Principles of NMR spectroscopy
Transcript of Principles of NMR spectroscopy
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Principles of NMR spectroscopy Atomic nuclei have spin angular momentum (spin) 𝑰𝑰 characterized by spin quantum number 𝐼𝐼
𝑰𝑰 = ℏ [𝐼𝐼 𝐼𝐼 + 1 ]12
gyromagnetic ratio
Nuclei with spin have magnetic dipole moment: 𝝁𝝁 = 𝛾𝛾 𝑰𝑰
=ℎ
2𝜋𝜋= 1.055 ∗ 10−34 𝐽𝐽𝐽𝐽 Planck constant
• Odd mass number: half-integer spins • Even mass number, even atomic number: spin=0 • Even mass number, odd atomic number: integer spin
Nuclei with spin > ½ also have electric quadrupole moment: faster relaxation, broad signals, not so good for NMR
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Cavanagh, J., Fairbrother, W.J., Palmer, A.G.III, Rance, M., Skelton, N.J. Protein NMR Spectroscopy: Principles and Practice, 2nd edition, 2007, Academic Press
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x
y
z
Nucleus with spin ½ in external magnetic field 𝐵𝐵0 along z
𝐵𝐵0 Two possible states: 𝐼𝐼𝑧𝑧 = ℏ 𝑚𝑚 𝐼𝐼𝑧𝑧 =
ℏ2
magnetic quantum number 𝑚𝑚 = −𝐼𝐼,−𝐼𝐼 + 1, … , 𝐼𝐼 for 𝐼𝐼 = 1
2:𝑚𝑚 = −1
2, + 1
2
𝐼𝐼𝑧𝑧 = −ℏ2
𝐸𝐸 = −𝝁𝝁 ∙ 𝑩𝑩 = −𝝁𝝁𝒛𝒛𝑩𝑩 = −𝛾𝛾 ℏ𝑚𝑚 𝐵𝐵
𝝁𝝁 = 𝛾𝛾 𝑰𝑰 𝜇𝜇𝑧𝑧 = 𝛾𝛾 𝐼𝐼𝑧𝑧 = 𝛾𝛾 ℏ𝑚𝑚
Energy of a magnetic dipole in external magnetic field
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Energy of a magnetic dipole in external magnetic field
𝐵𝐵
𝐸𝐸 = −𝝁𝝁 ∙ 𝑩𝑩 = −𝜇𝜇𝑧𝑧𝐵𝐵 = −𝛾𝛾 ℏ 𝑚𝑚 𝐵𝐵
𝝁𝝁 = 𝛾𝛾 𝑰𝑰 𝜇𝜇𝑧𝑧 = 𝛾𝛾 𝐼𝐼𝑧𝑧 = 𝛾𝛾 ℏ𝑚𝑚
𝐸𝐸 𝑚𝑚 = - ½ 𝐸𝐸 = + 𝛾𝛾ℏ𝛾𝛾
2 (𝛾𝛾 > 0)
𝑚𝑚 = + ½ 𝐸𝐸 = −𝛾𝛾ℏ𝛾𝛾2
∆𝐸𝐸 ∆𝐸𝐸 = 𝛾𝛾 ℏ ∆𝑚𝑚 𝐵𝐵 = 𝛾𝛾 ℏ 𝐵𝐵
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∆𝐸𝐸 = 𝛾𝛾 ℏ ∆𝑚𝑚 𝐵𝐵 = 𝛾𝛾 ℏ 𝐵𝐵
𝐸𝐸 = ℎ 𝜈𝜈 =ℎ2𝜋𝜋
2𝜋𝜋𝜈𝜈 = ℏ 𝜔𝜔
frequency in s-1=Hz 𝜔𝜔 = 2𝜋𝜋𝜈𝜈 angular velocity (angular frequency) in radians/s
Energy difference between spin states
Energy of a photon
ℏ 𝜔𝜔 = 𝛾𝛾 ℏ 𝐵𝐵
𝜈𝜈 =𝜔𝜔2𝜋𝜋
=𝛾𝛾𝐵𝐵2𝜋𝜋
𝜔𝜔 = 𝛾𝛾𝐵𝐵 Larmor frequency
∆𝐸𝐸 (and 𝜔𝜔) can be tuned as a function of 𝐵𝐵 ∆𝐸𝐸 is rather small → 𝜔𝜔 is radiofrequency
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Populations (Boltzmann distribution) of nuclei with 𝑚𝑚 = ± 12
𝜈𝜈(1H)=500 MHz: 𝐵𝐵 = 2𝜋𝜋𝜋𝜋𝛾𝛾
= (2𝜋𝜋)(500∗106𝑠𝑠−1)2.6752∗108𝑇𝑇−1𝑠𝑠−1
= 11.74 𝑇𝑇 Earth magnetic field ~ 30-60 μT
𝑁𝑁𝑚𝑚𝑁𝑁
=𝑒𝑒−
𝐸𝐸𝑚𝑚𝑘𝑘𝐵𝐵𝑇𝑇
∑ 𝑒𝑒−𝐸𝐸𝑚𝑚𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚
=𝑒𝑒𝛾𝛾ℏ𝑚𝑚𝛾𝛾𝑘𝑘𝐵𝐵𝑇𝑇
∑ 𝑒𝑒𝛾𝛾ℏ𝑚𝑚𝛾𝛾𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚
≈1 + 𝛾𝛾ℏ𝑚𝑚𝐵𝐵
𝑘𝑘𝛾𝛾𝑇𝑇
∑ 1 + 𝛾𝛾ℏ𝑚𝑚𝐵𝐵𝑘𝑘𝛾𝛾𝑇𝑇𝑚𝑚
=1 + 𝛾𝛾ℏ𝑚𝑚𝐵𝐵
𝑘𝑘𝛾𝛾𝑇𝑇2
=12
±𝛾𝛾ℏ𝐵𝐵
4𝑘𝑘𝛾𝛾𝑇𝑇
Boltzmann constant 𝑘𝑘𝛾𝛾 = 1.381 ∗ 10−23𝐽𝐽𝐾𝐾−1
~2 ∗ 10−5 for 𝜈𝜈(1H)=500 MHz and 25℃
