nuclear magnetic resonance - rutgers...

Nuclear Magnetic Resonance Principles

Nagarajan Murali

Rutgers, The State University of

New Jersey

References

Understanding NMR Spectroscopy

James Keeler

John Wiley & Sons (2006,2007)

Spin Dynamics Basics of Nuclear Magnetic Resonance

Malcolm H. Levitt

John Wiley & Sons (2007)

Overview

• Larmor Precession• Bulk Magnetization• Rotating Frame• RF Pulse in Rotating Frame• FID• Product Operators – Foundation for Future Lectures• pw calibration, pw calibration by Nutation• Simple 1D• Spin Echo• T2 – Relaxation• T1 - Relaxation• Spin Lock• 2D Correlation Spectroscopy (COSY)• 2D NOESY• 2D ROESY• pwx calibration• HMQC and HSQC

Nuclear Magnetic Moment

• The nucleus of many atoms behaves like a tiny bar magnet – we say the nucleus possesses a magnetic moment m.

• The magnetic moments arise from a fundamental property of the nucleus known as Spin which gives rise to an angular momentum I and given as

– gyromagnetic ratio specific for a nucleus and can be positive (m and I are parallel)or negative (m and I are anti-parallel) .

Iμ

Larmor Precession

> 0

The magnetic moment m in a magnetic field B0 experiences a torque

0Bμμ

dt

d

The size of the magnetic moment m is fixed - the effect of the torque is to rotate the magnetic moment around the magnetic field. This rotation is called Larmorprecession and the frequency is Larmor frequency.

02

10

00

B

B

rad s-1

Hz

Larmor frequency

Bulk Magnetization

• In NMR experiments we observe a large number of such nuclear magnetic moments.

• When an external magnetic field is applied to a sample the magnetic moments give a net contribution – called a magnetization along the direction of the applied magnetic field.

• At equilibrium the individual magnetic moments are predominantly oriented at an angle and precess on the surface of a cone at the Larmor frequency with no net magnetization perpendicular to the field.

Alignment of Nuclear Magnetic Moments

No external magnetic field

Net magnetization is ‘zero’

External Field induces net magnetization

kT

BIINM

3

)1( 022

00

m

N – number of spins, – gyromagnetic ratio, – Planks constant, I – spin quntum number, m0 –permeability of free space, B0 – Applied magnetic field strength (induction), K - Boltzman constant, T –temperature.

Alignment of Nuclear Magnetic Moments

External Field induces net magnetization

kT

BIINM

3

)1( 022

00

m

N – number of spins, – gyromagnetic ratio, – Planks constant, I – spin quntum number, m0 –permeability of free space, B0 – Applied magnetic field strength (induction), K - Boltzman constant, T –temperature.

It takes a finite time to induce the magnetization by the external field and the time constant T1 is known as longitudinal relaxation time.

Signal

02

10

00

B

B

rad s-1

Hz

> 0

If by some means the bulk magnetization is tilted, all individual magnetic moments are also tilted and as the moments precess around the field the bulk magnetization also precessesand induces a signal in a coil placed perpendicular to the applied field. Suppose the vector is tilted by an angle b from z axis towards the x axis then the observed signal will be along x axis

)cos(sin 00 tMM x b

Along a coil in y axis, it would be

)sin(sin 00 tMM y b

Radio Frequency Pulse

02

10

00

B

B

rad s-1

Hz

> 0

A radio frequency (RF) pulse at or near (resonance) the Larmor frequency applied along the x-axis can tilt the magnetization and is represented as

)cos(1 tB

B1 is the amplitude of rf field and is the frequency and t is the duration of the pulse.

0

Motion in the presence of RF

The motion of the magnetic moments is now complicated as the field is time-dependant due to the RF.

The dynamics can be simplified in a rotating frame where the field appears static.

))(( tdt

d10 BB

μ m

Rotating Frame

In the top figure a vector rotates in x-y plane, in bottom figure the x-y axes system is rotating and the vector is always along x-axis

Rotating FrameLet’s use the rotating frame to understand the effect of RF pulses. Say the applied field B0

is along Z-axis and an RF field B1cos(t) applied along the X-axis of the laboratory frame. Then the figures below show the fields in the rotating frame rotating at a frequency

given by the axes system (xyz).

effeff

eff

B

BBB

BBB

B

B

21

2

00

00

)(

If B1>>B then the effective field is along B1 and the magnetization vector will rotate about the x-axis.

