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Echoes in NMR

K.V.R. Charychary@tifr.res.in

Workshop on “NMR and it’s applications in Biological Systems”TIFR November 23, 2009

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•Spin-echo pulse sequence• The vector method

•Uncoupled spins•Homonuclear coupled spins•Heteronuclear couples spins

•Product operator formalism•Uncoupled spins•Homonuclear coupled spins•Heteronuclear couples spins

Lecture plan

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ECHO

echo: repitition of sound by reflection of sound waves, secondary sound so

produced; reflected radio or radar wave-Concise Oxford Dictionary

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Spin-Echo Experiment

Physics Today, 1953; E. Hahn, Physical Reivew, 80, 1950

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ECHO

echo: repitition of sound by reflection of sound waves, secondary sound so

produced; reflected radio or radar wave-Concise Oxford Dictionary

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Uncoupled Spin Transformations

CHCl3

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Vector representation of bulk magnetization

MZ

MX

MY

X

Z

Y

X

Z

Y

X

Z

Y

8

Evolution of spin operator under a pulse

1Η

X

Iz -Iy

Iy -Iz

(π/2)X Pulse:

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Evolution of spin operator under a pulse

1Η

Y

Iz Ix

-Ix -Iz

(π/2)Y Pulse:

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• Iz→ Iz cos(β) + Ix sin(β)

• Iz→ Iz cos(β) − Iy sin(β)

Evolution of spin operator under a pulse

βy

βx

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• Iz → Iz cos(β) + Ix sin(β)

• Iz → Ix

• Iz → -Iz

Evolution of spin operator under a pulse

βy

(π/2)y

(π)y

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• Iz → Iz cos(β) - Iy sin(β)

• Iz → -Iy

• Iz → -Iz

Evolution of spin operator under a pulse

βx

(π/2)x

(π)x

13

Y

X

Y

X

Ωtt1

My Sin (Ωt)

Mx Cos (Ωt) Mx

Evolution under Chemical Shift (Hδ = ΩΗIz)

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Rotation about the z-axis• I1x→ I1x cos(Ωt) + I1y sin(Ωt)

• I1y→ I1y cos(Ωt) − I1x sin(Ωt)

• I1z→ I1z

Evolution under Chemical Shift (Hδ = ΩΗIz)

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Evolution under Chemical Shift (Hδ = ΩΙIz)

1Η

y

Ix Iy

-Iy -Ix

X/Y Magnetization :

Hδ= ΩΙIz

Angle = ΩΙ t

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(π/2)φ (π)ϕ

τ τ t

2τ = 1/2J

τ is the refocussing delay.

Spin-echo pulse sequence

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Vector Method

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Spin-echo pulse sequence

X

Y

τ π x X

Y

Ω t

b a

X

Y

X

Y

Ω t

b aτX

Y

τ π x X

Y

Ω t

b a

X

Y

Ω t

b a

X

Y

X

Y

Ω t

b a

X

Y

Ω t

b aτ

Effect of a SE sequence on an uncoupled spin with precessional frequency Ω.

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•Spin-echo pulse sequence removes the dephasing caused by the field inhomogenity.

•Other processes which contribute to the decay of transverse magnetization continue to exert their influence.

How about in the case of coupled spin-systems ?

Spin-echo pulse sequence

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Coupled Spin Transformations

CHCl2-CHOA X

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αX βX αA βA

ΩA+π(JAX/2) ΩA−π(JAX/2) ΩX+π(JAX/2) ΩX−π(JAX/2)

NMR Spectrum of an AX spin system. Offset frequencies are depicted below the individual lines. The spin states (α and β) of the coupled partner associated with individual lines are depicted on top of the spectrum.

ΩA−π(JAX/2) ΩX−π(JAX/2)ΩA+π(JAX/2) ΩX+π(JAX/2)

AX spin system

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In the case of coupled spin-systems

• The spin-echo sequence refocuses the chemical shifts.

• It phase modulates the J-couplings that may be present.

Spin-echo pulse sequence

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Effect of (π)x pulse

•Tilts the magnetization vectors with respect to x-axis.

• Swaps the associated spin states of the coupled partner.

Spin-echo pulse sequence

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Effect on a J-coupled spin A. The αx and βx represent the spin states of the coupling partner spin X.

