Formalism and Interactions in NMR

(1)

EPJ Web of Conferences

30 , 01002 (2012)

DOI: 10.1051/epjconf/20123001002

© Owned by the authors, published by EDP Sciences, 2012

Organized by

Thibault Charpentier [email protected]

Patrick Berthault

[email protected]

Constantin Meis

[email protected]

Experiment and Modelling in Structural NMR

November 28th – December 2

nd

₂₀₁₁

INSTN – CEA Saclay, France

Hervé Desvaux

CEA, IRAMIS France

Formalism and Interactions

in NMR

[01002]

(2)

Formalism and interactions in NMR

Herv´e Desvaux

1,a

CEA, IRAMIS, UMR CEA

/

CNRS 3299 SIS2M, Laboratoire Structure et Dynamique par R´esonance Magn´etique, 91191

Gif sur Yvette, cedex

Abstract.

In this lecture, a fast overview of the formalism used in Nuclear Magnetic Resonance is provided,

with a stress on the density matrix approach. Its equivalence to the classical derivation for particular spin systems

is shown. The different interactions which are usually encountered are introduced. Finally, the case of randomly

time-dependent interactions inducing relaxation is addressed.

1 Overview of the presentation

Nuclear Magnetic Resonance has a theory which is well

established and very well validated. The aim of this

lec-ture was to provide a fast introduction to this theory, more

detailed and in-depth descriptions can be found in several

reference books [1,2,5,4,3,6,7]. The presentation is

orga-nized in such a way. First, after introducing the nuclear

spin, elements are provided explaining why quantum

me-chanics and statistical physics are needed to properly

de-scribe the evolution of the magnetization and thus predict

the result of an NMR experiment. In a second step, the

density matrix operator which allows such a treatment is

defined and the main theorems allowing the calculation of

its value at thermal equilibrium, its time-domain evolution,

its expression in a rotating frame and its reduction to the

spin system are given. Two simple examples of

calcula-tions (free evolution and rf excitation) performed in this

framework are given. The next section is devoted to the

classical approach based on the Bloch equations. The

ques-tion of sensitivity in NMR is stressed and the equivalence

between this approach and the density matrix one with

its limits is shown. Then, the di

ff

erent main interactions

present in conventional NMR (chemical shift, dipolar and

quadrupolar interactions, scalar coupling) are described.

The aspects of truncated Hamiltonian and interaction

sym-metries are considered. The last section is devoted to a fast

introduction to relaxation due to randomly time-dependent

Hamiltonian. The master equation of relaxation is derived

and the principal definitions (self and cross-relaxations,

auto-correlation and cross-correlation contribution to

re-laxation) are given. Finally the importance and relevance

of spectral density functions are discussed.

References

1. A. Abragam,

Principles of Nuclear Magnetism

(Claren-don Press, Oxford, 1961).

2. A. Abragam and M. Goldman,

Nuclear magnetism:

or-der and disoror-der

(Clarendon press, Oxford, 1982).

a

_e-mail:

_{[email protected]}

3. R. R. Ernst, G. Bodenhausen and A. Wokaun,

Princi-ples of nuclear magnetism resonance in one and two

di-mensions

(Clarendon Press, Oxford, 1987).

4. C. P. Slichter,

Principles of Magnetic Resonance

, 3

rd

Ed., (Springer–Verlag, Berlin, 1989).

5. M. Goldman,

Quantum description of high-resolution

NMR in liquids

(Clarendon Press, Oxford, 1988).

6. J. Cavanagh, W. J. Fairbrother, A. G. Palmer III and N.

J. Skelton,

Protein NMR spectroscopy

(Academic Press,

San Diego, 1995).

