basic principles of nmr - freethcgerm.free.fr/img/pdf/principedebase_dmarion.pdfbasic principles of...
TRANSCRIPT
![Page 1: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/1.jpg)
Basic principles of NMR
Dominique MarionInstitut de Biologie
StructuraleJean-Pierre Ebel
CNRS - CEA - UJFGrenoble
PowerPoint 2004
for MacOS
![Page 2: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/2.jpg)
Summary of the lecture
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Summary of the lecture Bloch vector model
Basic quantum mechanics
Product operator formalism
Spin hamiltonian
NMR building blocks
Coherence selection - phase cycling
Pulsed field gradients
![Page 4: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/4.jpg)
Nuclei observable by NMR
![Page 5: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/5.jpg)
Why some nuclei have no spin ?The proton is composed of 3 quarks stuck together by gluons
12C 13C 14NAtomic number 6 6 7Mass number 6+6 6+7 7+7Spin quantum number
0 1/2 1
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Why some nuclei have no spin ?
Isotopes with odd mass number
S = 1/2, 3/2 …
Isotopes with even mass number
Number of protons and neutron even
S = 0
Number of protons and neutron odd
S=1, 2, 3 …
(1H, 13C, 15N, 19F, 31P)
![Page 7: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/7.jpg)
Larmor frequency
Laboratory reference frame
Rotating reference frameat frequency ω
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Bloch equations without relaxation
B0 static magnetic fieldM macroscopic magnetization∧Cross-productB1 r.f. magnetic field
![Page 9: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/9.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Thermal motionPrecession in a fluctuating magnetic field
Non isotropic motion
Magnetization ⇒ Thermal equilibrium⇒ Fluctuating magnetic field
Spin-lattice relaxation T1
![Page 10: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/10.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 11: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/11.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Spin-spin relaxation T2
Precession in the transverse plane
The individual magnetic dipolesall have slightly differentprecession frequencies
True T2 relaxation
B0 inhomogeneity
![Page 12: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/12.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 13: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/13.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 14: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/14.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Subtitution
Incorporation of T1 and T2 relaxation times
![Page 15: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/15.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 16: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/16.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 17: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/17.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Longitudinal and transverse relaxation mechanisms areindependent
![Page 18: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/18.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
![Page 19: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/19.jpg)
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
rf pulses connectthe z axis with the transverse xy plane
![Page 20: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/20.jpg)
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetization
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Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1
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Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1
![Page 23: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/23.jpg)
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1Transverse magnetization
Coherence
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Bloch equations with relaxation
What are the limitations of the Bloch equations?
![Page 25: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/25.jpg)
Bloch equations with relaxation
What are the limitations of the Bloch equations?
Planes : no collision
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Bloch equations with relaxation
What are the limitations of the Bloch equations?
Planes : no collision Cars : collision
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The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
![Page 28: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/28.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
![Page 29: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/29.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
Vector Transformation
![Page 30: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/30.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
![Page 31: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/31.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, N
number of spins
![Page 32: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/32.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, N
number of spinsAdditional terms if I and S interact
![Page 33: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/33.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, N
![Page 34: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/34.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, NVector16 terms
![Page 35: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/35.jpg)
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, NTransformation
16x16 terms
Vector16 terms
![Page 36: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/36.jpg)
Basic Quantum Mechanics
Operator Performs some operation on a function
Ex: Dx derivative operator
Ex: 1 unity operator 1f(x) = f(x)
Commutation The effect of consecutive operationsmay depends on their order
Drive straight for 50 m Drive straight for 100 m
Turn left Turn left
Drive straight for 100 m Drive straight for 50 m
B{A( f(x) )} A{B( f(x) )} =?