Taylor expansion
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1H at 500 MHz and 25℃
𝑚𝑚 = - ½: population≈ 0.49998 (49.998 %)
𝑚𝑚 = ½: population≈ 0.50002 (50.002 %)
• Population difference → sample has bulk magnetic dipolar moment along z
• Population difference is very small → the magnetic moment is very small → low sensitivity
• Population difference ≈ 𝛾𝛾ℏ𝛾𝛾 2𝑘𝑘𝐵𝐵𝑇𝑇
(proportional to 𝛾𝛾 and 𝐵𝐵)
x
y
z 𝐵𝐵0
𝐼𝐼𝑧𝑧 =ℏ2
𝐼𝐼𝑧𝑧 = −ℏ2
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Chemical shift • Electrons surrounding the nucleus move under the influence of the applied
magnetic field • This movement of electrons generates additional magnetic field that usually
opposes the externally applied magnetic field (“shielding”)
𝐵𝐵 = (1 − 𝜎𝜎) 𝐵𝐵0 actual magnetic field perceived at the nucleus
external magnetic field
nuclear shielding
𝜔𝜔 = 𝛾𝛾𝐵𝐵 = 𝛾𝛾(1 − 𝜎𝜎) 𝐵𝐵0
Larmor frequency (𝜔𝜔) of each nucleus is unique and very sensitive to local electronic environment • Covalent bonds • Conformation • 3D structure, in particular ring currents, charges, hydrogen bonding...
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Chemical shift (continued)
𝛿𝛿 =𝜈𝜈 − 𝜈𝜈0𝜈𝜈0
∗ 106
chemical shift units: ppm (parts per million)
Reference frequency = frequency of a reference compound: tetramethylsilane (TMS) in organic solvents trimethylsilylpropanesulfonate = 4,4-dimethyl-4-silapentane-1-sulfonate (DSS) in aqueous solutions
𝜔𝜔 = 𝛾𝛾(1 − 𝜎𝜎) 𝐵𝐵0
𝜈𝜈(1H)=500 MHz (11.74T): 1 ppm = 500 𝑀𝑀𝑀𝑀𝑧𝑧106
= 500 ∗106 𝑀𝑀𝑧𝑧106
= 500 𝐻𝐻𝐻𝐻
𝜈𝜈(1H)=600 MHz (14.09T): 1 ppm = 600 𝑀𝑀𝑀𝑀𝑧𝑧106
= 600 ∗106 𝑀𝑀𝑧𝑧106
= 600 𝐻𝐻𝐻𝐻
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J-coupling (scalar coupling) Interaction between nuclear spins mediated by electrons that form the bond(s) between the nuclei • occurs over 1-4 bonds • 3-bond J-coupling depends on conformation
(dihedral angle)
𝐸𝐸 = ℏ𝜔𝜔𝐼𝐼𝑚𝑚𝐼𝐼 + ℏ𝜔𝜔𝑆𝑆𝑚𝑚𝑆𝑆 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 𝑚𝑚𝐼𝐼𝑚𝑚𝑆𝑆 scalar coupling constant
𝑚𝑚𝑆𝑆= +12
: ∆𝐸𝐸𝐼𝐼= ℏ𝜔𝜔𝐼𝐼 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆12
= ℏ𝜔𝜔𝐼𝐼 + 𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆
𝑚𝑚𝑆𝑆= −12
: ∆𝐸𝐸𝐼𝐼= ℏ𝜔𝜔𝐼𝐼 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 −12
= ℏ𝜔𝜔𝐼𝐼 − 𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆
∆𝐸𝐸𝐼𝐼 depends on spin S: ±𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 = ± 12ℎ 𝐽𝐽𝐼𝐼𝑆𝑆
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J-coupling (scalar coupling)
