The Quantum Mechanics of MRI Part 1: Basic concepts David Milstead Stockholm University

Background reading Fundamentals of Physics, Halliday, Resnick and Walker (Wiley) The Basics of NMR, J. Hornak (http://www.cis.rit.edu/htbooks/nmr/inside.htm )

Outline What is quantum mechanics Wave-particle duality Schrödinger’s equation Bizarreness uncertainty principle energy and momentum quantisation precession of angular momentum

What is quantum mechanics ?

Basic concepts What is a wave ? What is a particle ? What is electromagnetic radiation ? What happens to a magnet in a magnetic field ?

What is a wave ? Double slit diffraction Properties of waves: Superposition, no localisation (where is a wave??)

What is electromagnetic radiation ? EM radiation is made up of electromagnetic waves of various wavelengths and frequencies Radio frequency is of most interest to us

Circularly polarised light Rotating magnetic field Radiation can be produced and filtered to produce a rotating magnetic field.

Light is a wave!

Photon

Photons

Wave-particle Duality

We’ve just learned a basic result of quantum mechanics. Now we move onto some maths.

Fundamental equation of quantum mechanics

Free particle

Wave function of a confined particle Eg particle trapped in a tiny region of space. How do we model the wave function ? Solution to Schrödinger’s equation would be sum lots of sine waves with different wavelengths/momenta.

x x

Question A 12-g bullet leaves a rifle at a speed of 180m/s. a) What is the wavelength of this bullet? b) If the position of the bullet is known to an accuracy of 0.60 cm (radius of barrel), what is the minimum uncertainty in its momentum? c) If the accuracy of the bullet were determined only by the uncertainty principle (an unreasonable assumption), by how much might the bullet miss a pinpoint target 200m away?. 

Hydrogen atom Solve Schrödinger’s equation for an electron around a proton in a hydrogen atom. The electron is confined due to a Coulomb potential.

Where is the electron ? Wave functions

Current loops and magnetic dipoles + - Current loop N S Bar magnet

Orbiting electron as a current loop

An atom in a magnetic field l=1 and therefore 2l+1 states E (1) (2) (3)

l=1 l=0

z y x LZ U X

Angular momentum precession LZ constant and L precesses around z-axis

Question

The mathematics of spin angular momentum is identical to orbital

FN FS Magnetic field FN FS Magnetic field Ignore force not parallel to North-South axis.

Magnetic field Magnetic field

Magnetic field

Zeeman effect with orbital and spin angular momentum In the presence of a magnetic field, multiplicities of ”spectral lines” appear

Larmor precession for spin z y x SZ S S sinq

Summary Established basic quantum mechanics theory needed for NMR wave-particle duality Light is either photons or electromagnetic waves Schrödinger’s equation and the wave function at the heart of QM predictions Energy and angular momentum are quantised Larmor precession Angular momentum comes in two varieties (orbital and spin)

The Quantum Mechanics of MRI Part 2: Understanding MRI David Milstead Stockholm University

Outline Spin - reminder Fermions and bosons Nuclear energy levels

Spin z y x SZ S S sinq

Gyromagnetic ratio Why do they have different values ?

Fermions and bosons Fermions Bosons Spin 1/2, 3/2, 5/2 objects Electrons, protons and neutrons have spin 1/2 Tricky bit comes when combining their spins to form the spin of, eg, an atom or a nucleus Bosons Integer spin objects

Similar shell structure for nuclear physics as for atomic physics Need to fill up, shell by shell Pairs of protons and neutrons cancel each other’s spins. Pauli’s exclusion principle ensures that many shells are filled. Nuclei with uneven (even) atomic number have half-integer (integer) spin Nuclei with even atomic and mass numbers have zero spin. Unpaired neutrons/protons provide the spin for MRI.

Question Nature would prefer all electrons to be in the lowest shell and all nucleons protons/neutrons) in the lowest shell ? Why doesn’t this happen ?

Usefulness for MRI Need isotopes with unpaired protons (to produce signal for MRI) Most elements have isotopes with non-zero nuclear spin Natural abundance must be high enough for MRI to be performed.

Spins of various nuclei

Now we can understand MRI

No magnetic field

Apply an external magnetic field B0

Another look at the system Split into ”spin-zones”. For uniform system we can regard the macroscopic system as giving a single magnetisation.

Putting together what we’ve learned

Energy level population

Changing the spin populations

Question

Exciting the nuclei - Rf pulse Rotating magnetic field B1 Typical pulse duration ~1ms. Two ways to think about the pulse. Both are needed to understand MRI.

Excitation Rotating magnetic field B1 B0 B1

Rotating frame of reference B0 B1 X’ y’

Different types of pulses

Question

What happens next ?

Longitudinal relaxation

Relaxation times for different materials

Transverse relaxation

Free induction decay

Free induction decay

Summary Basic quantum mechanics at the heart of nuclear magnetic resonance Angular momentum quantisation Energy quantisation Features of a MRI experiment investigated.