David J. Keeble

1. Introduction to Magnetic Resonance David J. Keeble

2. 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 Mcarrying 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 momentumLis called the gyromagnetic ratio, g(or magnetomechanical ratio).

3. 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

4. Magnetic Resonance Magnetic Theelectron Magnetic moments? 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 define a ‘g-factor’ where now let Spin angular momentum 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

5. Magnetic Resonance Magnetic Theelectron Magnetic moments? 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

6. Magnetic Resonance Magnetic Magnetic moments? Theproton The proton is composed of three quarks (uud) Theintrinsic angular momentum of the proton Magnetic moment Magnetic moment Theneutron The neutron is composed of three quarks (udd) Spin angular momentum Spin angular momentum Theintrinsic angular momentum of the neutron

7. Magnetic Resonance Magnetic Magnetic moments? Theproton Experimental values Theneutron 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.

8. Magnetic Resonance Magnetic Theelectron Magnetic moments? Define a proton g–factor remember above we define S as a dimensionless number above Magnetic moment Magnetic moment Theproton Spin angular momentum Spin angular momentum

9. Magnetic Resonance Magnetic Theproton Define a proton g–factor Magnetic moment 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. Spin angular momentum

10. Magnetic Resonance Magnetic Magnetic moments? Nuclear moments

11. Magnetic Resonance Magnetic Theelectron Magnetic moments? Theproton Quantum Mechanics? ‘Observe’ magnetic moments magnetic moment OPERATOR (A special case of the Wigner – Eckhart theorem) If we assume the non-interacting ‘particles’ each have a total angular momentum 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.

12. 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.

13. 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 What effect does this have on the energy, E, of our particles carrying magnetic moments? Spin angular momentum

14. 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 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. Spin angular momentum

15. Magnetic Resonance Classical perfect magnetic dipole Let’s first go back to the classical caseand 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 : Magnetic moment The force on an infinitesimal loop, with dipole moment m, in a field B is: 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 :

16. Magnetic Resonance Resonance? Classical E&M Magnetic moment 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: Spin angular momentum

17. Magnetic Resonance Resonance? Magnetic moment 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 Spin angular momentum 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

18. Magnetic Resonance Resonance? unfortunately? No spin stands lone – 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 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’. Spin angular momentum

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

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

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

22. 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: It is the fact they are non-interacting, and hence distinguishable that’s key Why Boltzmann statistics? The differences in population of the levels means that energy can be absorbed, there can be a net moving of spins ‘up’

23. 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 lineshapeg(w)

24. 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.

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

26. Magnetic Resonance Resonance? Magnetic moment Spin angular momentum

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

28. 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

29. Hyperfine & Nuclear Zeeman Electron Magnetic Resonance Spectroscopy Symmetry Zeeman 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)

30. Hyperfine & Nuclear Zeeman Electron Paramagnetic Resonance (EPR) Zeeman Cu (d9): S = 1/2 PbTiO3 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 = 3/2m/mn = 2.22 65Cu 30.8 % I = 3/2m/mn= 2.38 2I+1 lines