Effect of Radio Frequency Pulse

Radio Frequency (RF) pulses at a frequency 0 and strength B1 rotates the magnetization.

900 rotation 1800 rotation

In the rotating frame, the static field along z-direction is zero on-resonance. The RF field appears static along x-axis. The rotation angle b increases with increasing pulse width for a given RF strength.

pwpw ttB 11 b

One Pulse Experiment

With a 900 pulse along x axis, the magnetization vector rotated from z to –y and evolves with a precession frequency rot 0

)sin(

)cos(

0

0

tMM

tMM

x

y

Free Induction Decay - FIDWith a 900 pulse along x axis, the magnetization vector rotated from z to –y then the observed FID is

2

2

2

/0

/0

/

)(

))sin()(cos()(

))()(()(

Ttti

Tt

Ttxy

eeMtS

etitMtS

etiMtMtS

My(t)

Mx(t)

Fourier Transform

T2 is the decay constant of the signal in the xy plane or the transverse relaxation time constant.

dtetSS ti )()(

Transverse relaxation – T2

12 TThe relaxation rate constant in terms of half the line width at half maximum

Line width (LW=2d) is usually measured in units of Hz, therefore

)(*

1

2

12

HzLWT

d

Proton NMR Spectrum

Proton NMR spectrum illustrating major functional groups. The solvent is deuterated dimethylsulfoxide.

Chemical Shift

TMS

TMS

d

TMS

TMSppm

d

610)(

Since the frequency increases with the field strength the chemical shift difference between two peaks is larger in frequency units. 1ppm at 400MHz = 400Hz; 1ppm at 500MHz = 500Hz.

TMS

TMS

TMS

TMSppmppm

dd

2616

21 1010)()(

06

21

621

216

10)()(

10)()(

)(10)(

d

d

d

ppm

ppm

ppm

TMS

TMS

Tools for Understanding NMR Experiments

To understand NMR experiments with more than one type of spins and many pulses we need more sophisticated tools developed based on quantum mechanics and is popularly known as Product Operator Formalism (POF).

The state of the magnetizations (spin states) of different species are represented by operators and their products to describe the time evolution of the spin states.

Operators Approach

Hamiltonian PartsIIz I – Spin Chemical ShiftSSz S– Spin Chemical ShiftJIS2 Iz Sz J – Coupling between Spins I & S

Iz Longitudinal MagnetizationIx Single Quantum Coherence

X Magnetization Iy Single Quantum Coherence

Y Magnetization2 Ix Sz Anti-phase X Coherence2 Iy Sz Anti-phase Y CoherenceE Identity operator

Multiple Quantum Coherence (MQC) 2 Ix Sx , 2 Iy Sy , 2 Ix Sy , 2 Iy Sx

Longitudinal 2- Spin Order2 Iz Sz

2 Ix Sx+ 2 Iy Sy Zero Quantum Coherence (ZQC)2 Ix Sx- 2 Iy Sy Double Quantum Coherence (DQC)2 Ix Sy -2 Iy Sx ZQC2 Ix Sy +2 Iy Sx DQC

Operators for two species I and S

Operators Approach

NMR Experiments can be understood by following the evolution of operators.

bb

bb

b

b

b

sincos

sincos

yzx

z

zyx

y

xx

x

III

III

II

bb

bb

b

b

b

sincos

sincos

xzy

z

yy

y

zxy

x

III

II

III

zy

z

xyz

y

yxz

x

II

III

III

b

b

b

bb

bb

sincos

sincos

Pulse Calibration

1 2

bx

t

zz SI bbbb sincossincos yzyz SSII bx

PW-Calibration by Nutation

11 B

11 222

)(4

11

2

2

11

1

Hzp

p

Ref: Rapid pulse length determination in high-resolution NMRPeter S.C. Wu, Gottfried Otting, J. Magn. Reson 176(1), 115, 2005

