Post on 19-Mar-2016
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A few fundamentals of NMR
Dieter Freude
Harry Pfeifer's NMR-Experiment 1951 in Leipzig
H. Pfeifer: Über den Pendelrückkoppelempfänger (engl.: pendulum feedback receiver) und die Beobachtungen von magnetischen Kernresonanzen, Diplomarbeit, Universität Leipzig, 1952
Nuclear spin I = 1/2 in an magnetic field B0 B0, z
y
x
L B0, z
y
x
L
Many atomic nuclei have a spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is
The component in the direction of an applied field is
Lz = Iz m = ½ for I = 1/2.
.)1( IIL
Atomic nuclei carry an electric charge. In nuclei with a spin, the rotation creates a circular current which produces a magnetic moment µ.
An external homogenous magnetic field B results in a torque T = µ B with a related energy of E = µ·B.
The gyromagnetic (actually magnetogyric) ratio is defined byµ = L.
The z component of the nuclear magnetic moment isµz = Lz = Iz m .
The energy for I = 1/2 is split into 2 Zeeman levelsEm = µz B0 = mB0 = B0/2 = L/2.
Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet.
Larmor frequency
Joseph Larmor described in 1897 the precession of electron orbital magnetization in an external magnetic field.
Classical model: the torque T acting on a magnetic dipole is defined as the time derivative of the angular momentum L. We get
By setting this equal to T = µ B , we see that
The summation of all nuclear dipoles in the unit volume gives us the magnetization. For a magnetization that has not aligned itself parallel to the external magnetic field, it is necessary to solve the following equation of motion:
.dd1
dd
ttμLT
.dd Bμμ
t
.dd BMM
t
B0, z M
y
x L
We define B (0, 0, B0) and choose M(t 0) |M| (sin, 0, cos). Then we obtain
Mx |M| sin cosLt, My |M| sin sinLt, Mz |M| cos with L = B0.
The rotation vector is thus opposed to B0 for positive values of . The Larmor frequency
is most commonly given as an equation of magnitudes: L = B0 or.
2 0L B
Macroscopic magnetization energy Em = ½
E = hL
Em = ½
Nm = ½
Nm = ½
hL « kT applies at least for temperatures above 1 K
and Larmor frequencies below 1 GHz. Thus, spontaneous transitions can be neglected, and the probabilities P for absorption and induced emission are equal. It follows P = B+½,½ wL= B½,+½ wL, where B
refers to the Einstein coefficients for inducedtransitions and wL is the spectral radiation density at the Larmor frequency. A measurable absorption (or emission) only occurs if there is a difference in the two occupation numbers N. In thermal equilibrium, the Boltzmann distribution applies to N and we have .expexp L0
2/1
2/1
kTh
kTB
NN
If L 500 MHz and T 300 K, hL/kT 8 10 is very small, and the exponential function can be expanded to the linear term:
.108 5L
2/1
2/12/1
kTh
NNN
Longitudinal relaxation time T1
All degrees of freedom of the system except for the spin (e.g. nuclear oscillations, rotations, translations, external fields) are called the lattice. Setting thermal equilibrium with this lattice can be done only through induced emission. The fluctuating fields in the material always have a finite frequency component at the Larmor frequency (though possibly extremely small), so that energy from the spin system can be passed to the lattice. The time development of the setting of equilibrium can be described after either switching on the external field B0 at time
t 0 (difficult to do in practice) with,1 1
0
T
t
enn
T1 is the longitudinal or spin-lattice relaxation time an n0 denotes the difference in
the occupation numbers in the thermal equilibrium. Longitudinal relaxation time because the magnetization orients itself parallel to the external magnetic field. T1 depends upon the transition probability P as
1/T1 = 2P 2B½,+½ wL.
To measure T1 by IR
The inversion recovery (IR) by -/2
1210Tenn
By setting the parentheses equal to zero, we get 0 T1 ln2 as the passage of zero.