[A,B] = AB - BA
Commutator
![Page 37: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/37.jpg)
Basic Quantum Mechanics
Matrix representation of operators
!! The matrix representation depend on the basis
Product of two operators A.B
Usual law for matrix multiplication
Inverse
AB = AB = 1
A = B–1
Hermitian operator
A = A†Adjoint
Aij = Bji*
A = B† Unitary operator
A–1 = A†
![Page 38: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/38.jpg)
Basic Quantum Mechanics
Eigenvalues
Change of basis Diagonal matrix
A |νi> = λi |νi>
Operator Eigenvalue( complex number)Eigenvector
![Page 39: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/39.jpg)
Basic Quantum Mechanics
Eigenvalues
Change of basis Diagonal matrix
A |νi> = λi |νi>
Operator Eigenvalue( complex number)Eigenvector
Hermitian operator
A = A†
Realeigenvalues
Orthogonaleigenvectors
![Page 40: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/40.jpg)
Basic Quantum Mechanics
Eigenvalues
Change of basis Diagonal matrix
A |νi> = λi |νi>
Operator Eigenvalue( complex number)Eigenvector
Hermitian operator
A = A†
Realeigenvalues
Orthogonaleigenvectors
If [A,B] = 0
i.e. A and B commute
∃ Basis such that
A and B diagonal
![Page 41: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/41.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
![Page 42: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/42.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
As [A,A]=0 A |νi> = λi |νi> An |νi> = λin |νi>
All power of an operator have the same eigenvector
![Page 43: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/43.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
![Page 44: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/44.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
Exponential of operators
For ordinary numbers
For operators
exp(A+B) = exp(A) . exp(B) only if [A,B]=0!
![Page 45: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/45.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
Exponential of operators
For ordinary numbers
For operators
![Page 46: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/46.jpg)
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
Exponential of operators
For ordinary numbers
For operators
Complex exponential of operators
For operators
A hermitian E unitaryA = A† E–1 = E†
![Page 47: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/47.jpg)
Basic Quantum Mechanics
Cyclic commutation
[A, B] = iC Definition [B, C] = iA [C, A] = iB
B
A
C
Sandwich formula
exp (-iθA) B exp (iθA) = B cos θ + C sin θ
Rotation angle
Cyclic permutation
![Page 48: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/48.jpg)
Basic Quantum Mechanics
Cyclic commutation
BA
C
Rotation around the 3 axes
exp (-iθA) B exp (iθA) = B cos θ + C sin θ
exp (-iθB) C exp (iθB) = C cos θ + A sin θ exp (-iθC) A exp (iθC) = A cos θ + B sin θ
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Liouville-von Neumann equation
Classical description
MagnetizationMagnetic field
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Liouville-von Neumann equation
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
![Page 51: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/51.jpg)
Liouville-von Neumann equation
|α>
|β>E
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
![Page 52: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/52.jpg)
Liouville-von Neumann equation
|α>
|β>E
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
|ψ> = cα |α> + cβ |β>
Single1/2 spin particle
Superposition state
Quantum indeterminacy
![Page 53: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/53.jpg)
Liouville-von Neumann equation
|α>
|β>E
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
|ψ> = cα |α> + cβ |β>
Single1/2 spin particle
Superposition state
Quantum indeterminacy
Ensemble of1/2 spin particles
Density matrix
Ensemble average
![Page 54: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/54.jpg)
Liouville-von Neumann equation
Quantum description
Density matrix Hamiltonian
![Page 55: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/55.jpg)
Liouville-von Neumann equation
Quantum description
Density matrix Hamiltonian
Hamiltonian:Time-independent part
Static magnetic field B0
Time-dependent part
Radiofrequency field B1 (pulses)
Scalar coupling
![Page 56: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/56.jpg)
Liouville-von Neumann equation
Quantum description
Density matrix Hamiltonian
Hamiltonian:Time-independent part
Static magnetic field B0
Time-dependent part
Radiofrequency field B1 (pulses)
Scalar coupling
Transformation that render the pulse Hamiltonian time-independent ?
Rotating frame
σr = U σ U-1
![Page 57: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/57.jpg)
Rotating frame
Rotating frame
σr = U σ U-1
![Page 58: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/58.jpg)
Summary of the lecture Bloch vector model
Basic quantum mechanics
Product operator formalism
Spin hamiltonian
NMR building blocks
Coherence selection - phase cycling
Pulsed field gradients
![Page 59: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/59.jpg)
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
![Page 60: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/60.jpg)
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
![Page 61: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/61.jpg)
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
[Ix,Iy] = i IzThe spin operators satisfy the commutation relation
![Page 62: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/62.jpg)
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
[Ix,Iy] = i IzThe spin operators satisfy the commutation relation
![Page 63: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/63.jpg)
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
[Ix,Iy] = i IzThe spin operators satisfy the commutation relation
![Page 64: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/64.jpg)
Matrix representation of the spin operators
![Page 65: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/65.jpg)
Matrix representation of the spin operators
The transverse coherence has a phase !
