The Product Operator Cookbook

Alec Ricciuti

BMB 601 – Fundamentals of MR

April 26, 2011

Outline

• Review

– Notation

– General product operators

• Levitt’s box notation

– Populations and coherences

– Physical interpretation

• Homonuclear two-spin system– Thermal equilibrium

– Radio-frequency pulses

– Free evolution• Chemical shift

• J-coupling

• Relaxation

– Observables

• Heteronuclear– Dipolar couplings

• Summary

Notation Review

• Eigenbasis of α, β states for a single spin-1/2 nuclei

• A given spin usually exists in a superposition ψ of these states, with normalized coefficients

• These can be used to construct angular momentum & half-unity operators

0

1α

1

0β

β

α

βα c

cβcαcψ

1**

β

α

βα c

cccψψ

01

10ˆ

21

21 αββαIx

10

01ˆ

21

21 ββααIz

01

10ˆ

i21

i21 αββαIy

10

011̂ 2

121

21 ββαα

Product Operator Review

• The most general recipe from previously

– For a two-spin system

– Two examples out of 16 vvs IIB 21ˆˆ2

N

k

akv

qs

skIB1

)1( )ˆ(2

0100

1000

0001

0010

i2

1

01-

101

01-

100

01-

100

01-

101

i2

1

01-

10

10

01ˆ1̂2ˆ

i21

221

2 yy II

0010

0001

1000

0100

2

1

10

010

10

011

10

011

10

010

2

1

10

01

01

10ˆˆ2 2

121 zx II

Levitt’s Box Notation

• These product operators provide a basis for the spin density operator , here in terms of the two-spin superposition ψ

ββ

βα

αβ

αα

βββααβαα

c

c

c

c

ββcβαcαβcααcψ

ρ̂

****

****

****

****

****ˆ

βββββαββαβββααββ

βββαβαβααββαααβα

ββαββααβαβαβαααβ

ββααβααααβαααααα

βββααβαα

ββ

βα

αβ

αα

cccccccc

cccccccc

cccccccc

cccccccc

cccc

c

c

c

c

ψψρ

ββββ

ββαα

βαβα

αααα

ρρρρ

ρρρρ

ρρρρ

ρρρρ

ρ̂

Populations and Coherences

• As a linear sum of product operators,

ββββ

ββαα

βαβα

αααα

yxz

ρρρρ

ρρρρ

ρρρρ

ρρρρ

IIcIbaρ 211ˆˆˆ1̂ˆ

– Diagonal elements define population distributions

1

1

1

1

2

11̂zI

2

2

2

1ˆˆ21 zz II

1

1

1

1

2

1ˆˆ21 zzII

αββα

αα

ββ

Populations and Coherences

• As a linear sum of product operators,

ββββ

ββαα

βαβα

αααα

yxz

ρρρρ

ρρρρ

ρρρρ

ρρρρ

IIcIbaρ 211ˆˆˆ1̂ˆ

– Off-diagonalsdefine coherences

1

1

1

1

2

1ˆˆ2 21 zxII

2

2

i2

1ˆˆ2ˆˆ2 2121 yxxy IIII

2

2

2

1ˆˆ2ˆˆ2 2121 yyxx IIII

αββα

αα

ββ

Physical Interpretation

The term Îkv in the density operator indicates that spins k tend to align with the v-axis

• Î1x indicates spins I1 align with the x-axis

• Î2z indicates spins I2 align with the z-axis

The term 2ÎkuÎlv in the density operator indicates that the u-components of spins k correlate with v-components of spins l

• 2Î1x Î2x indicates I1 are correlated with I2 on the x-axis

• 2Î1y Î2z indicates I1 y-axis polarizations are correlated with I2 z-axis polarizations

• -2Î1x Î2y indicates I1 x-axis polarizations are anticorrelated with I2 y-axis polarizations