X

Y

X

Y

τ ( π x)Aα Xβ X

X

Y α X

β Xβ X

X

Y

α X

τ( π x)X

α X

X

Y

β X

X

Y

X

Y

τ ( π x)Aα Xβ X

X

Y α X

β X

X

Y α X

β Xβ X

X

Y

α X

β X

X

Y

α X

τ( π x)X

α X

X

Y

β X

Spin-echo pulse sequence

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During the second delay time τ

• The vectors start precessing with exchanged frequencies, namely ΩΑ−2π(JAX/2) and ΩA+2π(JAX/2). Thus, the original vector associated with the spin state αx of spin X, which was precessing with a frequency ΩΑ+2π(JAX/2) during the first time interval τ, acquires a frequency ΩΑ-2π(JAX/2) after the (π)x pulse.

Spin-echo pulse sequence

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At end of the -τ-(πx)-τ- sequence

• The vector αx effectively spans an angle [-ΩΑ-2π(JAX/2)τ + ΩΑ+2π(JAX/2)τ] = 2πJAXτ.

• Its partner βx spans [-ΩΑ+2π(JAX/2)τ + ΩΑ-2π(JAX/2)τ] = -2πJAXτ.

• Thus, the spin-echo sequence in the case of coupled (homo-nuclear) spin system refocuses the chemical shifts, leaving behind the phase modulation due to only the J-couplings.

Spin-echo pulse sequence

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•When τ is set to 1/4JAX, the two frequencies align along the ±y-axes giving rise to an anti-phase doublet.

Spin-echo pulse sequence

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XαX/βXX

αX

βX

The effect on a J-coupled spin A doublet. a) when τ is set to 1/4JAX, and b) when τ is set to 1/2JAX.

a b

Spin-echo pulse sequenceY Y

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•When τ is set to 1/2JAX, the two frequencies align along the –x-axis giving rise to an inverted in-phase doublet.

•Such time delays are often used in homo-nuclear and hetero-nuclear higher dimensional NMR experiments which will be discussed later.

Spin-echo pulse sequence

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XαX/βXX

αX

βX

The effect on a J-coupled spin A doublet. a) when τ is set to 1/4JAX, and b) when τ is set to 1/2JAX.

a b

Spin-echo pulse sequenceY Y

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In the case of “unlike spins”

•The effect of the spin-echo sequence is different depending on whether the (π)x pulse is applied on both spins or only on one of them.

Spin-echo pulse sequence

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In the case of “unlike spins”

•When the (π)x pulse is applied on both the spins, we end up observing the same phase modulation as in the case of “like spins”.

(π/2)φ (π)ϕ

t

(π/2)φ (π)ϕ

τ τ t

Spin-echo pulse sequence

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However, when it is applied only along A spin, then there is no phase modulation of the hetero-nuclear JAX.

Spin-echo pulse sequence

(π/2)φ (π)ϕ

t

(π/2)φ (π)ϕ

τ τ t

A

X

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The effect of a spin echo sequence ((π/2)y)A-τ-(π x)A-τ on a J-coupled spin A. Note that pulses are applied only on the spin A.

Spin-echo pulse sequence

X

Y

τ ( π x)AX

Y

τX

Y

α X

β X

α X

X

Y

β X

X

Y

τ ( π x)AX

Y

τX

Y

α X

β X

X

Y

α X

β X

α X

X

Y

β X

α X

X

Y

β X

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When the (π)x pulse is applied only to X spin, the A spin gets frequency labelled and there is no phase modulation due to the hetero-nuclear JAX.

Spin-echo pulse sequence

(π/2)φ (π)ϕ

t

(π/2)φ (π)ϕ

τ τ t

A

X

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The effect of a spin echo sequence ((π/2)y)A-τ-(π x)X-τ on a J-coupled spin A. Note that (πx) pulse is applied only on the spin X.

Spin-echo pulse sequence

X

Y

τ ( π x)X τ

α X

X

Y

β X

β X

X

Y

α X

X

α X/ βXY

X

Y

τ ( π x)X τ

α X

X

Y

β X

α X

X

Y

β X

β X

X

Y

α X

β X

X

Y

α X

X

α X/ βXY

X

α X/ βXY

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Spin-echo sequences are extensively used in hetero-nuclear experiments. It enables one to retain the chemical shift information of spin X and simultaneously refocus the evolution under the hetero-nuclear coupling JAX.

