fundamentals of nuclear magnetic resonance
P. J. Grandinetti
Precessing top in a gravitational field The top will precess about the direction of the gravitational field, with a characteristic frequency determined by parameters such as the mass, moments of inertia and the gravitational field strength, g
g
P. J. Grandinetti
Precessing top without gravitational field If we eliminate g, place the top in a zero gravity environment, it would continue spinning but stop precessing.
P. J. Grandinetti
Magnetic top in a magnetic field A magnetic top and in a magnetic field would precess about the direction of the magnetic field with a precession frequency, ω0, that is linearly proportional to the magnetic field strength.
B
Larmor Frequency
ω0 N
g=0 S
P. J. Grandinetti
How to measure precession frequency? Measure voltage in surrounding loop of wire Faraday's Law tells us that the Electromotive Force (EMF, i.e. voltage)
B N
S
induced in the coil will be related to the change in magnetic flux.
P. J. Grandinetti
EMF of precessing magnetic dipole
ω0 t
Assignment: Calculate time dependent magnetic flux along x due to precessing magnetic dipole at origin.
μ
ψ
ξ X
P. J. Grandinetti
Z
Y
EMF of precessing magnetic dipole
ω0 t
Z
μ
ψ
ξ X
Y
Four Important Lessons Signal Amplitude is • proportional to the magnetic dipole moment strength • scaled by sin ψ • scaled by the inverse of the coil radius • increasing with increasing precession frequency.
P. J. Grandinetti
Imagine every NMR active nucleus as microscopic magnetic top The Atom Surrounding orbiting electrons
Nucleus (~ 10-14 m diameter) composed of protons (I=1/2) and neutrons (I=1/2) with a total nuclear spin angular momentum I
μ
B
Magnetic dipole moment of atomic nucleus precesses in a magnetic field
~10-10 m
Allowed Nuclear Multipole Moments as a function of Nuclear Spin I Nuclear Spin
P. J. Grandinetti
All Nuclei with Spin I ≥ 1/2 will Precess in a Magnetic Field
Different NMR active isotopes have different NMR frequencies precession frequency
gyromagnetic ratio
magnetic field strength 1
amplitude
H
14
N
17
O
13
C
Alanine Nuclear Precession (Larmor) Frequency
P. J. Grandinetti
Find NMR Larmor (precession) frequency tables anywhere
precession frequency
P. J. Grandinetti
gyromagnetic magnetic field ratio strength
Identical NMR active isotopes in different bonding environments have different NMR frequencies 1
amplitude
H
14
N
17
O
Zeeman Coupling
Interaction between nuclear magnetic dipole and magnetic field with uniform magnetic field along z
13
C
where
Larmor (Precession) Frequency
Nuclear Shielding
Surrounding electrons slightly shield the nucleus from full strength of the magnetic field, reducing the precession (Larmor) frequency.
typically on the order of 10-6. Nuclear shieldings depends on a number of factors, and often increases with increasing local electron density around a nucleus.
Chemical Shift
amplitude
Used by chemists to avoid complications of bulk magnetic susceptibilities. Defined as difference between the NMR frequency of a resonance in a reference compound and the NMR a given resonance:
13C Chemical Shift P. J. Grandinetti
Magnetic dipole couplings between nuclei split resonances Indirect electron mediated magnetic Dipole-Dipole Interaction
Direct Through Space magnetic Dipole-Dipole Interaction θ r
e-
μ1 μI
2πJ
ΩI
μS
φ
Averages to Zero in liquid samples, Not directly Observed in liquid spectrum Observed indirectly through relaxation
2πJ
ΩS
Used to Establish Through Bond Connectivities P. J. Grandinetti
μ2
ω
Splitting is directly observable in Spectra of Solid Samples
Used to Establish Through Space Distances
Atomic nuclei with spin I ≥ 1 will have an Electric Quadrupole Moment The nuclear electric quadrupole moment interacts with surrounding electric field gradients. These electric field gradients are generated by orbiting electrons as well as neighboring nuclei.