0
Rotating coordinate system and the offsetFor the case of a static external magnetic field B0 pointing in z-direction and the application of a rf field Bx(t) = 2Brf cos(t) in x-direction we have for the Hamilitonian operator of the external interactions in the laboratory sytem (LAB) H0 + Hrf = LIz + 2rf cos(t)Ix,where L = 2L = B0 denotes the Larmor frequency, and the nutation frequency rf is defined as rf = Brf.The transformation from the laboratory frame to the frame rotating with gives, by neglecting the part that oscillates with the twice radio frequency, H0 i + Hrf i = Iz + rf Ix, where = L denotes the resonance offset and the subscript i stays for the interaction representation.
B0
M
x
y
z
B0
M x’
y
z
Magnetization phases develop in this interaction representation in the rotating coordinate system like = rf or = t.
Quadratur detection yields value and sign of .
Bloch equation and stationary solutions
We define Beff (Brf, 0, B0 /) and introduce the Bloch equation:
1
0
2effd
dTMM
TMM
tzx zyyx eee
BMM
Stationary solutions to the Bloch equations are attained for dM/dt 0:
.
11
,21
,21
021
2rf
222
2L
22
2L
rf0rf21
2rf
222
2L
2
rf0rf21
2rf
222
2L
22L
MTTBT
TM
HMBTTBT
TM
HMBTTBT
TM
z
y
x
Correlation time c, relaxation times T1 and T2
tftfG
c
GG
exp0
2L
2L0
6
24
1 218
12
41
511
c
c
c
cIIrT
2
L2
L06
24
2 212
15
34
1511
c
c
c
cc
IIrT
T1
T2
ln T1,2
1/T
T1 min
T2 rigid
The relaxation times T1 and T2 as a function of the reciprocal absolute temperature
1/T for a two spin system with one correlation time. Their temperature dependency can be described by c 0 exp(Ea/kT).
It thus holds that T1 T2 1/c when Lc « 1 and T1 L2 c when Lc » 1.
T1 has a minimum of at Lc 0,612 or Lc 0,1.
Hahn echo B0
M
x
y
z B0
M x
y
z B0
x
y
z
5 4
1 2
3
B0
x
y
z
1 2
5 4
3
B0
M x
y
z
/2 pulse FID, pulsearound the dephasing around the rephasing echo y-axis x-magnetization x-axis x-magnetization
(r,t) = (r)·t (r,t) = (r,) + (r)·(t )
Line width and T2
A pure exponential decay of the free induction (or of the envelope of the echo, see next page) corresponds to
G(t) = exp(t/T2).
The Fourier-transform gives fLorentz = const. 1 / (1 + x2) with x = ( 0)T2, see red line. The "full width at half maximum" (fwhm) in frequency units is
.1
22/1 T
Note that no second moment exists for a Lorentian line shape. Thus, an exact Lorentian line shape should not be observed in physics.
Gaussian line shape has the relaxation function G(t) = exp(t2 M2 / 2) and a line
form fGaussian = exp (2/2M2), blue dotted line above, where M2 denotes the
second moment. A relaxation time can be defined by T22 = 2 / M2. Then we get
21/2=2/T2=1/2
0
fLorentz
1
1/2
( ) ( ) ( ) .Hz/×12.74ln
Hz/=s/
2=s/ 2
2/1
22
2/122
2-2
≈
TM
T2 and T2*
( ) 2
2
e= TG
( ) 2e= Tt
tG
EXSY, NOESY, stimulated spin echo
stimulated echo
0
t1 t2 tmix
t1
time
FID FID
after mixing
Magic-angle spinning
B0, external magnetic field
Rot
rotor with sample in the rf coil
o7,543
1cosarc
Rotation frequency should be much greater than:
heterogeneous line broadening
homogeneous line broadening
rate of mobility of the species
Magnetic resonance shifting with the geometry factor 3cos2 1 is caused by:
homo-nuclear dipole-dipole interaction,
heteronuclear interaction,
anisotropy of the chemical shift,
first-order quadrupole interaction,
sample microinhomogenieties.
The shift gives rise to a signal broadening in powder materials.