NMR-Primer for Chemists and Biologists
Shimon Vega & Yonatan Hovav
shimon.vega@weizmann.ac.il and yonatan.hovav@weizmann.ac.il
Content
November 2013
1. Basic Concepts of Nuclear Spins in a Magnetic Field
a. Angular momentum, magnetic moments, magnetization
b. Precession of the classical magnetization
c. RF irradiation, resonance and the rotating frame
d. Concepts of quantum mechanics*
e. T1 and T2 Relaxation
* Extension depending on students
3. Basic Concept of Pulsed NMR
a. The Bloch equations, NMR signals in the laboratory frame
b. NMR signals in the rotating frame, quadrature detection
c. Manipulating the magnetization and continuous wave spectroscopy
d. T1 and T2 measurement, Hahn echo
e. Free induction decays and Fourier transform
f. FFT, data sampling, spectral width and Nyquist Theorem
g. RF pulses, off-resonance effects and composite pulses
h. The NMR spectrometer
i. Phase cycling
j. Digital filtering, pulse programming, the magnet and field inhomogeneity
1
4. The NMR Interactions and 1D spectra
a. Chemical shift, isotropic and CSA-interactions
b. The vector model and two level system
c. Nuclear spin-spin interactions and spectral multiplets
d. INEPT and COSY
e. Decoupling
5. Two dimensional NMR
a. Basic principles
b. 2D COSY
c. Twisted peaks in 2D NMR, TPPI and STATES
d. Examples of 2D experiments
e. Nuclear Overhauser Effect and NOESY
6. Solid State NMR Basic principles*
* Depending on time left
www.rug.nl/zernike/research/groups/phynd/research/spinpolarizedtransport
2
Some books:
Modern NMR techniques for Chemistry Research
Nuclear Magnetic Resonance spectroscopy
Advanced:
Spin Dynamics
Understanding NMR Spectroscopy
Principles of NMR in 1 and 2 Dimensions
Principles Magnetic Resonance
Solid State NMR Spectroscopy
by A.E Derome
by R.K Harris
by M.H Levitt
by James Keeler
by R. R. Ernst,
G. Bodenhausen
A. Wokaun
by C. P. Slichter
by M Duer
Web:
1.
http://www.cis.rit.edu/htbooks/nmr/
2.
http://www-keeler.ch.cam.ac.uk/lectures/
3
Felix Bloch
Born
(1905-10-23)October
23, 1905
Zürich, Switzerland
Died
September 10,
1983(1983-09-10)
(aged 77)
Zürich, Switzerland
Citizenship
Swiss, American
Nationality
Swiss
Fields
Physics
Institutions
University of
California, Berkeley
Stanford University
Alma mater
ETH Zürich and
University of Leipzig
Doctoral advisor
Werner Heisenberg
Known for
NMR
Bloch wall
Bloch's Theorem
Bloch Function (Wave)
Bloch sphere
Notable awards
Nobel Prize for Physics
(1952)
Edward Purcell
Born
(1912-08-30)August
30, 1912
Taylorville, Illinois,
USA
Died
March 7, 1997(199703-07) (aged 84)
Cambridge,
Massachusetts, USA
Nationality
United States
Fields
Physics
Institutions
Harvard University
MIT
Alma mater
Purdue University
Harvard University
Doctoral advisor
Kenneth Bainbridge
Other
academic advisors
John Van Vleck
Doctoral students
Nicolaas
Bloembergen
George Pake
George Benedek
Charles Pence
Slichter
Known for
Nuclear magnetic
resonance (NMR)
Smith-Purcell effect
21 cm line
Notable awards
Nobel Prize for
Physics (1952)
4
1920's Physicists have great success with quantum theory
Quantum theory was used to explain phenomena where
classical mechanics failed. This theory, proposed by Bohr,
was particularly useful for the understanding of absorption
and emission spectra of atoms. These spectra showed
discrete lines which could be accounted for quantitatively
by quantum theory. However, this theory still could not
explain doublet lines found in high resolution spectra.
1921 Stern and Gerlach carry out atomic
and molecular beam experiments
The basis of quantum theory was confirmed by the atomic
beam experiment. A beam of silver atoms was formed
in high vacuum and passed through a magnetic field.
1925 Uhlenbeck and Goudsmit introduce
the concept of a spinning electron
The idea of a spinning electron with resultant angular
momentum gave rise to the magnetic dipole moment.
1926 Schrödinger/Heisenberg formulate quantum mechanics
This new branch of quantum physics replaced the old quantum
theory. Quantum mechanics was successful for understanding
many phenomena but still could not account for doublets in
absorption and emmision spectra.
1927 Pauli and Darwin include electron spin in quantum mechanics
1933 Stern and Gerlach measure the effect of nuclear spin