𝐸𝐸 = ℏ𝜔𝜔𝐼𝐼𝑚𝑚𝐼𝐼 + ℏ𝜔𝜔𝑆𝑆𝑚𝑚𝑆𝑆 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 𝑚𝑚𝐼𝐼𝑚𝑚𝑆𝑆 scalar coupling constant
𝑚𝑚𝐼𝐼= +12
: ∆𝐸𝐸𝑆𝑆= ℏ𝜔𝜔𝑆𝑆 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆12
= ℏ𝜔𝜔𝑆𝑆 + 𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆
𝑚𝑚𝐼𝐼= −12
: ∆𝐸𝐸𝑆𝑆= ℏ𝜔𝜔𝑆𝑆 + 2𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 −12
= ℏ𝜔𝜔𝑆𝑆 − 𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆
∆𝐸𝐸𝑆𝑆 depends on spin I: ±𝜋𝜋ℏ 𝐽𝐽𝐼𝐼𝑆𝑆 Same splitting as for ∆𝐸𝐸𝐼𝐼
Does not depend on magnetic field!
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Rule, G.S. and Hitchens, T.K. Fundamentals of Protein NMR Spectroscopy, 2006, Springer
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Cavanagh, J., Fairbrother, W.J., Palmer, A.G.III, Rance, M., Skelton, N.J. Protein NMR Spectroscopy: Principles and Practice, 2nd edition, 2007, Academic Press
- (upfield) + (downfield)
H2O
NMR spectrum of ubiquitin (76 amino acid residues, MW=8,565 Da)
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Wishart DS, Bigam CG, Holm A, Hodges RS, Sykes BD. 1H, 13C and 15N random coil NMR chemical shifts of the common amino acids. I. Investigations of nearest-neighbor effects. J Biomol NMR. 1995 Jan;5(1):67-81.
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Wishart DS, Bigam CG, Holm A, Hodges RS, Sykes BD. 1H, 13C and 15N random coil NMR chemical shifts of the common amino acids. I. Investigations of nearest-neighbor effects. J Biomol NMR. 1995 Jan;5(1):67-81.
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Semi-classical model (Bloch equations) Magnetic dipole 𝝁𝝁 experiences torque in magnetic field:
𝝁𝝁 𝑡𝑡 × 𝐁𝐁 =𝑑𝑑𝑰𝑰 𝑡𝑡𝑑𝑑𝑡𝑡
=1𝛾𝛾𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
torque
𝝁𝝁 = 𝛾𝛾 𝑰𝑰
𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= 𝝁𝝁 𝑡𝑡 × 𝛾𝛾𝐁𝐁 = −𝛾𝛾𝑩𝑩 × 𝝁𝝁(𝑡𝑡) 𝝎𝝎
𝝁𝝁(𝑡𝑡) rotates (precesses) with angular velocity 𝝎𝝎 = −𝛾𝛾𝑩𝑩 Larmor frequency
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x
y
z 𝑩𝑩𝟎𝟎 𝜔𝜔
x
y
z 𝑩𝑩𝟎𝟎
𝜔𝜔
Rotation not observable
𝝁𝝁 𝝁𝝁
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Application of perpendicular (horizontal) magnetic field 𝑩𝑩𝟏𝟏(𝑡𝑡) that oscillates with angular velocity 𝜔𝜔 = −𝛾𝛾𝐵𝐵:
𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= −𝛾𝛾(𝑩𝑩 + 𝑩𝑩𝟏𝟏 𝑡𝑡 ) × 𝝁𝝁(𝑡𝑡)
Switch to a coordinate frame that rotates with angular velocity 𝜔𝜔 = −𝛾𝛾𝐵𝐵:
𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= −𝛾𝛾𝑩𝑩𝟏𝟏 × 𝝁𝝁(𝑡𝑡)
𝝁𝝁(𝑡𝑡) rotates with angular velocity 𝝎𝝎𝟏𝟏 = −𝛾𝛾𝑩𝑩1 around 𝑩𝑩1 in the rotating frame!
