Quantum Mechanics of NMR

8/12/2019 Quantum Mechanics of NMR

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The Quantum Mechanics of MRI

Part 1: Basic concepts

David Milstead

Stockholm University


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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 )


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Outline What is quantum mechanics

Wave-particle duality

Schrdingers equation

Bizarreness

uncertainty principle energy and momentum quantisation

precession of angular momentum


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What is quantum mechanics ?


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Basic concepts What is a wave ?

What is a particle ? What is electromagnetic radiation ?

What happens to a magnet in a magneticfield ?


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What is a wave ?

Double slit diffraction

Properties of waves:Superposition, no localisation (where is a wave??)


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What is electromagnetic radiation ?

EM radiation is made up of electromagnetic wavesof various wavelengths and frequencies

Radio frequency is of most interest to us


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Circularly polarised light

Radiation can be produced and filtered to produce a rotating magneticfield.

Rotating magnetic field


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Light is a wave!


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Photon


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Photons

346.626 10

Quanta of light:

Energy

=Planck's constant= Js

=frequency

= =wavelength, speed

E hf

h

f

vv

f

=

=


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Wave-particle Duality

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Particles also show wave-like properties

Wave-length = Energy=

=frequency, =momentum, =6.6 Js Planck's constant.

hhf

p

f p h


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Weve just learned a basic result of quantum mechanics.Now we move onto some maths.


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Fundamental equation of quantummechanics

( ) wave function=particle energy

=potential energy of particle

xE

U

=


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( )Re( )x

x

Free particle

0

Well defined wavelength and momentum but no preferential

position.

"Uncertainty on position":

"Uncertainty on momentum":

x

p

=

=

( )

2 2

2

2

2

0

2

cos sin

2

2

; Fixed

-

constant - particle can be anywhere!

; ;

fixed wavelength and momentum

U E

Em x

A kx i kx

h pp k E

m

=

=

= +

=

= = =

A particle not confined or subject to forces.


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Wave function of a confined particle

p

x

p

Eg particle trapped in a tiny region of space. How do we model thewave function ?

Solution to Schrdingers equation would besum lots of sine waves with different wavelengths/momenta.


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x

x

Similarly:y z

p y h p z h > >


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QuestionA 12-g bullet leaves a rifle at a speed of 180m/s. a) What is the wavelength of thisbullet? b) If the position of the bullet is known to an accuracy of 0.60 cm (radius ofbarrel), 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?.

3434

34

31 1

3131

6.6 103.06 10

0.012 180

6.6 10 100.006

10200 200 9 10

0.12 180

(a) Wavelength m

(b) kgms

(c) Displacement m Tiny!

Because is so small, quantum effects are nev

h

p

hpx

p

p

h

= = =

er seen for macroscopicobjects.


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Hydrogen atomSolve Schrdingers equation for an electron around aproton in a hydrogen atom.The electron is confined due to a Coulomb potential.

2

04

proton chargee

U e

r

= =


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( )- , - 1 ,...., 1,

(2 1)

Component along one direction:

different states.

z l lL m m l l l l

l

= =

+

2

whereh

=


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Where is the electron ?Wave functions


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Orbiting electron as a current loop

2

1

2 2 2 2

2

Tiny current loop.

chargecurrent

period

Magnetic moment

area of loop

Angular momentum:

(parallel to angular momentum)

Gyrom

e

l

e

dq eI

dt T

IAn

A

v ev ev evRIA A R

T R R R

L m vr

eL

m

= = =

=

=

= = = = =

=

=

2agnetic ratio: =-l

e

e

L m

=

e

R

into the page.n


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Recall that the electron has fixed values of ZL

2 2

lzz z

emeLU B B

m m = =


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An atom in a magnetic field l=1 and therefore 2l+1 states

E

- -2

(1) 12 2

(2) 0 0 ; 0

(3) 12 2

; ( antiparallel with )

; ( parallel with )

Two states with

ll lz z lz

e

l lz z le e

l lz

l lz z z l

e e

emU B B

m

e em U B B

m m

m U

e em B U B B

m m

= = =

= = =

= = =

= = =

shifted energy one state with no energy shift.+

2 z zB(1)

(2)

(3)


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l=0

l=1


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( )( )

2

2 2 2 2 2 2

,

1

What happens when a circulating electron is placed

in a magnetic field.