Spin-echo pulse sequence

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Product Operator Formalism

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Let us consider a two-spin system (spin k and spin l; which form a weakly coupled AX spin system. In such a situation, the spectrum of each spin is a doublet. The equilibrium magnetization prior to the application of the first (π/2)y pulse is represented by the density operator at an instant of time 1 as

ρ1 = Ikz + Ilz

The (π/2)y pulse tilts the magnetization of both k and l to x axis, which is given as

ρ2 = +Ikx+Ilx

(π/2)φ (π)ϕ

τ τ t 1 2 3 4 5

Spin-echo pulse sequence

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ρ2 = +Ikx+Ilx

ρ2 evolves during time t1 in the transverse plane under what is called as a free Hamiltonian. The operator consists of two terms namely a chemical shift term (Hδ= ΩκIkz) and a coupling term (HJ=2πJklIkzIlz). As the evolution of both Iky and Ily is identical, we consider only one for our discussion purpose. Later we can extend the same for the other term.

A criterion:

If the two Hamiltonians Hδ and HJ contain commuting parts, then their effects can be considered independently.

[Ikz, IkzIlz] = [Ikz, Ikz]Ilz = 0

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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The term Ikx evolves under Hδ as follows:

ρ3 = Ikx cos(Ωkτ) + Iky sin(Ωkτ)

ρ3 is transformed into the following by the next nonselective (π)x pulse.ρ4 = Ikx cos(Ωkτ) - Iky sin(Ωkτ)

The term ρ4 further evolves under Hδ during the second τ as follows:

ρ5 = Ikx cos(Ωkτ) + Iky sin(Ωkτ) cos(Ωkτ) - Iky cos(Ωkτ) + Ikx sin(Ωkτ) sin(Ωkτ)

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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ρ5 = Ikx cos(Ωkτ) + Iky sin(Ωkτ) cos(Ωkτ) -

Iky cos(Ωkτ) + Ikx sin(Ωkτ) sin(Ωkτ) or

ρ5 = Ikx cos2(Ωkτ) +Ikx sin2(Ωkτ) = +Ikx

At the end of the second τ, the magnetization returns to Ikx

This is precisely what is known as refocussing

How about evolution under HJ ?

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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Two coupled spins

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Four operators each of spin1 (11, I1x, I1y, I1z) and spin2 (12, I2x, I2y, I2z)

Total number of operators (4 X 4=16)• Unity operator• I1x, I1y, I1z (Single quantum; in-phase)• I2x, I2y, I2z (Single quantum; in-phase)• 2I1xI2z, 2I1yI2z , 2I1zI2x, 2I1zI2y (Single quantum; anti-phase)• 2I1zI2z (Longitudinal 2-spin order)• 2I1xI2x, 2I1yI2y, 2I1xI2y, 2I1yI2x (Multiple Quantum)

Two-spin Product Operators

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• I1x→ I1x cos(πJ12t) + 2I1yI2z sin(πJ12t)

• I1y→ I1y cos(πJ12t) − 2I1xI2z sin(πJ12t)

• 2I1xI2z → 2I1xI2z cos(πJ12t) + I1y sin(πJ12t)

• 2I1yI2z → 2I1yI2z cos(πJ12t) − I1x sin(πJ12t)

Evolution under Couplings (HJ = 2πJ12I1zI2z)

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Evolution under Couplings (HJ = 2πJ12I1zI2z)

1Η

y

I1x 2I1yI2z

-2I1yI2z -I1x

X Magnetization :HJ=2πJ12I1zI2z

Angle =2πJ12t

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Evolution under Couplings (HJ = 2πJ12I1zI2z)

1Η

x

I1y 2I1xI2z

-2I1xI2z -I1y

Y Magnetization :HJ=2πJ12I1zI2z

Angle =2πJ12t

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Vector repre’tion of in-phase and anti-phase magnetization

I1X

2I1xI2z

I1Y

X

Z

Y

X

Z

Y

X

Z

Y

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Vector repre’tion of in-phase and anti-phase magnetization

I1X

2I1XI2Z

2I1YI2Z

X

Z

Y

X

Z

Y

X

Z

Y

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Observable CoherencesRules

• POs containing a single transverse component Ix or Iy lead to observable signals. (for example, one spin operators Ikx and Iky)

• POs with more than one transverse component (for example 2IkyIlx) lead to multiple quantum coherences and are not observed.

