Content

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

NMR-Primer for Chemists and BiologistsShimon Vega & Yonatan Hovav

shimon.vega@weizmann.ac.il and yonatan.hovav@weizmann.ac.il

November 2013

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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

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Some books:

Modern NMR techniques for Chemistry Research by A.E DeromeNuclear Magnetic Resonance spectroscopy by R.K Harris

Advanced:Spin Dynamics by M.H LevittUnderstanding NMR Spectroscopy by James KeelerPrinciples of NMR in 1 and 2 Dimensions by R. R. Ernst,

G. Bodenhausen A. Wokaun

Principles Magnetic Resonance by C. P. SlichterSolid State NMR Spectroscopy by M Duer

Web: 1. http://www.cis.rit.edu/htbooks/nmr/

2. http://www-keeler.ch.cam.ac.uk/lectures/

3

Born(1905-10-23)October 23, 1905Zürich, Switzerland

Died

September 10, 1983(1983-09-10) (aged 77)Zürich, Switzerland

Citizenship Swiss, American

Nationality Swiss

Fields Physics

InstitutionsUniversity of California, Berkeley

Stanford University

Alma mater ETH Zürich and University of Leipzig

Doctoral advisor Werner Heisenberg

Known for

NMRBloch wallBloch's TheoremBloch Function (Wave)Bloch sphere

Notable awards Nobel Prize for Physics (1952)

Born

(1912-08-30)August 30, 1912Taylorville, Illinois, USA

Died

March 7, 1997(1997-03-07) (aged 84)Cambridge, Massachusetts, USA

Nationality United States

Fields Physics

Institutions Harvard UniversityMIT

Alma mater Purdue UniversityHarvard University

Doctoral advisor Kenneth Bainbridge

Other academic advisors John Van Vleck

Doctoral students

Nicolaas Bloembergen

George PakeGeorge BenedekCharles Pence Slichter

Known for

Nuclear magnetic resonance (NMR)Smith-Purcell effect21 cm line

Notable awards Nobel Prize for Physics (1952)

Felix Bloch Edward Purcell

4

1920's Physicists have great success with quantum theoryQuantum 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 experimentsThe 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 electronThe idea of a spinning electron with resultant angularmomentum gave rise to the magnetic dipole moment. 1926 Schrödinger/Heisenberg formulate quantum mechanicsThis 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 mechanics1933 Stern and Gerlach measure the effect of nuclear spinStern 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 to the nuclear magnetic moment. 5

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 resonanceDuring 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, resonanceoccurred. 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 physicsOtto 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 physicsRabi was given this prize for his work on molecular beams, especially the resonance method.

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1945 Purcell, Torey and Pound observe NMR in a bulk materialAt 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 physicsThis 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 Fielda. 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: withI

2/5;2;2/3;1;2/1I

h = 6.6260755x1034 m2kg/sec and

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 +/- 0.0000022Neutron: g = -3.8260837 +/- 0.0000018

I For each nucleus the angular momentum vector and the magnetic moment vectorare related by its magnetogyric ratio .g

(from QM)

or

= g magnetogyric ratioB

Bh

BE

/

.

.

8

9

prvmrI )(

pr

I

m mR2/1

I

]sec)/()/[(/2

])/()/[()2/(1

msmRv

HzmsmRv

General comments about angular momentum:

Remember: conservation of angular momentum !

I

2mr

pr

r

v

`

q

Ai

q = chargeq/L = charge densityL = 2prv = velocityi = currentA = areaA = p2r

B1. The torque on the magnetic moment induced by a magnetic field B

2. Energy of a magnetic moment in a magnetic field

B

BE .

Minimizing energy

B

q

mAi

q

mei

q

rmrvmrI

222

vr

qi

2

moment of inertia

number of turn per second

number of radians per second

General comments about magnetic momentums:

e

xyyx

zxxz

yzzy

z

y

x

z

y

x

ab

baba

baba

baba

b

b

b

a

a

a

bacbaccba ,sinReminder:

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1 H Hydrogen ½ 300.130 2 H Deuterium 1 46.0733 H Tritium 1/2 320.128 3 He Helium 1/2 228.633 6 Li Lithium 1 44.167 7 Li Lithium 3/2 116.640 9 Be Beryllium 3/2 42.174 10 B Boron 3 32.24611 B Boron 3/2 96.258 13 C Carbon 1/2 75.46

Larmor Frequencies in MHz units: withBL TeslaB 0.7

100MHz - 2.3 T300MHz - 7.0 T

500MHz - 11.7 T800MHz - 18.8 T

900MHz - 21.1 Tesla(2.1 KHz - 0.5 Gauss)

1T = 10,000G

Proton NMR:

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Element/Name Isotope Symbol Nuclear Spin Sensitivity vs. 1H

Hydrogen 1H 1/2 1.000000Deuterium 2H or D 1 1.44 e-6Tritium 3H 1/2 - Helium-3 3He -1/2 - Lithium-6 6Li 1 0.000628Lithium-7 7Li 3/2 0.270175Beryllium-9 9Be -3/2 0.013825Boron-10 10B 3 0.00386Boron-11 11B 3/2 0.132281Carbon-13 13C 1/2 0.000175Nitrogen-14 14N 1 0.000998Nitrogen-15 15N -1/2 3.84E-06Oxygen-17 17O -5/2 1.07E-05Fluorine-19 19F 1/2 0.829825Neon-21 21Ne -3/2 6.3E-06Sodium-23 23Na 3/2 0.092105Magnesium-25 25Mg -5/2 0.00027Aluminum-27 27Al 5/2 0.205263Silicon-29 29Si -1/2 0.000367

