Cross polarization (CP)

28
Barth-Jan van Rossum: Solid-State NMR Cross polarization (CP) this approach is valid as long: 1. system contains a large number of spins 2. strong 1 H- 1 H dipolar couplings are present ‘classical’ description of cross-polarization uses concept of spin temperature thermodynamic approach: - polarization exchange between two reservoirs with different spin temperature coupled to a large reservoir (‘lattice’) - relaxation to equilibrium 13 C 1 H CP CP decoupling

Transcript of Cross polarization (CP)

Page 1: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

Cross polarization (CP)

this approach is valid as long:

1. system contains a large number of spins

2. strong 1H-1H dipolar couplings are present

‘classical’ description of cross-polarization

uses concept of spin temperature

thermodynamic approach:

- polarization exchange between two

reservoirs with different spin temperature

coupled to a large reservoir (‘lattice’)

- relaxation to equilibrium

13C

1H CP

CP

decoupling

Page 2: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

Zeeman interaction

spins rotate with theLarmor frequency ω0 …

z

y

x

B0

…or, spin up and spin down have an energy difference of ω0

ω0

ω0

Page 3: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

high Tlow T

low T à large population differences à high polarization

high T à small population differences à low polarization

“Boltzmann’s ingenious concept”

mhhh..

Page 4: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

large splitting à large population differences à high polarizationsmall splitting à small population differences à low polarization

large splitting

smallsplitting

equaltemperature

“Boltzmann’s ingenious concept”

Page 5: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

“Boltzmann’s ingenious concept”

à high magnetic fields give larger signal

à 1H (high γ) more sensitive than 13C or 15N (low γ) à CP

à e- (very high γ) used to enhance NMR signal à DNP

à low temperature larger signal

than high temperature

Page 6: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

Cross polarization (CP)

like the Zeeman interaction, the

spin-lock pulse gives rise to a

splitting (spin up, spin down)

pulse :spins rotate

spin-lock pulse:spins are trapped

during CP, both spin-types (1H and 13C)

are ‘spin-locked’

Page 7: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

Cross polarization (CP)

since B0, γH and γC are all

fixed, the Zeeman splitting

is different for 1H and 13C …

… however, the spin-lock

fields B1H and B1

C can be

chosen so that the splitting

for 1H and 13C becomes

equal

Hartmann-Hahn Matching

c

Page 8: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

ω0

ω1

‘real’ Boltzmanndistribution

apparentBoltzmanndistribution

in equilibrium:

levels separated by ω0

à occupation of levels according

to real Boltzmann distribution

during spin lock:

levels separated by ω1

with ω1 << ω0

à levels are ‘compressed’

à occupation of levels looks like

a Boltzmann distribution, but

one for much lower T

high T

Cross polarization (CP)

low T

Page 9: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

ω0

ω1

‘real’ Boltzmanndistribution

during spin lock:

relaxation back to the normal

temperature with T1ρ

high T

spin locking 1H

lowers the spin-temperature

low T

T1ρ

Cross polarization (CP)

apparentBoltzmanndistribution

Page 10: Cross polarization (CP)

Barth-Jan van Rossum: Solid-State NMR

(T1ρS ) T1ρH

Cross polarization (CP)

by matching energy splitting for 1H and 13C

(Hartmann-Hahn matching), polarization

can be exchanged with conservation of energyhigh T

S (13C)

low T

I (1H)

heteronuclear (1H - 13C) dipolar interaction

couples 1H and 13C polarization reservoirs

homonuclear interaction (mostly 1H- 1H)

provides coupling to lattice (T1ρ relaxation)

ambient T

equilibration lowers 13C spin temperature

à 13C polarization first increases (short CP)

coupling to lattice reduces spin temperature

à 13C polarization relaxes (long CP)

Page 11: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Overview

dipolar interaction

- Hamiltonian

- recoupling

- dipolar truncation

cross polarization (CP) - part II

DNP (dynamic nuclear polarization)

Page 12: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Dipolar interaction

α

zero for α = 90° max for α = 0°

the dipolar coupling is the interaction between

two magnetic moments µ1 and µ2

r r

same α !