We know that are fixed and known.

constant

By symmetry, same spread of results of

or when mea

Z

x y Z l

x y

L L

L L L L l l m

L L

+ = = + =

sin

1 1sin

sin sin

2

2

surements of made.

i.e. just as likely to find angular momentum and .

Magnetic field introduces a torque

=

i.e

l l

ll

e

le

X Y

dL

B B dt

Bd dLB

dt L dt L L

m eBL

e m

= = =

= = =

= =

.

2

. the angular momentum vector precesses around the magnetic

field ( -axis) with frequency This is Larmor precession.

Gyromagnetic ratio : e

z

e

B m

= =

LZ

UX

xy

z


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Angular momentum precession

LZconstant andL precesses around z-axis


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Question

.

Is it possible for an electron in a hydrogen atom to have ?

Show that this value can be approached for large values of

Would be possible using classical physics ?

No - it can never happen o

Z

Z

L L

l

L L

=

=

( )2 2 2 2 2

,

0

1

therwise we would "know" values of

i.e.

For largeMax

(comes close)

In classical physics there is no restriction on what we can and ca

x y

x y

Z l Z

Z

L L

L L

L l l l L l L l

L m L l

L L

= =

= + = == =

n't know.


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The mathematics of spin angular momentum is identical to orbital

angular momentum.


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FN

FS

FN

FS

Magnetic field

Magnetic field

Ignore force not parallel to North-South axis.


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Magnetic field

Magnetic field


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Zeeman effect with orbital and spin

angular momentum

In the presence ofa magnetic field,multiplicities of

spectral linesappear

10

2

2 1 2

Eg for ,

number of lines:

l s

n s

= =

= + =

2BB

Larmor precession for spin


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2

2

Identical to orbital case.

;

spin magnetic moment

Electron:

electron g-factor 2, electron mass.

Nucleus:

nuclear g-factor, proton mass.

(ne

s

s

e e

e

e e

N N p

p

S B

eg

m

g m

eg g m

m

= =

=

=

= =

= = =

xt lecture)

SZ

xy

z

Larmor precession for spin

S sin

S


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Summary Established basic quantum mechanics

theory needed for NMR wave-particle duality

Light is either photons or electromagnetic waves

Schrdingers equation and the wave functionat the heart of QM predictions

Energy and angular momentum are quantised Larmor precession

Angular momentum comes in two varieties(orbital and spin)


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Outline Spin - reminder

Fermions and bosons Nuclear energy levels

Spin


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2 2( 1)

2

2

Identical in form to orbital case.

spin magnetic moment

Electron:

electron g-factor 2, electron mass.

Nucleus:

nuclear g-factor,

z s

s

s

e e

e

e e

N N

p

S s s S m

B

S

eg

m

g m

eg g

m

= + =

=

=

=

=

= =

= =

proton mass.

(next lecture)

pm = SZ

xy

z

Spin

S sin

S

G ti ti


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Gyromagnetic ratio

Why do they have different values ?


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Fermions and bosons Fermions

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


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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 enoughfor MRI to be performed.


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Spins of various nuclei

Now we can understand MRI


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Now we can understand MRI

0

1

1 0?

We know the basics:

A uniform magnetic field B

A short pulse of a rf field BA system out of equilibrium

measurement of return to equilibrium.

Why is

(a)

(b) The rf pulse shorter than recovery

B B


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No magnetic field

Apply an external magnetic field


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Apply an external magnetic field

.

, 0

1

2

Spins precess at Larmorfrequency

Gyromagnetic ratio:

Precessions incoherent:

Total spin Z x y

Z

B

S S S S

S

=

= = =

=

B0

In fact, there are continual transitions

and interactions from thermal energy.


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Another look at the system

Split into spin-zones.For uniform system we can regard the macroscopic system as giving

a single magnetisation.

Conventional to talk about magnetisation :

; no. dipoles, volumes

M

NM N V

V

= = =

Putting together what weve learned


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0

0 0 0

,2

1

2 2 4

2

2

Need to know its energy splitting.

=spin angular momentum nuclear g-factor proton mass.

