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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.