Stern and Gerlach increased the sensitivity of their molecular beam
apparatus enabling them to detect nuclear magnetic moments.
They observed and measured the deflection of a beam of hydrogen
molecules. This has no contribution to the magnetic moment from
electron orbital angular momentum so any deflection would be due
5
to the nuclear magnetic moment.
1936 Gorter attempts experiments
using the resonance property of nuclear spin
The Dutch physicist, C.J.Gorter, used the resonance property
of nuclear spin in the presence of a magnetic field to study nuclear
paramagnetism. Although his experiment was unsuccessful,
the results were published and this brought attention to the
potential of resonance methods.
1937 Rabi predicts and observes nuclear magnetic resonance
During the 1930's, Rabi's laboratory in Columbia University became
a leading center for atomic and molecular beam studies.
One experiment involved passing a beam of LiCl through a strong
and constant magnetic field. A smaller oscillating magnetic field
was then applied at right angles to the initial field.
When the frequency of the oscillating field approached
the Larmor frequency of the nucleus in question, resonance
occurred. The absorption of energy was recorded as a dip in
the beam intensity as the magnetic current was increased.
1943 Stern awarded the Nobel prize for physics
Otto Stern was awarded this prize 'for his contribution
to the development of the molecular ray method and
discovery of the magnetic momentum of the proton'.
1944 Rabi awarded the Nobel prize for physics
Rabi was given this prize for his work on molecular beams,
especially the resonance method.
6
1945 Purcell, Torey and Pound observe NMR in a bulk material
At Harvard, Purcell, Torey and Pound assembled apparatus
designed to detect the transition between nuclear magnetic
energy levels using radiofrequency methods.
Using about 1kg of parrafin wax, the absorbance was predicted
and observed.
1951 Packard and Arnold observe that the chemical shift
due to the -OH proton in ethanol
varies with temperature.
It was later shown that the chemical shift for this proton
was also dependent on the solvent. These results were explained
by hydrogen bonding.
1952 Bloch and Purcell share the Nobel prize in physics
This prize was awarded 'for their development of new methods
for nuclear magnetic precision measurements and discoveries
in connection therewith'.
1953 A. Overhauser predicts that a small alteration
in the electron spin populations would produce
a large change in the nuclear spin polarisation.
This theory was later to be named the Overhauser effect and
is now a very important tool for the determination
of complex molecular structure.
1957 P. Lauterbur and C. Holm independently
record the first 13C NMR spectra.
Despite the low natural abundance of the NMR active isotope 13C,
the recorded spectra showed a signal to noise ratio as high as 50.
1961 Shoolery introduces
the Varian A-60 high-resolution spectrometer.
The Varian A-60 was used to study proton NMR at 60MHz
and proved to be the first commercial NMR spectrometer
to give highly reproducible results.
7
2. Basic Concepts of Nuclear Spins in a Magnetic Field
a. Angular momentum, magnetic moments: magnetization
We are dealing with the nuclei of atoms and in particular with their magnetic properties.
The nuclei are characterized by their “spin values”. These spins correspond to well-defined
angular momenta with values proportional to Planck’s constant:
h = 6.6260755x1034 m2kg/sec
I with
and
I  1/ 2 ; 1 ; 3 / 2 ; 2 ; 5 / 2
(from QM)
The protons and neutrons (fermions) composing the nuclei determine the nuclear spin value.
A nucleus with an odd mass number M has an half-integer spin and a nucleus of an even M
has an integer spin. Nuclei with an even number of protons and neutrons have nuclear
spin I=0.
Each nuclear spin has a magnetic moment proportional to its angular momentum-spin.
Proton: g = 5.5856912 +/- 0.0000022
Neutron: g = -3.8260837 +/- 0.0000018
or
  I
For each nucleus the angular momentum vector and the magnetic moment vector
are related by its magnetogyric ratio .
E    .B
  magnetogyric ratio
h    .B
   / B
8
Remember: conservation of angular momentum !
General comments about angular momentum:
I

