Basic principles of NMR

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Basic principles of NMR
Dominique Marion Institut de Biologie
Structurale Jean-Pierre Ebel CNRS - CEA -
UJF Grenoble
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Summary of the lecture
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Nuclei observable by NMR
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Why some nuclei have no spin ?
The proton is composed of 3 quarks stuck together
by gluons
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Why some nuclei have no spin ?
Isotopes with odd mass number
(1H, 13C, 15N, 19F, 31P)
S 1/2, 3/2
Isotopes with even mass number
Number of protons and neutron even
S 0
Number of protons and neutron odd
S1, 2, 3
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Larmor frequency
Rotating reference frame at frequency w
Laboratory reference frame
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Bloch equations without relaxation
B0 static magnetic field M macroscopic
magnetization ?Cross-product B1 r.f. magnetic
field
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Bloch equations with relaxation
90º pulse
Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization ?
Longitudinal magnetization ?
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Bloch equations with relaxation
90º pulse
Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization ?
Longitudinal magnetization ?
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Bloch equations with relaxation
90º pulse
Magnetization in the XY plane
Precession around B0
Recovery to the equilibrium state ?
Transverse magnetization ?
Longitudinal magnetization ?
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Longitudinal and transverse magnetization
Thermal equilibrium
Longitudinal magnetization
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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 equations
Suitable dimensionality for description
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The limitations of the Bloch equations
Suitable dimensionality for description
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Basic Quantum Mechanics
Operator
Performs some operation on a function
Ex Dx derivative operator
Ex 1 unity operator 1f(x) f(x)
The effect of consecutive operations may depends
on their order
Commutation
Commutator
BA( f(x) )
AB( f(x) )
A,B AB - BA
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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
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Basic Quantum Mechanics
Eigenvalues
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Basic Quantum Mechanics
Exponential operators
? Power of operators
A0 1
A2 AA
A1 A
A3 AAA
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Basic Quantum Mechanics
Cyclic commutation
A, B iC
B, C iA
C, A iB
? Definition
Rotation angle
? Sandwich formula
exp (-iqA) B exp (iqA) B cos q C sin q
Cyclic permutation
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Basic Quantum Mechanics
Cyclic commutation
? Rotation around the 3 axes
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Liouville-von Neumann equation
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Liouville-von Neumann equation
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Rotating frame
Rotating frame
sr U s U-1
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Summary of the lecture
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Matrix representation of the spin operators
We use the agt and bgt states of the spin as a
basis
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Matrix representation of the spin operators
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Matrix representation of the spin operators
The transverse coherence has a phase !
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Matrix representation of the spin operators
Bras / Kets
Bra notation (1?2 vectors)
Ket notation (2?1 vectors)
Matrix representation using different basis sets
can be interconverted using unitary transformation
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Multispin systems
Bloch model
Strictly applicable only to a system of
non-interacting spins
Quantum mechanics
Direct product space
The two spins are independent
Nb of basis vectors 2N
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Multispin systems
?gt ?1gt ? ?2gt
Operators
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Multispin systems
?gt ?1gt ? ?2gt
Operators
ABijgt (A?B)(igt ? jgt ) A igt ?Bjgt
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Multispin systems - product operators
Spectrum of a AX spin system
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Product operators - coherence /population
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Product operators - coherence /population
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Product operators - coherence /population
0 / 2 Quantum coherence
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Multispin systems - product operators
Spectrum of a AX spin system
Spectrum of A
Spectrum of X
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Multispin systems - product operators
In-phase coherence of A along y
Anti-phase coherence of A along y
Spectrum of A
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Multispin systems - product operators
Spectrum of A
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Commutation in coherence space
Rule 1
Ix,Iy i Iz
Iy,Iz i Ix
Rule 2
Iz,Ix i Iy
Iy,Ix i Iz
Rule 3
Ip,Iq 0 for (p,q) (x,y,z)
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Commutation in coherence space
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
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Commutation in coherence space (summary)
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Operator product
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Terms of the spin hamiltonian
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Terms of the spin hamiltonian (conflicts)
RF field
H w1 Ix cos (wt) - Iy sin(wt)
Zeeman interaction
H w0 Iz
Scalar interaction
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Terms of the spin hamiltonian (solutions)
RF field
H w1 Ix cos (wt) - Iy sin(wt)
Hypothesis short pulse The spins do not
precess during the pulse
Zeeman interaction
H w0 Iz
Scalar interaction
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Terms of the spin hamiltonian (solutions)
RF field
During the free precession
H w1 Ix cos (wt) - Iy sin(wt)
Hypothesis (1) weak coupling JIS ltlt wI - wS
Zeeman interaction
H w0 Iz
Scalar interaction
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Terms of the spin hamiltonian (solutions)
RF field
During the free precession
H w1 Ix cos (wt) - Iy sin(wt)
Hypothesis (2) the chemical shift evolution is
eliminated
Zeeman interaction
H w0 Iz
Isotropic mixing
Scalar interaction
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Evolution of the spin system
exp (-iqH) s0 exp (iqH) s0 cos q s1 sin q
s0 , H i s1
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Evolution of the spin system (chemical shift)
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Evolution of the spin system (radiofrequency)
(rotating frame)
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Evolution of the spin system (radiofrequency)
(rotating frame)
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Evolution of the spin system (scalar coupling)
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Summary of the lecture
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NMR building blocks (1)
Spin echoes in heteronuclear spin systems
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NMR building blocks (1)
Spin echoes in heteronuclear spin systems
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NMR building blocks (2)
Spin echoes in homonuclear spin systems
Chemical shift
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NMR building blocks (3)
Spin echoes in homonuclear spin systems
x
J-coupling
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NMR building blocks (4)
Spin echoes in homonuclear spin systems
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NMR building blocks (5)
Spin echoes in heteronuclear spin systems
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NMR building blocks (6)
Spin echoes in heteronuclear spin systems
X Chemical shift
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NMR building blocks (7)
Spin echoes in heteronuclear spin systems
J-coupling
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NMR building blocks (8)
Spin echoes in heteronuclear spin systems
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NMR building blocks (9)
Spin echoes in heteronuclear spin systems
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Coherence selection (1)
Pulse sequences with three 90º pulses
DQF COSY
Double quantum spectroscopy
NOESY
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Coherence selection (2)
Coherence order
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Coherence selection (3)
Coherence order
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Coherence selection (4)
Coherence order
Order 2 2 0
0
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Coherence selection (5)
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Coherence selection (6)
Phase cycling
f ? fDf
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Coherence selection (7)
Phase cycling for the selection of the ?p3
coherence pathway
recv phase
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Pulsed field gradients (1)
Homogeneous magnetic field (well shimmed magnet)
Inhomogeneous magnetic field (field gradient)
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Pulsed field gradients (2)
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Pulsed field gradients (3)
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Pulsed field gradients (4)
Imperfect 180º pulses
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The end