Lecture 5: Bloch equation and detection of magnetic resonance

Lecture aims to explain: 1. Bloch equations, transverse spin relaxation time T2 and T2*

2. Detection of Magnetic Resonance: Free Induction Decay

Bloch equations

Bloch equation describes the evolution of sample magnetisation in magnetic field (with a large static z-component) taking into account spin relaxation:

⊥−−+×= MkHMM2

z01 T

1)M(MT1

dtd γ

Bloch equation in a vector form

Important: (i) decay of the transverse and longitudinal spin components is assumed to be exponential (ii) Decay of the z and x,y components is described by different time constants T1 and T2

000 HχM =

In thermal equilibrium magnetisation will tend to align along H0. χ0 is the static magnetic susceptibility

Note, in contrast to longitudinal decay, transverse decay conserves energy in the static field

Bloch equations (explicit expressions for all components)

z1

z0z

2

yy

y

2

xx

x

)γ(T

MMdt

dMTM

)γ(dt

dMTM)γ(

dtdM

HM

HM

HM

×+−

=

−×=

−×=

T2 – the transverse spin relaxation time

Static field solutions for Bloch equations

)e-(1M(0)eM(t)M

t]ω(0)Mtω(0)[Me(t)M

t]ω(0)Mtω(0)[Me(t)M

11

2

2

t/T-0

t/Tzz

0x0yt/T

y

0y0xt/T

x

+=

−=

+=

−

−

−

sincos

sincosSolutions for Bloch equations in case H=H0k are given by:

0z

yx

M)(M)(M)(M

=∞

=∞=∞ 0The equilibrium or steady-state solutions are found from t→∞

Evolution of magnetization according to Bloch equations

Rough estimation of T2 in solids

if we use data for GaAs crystal: γ for 69Ga 6.438855×107 rad s-1 T-1

µ0=1.256×10-6 V·s/(A·m) r=0.25 nm

See also examples 3.1 and 3.2

Each nucleus experiences a “local” magnetic field from its neighbours given by (in SI units):

μT518≈== 30

30

loc rγμ

rμμH

mAN

msVTesla 2 ⋅=

⋅=

Use:

Random precession of different nuclei in this magnetic field will lead to transverse spin relaxation with time T2 of the order

μs30≈=loc

2 γH1T

How does transverse relaxation (or dephasing) work

0 0.5 1 1.5 2 2.5 3

x 10-7

-1

-0.5

0

0.5

1

Time (s)

Mag

netis

atio

n (a

rb. u

nits

)

Sine functions

0 0.5 1 1.5 2 2.5 3

x 10-7

-10

-5

0

5

10

Time (s)

Mag

netis

atio

n (a

rb. u

nits

)

Sine functions

Transverse spin components of different nuclei precess with different periods according to sin(ωt) law

The resultant magnetisation (the sum of all sine functions) quickly decays as described by Bloch equations using the relaxation constant T2

T2 versusT2*

There is an additional dephasing of the magnetization introduced by external field inhomogeneities, and also by inhomogeneities of the spin ensemble (for example due to the chemical or Knight shifts). This reduction in an initial decay of M⊥ can be characterised by a separate decay time T2’. Thus the total decay rate will be defined:

'T1

T1

T1

22*2

+=

Note, that the decay due to field or ensemble inhomogeneities is reversable (phase relationship between spins is recovarable) in “spin-echo” experiments. Decay due to T2 is not reversible.

Typical magnitudes of transverse spin relaxation time

Material/Tissue T1 (ms) T2 (ms)

Gray matter 950 100

White matter 600 80

Muscle 900 50

Cerebrospinal fluid 4500 2200

Fat 250 60

Blood 1200 100-200

GaAs crystal ~1000 ~0.1

Self-assembled semiconductor quantum dot

>106 ~1

Detection of Magnetic Resonance: Free Induction

Decay

Faraday’s law of induction

The induced electromotive force (EMF) in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit

dtdΦEMF −=

The magnetic flux through the circuit is defined as:

∫ ⋅=area coildSBΦ

Free induction decay

Example 5.1 Describe the evolution of nuclear spins after a π/2-pulse.

Motion of spins will be independent of the oscillating field H1 and will only be defined by the static external field H0

The angle of rotation in the plane normal to H0 is given by:

tH0γθ =

“Free” refers to free of the oscillating field H1

In a standard MRI experiment, the field associated with a precessing magnetization sweeps past fixed receiving coils

0H

Detection of free induction decay Once the magnetisation has a transverse component an electromotive force (emf) will be created in a coil, a consequence of Faraday’s law. The time-dependent form of this current carries the information that is eventually transformed into an image of the sample. Advantage of FID, voltage needed to create H1 is only applied for a short time. Note, FID signal decays with time

SUMMARY Bloch equations describe the evolution of sample magnetisation in magnetic field. Two spin relaxation times are explicitly introduced for longitudinal (T1) and transverse (T2) spin relaxation z

1

z0z

2

yy

y

2

xx

x

)γ(T

MMdt

dMTM

)γ(dt

dMTM)γ(

dtdM

HM

HM

HM

×+−

=

−×=

−×=

'T1

T1

T1

22*2

+=

There is an additional dephasing of the magnetization introduced by external field inhomogeneities, and also by inhomogeneities of the spin ensemble (for example due to the chemical or Knight shifts). Thus the total decay rate will be defined:

Free induction decay provides the simplest way for MR detection using a coil where the varying magnetic flux will produce emf