# Introduction to NMR Physics

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```							Introduction to NMR Physics
Terry M. Button, Ph.D.

Tiny Magnets
• Nucleons behave as small current carrying loops. • Such current carrying loops give rise to a small magnetic field.

Tiny Magnets
• Like nucleons pair such their net magnetic fields cancel.

• Only nuclei with unpaired nucleons have magnetic properties.

Nuclear Spin Quantum Number
• I is quantized in half units of ħ:
– 0, ½, 1, etc…

• Nuclear magnetic moment is proportional to I:
 = Iħ

Which nuclei are useful?
• Not useful for MRI (even-even, I =0):
• • •
4

He 12C 16O

• Useful for MRI (one unpaired):
• 1H • 13C • 31P • 129Xe

Magnetic Moment
N S

A current carrying loop (l by w) will experience a torque:

 = 2 (w/2) I dl x B
 = IA x B  =  x B, where  is the magnetic moment

Effect of Applied Field - Classical
• An external magnetic field (Bo) causes the proton to precess about it. • Larmor (precessional) frequency: fL = Bo/2. • For protons fL is approximately 42 MHz/Tesla.
B0

Magnetization
• A sample of protons will precess about an applied field. The sample will have:
– a net magnetization along the applied field (longitudinal magnetization). – no magnetization transverse to the applied field (transverse magnetization).

B0

M

Classical Picture of Excitation
• A second field (B1) at the fL and at right angles to Bo will cause a tipping of the longitudinal magnetization. • The result is a net transverse component; this is what is detected in MRI. • B1 is radiofrequency at fL.

RF Excitation for Transverse Magnetization
B0 B0

90o RF at fL

M

M

Signal from the Free Induction Decay
S exp(-t/T2*)

M

t

Longitudinal Relaxation
• Relaxation of the longitudinal component to its original length is characterized by time constant T1
– Spin lattice relaxation time – Tumbling neighbor molecules produce magnetic field components at the Larmor frequency resulting in relaxation. – following a 90o tip, T1 provides recovery to [11/e] or 63% of initial value.

T1

Transverse Relaxation
• Relaxation of the transverse magnetization to zero is characterized by time constant T2
– Spin-spin relaxation time. – following a 90o tip, reduction to 1/e or 37% of initial value. – T2* combined dephasing due to T2 and field inhomogeneity.

T2

In vivo Relaxation
• T1 > T2 > T2* • T1 increases with Bo • T2 is not strongly effected.

Relaxation

Application of FFT to S vs. t
• FT
– FFT provides real (a) and imaginary (bi) components at frequencies dictated by Nyquist sampling
• Magnitude: [a2 + b2]1/2 • Phase: arctan (b/a)

• The magnitude
– Has center frequency at the Larmor frequency – The decay is contained within an exp (-t/T2*) envelope:
• T2* determines the line width

Spectra
I long T2*

short T2*

f

Effect of Applied Field Quantum Mechanical
• Protons can be in one of two state:
– aligned with the field (low energy) – aligned against the field (high energy)

• The energy separation is: E = h fL.

Quantum Mechanical

E = hfL

•Protons moving from low to high energy state require radiofrequency. •Protons moving from high to low energy release radiofrequency.

State Population Distribution
-

.

•Boltzmann statistics provides population distribution these two states:
N-/N+ = e-E/kT where:

E is the energy difference between the spin states k is Boltzmann's constant (1.3805x10-23 J/Kelvin) T is the temperature in Kelvin.

•At physiologic temperature approximately only 1 in 10 6 excess protons are in the low energy state.

Chemical Shift
• Electrons in the molecule shield the nucleus under study:
Bobserved = Bapplied - B = Bapplied (1 - )

• The chemical shift is measured in frequency relative to some reference:
 = [(fsample – freference )/freference ]x106 ppm Usually freference is tetramethylsilane (TMS) for in vitro. In the body fat and water 3.5 ppm shift.

In Body
Fat and water have 3.5 ppm shift; at 1.5 T this amounts to 220 Hz.

I

water lipid
220Hz

f

Recovery of Rapid T2* Signal Loss Using Spin-Echo

Spin Echo
90o 180o echo

TE/2 Bo -  t=0

TE/2 Bo
Bo + 

t = TE/2
t = TE

Echo!

Multi Echo Decay – T2
exp(-t/T2)

exp(-t/T2*)

Introduction to Image Formation

Simple NMR Experiment
Bo
S

I
FFT

t fL f

Bo

Bo
S

I
FFT

t
f