~ cos αα

β

γ βγ

~ cos β

~ cos γ

r

Page 13: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Dipolar interaction

the dipolar coupling is the interaction between

two magnetic moments µ1 and µ2

(correspondence principle)

how does the dipolar coupling look in quantum mechanics?

r

Page 14: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Dipolar interaction

“dipolar alphabet”

Page 15: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Dipolar interaction

à ‘truncation’ or ‘quenching’

Only terms that commute with Zeeman interaction (Iz) remain

Zeeman interaction (~500 MHz) >> dipolar interaction (~50 kHz)

A, B – terms:

commute with Iz (“secular terms”) à no time evolution under Iz

C, D, E, F – terms:

do not commute with Iz (“non-secular terms”) à time evolution under Iz

à averaged to zero

Page 16: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Dipolar interaction

truncated

B-term only relevant when ωI ~ ωS

homonuclear case (e.g.1H-1H) à B-term

heteronuclear case (e.g. 1H-13C) à B-termr and θ : spatial parameters

Page 17: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

θ

 

D∝1

r3 3cos2 θ −1( )

 

3cos2 θ −1= 0 in isotropic liquids

0 in solids or oriented media

Dipolar interaction

the dipolar coupling depends on :

- the distance between the spins (r)

- the angle θ between the magnetic field B

and the vector connecting the spins

r

B

Page 18: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Recoupling

The time-average of the dipolar coupling is zero

but not the instantaneous coupling, it is oscillating around zero

Page 19: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Overview

dipolar interaction

- Hamiltonian

- recoupling

- dipolar truncation

cross polarization (CP) - part II

DNP (dynamic nuclear polarization)

Page 20: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

since B0, γH and γC are all

fixed, the Zeeman splitting

is different for 1H and 13C …

… however, the spin-lock

fields B1H and B1

C can be

chosen so that the splitting

for 1H and 13C becomes

equal

Hartmann-Hahn Matching

c

Cross polarization (CP) – part II

Page 21: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

the heteronuclear dipolar interaction between 1H (I-spins) and 13C (S-spins)

drives the polarization transfer:

H ~ b · Iz · Sz

B1

z

y

xduring CP, the spin lock pulses are along x-axis…

Cross polarization (CP) – part II

…transform to a frame that rotates around the x-axis

Page 22: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

in NMR, everything is rotating…

rotating frames

spins rotate with theLarmor frequency…

RF pulses are rotating withthe Larmor frequency…

z

y

x B1

z

y

x

B0

blerk…

observation from a frame that rotates itself, makes everything appear static

z

y

x B1

z

y

x

it is as if the Zeeman

interaction has ‘gone’…..

these kind of transformations are quite common in NMR (rotating frame, interaction frame…)

WTF…

Page 23: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

the heteronuclear dipolar interaction between 1H (I-spins) and 13C (S-spins)

drives the polarization transfer:

H ~ b · Iz · Sz

B1

z

y

xduring CP, the spin lock pulses are along x-axis…

…transform to a frame that rotates around the x-axis

with ω1H (for the I spins) and ω1

C (for the S spins)

B1

z

y

xH’ ~ b’ ·( Iz · Sz + Iy · Sy ) cos (ω1H- ω1

C )t =

= b’ ·( Iz · Sz + Iy · Sy ) [ for ω1H = ω1

C ]

Hartmann-Hahn matching !

Cross polarization (CP) – part II

Page 24: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

H’ ~ b’ ·( Iz · Sz + Iy · Sy )

the magnetization is along x-axis

à transform to a frame with the new z’ axis along the old x-axis

x à z’ or Ix à Iz

replace x,y,z as follows: y à y’ or Iy à Iy

z à -x’ or Iz à -Ix

H’’ ~ b’ ·( Ix · Sx + Iy · Sy )

z

y

x x'

y'

z'

Ix = ½ ( I+ + I– )

Iy = -½·i ( I+ – I– )rewrite, using raising and lowering operators:

Cross polarization (CP) – part II

Page 25: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

H’’ = b’ ·( Ix · Sx + Iy · Sy )

H’’ = b’ [ ½ ( I+ + I– ) · ½ ( S + + S – ) + (-½·i) ( I+ – I– ) · (-½·i) ( S + – S – ) ]

H’’ = ½·b’ ( I +·S – + I –·S + )

I –

I +

I –

I +

0

0

I – S +

I + S –

I S I S

all other combinations give zero

Cross polarization (CP) – part II

Page 26: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

1H

1H netpolarization random

13C

I – S +

I + S –

I S I S

net magnetization is ‘spin-up’ for 1H (I-spins)

Cross polarization (CP) – part II

Page 27: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

1H

1H netpolarization random

13C

I – S +

I + S –

I S I S

net magnetization is ‘spin-up’ for 1H (I-spins)

Cross polarization (CP) – part II

Page 28: Cross polarization (CP)

Barth-Jan van Rossum: Solid-state NMR, advanced concepts I

Cross polarization (CP) – part II

13C

1H

1H netpolarization random

13C

1H

I – S +

13C net-polarization