(3)

Formalism and Interactions in NMR

Herv´e Desvaux

Laboratoire de Structure & Dynamique par R´esonance Magn´etique

CEA/DSM/IRAMIS

Monday 28

th

November 2011

Herv´e Desvaux Formalism and Interactions Monday 28th November 2011 1/ 21

Introduction

Summary of the lecture

1 _{Quantum description of NMR}

The nuclear spin

Properties of density matrix

Evolution of the density matrix

2 Classical description

Key formula

Bloch equations

3 Interactions

Chemical shift

Expressions of other interactions

4 Principles of relaxation

(4)

Quantum description of NMR The nuclear spin

Origin of the nuclear spin

Nucleus constitution : not elementary particle

Protons (2 quark up, 1 quark down)

Neutrons (1 quarks up, 2 quarks down)

Nuclei as atoms have quantified levels

Nuclear spin

I

_⇔

symmetry of the ground state

It is a given value (0, 1/2, 1, 3/2, 2, 5/2,. . . )

This value is known for each isotope.

The large difference of nucleus energy levels warranties the proportionality

between the nuclear spin and the nuclear magnetic moment.

−

→

_µ

₌

_γ

−

→

_I

The energy of nuclear spin in a magnetic field

−

→

B

is provided by the

Zeeman

Hamiltonian

:

HZ

=

₋

γ

−

→

I

_·

−

→

B

H

in

units

Quantum description of NMR The nuclear spin

Properties of the spin

General properties

Definitions

[

I

x

, I

y

] =

ıI

z

[

I

y

, I

z

] =

ıI

x

[

I

z

, I

x

] =

ıI

y

I

+

=

I

x

+

ıI

y

I

−

=

I

x

−

ıI

y

I

2 =

I

x

2 +

I

y

2 +

I

z

2 =

I

+

I

−

+

I

z

2 −

I

z

=

I

−

I

+

I

z

2 +

I

z

So:

[

I

2 , I

x

] = [

I

2 , I

y

] = [

I

2 , I

z

] = 0

Eigenvectors

|

j, m

j

with

m

j

∈ {−

j,

−

j

+ 1

, . . . , j

}

:

I

2 |

j, m

j

=

j

(

j

+ 1)

|

j, m

j

;

I

z

|

j, m

j

=

m

j

|

j, m

j

;

I

±

|

j, m

j

∝ |

j, m

j

±

1 Case of spin

1 /

2 |

α

=

|

1 /

2 ,

1 /

2 and

_|

β

=

|

1 /

2 ,

−

1 /

2 :

I

z

=

1 /

2

0

0 −

1 /

2 I

x

=

0

1 /

2

1 /

2

0 I

y

=

0 ₋

ı/

2 ı/

2

0 I

+

=

0

1

0

0 I

−

=

0

1

0

(5)

Quantum description of NMR Properties of density matrix

A statistical quantum description

Need:

Spectroscopy

_⇒

Quantum mechanics

Nuclear spin

⇒

Quantum mechanics

Wavelength

⇒

Statistical physics

Coherence

_⇒

Quantum mechanics and statistical physics

How :

Initial state given by thermodynamics

Density matrix description

Time evolution given by Liouville-Von Neumann equation

Why is it so powerful?

Very weak coupling between nuclear spins and other parameters

⇒

1 st

reduction of the Hilbert space

Most of the time, interactions between molecules are negligible

⇒

2 nd

reduction of the Hilbert space

⇒

Expectation values can analytically be computed.

Density operator expression

Position of the problem

N

spins

1 /

2 means a Hilbert space of dimension

2 N

_⇒

Huge!

Information of a ket irrelevant: too complex and too many unknown!

Solution:

Use of superposition of states weighted by the probability of being

the representative ket.

How?

Through the density operator!

we search for the statistical mean of the measured values (expectation value)

A

associated to an operator

A

we call

P

(

ψ

)

dτ

the probability of the ket

_|

ψ

to describe the system

we call

{|

i

}

a basis of the Hilbert space;

1 =

_i

|

i

|

A

=

A

=

P

(

ψ

)

ψ

_|

A

_|

ψ

dτ

=

i,j

P

(

ψ

)

ψ

|

i

|

A

|

j

|

ψ

dτ

=

i,j

i

|

A

|

j

P

(

ψ

)

j

|

ψ

|

i

dτ

=

Tr

(

A

· ρ

)

Definition

Expectation value:

A

=

Tr

(

A

_·

ρ

)

where the density matrix

ρ

is:

ρ

=

P

(

ψ

)

|

ψ

· ψ

|

dτ

If

_|

ψ

=

_i

a

i

|

i

then

ρ

ij

=

i

|

ρ

|

j

=

P

(

ψ

)

a

i

a

∗

j

dτ

=

a

i

a

∗

_j

ρ

ii

probability of finding the state

|

i

(population)

(6)

Density operator expression

Position of the problem

N

spins

1 /

2 means a Hilbert space of dimension

2 N

⇒

Huge!

