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Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓):
𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= 𝝎𝝎−𝝎𝝎𝒓𝒓𝒓𝒓 + 𝝎𝝎𝟏𝟏 × 𝝁𝝁 𝑡𝑡 = −𝛾𝛾 𝑩𝑩 +𝝎𝝎𝒓𝒓𝒓𝒓
𝛾𝛾 + 𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡
𝜴𝜴 (offset) ∆𝑩𝑩 (residual vertical magnetic field in the rotating frame)
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
𝛼𝛼 𝑴𝑴
x
y
z 𝑩𝑩𝟎𝟎
𝜴𝜴
𝑴𝑴
1 2 3 4
Basic NMR experiment
Relaxation delay
90° pulse Free induction decay (FID)
Repeat 𝑛𝑛 times (to improve signal to noise ratio)
Free induction decay (FID) - Magnetization precesses in the x-y plane, this induces electrical current in a coil and is detected
90° pulse - Tips magnetization into the x-y plane
Relaxation delay - Allows magnetization to return to equilibrium along z
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
𝛼𝛼
Turn on 𝑩𝑩𝟏𝟏 for time 𝑡𝑡 such that 𝛼𝛼 = 𝜔𝜔1𝑡𝑡 = | − 𝛾𝛾𝐵𝐵1|𝑡𝑡 = 𝜋𝜋
2
𝑴𝑴
Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓)
90° pulse
x
y
z 𝑩𝑩𝟎𝟎
𝜴𝜴
Magnetization precesses in the x-y plane with frequency 𝜴𝜴 (and relaxes back to equilibrium)
𝑴𝑴
Free induction decay (FID)
1 2 3 4
Description and analysis of rotations (oscillations): Rotation with angular velocity 𝜴𝜴
x
y
𝑴𝑴
𝜴𝜴
x
y
𝑴𝑴 𝜴𝜴
𝑀𝑀𝑥𝑥
𝑀𝑀𝑦𝑦
time 𝑡𝑡
𝑀𝑀𝑥𝑥 = 𝑀𝑀0 𝑀𝑀𝑦𝑦 = 0
𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡
𝛼𝛼 = Ω𝑡𝑡
x
y
𝑴𝑴 𝜴𝜴
𝑀𝑀𝑥𝑥
𝑀𝑀𝑦𝑦
𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡
x
y
𝑴𝑴
−𝜴𝜴
𝑀𝑀𝑥𝑥
𝑀𝑀𝑦𝑦
𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cos(−Ω𝑡𝑡) =𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sin(−Ω𝑡𝑡) = −𝑀𝑀0 sinΩ𝑡𝑡
Need both 𝑀𝑀𝑥𝑥 and 𝑀𝑀𝑦𝑦 components to distinguish the direction (sign) of rotation
𝑀𝑀𝑥𝑥 and 𝑀𝑀𝑦𝑦 components formally treated like real and imaginary components of a complex number:
x
y
𝑴𝑴 𝜴𝜴
𝑀𝑀𝑥𝑥
𝑀𝑀𝑦𝑦
𝛼𝛼 = Ω𝑡𝑡
𝑀𝑀 = 𝑀𝑀𝑥𝑥 + 𝑖𝑖𝑀𝑀𝑦𝑦 = 𝑀𝑀0 cosΩ𝑡𝑡 + 𝑖𝑖𝑀𝑀0 sinΩ𝑡𝑡
𝑀𝑀 = 𝑀𝑀0 (cosΩ𝑡𝑡 + 𝑖𝑖 sinΩ𝑡𝑡) =𝑀𝑀0𝑒𝑒𝑖𝑖Ω𝑡𝑡
Return to equilibrium: relaxation • Longitudinal (spin-lattice) relaxation