• POs with one transverse component and rest of the components are z-terms, lead to signals with anti-phase character (for example, 2IkxIlz and 3IkxIlzImz).

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In-phase/anti-phase magnetization

Ikx Ilx

Ω k Ω l

Iky Ily

Ω k Ω l

α l β l α k β k

α l β l α k β k

2IkxIlz 2IkzIlx 2IkyIlz 2IkzIly

Ω k Ω l Ω k Ω l

α l β l α k β k

α l β l α k β k

Ikx Ilx

Ω k Ω l

Iky Ily

Ω k Ω l

α l β l α k β k

α l β l α k β k

Ikx Ilx

Ω k Ω l

Iky Ily

Ω k Ω l

Ikx Ilx

Ω k Ω l

Iky Ily

Ω k Ω l

α l β l α k β k

α l β l α k β k

α l β l α k β k

α l β l α k β k

2IkxIlz 2IkzIlx 2IkyIlz 2IkzIly

Ω k Ω l Ω k Ω l

α l β l α k β k

α l β l α k β k

2IkxIlz 2IkzIlx 2IkyIlz 2IkzIly

Ω k Ω l Ω k Ω l

α l β l α k β k

α l β l α k β k

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• I1x→ I1x cos(πJ12t) + 2I1yI2z sin(πJ12t)

• I1y→ I1y cos(πJ12t) − 2I1xI2z sin(πJ12t)

• 2I1xI2z → 2I1xI2z cos(πJ12t) + I1y sin(πJ12t)

• 2I1yI2z → 2I1yI2z cos(πJ12t) − I1x sin(πJ12t)

Evolution under Couplings (HJ = 2πJ12I1zI2z)

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Now let us consider the evolution of the term Ikx under HJ:

ρ3 = Ikx cos(πJklτ) + 2IkyIlz sin(πJklτ)

ρ3 is transformed into the following by the next nonselective (π)x pulse.ρ4 = Ikx cos(πJklτ) + 2IkyIlz sin(πJklτ)

The term ρ4 further evolves under HJ during the second τ as follows:

ρ5 = Ikx cos(πJklτ) + 2IkyIlz sin(πJklτ) cos(πJklτ) + 2IkyIlz cos(πJklτ) - Ikx sin(πJklτ) sin(πJklτ)

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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ρ5 = Ikx cos(πJklτ) + 2IkyIlz sin(πJklτ) cos(πJklτ) + 2IkyIlz cos(πJklτ) - Ikx sin(πJklτ) sin(πJklτ)

orρ5 = Ikx cos2(πJklτ) + 2IkyIlz sin(πJklτ)cos(πJklτ) +

2IkyIlz cos(πJklτ) sin(πJklτ) - Ikx sin2(πJklτ)

ρ5 = Ikx cos2(πJklτ) + 2IkyIlz sin(πJklτ)cos(πJklτ) + 2IkyIlz cos(πJklτ) sin(πJklτ) - Ikx sin2(πJklτ)

ρ5 = Ikx cos(2πJklτ) + 2IkyIlz sin(2πJklτ)

ρ5 ≠ ρ2

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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ρ5 = Ikx cos(2πJklτ) + 2IkyIlz sin(2πJklτ)

Thus, the spin-echo sequence in the case of coupled (homo-nuclear) spin system refocuses the chemical shifts, leaving behind the phase modulation due to only the J-couplings.

If τ is set to 1/4J,cos(2πJklτ) = 0; sin(2πJklτ) = 1

ρ5 = 2IkyIlz

y-magnetization of spin k anti-phase with respect to spin l

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

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ρ5 = Ikx cos(2πJklτ) + 2IkyIlz sin(2πJklτ)

If τ is set to 1/2J,

cos(2πJklτ) = -1; sin(2πJklτ) = 0

ρ5 = -Ikx

x-magnetization of spin k in-phase with respect to spin l

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

( π / 2 )φ ( π )ϕ

τ τ t1 2 3 4 5

57

XαX/βXX

αX

βX

The effect on a J-coupled spin A doublet. a) when τ is set to 1/4JAX, and b) when τ is set to 1/2JAX.

a b

Spin-echo pulse sequenceY Y

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In the case of unlike spins !!!