Information of a ket irrelevant: too complex and too many unknown!

Solution:

Use of superposition of states weighted by the probability of being

the representative ket.

How?

Through the density operator!

Definition

Expectation value:

A

=

Tr

(

A

_·

ρ

)

where the density matrix

ρ

is:

ρ

=

P

(

ψ

)

|

ψ

· ψ

|

dτ

If

_|

ψ

=

_i

a

i

|

i

then

ρ

ij

=

i

|

ρ

|

j

=

P

(

ψ

)

a

i

a

∗

j

dτ

=

a

i

a

∗

_j

ρ

ii

probability of finding the state

|

i

(population)

ρ

ij

coherent superposition of states

Density matrix at thermal equilibrium

General expression:

System at thermal equilibrium at temperature

T

,

{|

i

}

a basis on which

H

T

is

diagonal:

i

|

H

T

|

i

=

E

i

/

No coherence between states with different energies:

_∀

i

=

j, ρ

ij

= 0

The diagonal matrix elements are the state populations

p

i

∝

exp(

−

E

i

/kT

)

,

thus with

β

=

/kT

ρ

_∝

exp (

₋

β

HT

)

Reduction to the spin system:

Since

H

T

=

H

+

F

+

H

F

I

and there is weak coupling

(

H

F

I

)

:

ρ

=

σ

⊗

P

,

with

σ

the nuclear spin density matrix

Reduced trace and high temperature approximation

E

i

/kT

1 :

σ

= exp

−

β

H

1 −

β

H

Final expression in a high magnetic field:

Spin Hamiltonian expression:

H

−

molecules

spin

γB

0 I

z

k

In a

dilute

spin system

_⇒

average molecule

σ

=

1 +

k

(7)

Density matrix at thermal equilibrium

General expression:

ρ

_∝

exp (

₋

β

HT

)

Reduction to the spin system:

Since

H

T

=

H

+

F

+

H

F

I

and there is weak coupling

(

H

F

I

)

:

ρ

=

σ

⊗

P

,

with

σ

the nuclear spin density matrix

Reduced trace and high temperature approximation

E

i

/kT

1 :

σ

= exp

−

β

H

1 −

β

H

Final expression in a high magnetic field:

Spin Hamiltonian expression:

H

−

molecules

spin

γB

0 I

z

k

In a

dilute

spin system

_⇒

average molecule

σ

=

1 +

k

γB0

I

z

k

Time-domain evolution

Schr¨odinger equation:

d

_|

ψ

dt

=

−

ı

H

T

|

ψ

The

Liouville-Von Neumann equation

is obtained from the Schr¨odinger

equation and the definition of

ρ:

dρ

dt

=

−

ı

[

HT

, ρ

]

Evolution of one observable:

d

A

dt

=

d

A

dt

=

d

dt

Tr

(

ρ

· A

) =

Tr

dρ

dt

· A

= Tr (

₋

ıA

_·

[

H

T

, ρ

])

(8)

Quantum description of NMR Evolution of the density matrix

Free evolution of the magnetization

Key equations:

Initial state:

σ

=

1 −

k

γB0I

k

z

Time evolution:

d

dt

σ

=

−

ı

[

H

, σ

]

Observation:

A

=

Tr

(

A

_·

σ

)

Time evolution:

d

dt

A

=

−

ı

[

A,

H

]

Free evolution of

I

+

=

I

x

+

ıI

y

Hamiltonian

H

=

ω0

I

z

Evolution of

I

x

:

d

dt

I

x

=

−

ı

[

I

x

, ω0I

z

]

=

−

ω0

I

y

Evolution of

I

y

:

d

dt

I

y

=

−

ı

[

I

y

, ω0I

z

]

=

ω0

I

x

So:

d

dt

I+

=

ıω0

I+

⇒

I+

(

t

) =

I+

(0)

e

ıω

0

t

Precession around

Oz

at

ω

0 .