𝑀𝑀𝑧𝑧
𝑀𝑀𝑧𝑧 = 𝑀𝑀0(1 − 𝑒𝑒−𝑅𝑅1𝑡𝑡) = 𝑀𝑀0(1 − 𝑒𝑒−𝑡𝑡𝑇𝑇1)
𝑀𝑀0
0 𝑡𝑡
Return to equilibrium: relaxation • Transverse (spin-spin) relaxation:
𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω𝑡𝑡 = 𝑀𝑀0𝑒𝑒− 𝑡𝑡𝑇𝑇2 𝑒𝑒𝑖𝑖Ω𝑡𝑡
𝑀𝑀𝑥𝑥
https://en.wikipedia.org/wiki/Relaxation_(NMR)
𝑡𝑡 0
Fourier transform S ω = ∫ 𝑠𝑠(𝑡𝑡)𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡∞−∞ 𝑑𝑑𝑡𝑡
Oscillation 𝑠𝑠 𝑡𝑡 = 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω0𝑡𝑡
𝑆𝑆 𝜔𝜔 = � 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω0𝑡𝑡 𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡∞
0
𝑑𝑑𝑡𝑡
Converting oscillating signal to a frequency spectrum
𝑖𝑖(Ω0−𝜔𝜔 − 𝑅𝑅2]𝑡𝑡 = y 𝑖𝑖(Ω0−𝜔𝜔 − 𝑅𝑅2]𝑑𝑑𝑡𝑡 = dy
= 𝑀𝑀0𝑖𝑖(Ω0−𝑖𝑖)−𝑅𝑅2
�𝑒𝑒𝑦𝑦𝑑𝑑𝑑𝑑 = 𝑀𝑀0𝑖𝑖(Ω0−𝑖𝑖)−𝑅𝑅2
[𝑒𝑒𝑖𝑖 Ω0−𝑖𝑖 𝑡𝑡𝑒𝑒−𝑅𝑅2𝑡𝑡]𝑡𝑡=0
= � 𝑀𝑀0𝑒𝑒[𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2]𝑡𝑡
∞
0
𝑑𝑑𝑡𝑡 = �𝑀𝑀0𝑒𝑒𝑦𝑦 𝑑𝑑𝑦𝑦𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2
∞ �𝑒𝑒𝑦𝑦𝑑𝑑𝑑𝑑 = 𝑒𝑒𝑦𝑦 + 𝐶𝐶
(0-1)
𝑆𝑆 𝜔𝜔 = −𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2
= −𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2
−𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2−𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2
𝐵𝐵 + 𝐴𝐴 −𝐵𝐵 + 𝐴𝐴 = −𝐵𝐵2 + 𝐴𝐴2
= 𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 +𝑅𝑅2Ω0−𝑖𝑖 2+𝑅𝑅22
= 𝑀𝑀0[ 𝑅𝑅2Ω0−𝑖𝑖 2+𝑅𝑅22
+ 𝑖𝑖 Ω0−𝑖𝑖Ω0−𝑖𝑖 2+𝑅𝑅22
]
𝐴𝐴 𝜔𝜔 Absorption
𝐷𝐷(𝜔𝜔) Dispersion
Lorentzian line shape
𝜔𝜔 Ω0
∆𝜔𝜔 = 2𝑅𝑅2 (∆𝜈𝜈 = 𝑅𝑅2
𝜋𝜋 )
Full width at half height
2-dimensional (2D) NMR spectroscopy
Correlation between nuclear dipoles • Through-bond interactions (J-coupling) < 4 bonds
apart • Through-space interactions (nuclear Overhauser
effect, NOE) < 5 Å apart
Ω𝐴𝐴
Ω𝐵𝐵
Kumar A, Ernst RR, Wüthrich K. A two-dimensional nuclear Overhauser enhancement (2D NOE) experiment for the elucidation of complete proton-proton cross-relaxation networks in biological macromolecules. Biochem Biophys Res Commun. 1980 Jul 16;95(1):1-6.
