Rotating frame (angular velocity ππ )
Transcript of Rotating frame (angular velocity ππ )
Rotating frame (angular velocity ππππππ):
ππππ(π‘π‘)πππ‘π‘
= ππβππππππ + ππππ Γ ππ π‘π‘ = βπΎπΎ π©π© +ππππππ
πΎπΎ + π©π©ππ Γ ππ π‘π‘
π΄π΄ (offset) βπ©π© (residual vertical magnetic field in the rotating frame)
x
y
z π©π©ππ
π©π©ππ
ππ1
πΌπΌ π΄π΄
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y
z π©π©ππ
π΄π΄
π΄π΄
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Basic NMR experiment
Relaxation delay
90Β° pulse Free induction decay (FID)
Repeat ππ 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
90Β° pulse - Tips magnetization into the x-y plane
Relaxation delay - Allows magnetization to return to equilibrium along z
x
y
z π©π©ππ
π©π©ππ
ππ1
πΌπΌ
Turn on π©π©ππ for time π‘π‘ such that πΌπΌ = ππ1π‘π‘ = | β πΎπΎπ΅π΅1|π‘π‘ = ππ
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π΄π΄
Rotating frame (angular velocity ππππππ)
90Β° pulse
x
y
z π©π©ππ
π΄π΄
Magnetization precesses in the x-y plane with frequency π΄π΄ (and relaxes back to equilibrium)
π΄π΄
Free induction decay (FID)
1 2 3 4
Description and analysis of rotations (oscillations): Rotation with angular velocity π΄π΄
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y
π΄π΄
π΄π΄
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y
π΄π΄ π΄π΄
πππ₯π₯
πππ¦π¦
time π‘π‘
πππ₯π₯ = ππ0 πππ¦π¦ = 0
πππ₯π₯ = ππ0 cosΞ©π‘π‘ πππ¦π¦ = ππ0 sinΞ©π‘π‘
πΌπΌ = Ξ©π‘π‘
x
y
π΄π΄ π΄π΄
πππ₯π₯
πππ¦π¦
πππ₯π₯ = ππ0 cosΞ©π‘π‘ πππ¦π¦ = ππ0 sinΞ©π‘π‘
x
y
π΄π΄
βπ΄π΄
πππ₯π₯
πππ¦π¦
πππ₯π₯ = ππ0 cos(βΞ©π‘π‘) =ππ0 cosΞ©π‘π‘ πππ¦π¦ = ππ0 sin(βΞ©π‘π‘) = βππ0 sinΞ©π‘π‘
Need both πππ₯π₯ and πππ¦π¦ components to distinguish the direction (sign) of rotation
πππ₯π₯ and πππ¦π¦ components formally treated like real and imaginary components of a complex number:
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y
π΄π΄ π΄π΄
πππ₯π₯
πππ¦π¦
πΌπΌ = Ξ©π‘π‘
ππ = πππ₯π₯ + πππππ¦π¦ = ππ0 cosΞ©π‘π‘ + ππππ0 sinΞ©π‘π‘
ππ = ππ0 (cosΞ©π‘π‘ + ππ sinΞ©π‘π‘) =ππ0ππππΞ©π‘π‘
Return to equilibrium: relaxation β’ Longitudinal (spin-lattice) relaxation
πππ§π§
πππ§π§ = ππ0(1 β ππβπ π 1π‘π‘) = ππ0(1 β ππβπ‘π‘ππ1)
ππ0
0 π‘π‘
Return to equilibrium: relaxation β’ Transverse (spin-spin) relaxation:
πππ₯π₯π¦π¦ = ππ0ππβπ π 2π‘π‘ππππΞ©π‘π‘ = ππ0ππβ π‘π‘ππ2 ππππΞ©π‘π‘
πππ₯π₯
https://en.wikipedia.org/wiki/Relaxation_(NMR)
π‘π‘ 0
Fourier transform S Ο = β« π π (π‘π‘)ππβπππππ‘π‘βββ πππ‘π‘
Oscillation π π π‘π‘ = ππ0ππβπ π 2π‘π‘ππππΞ©0π‘π‘
ππ ππ = οΏ½ ππ0ππβπ π 2π‘π‘ππππΞ©0π‘π‘ ππβπππππ‘π‘β
0
πππ‘π‘
Converting oscillating signal to a frequency spectrum
ππ(Ξ©0βππ β π π 2]π‘π‘ = y ππ(Ξ©0βππ β π π 2]πππ‘π‘ = dy
= ππ0ππ(Ξ©0βππ)βπ π 2
οΏ½πππ¦π¦ππππ = ππ0ππ(Ξ©0βππ)βπ π 2
[ππππ Ξ©0βππ π‘π‘ππβπ π 2π‘π‘]π‘π‘=0
= οΏ½ ππ0ππ[ππ Ξ©0βππ βπ π 2]π‘π‘
β
0
πππ‘π‘ = οΏ½ππ0πππ¦π¦ πππ¦π¦ππ Ξ©0βππ βπ π 2
β οΏ½πππ¦π¦ππππ = πππ¦π¦ + πΆπΆ
(0-1)
ππ ππ = βππ0ππ Ξ©0βππ βπ π 2
= βππ0ππ Ξ©0βππ βπ π 2