μ
B
Q Magnetic dipole moment causes atomic nucleus to precess in a magnetic field
Vzz Electric quadrupole moment causes atomic nucleus to precess in an electric field gradient
Vyy Vxx
Like Dipolar Coupling, Quadrupolar couplings Average to Zero in liquid samples, Not directly Observed in liquid spectrum Observed indirectly through relaxation Results in directly observable Splitting in Spectra of Solid Samples
P. J. Grandinetti
Used to Establish local geometry and bonding around atom
Ensemble of I ≥ 1/2 nuclei will have small contribution to net sample magnetization At equilibrium I ≥ 1/2 nuclei have random precession phases and nearly random precession angles μ
μ
μ
B M
μ
B
μ
μ
Bulk Sample magnetization vector
μ
At equilibrium, slightly more cones along +z than -z P. J. Grandinetti
μ
Magnetic resonance approach for rotating magnetization vector Apply small oscillating magnetic field perpendicular to B0. If oscillation frequency matches precession frequency (resonance) then magnetization is rotated. Laboratory Frame Z
Z
B0
Beff(t)
Bx(t)
net magnetization vector
Y
B0 - magnetic field vector
Y X
2B1cos(|ωrf|t + ψrf)
X
Z
Z
Y
P. J. Grandinetti
Beff
B1
Effect of "90 degree" Pulse of resonant radio waves
X
B0
Z Beff
X
B1
B1
Z
magnetic resonance
Y X
Y
B1 X
B0
… Y
Precessing magnetization is detected with a coil of wire The Faraday Detector
Just a coil wrapped around sample π/2 Z
Z
Y
π/2 pulse
Y X
X
Precessing magnetization vector creates time dependent magnetic flux, Φ(t)
S(t) = - dΦ(t)/dt Signal Detected in Coil is time derivative of Time dependent magnetic flux.
P. J. Grandinetti
Real
Bloch Decay Experiment
S(t)
time
Imaginary
Fourier "NMR Spectrum" Transform Ω
Ω
Real Imaginary Absorption Mode
Dispersion Mode
2/T2
Precession (or Larmor) Frequency
ω
ω 2/T2
After pulse correlated nuclei precession phases lead to transverse precessing magnetization μ
μ
μ
B M
μ
B
μ
μ
Bulk Sample magnetization vector
μ
μ π/2
Immediately after pulse, slightly more vectors pointing toward +x Imaginary P. J. Grandinetti
Real
S(t)
time What causes signal decay?
Relax Relaxation processes destroy (randomize) the correlated phase relationship among precessing nuclear magnetic dipole vectors, causing spins to lose coherence and restore an equilibrium magnetization along the z-axis. Two time constants used to quantify these processes are:
P. J. Grandinetti
Relax ω0 t
Z
μ
ψ
ξ X
P. J. Grandinetti
Y
T1 relaxation
Log(τ)
P. J. Grandinetti
T2 relaxation
Log(τ)
P. J. Grandinetti
Bloch Decay Experiment π/2
Real
S(t)
time
Imaginary
Fourier "NMR Spectrum" Transform Ω
Ω
Real Imaginary Absorption Mode
Precession (or Larmor) Frequency P. J. Grandinetti
Dispersion Mode
2/T2
ω
ω 2/T2
Inhomogeneous magnetic fields and T2* A spatial variation in external magnetic field makes the precession frequency depend on the position in the sample
Total signal becomes an integral over the volume of the sample
2/T2
P. J. Grandinetti
Γ
ω
The Bloch equations
The magnetization vector The magnetization vector in NMR experiment is a vector sum of the individual nuclei's magnetic dipole moments
And can be expanded in terms of Cartesian components
unit vectors along Cartesian axes P. J. Grandinetti
Meet the Bloch equation In 1946 Felix Bloch proposed a phenomenological equation to describe the precession and relaxation of the net magnetization vector in the NMR experiment.
where
P. J. Grandinetti
Meet the Bloch equation
P. J. Grandinetti
Meet the Bloch equation
Describes the relaxation decay of x-y ``transverse'' magnetization and the growth of z ``longitudinal'' magnetization
P. J. Grandinetti
Meet the Bloch equation
Describes the change in M(t) as it precesses about the B(t) direction.
Describes the relaxation decay of x-y ``transverse'' magnetization and the growth of z ``longitudinal'' M(t) precesses about the instantaneous magnetization direction of the vector ω(t) with an instantaneous precession angular frequency specified by the length of the vector ω(t) P. J. Grandinetti
Solve the Bloch equation: free precession Solve the Bloch equation for the free evolution of the magnetization vector in the lab frame
P. J. Grandinetti
Solve the Bloch equation : free precession Solve the Bloch equation for the free evolution of the magnetization vector in the lab frame Solution
P. J. Grandinetti
The rotating frame How would the magnetization free precession appear if we stood on a turntable at the origin of our magnetization vector, and the turntable was spinning at the same angular velocity as the magnetization vector? -Y
-X Y
X
P. J. Grandinetti
Z
The rotating frame How would the magnetization free precession appear if we stood on a turntable at the origin of our magnetization vector, and the turntable was spinning at the same angular velocity as the magnetization vector? -Y
Z
-X Y
X Z* -X*
-Y*
Y*
X*
P. J. Grandinetti
The lab frame would appear (to us) to be rotating in the opposite direction, and the magnetization would appear stationary in this ``rotating frame''.