;

Frequency of light need

sN p

p

sZ sZ z N N

p p

p

sZ

p

e g S g mS m

e eE B S g g

m m

g eE B B

m

= = = =

= = = =

= = =

0 0

00 0

2

2

ed for excitation

Larmor frequency:

How many nuclei can be excited ?

p

p

N

p

g eB

m

eBB g

m

=

= = =

0

0

Energy level population


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e gy e e popu at o

Nature has a preference for the lowest energy states.

In thermal equilibrium the lowest states are a bit more

populated than the higher energy states.

2

1 ...2!

Taylor expansion: xx

e x= + + +


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23

0 0 0

6 26

0

2!

1

2

1.5 42.576 10 2 4.2 10

Boltzmann's constant = 1.38 10 J/K=room temperature 310 K

T J

B B

B

p

p

N E Ee

N k T k T

kT

g eE B B

m

B E

N

N

= +

=

= = =

= = =

0

1.000009

1

2 2

p

p B

g eN NN N N N B

N m k T

=

= =

Changing the spin populations


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Changing the spin populations

15

Tiny differences in, eg, 0.02ml of water

expect 610 more in parallel state.

Question


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QuestionChemists can excite certain samples using UV light. Would you expect

to be substantially different in this case ? What does this imply for

the relative size of the sample of a MRI scan and a test

N

N

88 16 1

8

34 16 17

17

23

3 1010 3 10

10

6.6 10 3 10 2 10

2 101 1 1 5000

1.4 10 300

made by

chemists with infrared light ?

UV-light m s

J

In fact, approximation no longer valid since it

B

cf

E

N E

N k T

= = =

=

+ + +

is no longer a

small difference!!

A chemist can use far smaller samples since the energy gap

is larger and there are far more in the lower energy state.

Exciting the nuclei - Rf pulse


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Rotating magnetic fieldB1

Typical pulse duration ~1ms.Two ways to think about the pulse. Both are needed to understand MRI.

Excitation


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Rotating magnetic fieldB1

B0

Pulse of rf-

1

0.

The rf pulse acts in two ways:

(1) The photons are absorbed, saturating the system and reducing(2) The rotating magnetic field acts on the magnetisation vector

to rotate into the complex plan

ZMB

1 1

1

1

1

e.

Individual dipole: ;

Use magnetisation: - simple form of the Bloch equations.

ss s

ddSB B

dt dt

dM M Bdt

= = = =

=

B1


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Rotating frame of reference

, ', '

'

Easier to understand if a rotating frame of reference

(with Larmor frequency) is used.

Rotate co-ordinates

Magnetic field constant on axis.

x y x y

x

B0

B1

X y

Different types of pulses


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90o 180o

1

1

1

5

1 3 6

0

2

2 42.58

1 110

10 42.58 10

1

Duration of pulse and size of B determine angle of rotation of magnetisation:

Spins precess around

Consider a full rotation of in 1ms

MHz/T

T

B

Bt

B

B T

= =

=

=

=

1 0. B B


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Question11 0

.

When applying a new magnetic field is obtained:

Why don't the protons just align with respect to that field ?

n

B

B B B= +

0B n

B

What happens next ?


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-

.

Consider a 90 rotation.

The magnetisation vector is in the plane.

The -pulse is turned off.

The system must return to equilibrium.

We have two components: and

o

Z xy

x y

rf

M M


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Longitudinal relaxation

Spin-lattice effect:

higher energy state interacts with lower energy state due and lose

energy through rotation and vibration .

Relaxation times for different materials


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Transverse relaxation


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In the laboratory frame

Dipole moments are initially in phase.

M precesses and decays.

As it precesses phase decoherence occurs.

Complicated process: contributing factor non-uniform magnetic

field ove

xy

r sample different precession rates for different regions.


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Free induction decay

*

2.

As the transverse magnetisation decays, a changing magnetic

field is produced:

=- an emf is produced in a receiver.

The signal is a decaying sinusoidal wave with lifetime

Measurement of gives

t

T

information on composition of sample.


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Free induction decay

Easier to interpret a single line on a "frequency" spectrum.

Use a Fourier transform to move from damped exponential to

signal.


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Summary Basic quantum mechanics at the heart of

nuclear magnetic resonance Angular momentum quantisation

Energy quantisation

Features of a MRI experimentinvestigated.

Documents

Quantum Mechanics of NMR