![Page 66: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/66.jpg)
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets
![Page 67: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/67.jpg)
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets Operator(square matrix)
Before
After
![Page 68: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/68.jpg)
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets
Bra ← adjoint → Ket
<n| = { |n> } †
Operator(square matrix)
Before
After
![Page 69: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/69.jpg)
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets
Bra ← adjoint → Ket
<n| = { |n> } †
Operator(square matrix)
Before
After
Orthonormal basis
![Page 70: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/70.jpg)
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets
Bra ← adjoint → Ket
<n| = { |n> } †
Matrix representation using differentbasis sets can be interconvertedusing unitary transformation
Operator(square matrix)
Before
After
Orthonormal basis
![Page 71: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/71.jpg)
Multispin systemsBloch model Strictly applicable only to a
system of non-interacting spins
Quantum mechanics Direct product space
Spins 1 2 3
Basis size 2 4 8
Nb of basis vectors = 2N
The two spins are independent
|ψ> = |ψ1> ⊗ |ψ2>
basis vectorfor spin #1
basis vectorfor spin #2
![Page 72: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/72.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 73: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/73.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Incorrect !
Operators
![Page 74: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/74.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 75: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/75.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 76: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/76.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 77: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/77.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 78: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/78.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 79: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/79.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 80: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/80.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
![Page 81: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/81.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Incorrect !
Dimension
2×2
4×4
Operators
![Page 82: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/82.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators AB|ij> = (A⊗B)(|i> ⊗ |j> ) = A |i> ⊗B|j>
![Page 83: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/83.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators AB|ij> = (A⊗B)(|i> ⊗ |j> ) = A |i> ⊗B|j>
Productoperator
A is an operator that acts on the i spin
B is an operator that acts on the j spin
AB= (A⊗B) = (A⊗E) (E⊗B)
![Page 84: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/84.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators AB|ij> = (A⊗B)(|i> ⊗ |j> ) = A |i> ⊗B|j>
Iz|αβ> = (Iz ⊗E)(|α> ⊗ |β> ) = Iz |α> ⊗E|β> = 1/2 |α> ⊗ |β> = 1/2 | αβ >
Ex:
Iz|αβ> = 1/2 |αβ>
Productoperator
A is an operator that acts on the i spin
B is an operator that acts on the j spin
AB= (A⊗B) = (A⊗E) (E⊗B)
![Page 85: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/85.jpg)
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators AB|ij> = (A⊗B)(|i> ⊗ |j> ) = A |i> ⊗B|j>
Iz|αβ> = (Iz ⊗E)(|α> ⊗ |β> ) = Iz |α> ⊗E|β> = 1/2 |α> ⊗ |β> = 1/2 | αβ >
Ex:
Iz|αβ> = 1/2 |αβ>
Iz Sz| αβ > = (Iz ⊗ Sz)(|α> ⊗ |β> ) = Iz |α> ⊗ Sz |β > = 1/2 |α> ⊗ –1/2 |β> = –1/4 | αβ >
Iz Sz| αβ > = –1/4 | αβ >
Productoperator
A is an operator that acts on the i spin
B is an operator that acts on the j spin
AB= (A⊗B) = (A⊗E) (E⊗B)
![Page 86: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/86.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
![Page 87: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/87.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Thermal equilibrium populations
![Page 88: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/88.jpg)
Product operators - coherence /population
Populations
Az XzAzXz
![Page 89: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/89.jpg)
Product operators - coherence /population
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
±1 Quantum coherence
Ax Xx
Ay Xy
![Page 90: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/90.jpg)
Product operators - coherence /population
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
0 / 2 Quantum coherence
AxXy
AyXx
AxXx
AyXy
![Page 91: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/91.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
|αα>
|αβ>
|ββ>
|βα>
![Page 92: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/92.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
![Page 93: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/93.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
![Page 94: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/94.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
![Page 95: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/95.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of ASpectrum of X
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
![Page 96: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/96.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of ASpectrum of X
X(β) X(α)A(β) A(α)
|αα>
|αβ>
|ββ>
|βα>
![Page 97: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/97.jpg)
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of ASpectrum of X
X(β) X(α)A(β) A(α)
|αα>
|αβ>
|ββ>
|βα>
![Page 98: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/98.jpg)
Multispin systems - product operators
![Page 99: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/99.jpg)
Multispin systems - product operators
Spectrum of A
![Page 100: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/100.jpg)
Multispin systems - product operators