RF excitation

Change of representations

Unitary transformation defined by

U

:

U

· U

†

=

U

†

· U

=

1 i.e.

U

†

=

U

−

1 With

_dt

d

U

(

t

) =

ıA

(

t

)

· U

(

t

)

with

A

(

t

)

hermitian.

d

σ

dt

=

d

dt

(

U

· σ

· U

†

_{) = ˙}

_U

· σ

_·

U

†

+

U

_·

dσ

dt

· U

†

₊

_U

· σ

_·

U

˙

†

=

−

ı

[

H

−

A

(

t

)

,

σ

] =

−

ı

[

H

effective,

σ

]

Example:

H

=

ω

0 I

z

+ 2

ω

1 cos

ωtI

x

U

(

t

) = exp(

ıωI

z

t

)

, A

(

t

) =

ωI

z

In the interaction frame:

d

σ

dt

=

−

ı

[(

ω0

−

ω

)

I

z

+

ω1

I

x

,

σ

]

=

−

ı

[Ω

I

Z

,

σ

]

Precession around the tilted effective field

tan

θ

=

ω1/

(

ω0

₋

ω

)

(9)

Simple forms

θ

pulse:

I

z

θy

−−−−−→

I

z

cos

θ

+

I

x

sin

θ

I

z

−−−−−→

θx

I

z

cos

θ

−

I

y

sin

θ

I

x,y

θx,y

−−−−−→

I

x,y

Ix

_{−−−−−→}

θy

Ix

cos

θ

₋

Iz

sin

θ

I

y

−−−−−→

θx

I

y

cos

θ

+

I

z

sin

θ

Chemical shift:

I

z

−−−−−→

ωtIz

I

z

Ix

_{−−−−−→}

ωtIz

Ix

cos

ωt

+

Iy

sin

ωt

Iy

_{−−−−−→}

ωtIz

Iy

cos

ωt

₋

Ix

sin

ωt

Scalar interactions:

I1z,2z

2πJ

_{−−−−−−−−→}

12tI1z I2z

I1z,2z

2 I1z

I2z

2πJ

_{−−−−−−−−→}

12tI1z I2z

2 I1z

I2z

2 I1x,1y

I2x,2y

2πJ

_{−−−−−−−−→}

12tI1z I2z

2 I1x,1y

I2x,2y

I1x

2πJ

−−−−−−−−→

12tI1z I2z

I1x

cos

πJt

+ 2

I1y

I2z

sin

πJt

I1y

2πJ

−−−−−−−−→

12tI1z I2z

I1y

cos

πJt

−

2 I1xI2z

sin

πJt

2 I1xI2z

2πJ

−−−−−−−−→

12tI1z I2z

2 I1xI2z

cos

πJt

+

I1y

sin

πJt

2 I1y

I2z

2πJ

−−−−−−−−→

12tI1z I2z

2 I1y

I2z

cos

πJt

−

I1x

sin

πJt

Classical description Key formula

Magnetization of a spin

1/2

Polarization at thermal equilibrium

P

=

N

α

−

N

β

N

α

+

N

β

= tanh

γ

B

0

2 kT

γ

B

0

2 kT

Magnetization

M

z

=

P µN

γ

2

2 N B

0

4 kT

Magnetization evolution

d

−→

M

dt

=

γ

−→

M

∧

−

→

B

0 Detection

d

M

_⊥

dt

=

γB

0 M

⊥

=

1

4 γ

3

2 _B

2

0 N

kT

Signal-to-noise ratio

Noise

_∝

√

γB

0 SNR

_∝

N γ

5 /

2 _B

3 /

2

0

(10)

Classical description Key formula

Equivalence to the quantum description

Hamiltonian:

