Todays Lecture

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Todays Lecture
8) Wed, Oct 18 Bloch Equations and
Relaxation a. Transverse and longitudinal
magnitization b. Mechanisms of relaxation c.
Measuring relaxation
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Individual magnetic moments
Bulk Magnetization
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Case 1 At equilibrium in a magnet
Case 2 After a radiofrequency pulse moves M away
from equilibirum
?
This describes precession in the x-y plane, but
there is no mechanism to return the magnetization
back to equilibrium along z.
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FTNMR

• Sample is in the magnetic field at equilibrium
  (Case 1)? nothing is happening
• A strong rf field is applied for long enough to
  take the magnetization from the z-axis to the x
  (or y) axis
• We observe the evolution in the x-y plane (Case 2)

The signal is decaying as the system returns to
equilibrium
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Bloch Equations
In order to allow the system to return to
equilibrium, Felix Bloch made the following
modifications to the basic equation
Empirical modification in which a relaxation
matrix R acts on magnetization that is different
from the equilibrium state, M0
Note unfortunately in NMR, both relaxation
and rotation matrices are sometimes denoted by R.
Usually context will tell you which one you are
looking at.
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Bloch Equations
This equation is easiest to understand broken
into its matrix components.
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Bloch Equations in the Rotating Frame
Substituting Dw-gB0-wrf (where B0Bz and is not
time-dependent) into the Bloch equations yields
B1 refers to the rf field in the rotating frame
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Bloch Equations
In the Bloch equations, magnetic fields along the
x and y axes create B1 fields or pulses. These
are typically applied for short durations, and
the length of time the pulse is turned on is
adjusted to give a desired rotation (such as 90
or 180 degrees).
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RF Pulses
Remember from an earlier lecture we try to apply
the radiofrequency pulses at frequencies close to
the Larmor frequency. The amplitude of the pulse
determines how quickly the magnetization is
rotated and is denote B1. The relationship
between rf field strength and pulse length is
simply
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Precession and relaxation
In most NMR experiments, the pulses are short and
the relaxation times are relatively long. We
mainly worry about relaxation after the pulses
are applied.
T2
T2
My
Mx
t
t
Mz
Mz
T1
Mx
t
My
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How does relaxation affect the signal?
T1
Putting the sample into a magnetic field Or after
the magnetization is in the x-y plane
Taking the sample out of a magnetic field
? One has to wait 5xT1 to get the signal back

• A lot of time in conventional NMR is spent
  waiting for relaxation.
• Initial experiments to observe NMR signals were
  hampered by not knowing T1

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What does relaxation look like in a spectrum?
T2
Decreasing T2 (increasing relaxation)?
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Sources of T1

• Incoherent molecular fluctuations on the order of
  the Larmor frequency
• T1 has a field dependent inflection point
• Historically called spin-lattice relaxation (heat
  lost to the surroundings)
• In NMR this is known as LONGITUDINAL relaxation
  due to our frame of reference

Usual experiment to measure T1 Inversion-Recovery
Measured signal
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One pulse experiment
For pulse sequences requiring 90 degree pulses
Optimize your repetition time
REPEAT
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For a one pulse experiment
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Sources of T2
Inhomogeneous broadening variations in the
macroscopic magnetic field

• Instrument limitations
• Magnetic susceptibility

Homogeneous broadening fluctuating microscopic
magnetic fields

• Molecular dynamics and spin-spin interactions ?
  more details later
• Chemical exchange
• Historically called spin-spin relaxation
• In NMR we call it TRANSVERSE relaxation? loss of
  signal in the x-y plane

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Back to pulse sequences
In a 1-D NMR spectrum you can see that the
various spin interactions lead to each type of
nucleus have a slightly different Larmor
frequency. This is wonderful for getting
chemical information out, but we need ways to get
back our coherences if we want to control the
magnetization. We do this using what is called a
spin echo.
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For a single frequency slightly off resonance
RF pulses are only on a certain points relaxation
and spin interactions are ALWAYS there
Levitt
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Several frequencies at different resonances
During the first t/2 delay
Levitt
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During the py pulse
During the second t/2 delay
Levitt
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Levitt
At this point we can either observe the signal or
apply another p/2 pulse and it will affect all
the spins equally.
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Relationship between T1 and T2
Simple, small molecule
Rigid, frozen solid Broad lines
Fast, tumbling liquid Narrow lines
tc refers to the correlation time of
tumbling Close to the inflection point,
experiments can be done quickly
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Homework

• 1)If you take 1024 scans and your spectrum still
  looks poor, what would you do? (Your T1 is 5
  seconds, you have more sample in the lab thats
  soluble, and you have 2 hours total of instrument
  time) What should your tip angle and repetition
  rate be? (Assume you are not comparing signals
  with different T1s and choose the curve on the
  relative S/N figure that will give you the most
  signal per unit time.)
• 2)Use a vector diagram to show what is happening
  in the inversion-recovery experiment to measure
  T1. Show this for three cases t 0, t
  ln2T1, t gtgt 5xT1

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1
2
3
3)Show the product operator version of the T1
relaxation experiment leaving out relaxation but
including chemical shift. 4) Show the product
operator version of the spin echo experiment with
only RF and chemical shift (no relaxation or
J-couplings).