Spectrum of A
In-phase coherence of A along y
![Page 101: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/101.jpg)
Multispin systems - product operators
Spectrum of A
In-phase coherence of A along y
Anti-phase coherence of A along y
![Page 102: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/102.jpg)
Multispin systems - product operators
Spectrum of A
In-phase coherence of A along y
Anti-phase coherence of A along y
with respect to X
![Page 103: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/103.jpg)
Multispin systems - product operators
Spectrum of A
![Page 104: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/104.jpg)
Multispin systems - product operators
Spectrum of A
![Page 105: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/105.jpg)
Multispin systems - product operators
Spectrum of A
![Page 106: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/106.jpg)
Multispin systems - product operators
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
![Page 107: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/107.jpg)
Multispin systems - product operators
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
![Page 108: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/108.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
Quantum description
Density matrix Hamiltonian
![Page 109: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/109.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
Quantum description
Density matrix Hamiltonian
![Page 110: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/110.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy
Quantum description
Density matrix Hamiltonian
![Page 111: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/111.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Quantum description
Density matrix Hamiltonian
![Page 112: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/112.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Quantum description
Density matrix Hamiltonian
![Page 113: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/113.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Rule 2:
Quantum description
Density matrix Hamiltonian
![Page 114: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/114.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Rule 2:
[Iy,Ix] = – i Iz
Quantum description
Density matrix Hamiltonian
![Page 115: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/115.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
Quantum description
Density matrix Hamiltonian
![Page 116: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/116.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
[Iz,Ix] = i Iy [Sx,Sy] = i Sz
[Sy,Sz] = i Sx
[Sz,Sx] = i Sy
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Quantum description
Density matrix Hamiltonian
![Page 117: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/117.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
![Page 118: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/118.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
![Page 119: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/119.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
![Page 120: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/120.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
![Page 121: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/121.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
Commuting operators
![Page 122: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/122.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
![Page 123: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/123.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
[Ip Sq , Ir] = Ip Ir Sq – Ir Ip Sq
![Page 124: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/124.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
[Ip Sq , Ir] = Ip Ir Sq – Ir Ip Sq
Rule 5:
[Ip Sq , Ir Ss ] =
![Page 125: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/125.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
[Ip Sq , Ir] = Ip Ir Sq – Ir Ip Sq
Rule 5:
[Ip Sq , Ir Ss ] =
0if p≠r and q≠s
![Page 126: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/126.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
[Ip Sq , Ir] = Ip Ir Sq – Ir Ip Sq
Rule 5:
[Ip Sq , Ir Ss ] =
0if p≠r and q≠s
1/4[Sq , Ss ]if p=r
![Page 127: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/127.jpg)
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
[Ip Sq , Ir] = Ip Sq Ir – Ir Ip Sq
[Ip Sq , Ir] = Ip Ir Sq – Ir Ip Sq
Rule 5:
[Ip Sq , Ir Ss ] =
0if p≠r and q≠s
1/4[Sq , Ss ]if p=r
1/4[Ip , Ir]if q=s
![Page 128: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/128.jpg)
Commutation in coherence space (summary)
[Ix,Iy] = i Iz
Rule 1:
Rule 2:
[Iy,Ix] = – i Iz
Rule 3:
[Ip,Iq] = 0 for (p,q) = (x,y,z)
Rule 4:
[Ip Sq , Ir] = [Ip , Ir] Sq
Rule 5:
[Ip Sq , Ir Ss ] =
0if p≠r and q≠s
1/4[Sq , Ss ]if p=r
1/4[Ip , Ir]if q=s
![Page 129: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/129.jpg)
Operator product
![Page 130: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/130.jpg)
Operator product
Any operator commuteswith itself
![Page 131: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/131.jpg)
Operator product
![Page 132: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/132.jpg)
Operator product
[Iz,Ix] ≠ 0They do not
commute
![Page 133: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/133.jpg)
Operator product
![Page 134: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/134.jpg)
Operator product
Any operator of Icommutes with
any operator of S
![Page 135: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/135.jpg)
Operator product
![Page 136: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/136.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
![Page 137: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/137.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
![Page 138: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/138.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
Shielding tensor
(fast tumbling in liquid)
![Page 139: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/139.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