Simple 1D

1 2 3

90x

tI S

J

1

2

3

zz SI

90x

yy SI

)sin()cos(

)sin()cos(

tStS

tItI

SxSy

IxIy

tSI zSzI )(

tSIJ zzIS )2()sin()sin(2)cos()sin(

)sin()cos(2)cos()cos(

)sin()sin(2)cos()sin(


tJtIStJtS

tJtIStJtS

tJtSItJtI

tJtSItJtI

ISSzyISSx

ISSzxISSy

ISIzyISIx

ISIzxISIy

Spin Echo90x

180y

zI90x

yI )sin()cos( IxIy II

zI I

)sin()cos( IxIy II 180y

yI

))(sin)((cos)sin()sin()cos()sin(

)sin()cos()cos()cos(22

IIy

IIyIIx

IIxIIyI

II

II

zI I

At the end of 2 period chemical shift evolution is refocused.

y y y

y

Spin Echo –J Evolution90x

180y

)2sin(2)2cos( ISzxISy JSIJI yI zzIS SIJ 2)2(

:4

1

J zxSI2

At the end of 2 period the anti-phase coherence is generated.

SI

J

)sin()sin()cos()sin(2

)sin()cos()cos()cos(2

JttIJttSI

JttIJttSI

IxIzy

IyIzx

t

FID

Transverse relaxation – T2

12 TThe relaxation rate constant in terms of half the line width at half maximum

Line width (LW=2d) is usually measured in units of Hz, therefore

)(*

1

2

12

HzLWT

d

T2 By Spin Echo Method

Spin Echo principle is used to measure the transverse relaxation time T2

2/20)2(

TteMM

T2 Relaxationindex freq(ppm) intensity

1 9.42982 66.76232 9.39694 62.1995

Exponential data analysis:

peak T2 error1 0.2626 0.019052 0.2472 0.01599

peak number 1T2 = 0.263 error = 0.019

time observed calculated difference0.025 66.8 65.3 1.430.05 58.7 59.4 -0.7560.1 46.5 49.1 -2.580.2 36.7 33.6 3.110.4 14.2 15.8 -1.520.8 3.81 3.53 0.2841.6 0.41 0.279 0.1313.2 -0.01 0.117 -0.1276.4 0.02 0.117 -0.0969


time observed calculated difference0.025 62.2 64.1 -1.940.05 59.5 58 1.470.1 49.7 47.4 2.30.2 29.5 31.7 -2.170.4 14.2 14.2 0.03610.8 3.62 2.96 0.6551.6 0.29 0.292 -0.002423.2 -0.02 0.183 -0.2036.4 0.02 0.183 -0.163

T1 Relaxation – Inversion Recovery

)21()( 1/0

TeMM

nullnull

null

T

T

Tnull

T

T

e

MeM

eMM

null

null

null

44.12ln

2ln

2

1

2

)21(0)(

1

1

/

0/

0

/0

1

1

1

T1 Relaxationindex freq(ppm) intensity

1 7.75198 50.48512 7.73657 54.2945

Exponential data analysis:

peak T1 error1 19.67 0.57212 19.75 0.4794


time observed calculated difference0.0625 -42.2 -41.3 -0.8590.125 -42.2 -41 -1.20.25 -40.4 -40.5 0.03240.5 -39.1 -39.3 0.28

1 -37.1 -37.1 0.01732 -32.3 -32.8 0.4984 -23.9 -24.9 1.018 -10.2 -11.3 1.05

16 9.19 8.94 0.24832 30 31.4 -1.3864 44.7 45.7 -1.02

128 50.5 49.1 1.41


time observed calculated difference0.0625 -45 -44.1 -0.9180.125 -43.7 -43.8 0.1550.25 -42.5 -43.2 0.6830.5 -42.6 -42 -0.558

1 -40.1 -39.6 -0.4562 -35.1 -35 -0.04564 -25.9 -26.6 0.6438 -10.9 -11.9 1.04

16 10.2 9.73 0.45132 32.7 33.8 -1.1564 48.2 49.3 -1.06

128 54.3 53 1.31

Spin Lock

90x SLy

zI yI yI

yI1

During the delay the Iy coherences of the spins are locked along the y axis along which RF is applied. Other components rotate about y-axis (nutation) and decay.

