Introduction to Magnetic Resonance - PowerPoint Presentation

Slide1

Introduction to Magnetic Resonance

David

J. KeebleSlide2

Magnetic ResonanceMagnetic moments?

MagneticWhat matters is matter with moments

Matter: Leptons and quarks The simplest fundamental particle is the lepton the electron

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?

Classical Physics:

T

he

ratio

of the

magnetic moment

,

m

,

to the angular momentum L is called the gyromagnetic ratio, g (or magnetomechanical ratio). Slide3

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?

zSlide4

Magnetic ResonanceMagnetic

The electron

Classical Physics:

For an electron

Quantum Mechanics

tells us there is an intrinsic angular momentum of .

Dirac’s Relativistic Quantum Mechanics

d

efine a ‘

g

-factor’

w

here 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 moment

Spin angular momentum

Magnetic moments?

The Bohr magnetonSlide5

Magnetic ResonanceMagnetic

The electron

Dirac’s Relativistic Quantum Mechanics:

w

here here we let

Feynman, Schwinger, and

Tomonaga

applied quantum electrodynamics :

Magnetic moments?

The most precisely known quantity

NB: change of units – try using dimensional analysis to check thisSlide6

Magnetic ResonanceMagnetic

The proton

The proton is composed of three quarks (uud)

Quark

Charge

Spin

Up, u

+2/3

1/2

Down, d

-1/2

1/2

The

neutron

The neutron is composed of three quarks (udd)

The intrinsic angular momentum of the proton

The

intrinsic angular momentum of the neutron

Magnetic moment

Spin angular momentum

Magnetic moment

Spin angular momentum

Magnetic moments?Slide7

Magnetic ResonanceMagnetic

The proton

The

neutron

The Bohr magneton

The nuclear magneton

Magnetic moments?

We define a similar quantity, the nuclear magneton where we substitute the mass of the proton, rather than the electron.

Experimental values

Comparing the measured magnetic moment values for the proton and neutron with the nuclear magnetron we see they are roughly of the same order.Slide8

Magnetic ResonanceMagnetic

Magnetic moment

Spin angular momentum

r

emember above we define

S

as a dimensionless number above

The

electron

Magnetic moments?

The

proton

Magnetic moment

Spin angular momentum

Define a

proton

g

–factor Slide9

Magnetic ResonanceMagnetic

The

proton

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. Slide10

Magnetic ResonanceMagnetic

Nuclear moments

Magnetic moments?IsotopeAtomic mass (ma/u)Natural abundance

(

atom %)

Nuclear spin (I)

Magnetic moment (

μ/μ

N

)

46Ti45.9526294 (14)8.25 (3)0

47Ti

46.9517640 (11)7.44 (2)

5/2

-0.7884848Ti

47.9479473 (11)73.72 (3)0

49Ti

48.9478711 (11)5.41 (2)7/2

-1.1041750Ti

49.9447921 (12)5.18 (2)

0IsotopeAtomic mass (ma/u)Natural abundance (atom %)Nuclear spin (I)Magnetic moment (μ/μN)63Cu62.9295989 (17)

69.17 (3)3/2

2.2233

65

Cu

64.9277929 (20)

30.83 (3)

3

/

2

2.3817

Isotope

Atomic mass (m

a

/u)

Natural abundance (atom %)

Nuclear spin (I)

Magnetic moment (

μ/μ

N

)

14

N

14.003 074 005 2(9)

99.632 (7)

1

0.4037607

15

N

15.000 108 898 4(9)

0.368 (7)

1

/

2

-0.2831892Slide11

Magnetic ResonanceMagnetic

The proton

The

electron

Magnetic moments?

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. Slide12

Magnetic ResonanceMagnetic

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

.Slide13

Magnetic Resonance

Magnetic moment

Spin angular momentum

Resonance?

So with an

a

ssemble of electron spins, or protons……..

Let’s put our

magnetic moments

into an external

magnetic field

,

B

What effect does this have on the

energy

,

E

, of our particles carrying magnetic moments?Slide14

Magnetic Resonance

Magnetic moment

Spin angular momentum

Resonance?

So with an

a

ssemble of electron spins, or protons……..

Let’s put our

magnetic moments

into an external

magnetic field

,

B

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.Slide15

Magnetic

Resonance

Magnetic moment

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 :

Here we’ve moved the dipole in from infinite and rotated it. Then as long as

B

is zero at

infinity

t

he

energy associated with the torque is

:

The force on an infinitesimal loop, with dipole moment

m

, in a field

B

is:Slide16

Magnetic

Resonance

Magnetic moment

Spin angular momentum

Resonance?

Classical E&M

For a

‘

static’

(it can rotate, but let’s not deal with translation) dipole moment

m

, in a field

B

we now have:

Quantum MechanicsSlide17

Magnetic

Resonance

Magnetic moment

Spin angular momentum

Resonance?

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 :

T

he eigenfunctions of orbital angular momentum are

spherical harmonics

we get from solving the differential equations that we get from the Schrödinger time-independent equation

T

he 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 Slide18

Magnetic

Resonance

Magnetic moment

Spin angular momentum

Resonance?

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.

u

nfortunately?

Simple, elegant, understandable –

but we’d be out of a job!

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’.Slide19

Magnetic Resonance

Resonance?

Let’s consider an

I

=

3/2

nucleus placed in a magnetic field

B

.

Magnetic moment

Spin angular momentumSlide20

Magnetic Resonance

Resonance?

Let’s consider an

I

=

3/2

nucleus placed in a magnetic field

B

.

Magnetic moment

Spin angular momentumSlide21

Magnetic Resonance

Resonance?

Let’s consider an

I

=

3/2

nucleus placed in a magnetic field

B

.

Magnetic moment

Spin angular momentumSlide22

Magnetic ResonanceResonance?

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

E

i

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’Slide23

Magnetic ResonanceResonance?

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’:

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

)

B

1

is the magnitude of

a

magnetic field oscillating at frequency

w

perpendicular to

B

0Slide24

Magnetic ResonanceResonance?

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.Slide25

Magnetic ResonanceResonance?

Magnetic moment

Spin angular momentum

Two

eigenstates

How about a single electron,

S

=

1

/2

,

placed in a magnetic field

B

.Slide26

Magnetic ResonanceResonance?

Magnetic moment

Spin angular momentumSlide27

Electron Paramagnetic Resonance (EPR)

Zeeman

9.5 GHz

0.34

34 GHz

1.22

94 GHz

3.36

B (T)

S = 1/2

g

= 2

Quantitative, Sensitivity ~ 10

10

spinsSlide28

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

g

e

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 2.0023Slide29

Electron Magnetic Resonance Spectroscopy

Zeeman

Symmetry

Hyperfine & Nuclear 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)Slide30

Electron Paramagnetic Resonance (EPR)

Zeeman

Hyperfine & Nuclear Zeeman

63

Cu

69.

2 %

I =

3/2

m

/

m

n

=

2.22

65

Cu

30.8 % I = 3/2 m/mn = 2.38Pb

TiO3

Cu (d

9

): S = 1/2

Here is an example of a real EPR spectrum from a very low concentration of Cu

2+

impurity ions substituting for Ti in the

perovskite

oxide PbTiO

3

.

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.

2

I

+1 lines