![Page 140: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/140.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
![Page 141: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/141.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction ( J )
H = J I . S = J (IxSx + IySy+ IzSz)
![Page 142: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/142.jpg)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction ( J )
H = J I . S = J (IxSx + IySy+ IzSz)
Dipolar interaction ( D )
→ 0 in isotropic liquids
![Page 143: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/143.jpg)
Terms of the spin hamiltonian (conflicts)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
![Page 144: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/144.jpg)
Terms of the spin hamiltonian (conflicts)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
[Iz,Ix] ≠ 0 [Iz,Iy] ≠ 0
![Page 145: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/145.jpg)
Terms of the spin hamiltonian (conflicts)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
![Page 146: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/146.jpg)
Terms of the spin hamiltonian (conflicts)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
[Iz,IxSx] ≠ 0 [Iz,IySy] ≠ 0
![Page 147: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/147.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
During the pulses
![Page 148: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/148.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis: short pulseThe spins do not precessduring the pulse
During the pulses
![Page 149: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/149.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis: short pulseThe spins do not precessduring the pulse
During the pulses
![Page 150: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/150.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis: short pulseThe spins do not precessduring the pulse
During the pulses
Trajectories of magnetizations
RF field strength = 1000 Hz
Offsets = 100, 250, 500 Hz
![Page 151: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/151.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
During the freeprecession
![Page 152: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/152.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (1) : weak coupling
JIS << | ωI - ωS|
During the freeprecession
![Page 153: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/153.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (1) : weak coupling
JIS << | ωI - ωS|
During the freeprecession
![Page 154: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/154.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (1) : weak coupling
JIS << | ωI - ωS|
During the freeprecession
H = JIS IzSz
![Page 155: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/155.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
During the freeprecession
![Page 156: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/156.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (2) : the chemical shift evolution is eliminated
During the freeprecession
![Page 157: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/157.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (2) : the chemical shift evolution is eliminated
During the freeprecession
![Page 158: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/158.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (2) : the chemical shift evolution is eliminated
During the freeprecession
[IzSz,IxSx] = 0
[IxSx,IySy] = 0
[IzSz,IySy] = 0
![Page 159: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/159.jpg)
Terms of the spin hamiltonian (solutions)
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (ωt) - Iy sin(ωt)]
Scalar interaction
H = J I . S = J (IxSx + IySy+ IzSz)
Hypothesis (2) : the chemical shift evolution is eliminated
During the freeprecession
[IzSz,IxSx] = 0
[IxSx,IySy] = 0
[IzSz,IySy] = 0
Isotropicmixing
![Page 160: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/160.jpg)
Evolution of the spin system
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (φ) - Iy sin(φ)]
Scalar interaction
H = JIS IzSz
exp (-iθH) σ0 exp (iθH) = σ0 cos θ + σ1 sin θ
[σ0 , H] = i σ1
![Page 161: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/161.jpg)
Evolution of the spin system
Zeeman interaction
H = – ω0 Iz
RF field
H = – ω1[ Ix cos (φ) - Iy sin(φ)]
Scalar interaction
H = JIS IzSz
exp (-iθH) σ0 exp (iθH) = σ0 cos θ + σ1 sin θ
Quantum description
Density matrix Hamiltonian
[σ0 , H] = i σ1
![Page 162: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/162.jpg)
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
![Page 163: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/163.jpg)
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
![Page 164: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/164.jpg)
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz Ix
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
![Page 165: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/165.jpg)
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz Ix Ix cos ω0t
Iy sin ω0t
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
![Page 166: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/166.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)
![Page 167: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/167.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)Phase of the rf
![Page 168: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/168.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)
H = – ω1 Ix
Pulse around x
Phase of the rf
![Page 169: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/169.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)
H = – ω1 Ix
Pulse around x
H = – ω1 Iy
Pulse around y
Phase of the rf
![Page 170: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/170.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
![Page 171: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/171.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz
![Page 172: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/172.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz Iz cos ω1t
![Page 173: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/173.jpg)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz Iz cos ω1t
– Iy sin ω1t
![Page 174: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/174.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
![Page 175: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/175.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
![Page 176: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/176.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
![Page 177: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/177.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
![Page 178: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/178.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
Ix
![Page 179: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/179.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
Ix Ix cos πJt
![Page 180: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/180.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
Ix Ix cos πJt
2IySz sin πJt
![Page 181: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/181.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
Ix Ix cos πJt
2IySz sin πJt
J
Definition of J(chemistry)
![Page 182: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/182.jpg)
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
x y
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
Ix Ix cos πJt
2IySz sin πJt
J
Definition of J(chemistry)
J
![Page 183: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/183.jpg)
Summary of the lecture Bloch vector model
Basic quantum mechanics
Product operator formalism
Spin hamiltonian
NMR building blocks
Coherence selection - phase cycling
Pulsed field gradients
![Page 184: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/184.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
Δ Δ
![Page 185: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/185.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
Δ Δ
Echo
![Page 186: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/186.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 187: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/187.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 188: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/188.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 189: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/189.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
Δ Δ
![Page 190: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/190.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
Δ Δ
Echo
![Page 191: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/191.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 192: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/192.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 193: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/193.jpg)
NMR building blocks (1)Spin echoes in heteronuclear spin systems
1H
X
1H-15N
1J ≈ 70 Hz
1H-13C
1J ≈ 120-140 Hz
180º pulses
Δ Δ
Echo
![Page 194: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/194.jpg)
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Chemical shift
![Page 195: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/195.jpg)
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Ix cos ω0Δ
–Iy sin ω0Δ
Chemical shift
![Page 196: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/196.jpg)
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Ix cos ω0Δ
–Iy sin ω0Δ
Ix cos ω0Δ cos ω0Δ
–Iy sin ω0Δ cos ω0Δ
Iy cos ω0Δ sin ω0Δ
+Ix sin ω0Δ sin ω0Δ
Chemical shift
![Page 197: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/197.jpg)
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Ix cos ω0Δ
–Iy sin ω0Δ
Ix cos ω0Δ cos ω0Δ
–Iy sin ω0Δ cos ω0Δ
Iy cos ω0Δ sin ω0Δ
+Ix sin ω0Δ sin ω0Δ
Chemical shift
![Page 198: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/198.jpg)
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Ix cos ω0Δ
–Iy sin ω0Δ
Ix cos ω0Δ cos ω0Δ
–Iy sin ω0Δ cos ω0Δ
Iy cos ω0Δ sin ω0Δ
+Ix sin ω0Δ sin ω0Δ
Chemical shift
Ix
![Page 199: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/199.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
![Page 200: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/200.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
2(–Iy)(–Sz) sin πJt
![Page 201: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/201.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
![Page 202: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/202.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
2IySz sin πJt cos πJt
–Ix sin πJt sin πJt
![Page 203: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/203.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
2IySz sin πJt cos πJt
–Ix sin πJt sin πJt
Ix cos πJt cos πJt – sin πJt sin πJt
2IySz sin πJt cos πJt + cos πJt sin πJt
![Page 204: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/204.jpg)
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
2IySz sin πJt cos πJt
–Ix sin πJt sin πJt
Ix cos πJt cos πJt – sin πJt sin πJt
2IySz sin πJt cos πJt + cos πJt sin πJt
Ix cos 2πJt
2IySz sin 2πJt
![Page 205: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/205.jpg)
NMR building blocks (4)Spin echoes in homonuclear spin systems
Δ Δ
Chemical shift
Ix Ix
J-coupling
Ix cos 2πJt
2IySz sin 2πJt
Ix
![Page 206: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/206.jpg)
NMR building blocks (4)Spin echoes in homonuclear spin systems
Δ Δ
Chemical shift
Ix Ix
J-coupling
Ix cos 2πJt
2IySz sin 2πJt
Ix
![Page 207: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/207.jpg)
NMR building blocks (5)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
![Page 208: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/208.jpg)
NMR building blocks (5)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
Δ Δ
![Page 209: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/209.jpg)
NMR building blocks (5)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
Δ Δ
These two cases are not possiblefor homonuclear spin-systems(band selective pulses!)
