum that is not purely spin. This is messy so magnetic resonance experimentalists rapidly adopted the spin-Hamiltonian concept. The point of the spin-Hamiltonian is that you keep assuming that you are working with pure spin functions , you fold the nasty complications into the parameters – in this case you define a g-matrix that departs from ge in a way that allows you to still use those spin functions that we can express as simple column vectors. The departure from ‘free’ is now characterized by the values in the g-matrix, the bonding character of the electronic state may now manifests itself as a g-value different from 2.0023

Electron Magnetic Resonance Spectroscopy Hyperfine & Nuclear Zeeman Zeeman Symmetry So armed with this spin-Hamiltonian concept we can develop terms which describe other important interactions between spins, for example the hyperfine interaction between magnetic nuclei and our electron spin(s)

Electron Paramagnetic Resonance (EPR) Zeeman Hyperfine & Nuclear Zeeman PbTiO3 Cu (d9): S = 1/2 Here is an example of a real EPR spectrum from a very low concentration of Cu2+ impurity ions substituting for Ti in the perovskite oxide PbTiO3. At this orientation of the magnetic field with the crystal axes the g-value is ~ 2.34. It’s determining what the center field of the spectrum is. The hyperfine interaction with the magnetic Cu nuclei is defining the number of lines and the separation. 63Cu 69.2 % I = 3/2 m/mn = 2.22 65Cu 30.8 % I = 3/2 m/mn = 2.38 2I+1 lines