I
I  
I  r  (mv )  r  p
r
  1 / 2mR
p
m
v
 r 
   
moment of inertia
  v /(2R) [(m / s) /(m)  Hz]
rp
m r2
number of turn per second
  2  v / R [(m / s) /(m)  sec1 ]
number of radians per second
General comments about magnetic momentums:
q = charge
q/L = charge density
L = 2r
v = velocity
i = current
A = area
A = 2r

i
e
q`
q
2 r
I  r  mv  r m
 i A
v
2 r
2m
2m
i e 
iA

q
q
q

Minimizing energy
1.
The torque on the magnetic moment
induced by a magnetic field B
   B
2.
Energy of a magnetic moment in
a magnetic field
E   .B
Reminder:
a b  c
c  a b sin  ab
B

B



 a x   bx   a y bz  a z by 

    
c  a , b  a y    by    a z bx  a x bz 
 a   b   a b  a b9 
y x
 z  z  x y
9
Larmor Frequencies in MHz units:
 L with
B
B  7.0 Tesla
1T = 10,000G
1H
2H
3H
3 He
6 Li
7 Li
9 Be
10 B
11 B
13 C
Proton NMR:
100MHz - 2.3 T
300MHz - 7.0 T
Hydrogen
Deuterium
Tritium
Helium
Lithium
Lithium
Beryllium
Boron
Boron
Carbon
½
1
1/2
1/2
1
3/2
3/2
3
3/2
1/2
500MHz - 11.7 T
800MHz - 18.8 T
300.130
46.073
320.128
228.633
44.167
116.640
42.174
32.246
96.258
75.46
900MHz - 21.1 Tesla
(2.1 KHz - 0.5 Gauss)
10
Element/Name Isotope Symbol Nuclear Spin
Hydrogen
Deuterium
Tritium
Helium-3
Lithium-6
Lithium-7
Beryllium-9
Boron-10
Boron-11
Carbon-13
Nitrogen-14
Nitrogen-15
Oxygen-17
Fluorine-19
Neon-21
Sodium-23
Magnesium-25
Aluminum-27
Silicon-29
1H
2H
or D
3H
3He
6Li
7Li
9Be
10B
11B
13C
14N
15N
17O
19F
21Ne
23Na
25Mg
27Al
29Si
Phosphorus-31 31P
33S
Sulfur-33
33Cl
Chlorine-33
37Cl
Chlorine-37
Potassium-39 39K
Potassium-41 41K
43K
Calcium-43
Scandium-45 45Sc
Titanium-47 47Ti
Titanium-49 49Ti
Vanadium-50 50V
Vanadium-51 51V
Chromium-53 53Cr
Manganese-55 55Mn
57Fe
Iron-57
59Co
Cobolt-59
61Ni
Nickel-61
63Cu
Copper-63
65Cu
Copper-65
67Zn
Zinc-67
1/2
1
1/2
-1/2
1
3/2
-3/2
3
3/2
1/2
1
-1/2
-5/2
1/2
-3/2
3/2
-5/2
5/2
-1/2
1/2
3/2
3/2
3/2
3/2
3/2
-7/2
7/2
-5/2
-7/2
6
7/2
-3/2
5/2
1/2
7/2
-3/2
3/2
3/2
5/2
Sensitivity vs.
1H
1.000000
1.44 e-6
0.000628
0.270175
0.013825
0.00386
0.132281
0.000175
0.000998
3.84E-06
1.07E-05
0.829825
6.3E-06
0.092105
0.00027
0.205263
0.000367
0.06614
1.71E-05
0.003544
0.000661
0.000472
5.75E-06
9.25E-06
0.3
0.00015
0.00021
0.00013
0.37895
8.6E-05
0.174386
7.37E-07
0.275439
4.21E-05
0.064035
0.035263
0.000117
Frequency (MHz)
Receptivity : (natur.abund.-%) x  x I(I+1)
1/2
Tesla 5 x10-5
2.35
7.05
9.40
11.75
18.80
21.15
1
3/5
5/2
7/2
MHz
2.1 x10-3
100
300
400
500
800
900
11
2b. The classical precession of the magnetization
B

Suppose we apply a magnetic field on our magnetization:
as a result a torque tries to rotate the direction of
the angular momentum.
I
A torque (   r  F
) perpendicular to an angular momentum causes a precession motion:
Example:
top view:
From http://hyperphysics.phy-astr.gsu.edu
Remember the motion of a top:
(gravitation + top)
12