Spin-echo pulse sequence

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(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

Spin-echo pulse sequence

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Let us consider a heteronuclear two-spin system (spin I and spin S); Automatically they form a weakly coupled AX spin system. In such a situation, the spectrum of each spin is a doublet. The equilibrium magnetization prior to the application of the first (π/2)y pulse is represented by the density operator at an instant of time 1 as

ρ1 = Iz

Effect of (π/2)y pulse tilts the magnetization to x-axis, which is given as

ρ2 = +Ix

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

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ρ2 = +IX

ρ2 evolves during time τ in the transverse plane under what is called as a free Hamiltonian. The operator consists of two terms namely a chemical shift term (Hδ= ΩΙIz) and a coupling term (HJ=2πJISIzSz).

A criterion:

If the two Hamiltonians Hδ and HJ contain commuting parts, then their effects can be considered independently.

[Iz, IzSz] = [Iz, Iz]Sz = 0

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

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The term Ikx evolves under Hδ as follows:

ρ3 = Ix cos(ΩIτ) + Iy sin(ΩIτ)

ρ3 is transformed into the following by the simultaneous (π)x pulses applied on both I and S spinsρ4 = Ix cos(ΩIτ) - Iy sin(ΩSτ)

The term ρ4 further evolves under Hδ during the second τ as follows:

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) - Iy cos(ΩIτ) - Ix sin(ΩIτ) sin(ΩIτ)

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

63

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) - Iy cos(ΩIτ) + Ix sin(ΩIτ) sin(ΩIτ)

or

ρ5 = Ix cos2(Ωiτ) +Ix sin2(Ωiτ) = +Ix

At the end of the second τ, the magnetization returns to Ikx

How about evolution under HJ ?

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

64

Now let us consider the evolution of the term Ix under HJ:

ρ3 = Ix cos(πJISτ) + 2IySz sin(πJISτ)

ρ3 is transformed into the following by the simultaneous (π)x pulses on both the spins I and S

ρ4 = Ix cos(πJISτ) + 2IySz sin(πJISτ)

The term ρ4 further evolves under HJ during the second τ as follows:

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) + 2IySz cos(πJISτ) - Ix sin(πJISτ) sin(πJISτ)

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

65

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) + 2IzSz cos(πJISτ) - Ix sin(πJISτ) sin(πJISτ)

orρ5 = Ix cos2(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) +

2IySz cos(πJISτ) sin(πJISτ) - Ix sin2(πJISτ)

ρ5 = Ikx cos(2πJISτ) + 2IkyIlz sin(2πJISτ)

ρ5 ≠ ρ2

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

66

ρ5 = Ix cos(2πJISτ) + 2IySz sin(2πJISτ)

Thus, the spin-echo sequence in the case of coupled (hetero-nuclear) spin system refocuses the chemical shifts, leaving behind the phase modulation due to only the J-couplings.

If τ is set to 1/4J,

cos(2πJISτ) = 0; sin(2πJISτ) = 1

ρ5 = 2IySz

y-magnetization of spin I anti-phase with respect to spin S

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

67

ρ5 = Ix cos(2πJISτ) + 2IySz sin(2πJISτ)

If τ is set to 1/2J,

cos(2πJISτ) = -1; sin(2πJISτ) = 0

ρ5 = -Ix

x-magnetization of spin I in-phase with respect to spin S

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

68

XαX/βXX

αX

βX

The effect on a J-coupled spin A doublet. a) when τ is set to 1/4JIS, and b) when τ is set to 1/2JIS.

a b

Spin-echo pulse sequenceY Y

69

ρ1 = Iz

ρ2 = Ix

ρ3 = Ix cos(ΩIτ) + Iy sin(ΩIτ)

ρ4 = Ix cos(ΩIτ) - Iy sin(ΩSτ)

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) -

Iy cos(ΩIτ) - Ix sin(ΩIτ) sin(ΩIτ)

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

70

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) - Iy cos(ΩIτ) - Ix sin(ΩIτ) sin(ΩIτ)

or

ρ5 = Ix cos2(Ωiτ) +Ix sin2(Ωiτ) = +Ix

How about evolution under HJ ?