Receptivity : (natur.abund.-%) x g x I(I+1)

Phosphorus-31 31P 1/2 0.06614Sulfur-33 33S 3/2 1.71E-05Chlorine-33 33Cl 3/2 0.003544Chlorine-37 37Cl 3/2 0.000661Potassium-39 39K 3/2 0.000472Potassium-41 41K 3/2 5.75E-06Calcium-43 43K -7/2 9.25E-06Scandium-45 45Sc 7/2 0.3Titanium-47 47Ti -5/2 0.00015Titanium-49 49Ti -7/2 0.00021Vanadium-50 50V 6 0.00013Vanadium-51 51V 7/2 0.37895Chromium-53 53Cr -3/2 8.6E-05Manganese-55 55Mn 5/2 0.174386Iron-57 57Fe 1/2 7.37E-07Cobolt-59 59Co 7/2 0.275439Nickel-61 61Ni -3/2 4.21E-05Copper-63 63Cu 3/2 0.064035Copper-65 65Cu 3/2 0.035263Zinc-67 67Zn 5/2 0.000117

1/2 1 3/5 5/2 7/2Fr

eque

ncy

(MH

z)

Tesla - MHz 5 x10-5 2.1 x10-3

2.35 100 7.05 300 9.40 400 11.75 500 18.80 800 21.15 900

11

2b. The classical precession of the magnetization

Suppose we apply a magnetic field on our magnetization:

B

as a result a torque tries to rotate the direction ofthe angular momentum.

I

A torque ( ) perpendicular to an angular momentum causes a precession motion:

Example: top view:

From http://hyperphysics.phy-astr.gsu.edu

Fr

Remember the motion of a top: (gravitation + top)

12

dtpdamF /

xy

z

00

cos)sin(

sin)sin(

cos

sinsin

cossin

)2/sin(

;;

0

0

0

0

0

,

x

y

z

y

x

yx

I

I

I

I

I

I

I

dt

d

I

I

I

dt

d

IBIBdt

Id

I

BB

IIIdt

Id

B 0The Larmor frequency

The precession of the magnetizationaround the magnetic field directionis independent of the orientation of

(in analogy with )

B

z

LyLx

LyLx

z

y

x

I

tItI

tItI

t

I

I

I

cossin

sincos

)(

FI

B

13

2c. RF irradiation, resonance and the rotating frame

xyyx

xzzx

yzzy

z

y

x

L

tdt

d

tBtdt

td

)(

)()()(

Let

Let us now consider a special time-dependent magnetic field:

0

1

1

10 sin

cos

)sin(cos

t

t

ytxtBzBB

The equation of motion for the magnetization in an external magnetic field

0B

1B

xy

z )(

Laboratory frame

)(tBL ;

x

y

z

00 B 11 || B

How does magnetic moment respond?

14

)(

cos)(sin)(

sin)(cos)(

)(

)(

)(

100

0cossin

0sincos

)(

)(

)(

t

tttt

tttt

t

t

t

tt

tt

t

t

t

z

yx

yx

z

y

x

RoFz

RoFy

RoFx

))(/(

sincos))(/(cossin))(/(

cossin))(/(sincos))(/(

)(

)(

)(

tdtd

tttdtdtttdtd

tttdtdtttdtd

t

t

t

dt

d

z

yyxx

yyxx

RoFz

RoFy

RoFx

To follow the response of the magnetization let us rotate the coordinate system:

Then we get the equation of motion:

)()sin()()cos(

)()cos()(

)()sin()(

)(

)(

)(

11

10

10

tttt

ttt

ttt

t

t

t

dt

d

xy

zx

zy

z

y

x

and insertion of the original equation of motion:

)sincos(

sincos)cos(cossin)sin(

cossin)cos(sincos)sin(

)(

)(

)(

11

1010

1010

tt

tttttt

tttttt

t

t

t

dt

d

xy

yzxxzy

yzxxzy

RoFz

RoFy

RoFx

we get

RoFy

RoFz

RoFx

RoFy

RoFz

RoFy

RoFx

dt

d

1

10

0

)(

)(

Thus the equation of motionin the rotating frame becomes:

15

Thus in the rotating frame the magnetic field becomes time-independentwhile the z-magnetic field component is reduced by the frequency of rotation

1

RoF

xRoF

yRoF

zRoF

On-resonance, when , there is only an x-components to the field.In such a case the magnetization performs a precession around the x-directionwith a rotation frequency .

0)( 0

rotating frame

1

How to generate this B1 RF irradiation field in the laboratory frame:

x

y

xt cos2 1

)sin(cos)sin(cos 11 ytxtytxt

xy

z

Ignore because it is off-resonance!

Top view

B0

tItI cos)( 1

tBtB cos)( 11

16

u-of-o-nmr-facility.blogspot.com/2008/03/prob...

Doty Scientific

National High Magnetic Field Laboratory

Bird cage

Thus the magnetic field in the laboratory frame :

Becomes in the rotating frame:

yxz

ytxtz

RoF

LAB

sincos

)sin()cos(

11

110

ttttt yRoFxRoFLABx sin)(cos)()(

Although the signal detection in the laboratory frame is along the direction of the coil:

In NMR we measure the magnetization in the rotating frame: )(

)(

,

,

t

t

yRoF

xRoF

in:

out:

A sample with an overall )(tM the S/N voltage at the coil is:

0

2/1

2

1/ M

kT

VQ

fNS s

f =noise of apparatus h =filling factor n =frequency Dn=band width Q =quality factor Vs =sample volume 17