The precession of the magnetization
around the magnetic field direction
is independent of the orientation of
F  ma  dp / dt

B
(in analogy with
)
F
z
I
B

0  B

The Larmor frequency
y
dI
;
dt
   B
 
x
I I ;  I
 x , y   B sin( / 2)
  I
dI
   B  I  B  0  I
dt
 Ix 
 I sin  cos     (0 I sin  ) sin     0 I y 


 
 

d
d 
 I y    I sin  sin     (0 I sin  ) cos     0 I x 
dt   dt 
 
  0 
0
 Iz 
 I cos   
 

 Ix 
 I x cos L t  I y sin  L t 
 


 I y (t )    I x sin  L t  I y cos L t 
I 


I
z
 z


13
2c. RF irradiation, resonance and the rotating frame
The equation of motion for the magnetization in an external magnetic field
Let
d (t )
  (t )  B   L   (t )
dt
  y  z  z  y
 x 

d  
  y (t )     x  z   z  x
dt  
    

y x
 z
 x y





z
L ; B
 (t )
y
x
Let us now consider a special time-dependent magnetic field:
 1 cost 


 B    B0 z  B1 (cos t x  sin t y )   1 sin t 
 

0


 B0  0
| B1 | 1
z
B0
()

How does magnetic moment respond?
x
B1
y
Laboratory frame
14
To follow the response of the magnetization let us rotate the coordinate system:
  xRoF (t )   cost sin t 0   x (t )    x (t ) cost   y (t ) sin t 
 RoF  

 

  y (t )     sin t cost 0   y (t )      x (t ) sin t   y (t ) cost 
  RoF (t )  
  (t )  

0
0
1

(
t
)
z
z
z








Then we get the equation of motion:
  xRoF (t )   (d / dt x (t )) cost  x sin t  (d / dt y (t )) sin t  y cost 

d  RoF  
  y (t )     (d / dt x (t )) sin t  x cost  (d / dt y (t )) cost  y sin t 
dt  RoF  

(d / dt z (t ))

  z (t )  
and insertion of the original equation of motion:
we get
  x (t )    0  y (t )  (1 sin t )  z (t ) 
 

d 
  y (t )    0  x (t )  (1 cost )  z (t )

dt 
  ( cost )  (t )  ( sin t )  (t ) 

(
t
)
y
1
x

 z   1
  xRoF (t )   (0  y  1 z sin t ) cost  x sin t  (0  x  1 z cost ) sin t  y cost 

d  RoF  
  y (t )     (0  y  1 z sin t ) sin t  x cost  (0  x  1 z cost ) cost  y sin t 
dt  RoF  

(1 y cost  1 x sin t )

  z (t )  
Thus the equation of motion
in the rotating frame becomes:
  xRoF  

 (0   )  yRoF

d  RoF  
RoF
RoF
  y    (0   )  x  1 z 
dt  RoF  

RoF



1 y
 z  

15
Thus in the rotating frame the magnetic field becomes time-independent
while the z-magnetic field component is reduced by the frequency of rotation
zRoF
 RoF
1
xRoF
yRoF
rotating frame
(0  ) ,there
0 is only an x-components to the field.
On-resonance, when
In such a case the magnetization performs a precession around the x-direction
with a rotation frequency .1
How to generate this B1 RF irradiation field in the laboratory frame:
B0
B1 (t )  B1 cost
I (t )  I1 cost
21 cost x
y
z
Ignore because it is off-resonance!
1 (cost x  sin t y)  1 (cost x  sin t y )
x
Top view
y
x
16
Bird cage
National High Magnetic
Field Laboratory
Doty Scientific
u-of-o-nmr-facility.blogspot.com/2008/03/prob...
in:
Thus the magnetic field in the laboratory frame :
LAB  0 z  1 cos(t   ) x  1 sin(t   ) y
Becomes in the rotating frame:
RoF   z  1 cos  x  1 sin  y
out:
 RoF , x (t )
In NMR we measure
the magnetization
in the rotating frame:
 RoF , y (t )
Although the signal detection in the laboratory frame is along the direction of the coil:
LABx (t )  xRoF (t ) cost  yRoF (t ) sin t
A sample with an overall
M (t )
the S/N voltage at the coil is:

S/N 
f
1/ 2
Vs 
1
    Q  M 0
kT 
2
f =noise of apparatus
 =filling factor
 =frequency
=band width
Q =quality factor
Vs =sample volume
17