Spin-echo pulse sequence( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

( π / 2 )φ ( π )ϕ

τ τ t

1 2 3 4 5

I

S

71

Now let us consider the evolution of the term Ix under HJ:

ρ3 = Ix cos(πJISτ) + 2IySz sin(πJISτ)

ρ3 is transformed into the following by the next nonselective (π)x pulse (applied only on the spin I).

ρ4 = Ix cos(πJISτ) - 2IySz sin(πJISτ)

The term ρ4 further evolves under HJ during the second τ as follows:

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) - 2IySz cos(πJISτ) - Ix sin(πJISτ) sin(πJISτ)

Spin-echo pulse sequence(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

72

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) - 2IySz cos(πJISτ) + Ix sin(πJISτ) sin(πJISτ)

orρ5 = Ix cos2(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) -

2IySz cos(πJISτ) sin(πJISτ) + Ix sin2(πJISτ) ρ5 = Ikx ρ5 = ρ2

Thus, in the case of heteronuclear coupled spin system the spin-echo sequence with the application of the π pulse only on the I- spin refocuses both the chemical shifts as well as the heteronuclear J-couplings.

Spin-echo pulse sequence(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

73

ρ5 = Ix cos(2πJISτ) + 2IySz sin(2πJISτ)

Thus, the spin-echo sequence in the case of coupled (homo-nuclear) spin system refocuses the chemical shifts, leaving behind the phase modulation due to only the J-couplings.

If τ is set to 1/4J,

cos(2πJISτ) = 0; sin(2πJISτ) = 1

ρ5 = 2IySz

y-magnetization of spin I anti-phase with respect to spin S

Spin-echo pulse sequence(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

74

ρ5 = Ix cos(2πJISτ) + 2IySz sin(2πJISτ)

If τ is set to 1/2J,

cos(2πJISτ) = -1; sin(2πJISτ) = 0

ρ5 = -Ix

x-magnetization of spin I in-phase with respect to spin S

Spin-echo pulse sequence(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

75

ρ1 = Iz

ρ2 = Ix

ρ3 = Ix cos(ΩIτ) + Iy sin(ΩIτ)

ρ4 = Ix cos(ΩIτ) + Iy sin(ΩSτ)

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) +

Iy cos(ΩIτ) - Ix sin(ΩIτ) sin(ΩIτ)

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

Spin-echo pulse sequence

76

ρ5 = Ix cos(ΩIτ) + Iy sin(ΩIτ) cos(ΩIτ) + Iy cos(ΩIτ) - Ix sin(ΩIτ) sin(ΩIτ)

or

ρ5 = Ix cos(2Ωiτ) +Iy sin(2Ωiτ)

How about evolution under HJ ?

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

Spin-echo pulse sequence

77

Now let us consider the evolution of the term Ix under HJ:

ρ3 = Ix cos(πJISτ) + 2IySz sin(πJISτ)

ρ3 is transformed into the following by the next nonselective (π)x pulse (applied only on the spin S).

ρ4 = Ix cos(πJISτ) - 2IySz sin(πJISτ)

The term ρ4 further evolves under HJ during the second τ as follows:

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) - 2IySz cos(πJISτ) - Ix sin(πJISτ) sin(πJISτ)

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

Spin-echo pulse sequence

78

ρ5 = Ix cos(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) - 2IySz cos(πJISτ) + Ix sin(πJISτ) sin(πJISτ)

orρ5 = Ix cos2(πJISτ) + 2IySz sin(πJISτ) cos(πJISτ) -

2IySz cos(πJISτ) sin(πJISτ) + Ix sin2(πJISτ) ρ5 = Ikx ρ5 = ρ2

Thus, in the case of heteronuclear coupled spin system the spin-echo sequence with the application of the π pulse only on the S- spin refocuses heteronuclear J-couplings but the transverse I-magnetization gets frequency labeled.

Spin-echo pulse sequence(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

(π/2)φ (π)ϕ

τ τ t

1 2 3 4 5

I

S

79

A double resonance experiment2D-HSQC

HSQC