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Basic principles of NMR
Dominique MarionInstitut de Biologie
StructuraleJean-Pierre Ebel
CNRS - CEA - UJFGrenoble
PowerPoint 2004
for MacOS
Summary of the lecture
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
Nuclei observable by NMR
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
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)
Larmor frequency
Laboratory reference frame
Rotating reference frameat frequency ω
Bloch equations without relaxation
B0 static magnetic fieldM macroscopic magnetization∧Cross-productB1 r.f. magnetic field
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
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
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
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
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
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
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
Bloch equations with relaxation90º pulse Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization Longitudinal magnetization
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
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetization
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1
Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetizationAt room temperature « 1Transverse magnetization
Coherence
Bloch equations with relaxation
What are the limitations of the Bloch equations?
Bloch equations with relaxation
What are the limitations of the Bloch equations?
Planes : no collision
Bloch equations with relaxation
What are the limitations of the Bloch equations?
Planes : no collision Cars : collision
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
Vector Transformation
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
number of spins
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
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
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, N
The limitations of the Bloch equationsSuitable dimensionality for description
Ix, Iy, Ix, N
z
yx
I
z
yx
S
Sx, Sy, Sx, NVector16 terms
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
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
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†
Basic Quantum Mechanics
Eigenvalues
Change of basis Diagonal matrix
A |νi> = λi |νi>
Operator Eigenvalue( complex number)Eigenvector
Basic Quantum Mechanics
Eigenvalues
Change of basis Diagonal matrix
A |νi> = λi |νi>
Operator Eigenvalue( complex number)Eigenvector
Hermitian operator
A = A†
Realeigenvalues
Orthogonaleigenvectors
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
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
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
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
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!
Basic Quantum Mechanics
Exponential operators
A0 = 1 A2 =AAA1 =A A3 =AAA
Power of operators
Exponential of operators
For ordinary numbers
For operators
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†
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
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
Liouville-von Neumann equation
Classical description
MagnetizationMagnetic field
Liouville-von Neumann equation
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
Liouville-von Neumann equation
|α>
|β>E
Classical description
MagnetizationMagnetic field
Quantum description
Density matrix Hamiltonian
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
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
Liouville-von Neumann equation
Quantum description
Density matrix Hamiltonian
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
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
Rotating frame
Rotating frame
σr = U σ U-1
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
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
Matrix representation of the spin operators
|α>|β>
E
We use the |α> and |β> states of the spin as a basis
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
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
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
Matrix representation of the spin operators
Matrix representation of the spin operators
The transverse coherence has a phase !
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets
Matrix representation of the spin operators
Bra notation (1×2 vectors)
Ket notation (2×1 vectors)
Bras / Kets Operator(square matrix)
Before
After
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
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
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
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
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Incorrect !
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Incorrect !
Dimension
2×2
4×4
Operators
Multispin systems
|ψ> = |ψ1> ⊗ |ψ2>
Operators AB|ij> = (A⊗B)(|i> ⊗ |j> ) = A |i> ⊗B|j>
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)
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)
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)
Multispin systems - product operators
Spectrum of a AX spin system
Multispin systems - product operators
Spectrum of a AX spin system
Thermal equilibrium populations
Product operators - coherence /population
Populations
Az XzAzXz
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
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
Multispin systems - product operators
Spectrum of a AX spin system
|αα>
|αβ>
|ββ>
|βα>
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of ASpectrum of X
X(β) X(α)
|αα>
|αβ>
|ββ>
|βα>
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>
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>
Multispin systems - product operators
Multispin systems - product operators
Spectrum of A
Multispin systems - product operators
Spectrum of A
In-phase coherence of A along y
Multispin systems - product operators
Spectrum of A
In-phase coherence of A along y
Anti-phase coherence of A along y
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
Multispin systems - product operators
Spectrum of A
Multispin systems - product operators
Spectrum of A
Multispin systems - product operators
Spectrum of A
Multispin systems - product operators
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
Multispin systems - product operators
Spectrum of A
|αα>
|αβ>
|ββ>
|βα>
|αα>
|αβ>
|ββ>
|βα>
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
Quantum description
Density matrix Hamiltonian
Commutation in coherence space
[Ix,Iy] = i Iz
Rule 1:y
xz
[Iy,Iz] = i Ix
Quantum description
Density matrix Hamiltonian
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
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
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
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
