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(
)sin()cos(2)cos()cos(
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
peak number 2T2 = 0.247 error = 0.016
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
peak number 1T1 = 19.7 error = 0.572
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
peak number 2T1 = 19.8 error = 0.479
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
)sin()sin(2)cos()sin(
)sin()cos(2)cos()cos(
1111
1111
tJtSItJtI
tJtSItJtI
ISIzyISIx
ISIzxISIy
90x
)sin()sin(2)cos()sin(
)sin()cos(2)cos()cos(
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
)sin()cos( 11 tItI IxIy
1)( tI zI
1)2( tSIJ zzIS
)sin()sin(2)cos()sin(
)sin()cos(2)cos()cos(
1111
1111
tJtSItJtI
tJtSItJtI
ISIzyISIx
ISIzxISIy
90x
)sin()sin(2)cos()sin(
)sin()cos(2)cos()cos(
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
)sin()cos( 11 tItI IxIy
1)( tI zI
1)2( tSIJ zzIS
)sin()sin(2)cos()sin(
)sin()cos(2)cos()cos(
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.