lecture 5: bloch equation and detection of magnetic resonance/file/... · 2012-03-08 · = m×h +...
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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