![Page 210: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/210.jpg)
NMR building blocks (6)Spin echoes in heteronuclear spin systems
Ix Ix cos 2ω0(2Δ)
Iy sin ω0(2Δ)
X Chemical shift
1H
X
Δ Δ
![Page 211: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/211.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
![Page 212: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/212.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
Ix cos πJt
–2IySz sin πJt
![Page 213: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/213.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
Ix cos πJt
–2IySz sin πJt
Ix cos πJt
–2IySz sin πJt
![Page 214: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/214.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
Ix cos πJt
–2IySz sin πJt
Ix cos πJt
–2IySz sin πJt
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
Ix sin πJt sin πJt
–2IySz sin πJt cos πJt
![Page 215: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/215.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
Ix cos πJt
–2IySz sin πJt
Ix cos πJt
–2IySz sin πJt
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
Ix sin πJt sin πJt
–2IySz sin πJt cos πJt
![Page 216: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/216.jpg)
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
Ix cos πJt
–2IySz sin πJt
Ix cos πJt
–2IySz sin πJt
Ix cos πJt cos πJt
2IySz cos πJt sin πJt
Ix sin πJt sin πJt
–2IySz sin πJt cos πJtIx Ix
![Page 217: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/217.jpg)
NMR building blocks (8)Spin echoes in heteronuclear spin systems
Ix Ix
J-coupling
1H
X
Δ Δ
Chemical shift
Ix Ix cos 2ω0(2Δ)
Iy sin ω0(2Δ)
![Page 218: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/218.jpg)
NMR building blocks (8)Spin echoes in heteronuclear spin systems
Ix Ix
J-coupling
1H
X
Δ Δ
Chemical shift
Ix Ix cos 2ω0(2Δ)
Iy sin ω0(2Δ)
![Page 219: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/219.jpg)
NMR building blocks (9)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
Chemical shift
Ix Ix cos 2ω0(2Δ)
Iy sin ω0(2Δ)
J-coupling
Ix cos 2πJt
2IySz sin 2πJt
Ix
Δ Δ
![Page 220: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/220.jpg)
Coherence selection (1)
Pulse sequences with three 90º pulses
![Page 221: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/221.jpg)
Coherence selection (1)
t1t2
DQF COSY
Pulse sequences with three 90º pulses
![Page 222: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/222.jpg)
Coherence selection (1)
t1t2
τ t1t2τ
DQF COSY
Double quantumspectroscopy
Pulse sequences with three 90º pulses
![Page 223: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/223.jpg)
Coherence selection (1)
t1t2
τ t1t2τ
t1 t2τ
DQF COSY
Double quantumspectroscopy
NOESY
Pulse sequences with three 90º pulses
![Page 224: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/224.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
![Page 225: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/225.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
![Page 226: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/226.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
![Page 227: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/227.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
![Page 228: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/228.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
![Page 229: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/229.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
![Page 230: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/230.jpg)
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
φ0×φ
1×φ
![Page 231: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/231.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φ
![Page 232: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/232.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
![Page 233: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/233.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
![Page 234: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/234.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
Rising / lowering operators
I+ Ix Ix+ i=
I– Ix Ix– i=
![Page 235: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/235.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
![Page 236: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/236.jpg)
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
I+ I+φ
exp(–i φ)
![Page 237: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/237.jpg)
Coherence selection (4)
Coherence order
z
yx
Mag
netic
fiel
d
φ
![Page 238: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/238.jpg)
Coherence selection (4)
Coherence order
0 / 2 Quantum coherences
AxXy
AyXx
AxXx
AyXy
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
z
yx
Mag
netic
fiel
d
φ
![Page 239: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/239.jpg)
Coherence selection (4)
Coherence order
0 / 2 Quantum coherences
AxXy
AyXx
AxXx
AyXy
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>2IxSx
I–S–I+S+ I–S+I+S–
=
+ + + )1/2 (
Order: 2 2 0 0
z
yx
Mag
netic
fiel
d
φ
![Page 240: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/240.jpg)
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
![Page 241: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/241.jpg)
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