The first 2D spectrum of a protein (BPTI)
2D NMR spectroscopy
Relaxation delay
90°y FID
𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
90°y 90°-y
x
y
z
𝑩𝑩𝟎𝟎
F B C D E A G
A
B
A: 𝑀𝑀𝑧𝑧 = 𝑀𝑀0 B: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0
2D NMR spectroscopy
Relaxation delay
90°y FID
𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
90°y 90°-y
x
y
z
𝜴𝜴
F B C D E A G
B C
C: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡1 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡1
x
𝜴𝜴
𝑀𝑀𝑥𝑥
𝑀𝑀𝑦𝑦
𝛼𝛼 = Ω𝑡𝑡1
C 𝑩𝑩𝟎𝟎 y
2D NMR spectroscopy
Relaxation delay
90°y FID
𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
90°y 90°-y
x
y
z
F B C D E A G
D
C
D: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 cosΩ𝑡𝑡1 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡1 (ignore 𝑀𝑀𝑦𝑦, it will get cancelled by phase cycling and/or gradients
𝑀𝑀𝑥𝑥 𝑀𝑀𝑦𝑦
𝑩𝑩𝟎𝟎
E: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 a cosΩ𝑡𝑡1 Relaxation during 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
2D NMR spectroscopy
Relaxation delay
90°y FID
𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
90°y 90°-y
x
y
z
F B C D E A G
E
F
𝑩𝑩𝟎𝟎 E: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 a cosΩ𝑡𝑡1
F: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 a cosΩ𝑡𝑡1
G: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2 cosΩ𝑡𝑡2 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2 sinΩ𝑡𝑡2
𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝑡𝑡2
𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1 𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝑡𝑡2 Amplitude modulation
• Acquire data with increasing 𝑡𝑡1 • Fourier transform each FID (in 𝑡𝑡2) • Fourier transform the spectra in 𝑡𝑡1
Rule, G.S. and Hitchens, T.K. Fundamentals of Protein NMR Spectroscopy, 2006, Springer
250
250
2-dimensional (2D) NMR spectroscopy
Relaxation delay
90°y FID
𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥
90°y 90°-y
F B C D E A G
𝑀𝑀𝑧𝑧 = −𝑀𝑀𝐴𝐴 cosΩ𝐴𝐴𝑡𝑡1 Nucleus A
D: 𝑀𝑀𝑧𝑧 = −𝑀𝑀𝐵𝐵 cosΩ𝐵𝐵𝑡𝑡1 Nucleus B
−𝑀𝑀𝐴𝐴 𝑎𝑎𝐴𝐴𝐴𝐴cosΩ𝐴𝐴𝑡𝑡1 E: −𝑀𝑀𝐵𝐵 𝑎𝑎𝐵𝐵𝐵𝐵cosΩ𝐵𝐵𝑡𝑡1
−𝑀𝑀𝐵𝐵 𝑎𝑎𝐵𝐵𝐴𝐴cosΩ𝐵𝐵𝑡𝑡1 −𝑀𝑀𝐴𝐴 𝑎𝑎𝐴𝐴𝐵𝐵cosΩ𝐴𝐴𝑡𝑡1
G: ( )𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝐴𝐴𝑡𝑡2 ( )𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝐵𝐵𝑡𝑡2
Ω𝐴𝐴
Ω𝐵𝐵
Ω𝐵𝐵
Ω𝐴𝐴
Rankin, Naomi & Preiss, David & Welsh, Paul & Burgess, Karl & Nelson, Scott & Lawlor, Debbie & Sattar, Naveed. (2014). The emergence of proton nuclear magnetic resonance metabolomics in the cardiovascular arena as viewed from a clinical perspective. Atherosclerosis. 237. 287–300.
Instrumentation
600 MHz Agilent VNMRS spectrometer: • four channels • z-axis gradients • HCN triple resonance probe 500 MHz Agilent VNMRS spectrometer: • three channels • three-axis gradients • HCN triple resonance probe • direct and indirect broadband
probes • accessories for MAS
University of Montana