H

=

₋

γ

−

→

I

_·

−

→

B

0 Evolution:

d

−→

M

dt

=

−

ı

−→

M

,

H

=

ıγ

2 →

−

I ,

−

→

I

_·

−

→

B

0 d

M

z

dt

=

ıγ

2 _I

z

,

−

→

I

· −

→

B

0 =

γ

2 B

0 y

I

x

−

B

0 x

I

y

=

γ

2 −

→

I

_∧

−

→

B

0 z

=

γ

−→

M

∧

−

→

B

0 _z

Classical description Bloch equations

Bloch equations

Assumptions:

Homogeneous magnetization and static magnetic field

No interaction

One spin species

Expressions:

d

M

x

(

t

)

dt

=

−

ω

0 M

y(

t

)

−

M

x

(

t

)

T

2 d

M

y(

t

)

dt

=

ω

0 M

x

(

t

)

−

M

y

(

t

)

T

2 d

M

z(

t

)

dt

=

−

M

z(

t

)

₋

M

0 T

1 T

1 is the longitudinal self-relaxation time.

T

2 is the transverse self-relaxation time.

(11)

Interactions Chemical shift

Origin of the chemical shift

Information on the chemical function

Results from the induced magnetic fields (electronic

spins, electronic current,. . . )

−

→

B

k

=

−

→

B

S

+

−

→

B

0 



B

Sx

Sy

B

Sz





=

₋







σ

xx

(

k

)

σ

xy

(

k

)

σ

xz

(

k

)

σ

yx

(

k

)

σ

yy

(

k

)

σ

yz

(

k

)

σ

zx

(

k

)

σ

zy

(

k

)

σ

zz

(

k

)







· 



0

0 B

0 



The Hamiltonian is computed using

−

→

B

k

By neglecting non-secular terms:

H

cs

=

k

γB

0 (1

−

σ

(

zz

k

)

I

z

k

Interactions Chemical shift

Tensorial expression of the chemical shift

A particular molecular frame defines the principal axis system:

σ

(

k

)

=







σ

(

₁₁

k

)

0

0 σ

(

₂₂

k

)

0

0 σ

(

₃₃

k

)







Thus

σ

(k)

=

σ

(k)_iso





1

0

1

0

1 



+

σ

(k) 11

+

σ

(k) 22

−

2 σ

(k) 33

6 



1

0

1

0

0 −

2 



+

σ

(k) 11

−

σ

(k) 22

2 



1

0 −

0

1

0

0 



Isotropic chemical shift

σ

(

_iso

k

)

=

σ

(

k

)

11 +

σ

(

k

)

22 +

σ

(

k

)

33

3

(12)

Interactions Expressions of other interactions

Dipolar interactions

Structural information (distance, angle)

Dipolar interaction between two nuclear spins

H

D

=

µ

0

4 π

γ

2 r

3

12 ₋

_→

I

1 _·

−

→

I

2 ₋

3 I

_Z

1 I

_Z

2 Truncated dipolar interactions:

H

D

=

µ

0

4 π

γ

2 r

₁₂

3

1 ₋

3 cos

2 θ

2

·

2 I

_z

2 I

_z

2 ₋

1

2 (

I

1 +

I

−

2 +

I

−

1 I

+

2 )

Only second-rank tensor contribution

_⇒

no

effect on resonance frequency in liquids.

Scalar couplings

Information on the connectivity and on

dihedral angles

Interactions between two nuclear spins

through chemical bonds (via electrons)

Mainly isotropic interaction

HJ

= 2

πJ

−

→

I

1 _·

−

→

I

2

(13)

Quadrupolar interactions

Structural information

All nuclei of spin larger than

1 /

2 have a quadrupolar contribution.

Interaction of the electric

quadrupole moment

(

eQ

)

with the

electric field gradient

V .

Quadrupolar coupling

C

Q

= (

eQV

ZZ

)

/h

Amplitude from 0 to 30MHz

⇒

H

=

H

Zeeman

+

H

Q

.

C

Q

values as a function of Al-site

symmetry.

