KEMM17 VT-10, L4 1

L3 Repetition

!

x

!

y

!

"

!

r

!

" t( ) = # $ t ( )d $ t

0

t

%

!

S t( ) = I "( )ei"td"#$

$

%!

ei"t

!

ei"

!

e"Rt

KEMM17 VT-10, L4 2

L4: Spin dynamics• Spin packets• RF pulses• The rotating frame• Signal detection• Magnetic field gradients• Spin echoes• Relaxation

Literature: Hore chap 6.2-6.3; Hornak MRI chap 3,4

KEMM17 VT-10, L4 33

Spin packetEnsemble of spins experiencing the same

magnetic field

!

m = µi

i

"

!

µi

Each µi obeys quantum laws,but m behaves classically!

magnetizationvector, m

KEMM17 VT-10, L4 4

Free precession

!

"0

= #$B0

!

dm t( )dt

= "#B0$m t( )

cf. spinning top

Note sign!Right-handedrotation positive

!

B0

KEMM17 VT-10, L4 5

Radiofrequency (RF) field, B1

• Magnetic field B1(B1<<B0) rotating inxy-plane withfrequency ωRF

• Produced by the RFcoil

!

B0

KEMM17 VT-10, L4 6

Resonance

!

dm t( )dt

= "#B t( ) $m t( )

B t( ) = B0

+ B1t( )

• m tilted from z-axisif ωRF ≈ ω0

• Resonance!freq. of perturbation =

natural freq. of thesystem

!

B0

KEMM17 VT-10, L4 7

Rotating frame - lab view• Reference frame

rotating in xy-planewith frequency ωRF

seen from the lab

!

B0

KEMM17 VT-10, L4 8

Step into the rotating frame• Motion of m appears

simpler: rotation of maround B1 with freq.ω1

!

"1

= #$B1

nutation frequency, ω1

seen from therotating framerotating frame often used implicitly

!

B0

KEMM17 VT-10, L4 9

RF pulseShort burst of RF

radiation (a few µs)

!

" =#1tRF

nutation angle, αpulse length, tRF

!

90°x

flip angle (-α)

RF phase(i.e. axis of rotationin the rotating frame)

KEMM17 VT-10, L4 10

The following RF pulses are applied to thermalequilibrium magnetization: a) 90°x, b) 90°y, c)180°x, d) 180°y. Calculate the Cartesiancomponents of the magnetization after thepulse.

Calculate the Cartesian components of themagnetization after the following RF pulsesapplied to thermal equilibrium magnetization:a) 90°xb) 90°y,c) 180°xd) 180°y

KEMM17 VT-10, L4 11

What RF pulse would give the followingrotation: a) (1, 0, 0) → (-1, 0, 0), b) (1, 0, 0) →(0, 0, 1) c) (1, 0, 0) → (0, 1, 0).

What RF pulse would give the followingrotation?a) (1, 0, 0) → (-1, 0, 0)b) (1, 0, 0) → (0, 0, 1)c) (1, 0, 0) → (0, 1, 0)

KEMM17 VT-10, L4 12

Macroscopic magnetization, M

!

M t( ) = " r( )m r,t( )dr#

!

m r, t( )time, tposition, rspin density, ρ

integral over entire sample

KEMM17 VT-10, L4 13

Signal detection• Rotating magnetization

=> alternating voltagein the coil

!

S"Mxy

signal, Stransverse magnetization, Mxy

volta

getime

KEMM17 VT-10, L4 14

Detection in the rotating framevo

ltage

time

!

"0

!

"#0

=#0$#

RF

demodulation

offset frequency, Δω0

realimag

quadrature

• Real and imaginary parts of the signal Scorrespond to Mx and My in the rotating frame

!

S t( )"Mx t( ) + iMy t( )

KEMM17 VT-10, L4 15

Loss of coherence• Different Δω0 for spin packets

experiencing different B0 or σ

time

fast

slow

KEMM17 VT-10, L4 16

Refocusing by 180° pulses• Refocused

• inhomogeneous B0• chemical shift

• Spin echo!

KEMM17 VT-10, L4 17

Magnetic field gradients, GInhomogeneous magnetic field

!

B0r( ) = " B

0+G # r

!

"#0z( ) = $%Gz

homogeneous component

gradient vector

!

B0

z( ) = " B 0

+ Gz

!

"RF

= #$ % B 0

!

B0z( )

!

z

!

" B 0

KEMM17 VT-10, L4 1818

Spin evolution in a gradient

z

time

!

"#0z( ) = $%Gz

KEMM17 VT-10, L4 19

Polarization and coherence• Polarization: mz ≠ 0• Coherence: mx,y ≠ 0

• Polarization convertedto coherence by RF

z

x

y

KEMM17 VT-10, L4 20

RelaxationApproach to the equilibrium state M0 after

perturbation

B0

M0

KEMM17 VT-10, L4 21

Molecular dynamicstime scale

s ms µs ns ps fs

macroscopicdiffusion,

flow

chemical exchange

molecularrotations

molecularvibrations

NMR

KEMM17 VT-10, L4 22

Bloch equations I: Relaxation

!

dMx

dt= "

Mx

T2

dMy

dt= "

My

T2

dMz

dt= "

Mz "M0( )T1

longitudinal relaxation time, T1 transverse relaxation time, T2

Felix BlochNobel Prize 1946

KEMM17 VT-10, L4 23

Exponential approach toequilibrium

!

Mx = M0 exp " t T2( )

My = 0

Mz = M0 1" exp " t T1( )[ ]

time

KEMM17 VT-10, L4 24

Bloch equations II: +precession

!

dMx

dt= "#$

0My "

Mx

T2

dMy

dt= #$

0Mx "

My

T2

dMz

dt= "

Mz "M0( )T1

resonance offset, Δω0

KEMM17 VT-10, L4 25

Exponential decay andoscillation

!

Mx = M0 cos "#0t( )exp $ t T2( )

My = M0 sin "#0t( )exp $ t T2( )

Mz = M0 1$ exp $ t T1( )[ ]

time

KEMM17 VT-10, L4 26

Free induction decay (FID)

!

S t( )"M0 exp i#$0t %t

T2

&

' (

)

* +

• Time-domain signal after an excitation RFpulse

• a.k.a Bloch decay

real

imag

KEMM17 VT-10, L4 27

Under what conditions would the spins behavein the following way? Sketch the phaseevolution for some representative spins!