βππ Ξ©0βππ βπ π 2βππ Ξ©0βππ βπ π 2
π΅π΅ + π΄π΄ βπ΅π΅ + π΄π΄ = βπ΅π΅2 + π΄π΄2
= ππ0ππ Ξ©0βππ +π π 2Ξ©0βππ 2+π π 22
= ππ0[ π π 2Ξ©0βππ 2+π π 22
+ ππ Ξ©0βππΞ©0βππ 2+π π 22
]
π΄π΄ ππ Absorption
π·π·(ππ) Dispersion
Lorentzian line shape
ππ Ξ©0
βππ = 2π π 2 (βππ = π π 2
ππ )
Full width at half height
2-dimensional (2D) NMR spectroscopy
Correlation between nuclear dipoles β’ Through-bond interactions (J-coupling) < 4 bonds
apart β’ Through-space interactions (nuclear Overhauser
effect, NOE) < 5 Γ apart
Ξ©π΄π΄
Ξ©π΅π΅
Kumar A, Ernst RR, WΓΌthrich 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
90Β°y FID
π‘π‘1 π‘π‘2 π‘π‘πππππ₯π₯
90Β°y 90Β°-y
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y
z
π©π©ππ
F B C D E A G
A
B
A: πππ§π§ = ππ0 B: πππ₯π₯ = ππ0
2D NMR spectroscopy
Relaxation delay
90Β°y FID
π‘π‘1 π‘π‘2 π‘π‘πππππ₯π₯
90Β°y 90Β°-y
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y
z
π΄π΄
F B C D E A G
B C
C: πππ₯π₯ = ππ0 cosΞ©π‘π‘1 πππ¦π¦ = ππ0 sinΞ©π‘π‘1
x
π΄π΄
πππ₯π₯
πππ¦π¦
πΌπΌ = Ξ©π‘π‘1
C π©π©ππ y
2D NMR spectroscopy
Relaxation delay
90Β°y FID
π‘π‘1 π‘π‘2 π‘π‘πππππ₯π₯
90Β°y 90Β°-y
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F B C D E A G
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C
D: πππ§π§ = βππ0 cosΞ©π‘π‘1 πππ¦π¦ = ππ0 sinΞ©π‘π‘1 (ignore πππ¦π¦, it will get cancelled by phase cycling and/or gradients
πππ₯π₯ πππ¦π¦
π©π©ππ
E: πππ§π§ = βππ0 a cosΞ©π‘π‘1 Relaxation during π‘π‘πππππ₯π₯
2D NMR spectroscopy
Relaxation delay
90Β°y FID
π‘π‘1 π‘π‘2 π‘π‘πππππ₯π₯
90Β°y 90Β°-y
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z
F B C D E A G
E
F
π©π©ππ E: πππ§π§ = βππ0 a cosΞ©π‘π‘1
F: πππ₯π₯ = ππ0 a cosΞ©π‘π‘1
G: πππ₯π₯ = ππ0 a cosΞ©π‘π‘1ππβπ π 2π‘π‘2 cosΞ©π‘π‘2 πππ¦π¦ = ππ0 a cosΞ©π‘π‘1ππβπ π 2π‘π‘2 sinΞ©π‘π‘2
πππ₯π₯π¦π¦ = ππ0 a cosΞ©π‘π‘1ππβπ π 2π‘π‘2ππππΞ©π‘π‘2
πππ₯π₯π¦π¦ = ππ0 a cosΞ©π‘π‘1 ππβπ π 2π‘π‘2ππππΞ©π‘π‘2 Amplitude modulation
β’ Acquire data with increasing π‘π‘1 β’ Fourier transform each FID (in π‘π‘2) β’ Fourier transform the spectra in π‘π‘1
Rule, G.S. and Hitchens, T.K. Fundamentals of Protein NMR Spectroscopy, 2006, Springer
250
250
2-dimensional (2D) NMR spectroscopy
Relaxation delay
90Β°y FID
π‘π‘1 π‘π‘2 π‘π‘πππππ₯π₯
90Β°y 90Β°-y
F B C D E A G
πππ§π§ = βπππ΄π΄ cosΞ©π΄π΄π‘π‘1 Nucleus A
D: πππ§π§ = βπππ΅π΅ cosΞ©π΅π΅π‘π‘1 Nucleus B
βπππ΄π΄ πππ΄π΄π΄π΄cosΞ©π΄π΄π‘π‘1 E: βπππ΅π΅ πππ΅π΅π΅π΅cosΞ©π΅π΅π‘π‘1
βπππ΅π΅ πππ΅π΅π΄π΄cosΞ©π΅π΅π‘π‘1 βπππ΄π΄ πππ΄π΄π΅π΅cosΞ©π΄π΄π‘π‘1
G: ( )ππβπ π 2π‘π‘2ππππΞ©π΄π΄π‘π‘2 ( )ππβπ π 2π‘π‘2ππππΞ©π΅π΅π‘π‘2
Ξ©π΄π΄
Ξ©π΅π΅
Ξ©π΅π΅
Ξ©π΄π΄
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. 287β300.
Instrumentation
600 MHz Agilent VNMRS spectrometer: β’ four channels β’ z-axis gradients β’ HCN triple resonance probe 500 MHz Agilent VNMRS spectrometer: β’ three channels β’ three-axis gradients β’ HCN triple resonance probe β’ direct and indirect broadband
probes β’ accessories for MAS
University of Montana