Fast Course in NMR
Lecture 2
Jan/Feb, 2016
Behavior of nuclear spins in a magnetic field II
• Bloch equations
• Phenomenological introduction to T1 and T2
• RF Pulses
A. S. Edison
University of Georgia
2016
Now that we are experts in spin,
What happens when we put a
nuclear spin into a magnetic field?
A. S. Edison
University of Georgia
2016
Bulk Magnetization
Recall from lecture 1:
The magnetic moment (m) is a vector parallel
to the spin angular momentum (I). The
gyromagneto (or magnetogyro) ratio (g) is a
physical constant particular to a given nucleus.
Unfortunately, the vast majority of the magnetic moments cancel one another. The
“Boltzmann excess” in the a state add together to create bulk angular momentum
(J) and magnetization (M).
J = åI
M = åm
A. S. Edison
University of Georgia
2016
Gyroscopes!
Classical physics tells us about the
motion of a magnet in a magnetic field
dJ(t )
= M(t ) ´ B(t )
dt
This is very similar to the motion of a spinning
gyroscope or top in a gravitational field
dL(t )
= r ´ mg
dt
L(t) is the gyroscope’s
angular momentum, r its
radius from the fixed point
of rotation, m its mass and
g the force of gravity.
A. S. Edison
University of Georgia
2016
Bloch Equations
We can make the equations easier to
deal with by multiplying both sides by g
dJ(t )
= M(t ) ´ B(t )
dt
Multiply
by g
dM(t )
= M(t ) ´ gB(t )
dt
(Remember that m=gI)
A. S. Edison
University of Georgia
2016
Bloch Equations
dM(t )
= M(t ) ´ gB(t )
dt
What does this equation
describe?
After suitable choices for B, this equation predicts that nuclear
magnetization will precess at a frequency w0=gB0 FOREVER.
Nothing in the equation is a “restoring force” to cause the
magnetization to relax back to equilibrium. However, real-life NMR
experiments relax.
A. S. Edison
University of Georgia
2016
Bloch Equations
Therefore, Felix Bloch made the following
modifications to the basic equation
dM(t )
= M(t ) ´ gB(t ) - R(M(t ) - M0 )
dt
Empirical modification in which a
“relaxation matrix” R acts on
magnetization that is different
from the equilibrium state, M0
A. S. Edison
University of Georgia
2016
Bloch Equations
dM(t )
= M(t ) ´ gB(t ) - R(M(t ) - M0 )
dt
This equation is easiest to
understand broken into its
matrix components.
M (t ) - M 0
dM z (t )
= g [ M x (t ) B y (t ) - M y Bx (t )] - z
dt
T1
Magnetization
along the z-axis
dM x (t )
M (t )
= g [ M y (t ) Bz (t ) - M z B y (t )] - x
dt
T2
Magnetization
along the x-axis
dM y (t )
dt
= g [ M z (t ) Bx (t ) - M x Bz (t )] -
M y (t )
T2
Magnetization
along the y-axis
A. S. Edison
University of Georgia
2016
Changing the frame of reference
PROBLEM: The Bloch equations we have shown so far are helpful but still
too complicated. The problem is that as soon as magnetization is put into
the x-y plane, it starts to precess (we will see that soon) at NMR frequencies
(e.g. 500 MHz). Thus, the actual trajectory of the motion is VERY
COMPLICATED.
SOLUTION: We will define a coordinate system that rotates around the z-axis at
the same NMR frequency. This is accomplished by defining the following:
W = -gB0 - w rf
wrf is the frequency of the NMR transmitter and gB0 is the
frequency of the peak we are interested in observing. If the two
are the same, this is called “on-resonance”. In this case, the
“effective” magnetic field strength along the z-axis is 0.
A. S. Edison
University of Georgia
2016
Bloch Equations in the
Rotating Frame
Substituting W=-gB0-wrf (where B0=Bz and is not time-dependent) into the
Bloch equations yields:
M (t ) - M 0
dM z (t )
= g [ M x (t ) Byr (t ) - M y Bxr (t )] - z
dt
T1
dM x (t )
M (t )
= -WM y (t ) - gM z Byr (t ) - x
dt
T2
dM y (t )
dt
= gM z (t ) B (t ) + WM x r
x
M y (t )
The “r” superscript
refers to a magnetic
field in the rotating
frame
Try to simulate
these to see what
they mean!
T2
A. S. Edison
University of Georgia
2016
T1 and T2 are dependent on the tumbling rate
http://www.chem.wisc.edu/areas/reich/nmr/notes-8-tech-1-relax.pdf
A. S. Edison
University of Georgia
2016
Self Study
• Solve the Bloch equations in some
computational package.
• Play with the variables.
• See what happens.
A. S. Edison
University of Georgia
2016
Next Lecture
Introduction to NMR Parameters
• Review of Bloch Equations
• Chemical shift—BMRB database
• J coupling—Karplus equation
• T1
• T2
A. S. Edison
University of Georgia
2016