Thermal Equilibrium

• Spin density operator estimation generally assumes no coherences and Boltzmann-distributed populations– High field and high temperature approximations allow

for simpler expressions

– Complete density operator is a starting point for spin dynamics calculations

Tk

Bγ

Tk

ω

e

eρ αα

s

Tkω

Tkω

αααα

αα

B

0

B/

/eq

44

11

4

1ˆ

B

B

4

1ˆˆ eqeq βααβ ρρ Tk

Bγρ ββ

B

0eq

44

1ˆ

TkBγ

TkBγ

ρ

B

0

B

0

441

41

41

441

eqˆ

Radio Frequency Pulses• βp = ωnutτp

– Duration assumed too short for resonance offsets and coupling to affect evolution

– Rotation operator with respect to v-axis is simple• Arbitrary rotation phase sandwiches this between z-axis rotations in and

out of phase

– Application to density operators also uses the sandwich formula • Commutation relationships simplify rotation of the density operators

vIβ

pv eβR 1pˆi

1 )(ˆ )(ˆˆ)(ˆˆ

12 pφpφ βRρβRρpp

)(ˆ)(ˆˆˆ2)(ˆ)(ˆ)(ˆˆˆ2)(ˆ12212121 pxpxyxpxpxpxyxpx βRβRIIβRβRβRIIβR

)(ˆˆ)(ˆ)(ˆˆ)(ˆ2 222111 pxypxpxxpx βRIβRβRIβR

Free Evolution• Under free evolution, the product operators that

comprise a given density operator can be calculated between any time τ2 and τ3

– The sandwich formula applies the effects of the spin Hamiltonian Ĥ0, which contains the chemical shift and J-coupling evolutions

– The Hamiltonian terms commute, so they can be layered in the sandwich in any order

– We’re ignoring the effects of relaxation and dipole-dipole coupling for now

1

23 )(ˆˆ)(ˆˆ τUρτUρ τHeτU0ˆi)(ˆ

zzzz IIJπIIH 21122021

01

0 ˆˆ2ˆΩˆΩˆ

Chemical Shift Evolution

• Chemical shift transformations are similar to those under r.f. pulses– Rotations are about the z-axis only– The different rotation angles of each spin must be

applied where relevant

J-Coupling Evolution

• J-coupling mediates the creation and destruction of correlations between spin polarizations

– Transverse polarizations of spin I1 develop a correlation with longitudinal polarizations of spins I2

– Double- and zero-quantum coherences do not evolve

Relaxation

• Simplifying assumption for product operators involving coherences– Multiply each operator by the decay factor e-λτ

– This assumption does not apply if relaxation mechanisms are cross-correlated

• Formal treatment of relaxation is important to study motional processes and ‘hidden’ spin interactions such as dipole-dipole coupling in isotropic liquids– Teasing apart different relaxation mechanisms is not

within our scope here

Observables

• All product operators containing a single transverse component and an arbitrary number of longitudinal components are observable

– Assumes all couplings are resolved

• Example

)s in(ˆ)cos(ˆˆ2ˆˆ2 1211221

ˆˆ221

2112 τJπIτJπIIII yzxIIτJπ

zxzz

Dipolar Couplings

• For a heteronuclear two-spin system

– Larmor frequencies in the spin Hamiltonian depend on the angle of the molecule Θ with respect to the magnetic field

– We can get angular information, but in general this doesn’t contribute to predictive uses of product operators

zzISzSzI SIωSωIωH ˆˆ2ˆˆˆ 00

Θ100 IzzII δBγω Θ100 S

zzSS δBγω

1Θcos32

1 2 ISISIS bd

Summary

• Matrix representations of product operators provide intuitive insight into pulse sequences and chemical shift, J-coupling evolutions

• The predictive nature of product operators for these processes can be exploited to promote observable magnetization from antiphase coherence

Questions?

References:• Donne, D.G. & Gorenstein, D.G. 1997, "A pictorial representation of product operator formalism:

nonclassical vector diagrams for multidimensional NMR", Concepts in Magnetic Resonance, vol. 9, no. 2, pp. 95-111.

• Levitt, M.H. 2001, Spin dynamics: basics of nuclear magnetic resonance, Wiley.• Sorensen, O.W., Eich, G.W., Levitt, M.H., Bodenhausen, G. & Ernst, R.R. 1983, "Product operator

formalism for the description of NMR pulse experiments", Progress in Nuclear Magnetic Resonance Spectroscopy, vol. 16, no. 2, pp. 163-192.