2D Correlated Spectroscopy - COSY

zI

90x

yI

)sin()cos( 11 tItI IxIy

1)( tI zI

1)2( tSIJ zzIS



1111

1111

tJtSItJtI

tJtSItJtI

ISIzyISIx

ISIzxISIy

90x



1111

1111

tJtSItJtI

tJtSItJtI

ISIyzISIx

ISIyxISIz

Diagonal Peak Cross Peak

90x 90x

t1 t2

2D Spectrum

11

11

)sin()sin(2

1

)cos()sin(

tJtJI

tJtI

ISIISIx

ISIx

Diagonal Peaks

11

11

)cos()cos(2

12

)sin()sin(2

tJtJSI

tJtSI

ISIISIyz

ISIyz

Cross Peaks

I

S

I S

2

1

2D COSY Spectrum Line Shape

I

S

I S

Diagonal Peaks

Cross Peaks

2D Nuclear Overhauser Spectroscopy -NOESY

90x90x 90x

t1 t2m

zI

90x

yI


1)( tI zI

1)2( tSIJ zzIS



1111

1111

tJtSItJtI

tJtSItJtI

ISIzyISIx

ISIzxISIy

90x



1111

1111

tJtSItJtI

tJtSItJtI

ISIyzISIx

ISIyxISIz

Relaxation in m

)()cos()cos(

)()cos()cos(

11

11

mISISIz

mIIISIz

atJtS

atJtI

90x

)()cos()cos(

)()cos()cos(

11

11

mISISIy

mIIISIy

atJtS

atJtI

I

S

I S

2

1

2D Rotating Frame OverhauserSpectroscopy - ROESY

SLy

90x

t1 t2m

zI

90x

yI


1)( tI zI

1)2( tSIJ zzIS



1111

1111

tJtSItJtI

tJtSItJtI

ISIzyISIx

ISIzxISIy

SLy Relaxation in m

)()cos()cos(

)()cos()cos(

11

11

mrISISIy

mrIIISIy

atJtS

atJtI

I

S

I S

2

1

X-Nucleus PW Calibration

J2

1

90x

180y

bx

zx

ISzxISy

SI

JSIJI

2

)sin(2)cos(

yI

zzIS SIJ 2)(

yx IS b

bb sin2cos2 yxzx SISI

zzIS SIJ 2)(

bb sin2cos yxy SII

b=0 the signal is maximum.b =900 there is no signal as the multiple quantum coherence is unobservable.

HMQC - type

J2

1

90x

180y

bx

zzIS SIJ 2)(

bbb 2cossin2 SinISI yyx

b=0 the signal is maximum.b= 900 the signal is maximum.

bx

bbb 2sin2cossin2 zxyx SISI

zx

ISzxISy

SI

JSIJI

2

)sin(2)cos(

yI

zzIS SIJ 2)(

xSb

bb sin2cos2 yxzx SISI

yx IS b

PWX-Calibration by Nutation

)(4

11

2

2

11

1

Hzp

p

Ref: IDEAL- A fast single scan method for X pulse width calibrationNagarajan Murali, J. Magn. Reson 183, 142, 2006

2D-HMQC

zzIS SIJ 2)( )(2)cos(2 11 tSinSItSI sxxszx

zx

ISzxISy

SI

JSIJI

2

)sin(2)cos(

yI

zzIS SIJ 2)(

yxSI2

xS2

ISJ2

1

t1

1)(, tSI zsy

xS2

90x

180y

90x 90x

Decoupling RF

t2

)sin(2)cos(2 11 tSItSI sxxsyx

yI

2D-HSQC

zzIS SIJ 2)2( )(2)cos(2 11 tSinSItSI sxyszy

zx

ISzxISy

SI

JSIJI

2

)2sin(2)2cos(

yI

zzIS SIJ 2)2(

yzSI2

xy SI22

ISJ4

1

t1

1)(, tSI zsx

xx SI22

90x

180x

90x 90x

Decoupling RF

t2

)sin(2)cos(2 11 tSItSI sxzsyz

xI

180x

90y 180x 90x

180x

180x

xx SI

xx SI

2D-HSQC/HMQC

S

I

For each IS pair there will be one peak.

If there are homonuclear coupling that will split the lines along the Proton (I) dimension.

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