H

Q

=

eQV

ZZ

4 I

(2

I

₋

1)

3 I

_Z

2 ₋

I

(

I

+ 1) +

η

(

I

_x

2 ₋

I

_y

2 )

with

η

=

V

XX

−

V

Y Y

V

ZZ

Principles of relaxation

Master equation of relaxation

Hamiltonian:

H

=

H

0 +

H

1 (

t

)

where

H

1 (

t

) =

_α

V

α

F

α

(

t

)

and

F

α

(

t

) = 0

In the interaction frame

U

=

e

ı

H

0t

:

_dt

d

σ

=

₋

ı

[

H

1 (

t

)

,

σ

]

,

V

α(

t

) =

e

ıωα

t

V

α

Double integration and averaging:

σ

(

t

) =

σ

(0)

₋

ı

t

0 [

H

1 (

t

)

,

σ

(

t

)]

dt

d

dt

σ

(

t

) =

−

ı

[

H

1 (

t

)

,

σ

(0)]

−

t

0 H

1 (

t

)

,

[

H

1 (

t

)

,

σ

(

t

)]

dt

d

dt

σ

(

t

) =

−

t

0 H

1 (

t

)

,

[

H

1 (

t

)

,

σ

(

t

)]

dt

Assumptions

τ

c

correlation time

Slow evolution for time short compared

τ

c

Time

t

very large when compared to

τ

c

Replacement of

σ

by

σ

−

σeq

F

α

(

t

)

F

_β

∗

(

t

)

only depends on

|

t

−

t

|

Secular approximation

Master equation of relaxation

d

dt

σ

(

t

) =

−

α

V

α

,

[

V

α

†

,

(

σ

(

t

)

−

σ

eq)]

J

α

(

ω

α

)

with

J

α

the spectral density:

J

α

(

ω

) =

∞

0

(14)

Master equation of relaxation

Hamiltonian:

H

=

H

0 +

H

1 (

t

)

where

H

1 (

t

) =

_α

V

α

F

α

(

t

)

and

F

α(

t

) = 0

Double integration and averaging:

d

dt

σ

(

t

) =

−

t

0 H

1 (

t

)

,

[

H

1 (

t

)

,

σ

(

t

)]

dt

Assumptions

Master equation of relaxation

d

dt

σ

(

t

) =

−

α

V

α

,

[

V

α

†

,

(

σ

(

t

)

−

σ

eq)]

J

α

(

ω

α

)

with

J

α

the spectral density:

J

α

(

ω

) =

_∞

0 F

α

(

t

)

F

α

∗

(

t

+

τ

)

e

ıωτ

dτ

Key features of relaxation

Evolution due to relaxation of one operator:

d

dt

Q

(

t

) =

−

α

[

Q, V

α

]

, V

α

†

(

t

)

₋

[

Q, V

α

]

, V

α

†

eq

J

α

(

ω

α

)

Self-relaxation

and

cross-relaxation

d

dt

Q

(

t

) =

−

λ

Q

(

t

)

−

µ

Q

₍

_t

_{) +}

_{· · ·}

Autocorrelation twice the same interaction, cross-correlation two different

H

1 (

t

) =

k

H

1 (

k

)

(

t

)

and

d

dt

σ

(

t

) =

−

∞

0 H

1 (

t

)

,

[

H

1 (

t

)

,

σ

(

t

)

−

σ

eq

]

dt

Spectral density contains all structural and dynamic pieces of information

J

α

(

ω

) =

∞

0

(15)

Key features of relaxation

Evolution due to relaxation of one operator:

d

dt

Q

(

t

) =

−

α

[

Q, V

α

]

, V

α

†

(

t

)

₋

[

Q, V

α]

, V

_α

†

eq

J

α

(

ω

α

)

Self-relaxation

and

cross-relaxation

Autocorrelation twice the same interaction, cross-correlation two different

Spectral density contains all structural and dynamic pieces of information

J

α(

ω

) =

∞

0 F

α(

t

)

F

∗

α

(

t

+

τ

)

e

ıωτ

dτ