The rotating frame transformation Z
-X -Y
Y X
Precession frequency is slower (or zero) in the rotating frame P. J. Grandinetti
Solve the Bloch equation: free precession Solve the Bloch equation for the free evolution of the magnetization vector in the rotating frame
and P. J. Grandinetti
are magnetization vector components in the rotating frame
Cartesian to spherical basis
P. J. Grandinetti
Cartesian to spherical basis
unit vectors in spherical basis
P. J. Grandinetti
p orbitals : Cartesian basis z
x
Px
P. J. Grandinetti
y
Py
Pz
p orbitals : Spherical basis z
x
P-1
P. J. Grandinetti
y
P0
P+1
Solve the Bloch equation: free precession Solve the Bloch equation for the free evolution of the magnetization vector in the rotating frame in the spherical basis
P. J. Grandinetti
Solve the Bloch equation: free precession Solve the Bloch equation for the free evolution of the magnetization vector in the rotating frame in the spherical basis
P. J. Grandinetti
NMR Signal The NMR signal, written in terms of the spherical magnetization components in the rotating frame, is given by
constant that depends on the receiver coil geometry and the receiver frequency
P. J. Grandinetti
Receiver Phase
A radio frequency pulse Add magnetic field oscillating at a angular frequency Z B0
Beff(t)
Bx(t)
Z*
Laboratory Frame
B1
Y X
X*
φ
Rotating Frame ωeff
Y*
Define nutation frequency as
B1 time
P. J. Grandinetti
2B1cos(|ωrf|t + ψrf)
Define rf phase as to make independent of sign of
Solve the Bloch equation: rf pulse along x Solve the Bloch equation during an rf pulse in the rotating frame in the Cartesian basis
P. J. Grandinetti
A 90° pulse along x
Z*
Z*
Y* X*
ω1 P. J. Grandinetti
ω1 t = π/2
Y* X*
A 180° pulse along x
Z*
Z*
Y* X* P. J. Grandinetti
ω1
ω1 t = π
Y* X*
A pulse that takes the magnetization from +z to -z is called an inversion pulse
Solve the Bloch equation: rf pulse along y Solve the Bloch equation during an rf pulse in the rotating frame in the Cartesian basis
P. J. Grandinetti
A 90° pulse along y
Z*
Z*
Y* X* P. J. Grandinetti
ω1
ω1 t = π/2
Y* X*
Solve the Bloch equation: 90° pulse Solve the Bloch equation during an rf pulse in the rotating frame in the spherical basis
P. J. Grandinetti
Solve the Bloch equation: 180° pulse Solve the Bloch equation during an rf pulse in the rotating frame in the spherical basis
P. J. Grandinetti
Bloch decay experiment
P. J. Grandinetti
Spin echo experiment
P. J. Grandinetti
Spin echo experiment
P. J. Grandinetti
Spin echo experiment When
The magnetization as a function of becomes independent of
Signal as a function can be used to measure T2 regardless of magnetic field homogeneity However: • Chemical shift resolution is lost (refocused along with field homogneneity) • Unrefocused frequency contributions (e.g. J coupling) complicates T2 measurement • If nucleus spatially diffuses to another region of sample, where frequency is different, the echo doesn’t refocus and decays faster as a function of But all these issues can be turned into opportunities for other types of spin echo experiements P. J. Grandinetti
Measuring translational diffusion coefficients The dependence of the echo intensity on molecular diffusion can be exploited as a means to measure translational diffusion coefficients rf
π/2
π
τ
By applying a linear magnetic field gradient across the sample during the spin echo experiment τ
Gradient
the modified Bloch equation can be solved
to obtain expression for echo intensity
P. J. Grandinetti
Diffusion coefficients with pulsed gradients A more robust method for measuring diffusion coefficients employs pulsed field gradients instead of a static field gradient. π/2 rf
Gradient
π
τ
τ δ
δ
G
G
Δ
High resolution spectrum with higher sensitivity is obtained since signal is detected in a homogeneous magnetic field. Very useful for mixtures of molecules.
Can even measure flow!
P. J. Grandinetti
Measuring spin-lattice relaxation: T1 Saturation Recovery
( ) π/2
τ
n
t1
π/2
Inversion Recovery
t2
equilibration time
t1
π/2
t2
repeat sequence n times. Meq
100% 80%
Meq
100% 50%
60%
99.3% recovery at 5 T1
Mz(t1)
40%
Mz(t1) 0% -50%
20% 0%
π
0
2T1
4T1
6T1
8T1
10T1
2T1
4T1
-100%
Solve Bloch Equations:
Solve Bloch Equations:
Initial condition:
Initial condition:
P. J. Grandinetti
6T1
8T1
10T1
Bloch equation homework
P. J. Grandinetti
Limitations of the Bloch equation The Bloch equation was designed to describe an ensemble of such spin 1/2 nuclei, each having only three internal degrees of freedom. The Bloch equation does not treat degrees of freedom associated with (1) magnetic dipole coupling between nuclei μI
μS
Surrounding orbiting electrons
(2) the internal structure of nuclei with spin I>1/2 Allowed Nuclear Multipole Moments as a function of Nuclear Spin I Nuclear Spin
P. J. Grandinetti