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
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
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
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)
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:
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
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
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
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
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
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 ] =
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
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
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
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
Operator product
Operator product
Any operator commuteswith itself
Operator product
Operator product
[Iz,Ix] ≠ 0They do not
commute
Operator product
Operator product
Any operator of Icommutes with
any operator of S
Operator product
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
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)
Terms of the spin hamiltonian
Spins
B0 (static field)
B1 (rf field)
Other spins
Zeeman interaction
H = – (1 — σiso) B0 Iz
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)]
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)
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
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)
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
Evolution of the spin system (chemical shift)
Zeeman interaction
H = – ω0 Iz Ix
[Iy,Iz] = i Ix
[Ix,Iz] = –i Ix
[Iz,Iz] = 0
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
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1[ Ix cos (φ) – Iy sin(φ)]
(rotating frame)Phase of the rf
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
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
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz Iz cos ω1t
Evolution of the spin system (radiofrequency)
RF field
H = – ω1 Ix
(rotating frame)
Iz Iz cos ω1t
– Iy sin ω1t
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
Evolution of the spin system (scalar coupling)
Scalar interaction
H = JIS IzSz
[Ix, 2IzSz] = i 2IySz
[Iy, 2IzSz] = – i 2IySz
[Iz, 2IzSz] = 0
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
NMR building blocks (2)Spin echoes in homonuclear spin systems
Δ Δ
Ix Ix cos ω0Δ
Iy sin ω0Δ
Chemical shift
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
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
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
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
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
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
NMR building blocks (3)Spin echoes in homonuclear spin systems
t tJ-coupling
Ix Ix cos πJt
2IySz sin πJt
x
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
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
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
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
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
NMR building blocks (5)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
NMR building blocks (5)Spin echoes in heteronuclear spin systems
1H
X
Δ Δ
Δ Δ
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!)
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
NMR building blocks (7)Spin echoes in heteronuclear spin systems
J-coupling
Ix Ix cos πJt
2IySz sin πJt
1H
X
Δ Δ
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
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
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
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
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
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)
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)
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
Coherence selection (1)
Pulse sequences with three 90º pulses
Coherence selection (1)
t1t2
DQF COSY
Pulse sequences with three 90º pulses
Coherence selection (1)
t1t2
τ t1t2τ
DQF COSY
Double quantumspectroscopy
Pulse sequences with three 90º pulses
Coherence selection (1)
t1t2
τ t1t2τ
t1 t2τ
DQF COSY
Double quantumspectroscopy
NOESY
Pulse sequences with three 90º pulses
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
90º
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
Coherence selection (2)
Coherence order
z
yx
Mag
netic
fiel
d
z
yx
Longitudinalmagnetization
z
yx
Transversecoherence
φ0×φ
1×φ
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φ
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
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=
Coherence selection (3)
Coherence order
z
yx
Mag
netic
fiel
d
φIz Iz
φ
Ix Ixφ
Ix
cos φ
sin φ+
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)
Coherence selection (4)
Coherence order
z
yx
Mag
netic
fiel
d
φ
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
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
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
τ t1t2τ
Double quantumspectroscopy
Coherence selection (5)t1
t2
DQF COSY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
τ t1t2τ
Double quantumspectroscopy
t1 t2τ
NOESY
Coherence selection (6)
Phase cycling
Coherence selection (6)
Phase cycling
φ → φ+Δφ
Coherence selection (6)
Phase cycling
φ → φ+Δφ
Δp
Coherence selection (6)
Phase cycling
φ → φ+Δφ
Δp
Coherence phase shift:
Δp × Δφ
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Coherence selection (7)Phase cycling for the selection of
the Δp=–3 coherence pathway
+2+1
0–1
–2
Δp= –3
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
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
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
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
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
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
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
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
Pulsed field gradients (1)
B0eff
Homogeneous
magnetic field(well shimmed magnet)
Pulsed field gradients (1)
B0eff
B0eff
Homogeneous
magnetic field(well shimmed magnet)
Inhomogeneous magnetic field(field gradient)
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)
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)
Pulsed field gradients (2)
I+
I+
γΙ G(z)
exp(–i γΙ G(z)t)
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
Pulsed field gradients (3)
rf
pfg g1 g2
τ1 τ2
p1
p2
Pulsed field gradients (3)
rf
pfg g1 g2
τ1 τ2
p1
p2
Refocusing condition
g1τ1
g2τ2
–p1
–p2
=
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
rf
pfg g gτ τ
+p
–p
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
Refocusing pulseInversion pulse
Pulsed field gradients (4)
Imperfect 180º pulses
I
pfg+g
–gτ
τ
S
pI
pS
The end…
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