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x
y
z 𝑩𝑩𝟎𝟎
Rotating frame (angular velocity 𝝎𝝎 = −𝛾𝛾𝑩𝑩):
𝑩𝑩𝟏𝟏
𝜔𝜔1
𝑩𝑩𝟏𝟏 along x: Rotation in the y-z plane with angular velocity 𝝎𝝎𝟏𝟏 𝛼𝛼 = 𝜔𝜔1 𝑡𝑡
𝛼𝛼
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
90° pulse: 𝛼𝛼 = 𝜔𝜔1𝑡𝑡 =𝜋𝜋2
𝑀𝑀𝑧𝑧 = 𝑀𝑀0 → −𝑀𝑀𝑦𝑦 = 𝑀𝑀0
𝛼𝛼 𝝁𝝁 𝝁𝝁
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Rotating frame (angular velocity 𝝎𝝎 = −𝛾𝛾𝑩𝑩):
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
180° pulse: 𝛼𝛼 = 𝜔𝜔1𝑡𝑡 = 𝜋𝜋 𝑀𝑀𝑧𝑧 = 𝑀𝑀0 → −𝑀𝑀𝑧𝑧 = 𝑀𝑀0
𝛼𝛼 𝝁𝝁
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Rotating frame (angular velocity 𝝎𝝎 = −𝛾𝛾𝑩𝑩):
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
𝛼𝛼
𝑩𝑩𝟏𝟏 along y: Rotation in the x-z plane with angular velocity 𝝎𝝎𝟏𝟏 𝛼𝛼 = 𝜔𝜔1 𝑡𝑡
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝜔𝜔1
𝛼𝛼
90° pulse: 𝛼𝛼 = 𝜔𝜔1𝑡𝑡 =𝜋𝜋2
𝑀𝑀𝑧𝑧 = 𝑀𝑀0 → 𝑀𝑀𝑥𝑥 = 𝑀𝑀0
𝝁𝝁 𝝁𝝁
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When the angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓 of the perpendicular magnetic field 𝑩𝑩𝟏𝟏(𝑡𝑡) differs (somewhat) from Larmor frequency 𝝎𝝎 = −𝛾𝛾𝑩𝑩: Pulses still work as long as |𝛾𝛾𝑩𝑩𝟏𝟏| ≫|𝝎𝝎−𝝎𝝎𝒓𝒓𝒓𝒓|
Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓): 𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= −𝛾𝛾 𝑩𝑩 + 𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡 − 𝝎𝝎𝒓𝒓𝒓𝒓 × 𝝁𝝁(𝑡𝑡)
= −𝛾𝛾𝑩𝑩 −𝝎𝝎𝒓𝒓𝒓𝒓 − 𝛾𝛾𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡 𝝎𝝎 𝝎𝝎𝟏𝟏
Rotation around 𝑩𝑩 with Larmor frequency
Rotation around 𝐵𝐵1 (stationary in the rotating frame)
Correction for rotation of the rotating frame
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Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓):
Rotation around a tilted axis 𝝎𝝎𝒆𝒆𝒓𝒓𝒓𝒓 = 𝛀𝛀 + 𝝎𝝎𝟏𝟏
𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= 𝝎𝝎−𝝎𝝎𝒓𝒓𝒓𝒓 + 𝝎𝝎𝟏𝟏 × 𝝁𝝁 𝑡𝑡
𝜴𝜴 (offset)
x
y
z 𝑩𝑩𝟎𝟎
𝝎𝝎𝟏𝟏
𝝎𝝎𝒆𝒆𝒓𝒓𝒓𝒓 = 𝛀𝛀 + 𝝎𝝎𝟏𝟏
𝜴𝜴
𝝁𝝁
When |𝝎𝝎𝟏𝟏| ≫|𝛀𝛀| 𝝎𝝎𝒆𝒆𝒓𝒓𝒓𝒓 ≈ 𝝎𝝎𝟏𝟏 Rotation around axis in a horizontal plane
![Page 24: Principles of NMR spectroscopy](https://reader036.fdocuments.us/reader036/viewer/2022081623/6156f84da097e25c764f8951/html5/thumbnails/24.jpg)
Another way to look at it: Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓): 𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡
= −𝛾𝛾 𝑩𝑩 + 𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡 − 𝝎𝝎𝒓𝒓𝒓𝒓 × 𝝁𝝁(𝑡𝑡)
= −𝛾𝛾 𝑩𝑩 +𝝎𝝎𝒓𝒓𝒓𝒓
𝛾𝛾 + 𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡
∆𝑩𝑩 (residual vertical magnetic field in the rotating frame)
x
y
z 𝑩𝑩𝟎𝟎
𝑩𝑩𝟏𝟏
𝑩𝑩𝒆𝒆𝒓𝒓𝒓𝒓 = ∆𝑩𝑩 + 𝑩𝑩𝟏𝟏
∆𝑩𝑩
𝝁𝝁
Rotation around 𝑩𝑩𝒆𝒆𝒓𝒓𝒓𝒓 with angular velocity 𝝎𝝎𝒆𝒆𝒓𝒓𝒓𝒓 = −𝛾𝛾𝑩𝑩𝒆𝒆𝒓𝒓𝒓𝒓