![Page 242: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/242.jpg)
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
τ t1t2τ
Double quantumspectroscopy
![Page 243: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/243.jpg)
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
![Page 244: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/244.jpg)
Coherence selection (6)
Phase cycling
![Page 245: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/245.jpg)
Coherence selection (6)
Phase cycling
φ → φ+Δφ
![Page 246: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/246.jpg)
Coherence selection (6)
Phase cycling
φ → φ+Δφ
Δp
![Page 247: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/247.jpg)
Coherence selection (6)
Phase cycling
φ → φ+Δφ
Δp
Coherence phase shift:
Δp × Δφ
![Page 248: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/248.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
![Page 249: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/249.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –3
![Page 250: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/250.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360º
![Page 251: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/251.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360º
![Page 252: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/252.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
![Page 253: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/253.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
![Page 254: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/254.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
![Page 255: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/255.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
![Page 256: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/256.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
1
2
3
4
0º
90º
180º
270º
0º
180º
360º
540º
0º
180º
0º
180º
Step Δφ 2 × Δφmod 360º
–2
![Page 257: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/257.jpg)
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –31
2
3
4
0º
90º
180º
270º
0º
270º
540º
810º
0º
270º
180º
90º
Step Δφ 3 × Δφmod 360ºrecv
phase
1
2
3
4
0º
90º
180º
270º
0º
180º
360º
540º
0º
180º
0º
180º
Step Δφ 2 × Δφmod 360º
–2
![Page 258: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/258.jpg)
Pulsed field gradients (1)
B0eff
Homogeneous
magnetic field(well shimmed magnet)
![Page 259: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/259.jpg)
Pulsed field gradients (1)
B0eff
B0eff
Homogeneous
magnetic field(well shimmed magnet)
Inhomogeneous magnetic field(field gradient)
![Page 260: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/260.jpg)
Pulsed field gradients (1)
B0eff
B0eff
Homogeneous
magnetic field(well shimmed magnet)
Inhomogeneous magnetic field(field gradient)
I+ I+ω0 exp(–i ω0t)
I+
I+
ω0 + γ G(z)
exp(–i [ ω0 + γ G(z)]t)
![Page 261: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/261.jpg)
Pulsed field gradients (1)
B0eff
B0eff
Homogeneous
magnetic field(well shimmed magnet)
Inhomogeneous magnetic field(field gradient)
I+ I+ω0 exp(–i ω0t)
I+
I+
ω0 + γ G(z)
exp(–i [ ω0 + γ G(z)]t)
![Page 262: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/262.jpg)
Pulsed field gradients (2)
I+
I+
γΙ G(z)
exp(–i γΙ G(z)t)
![Page 263: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/263.jpg)
Pulsed field gradients (2)
I+
I+
γΙ G(z)
exp(–i γΙ G(z)t)
I+S+ (γΙ +γS )G(z)
exp(–i (pI γΙ +pS γS ) G(z) t)I+S+
pI coherence order
associated with spin I
![Page 264: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/264.jpg)
Pulsed field gradients (3)
rf
pfg g1 g2
τ1 τ2
p1
p2
![Page 265: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/265.jpg)
Pulsed field gradients (3)
rf
pfg g1 g2
τ1 τ2
p1
p2
Refocusing condition
g1τ1
g2τ2
–p1
–p2
=
![Page 266: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/266.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 267: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/267.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 268: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/268.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 269: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/269.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 270: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/270.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 271: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/271.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 272: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/272.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 273: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/273.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
rf
pfg g gτ τ
+p
–p
![Page 274: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/274.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
![Page 275: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/275.jpg)
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
I
pfg+g
–gτ
τ
S
pI
pS
![Page 276: Basic principles of NMR - Freethcgerm.free.fr/IMG/pdf/PrincipeDeBAse_DMarion.pdfBasic principles of NMR Dominique Marion Institut de Biologie Structurale Jean-Pierre Ebel CNRS - CEA](https://reader034.vdocument.in/reader034/viewer/2022042606/5f9f872e7586f82b50171a34/html5/thumbnails/276.jpg)
The end…