Perturbations, Rotating Frame,

FIDs- Chapter 4, on web site: http://www.chem.ubc.ca/faculty/straus/c518_09.html

A) Local magnetic field, B1:

1) How do we generate it?2) How does it affect the spins?

B) Rotating Frame

1) Ideal pulses2) Real pulses

C) Signal detected in NMR: FID

CW-NMR

Pulsed-NMR

e.g. flip magnetization by 90 degrees

A) Local magnetic field, B1:

1) How do we generate it?

Spins align with magnetic field need to generate a local magnetic field

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

But excitation needs to be at a specific frequency...

Other coil geometries:

Scroll coil

Birdcage coil

Example of a double resonance probe circuit:

Why do many of these circuit diagrams contain variable capacitors?

2) How does it affect the spins?

Recall:

time-dependence makes this difficult to solve!

B) Rotating frame http://www.youtube.com/watch?v=MIV0HlY_CVs

Magnetization and one component of the B1field will resonate.

The other component will be non-resonant and can usually be ignored.

Cases where the non-resonant component cannot be dropped: Bloch-Siegert shift

Now we can start working with the Bloch equations from the previous chapter:

ROTATING FRAME

Need to transform into rotating frame:

Bloch equations in rotating frame:

Some definitions:

Bγ= ∆

FT

* =tp

Pulse length (time, tp)

Source: bionmr-c1.unl.edu/921/Lectures/chapter-6-NMR-pulse.ppt

2) Real pulses

Make use of two facts:

1) During the pulse, the Bloch equations are valid2) Relaxation is negligible during the pulse, because

1 2,T Tτ

Examine 2 cases: 1) on-resonance2) off-resonance

FT

On-resonance:

If

PW (µs)

Peak

Inte

nsity

180o pulse

Source: bionmr-c1.unl.edu/921/Lectures/chapter-6-NMR-pulse.ppt

Off-resonance:

Can no longer describe analytically

Trajectory of (90o)x(180o)y(90o)x composite pulse with an incorrect 180o pulse length, where the effective pulse is 160o.

Even with the significant error, the net magnetization still winds up very close to -z

Off-resonance effects can be compensated by using composite pulses:

180 degree pulse: off-resonance

Summary

- Perturb spin system by applying a field- The time dependence requires us to transform into a

rotating frame- We only consider the component of B1 which is resonant

with the magnetization vector

- Looked at the on-resonance case and the off-resonance case

Trajectory of (90o)x(180o)y(90o)x composite pulse with an incorrect 180o pulse length, where the effective pulse is 160o.

Even with the significant error, the net magnetization still winds up very close to -z

Off-resonance effects can be compensated by using composite pulses:

180 degree pulse: off-resonance

NMR Pulse (spin gymnastics)Decoupling

a) MLEV-4 composite pulse decoupling schemei. Based on the composite pulse:

(90o)x(270o)y(90o)x = RMLEV-16 decouples efficiently ± 4.5 kHz

b) WALTZ-16i. Based on the composite pulse:

(90o)x(180o)-x(270o)x decouples efficiently over ± 6 kHz Corrects imperfections in MLEV 90o ~ 100µs reduces sample heating 1 = 90o, 2=180o, 3=270o, 4=360o

c) GARPi. Computer optimized using non-90o flip angles

Effective decoupling bandwidth of ± 15 kHz 90o ~ 70 µs

)( phasereverseRRRRR −

Trajectory of 1H nuclei after two R MLEV-4 pulses results inan effective 360O pulse. Results is improved slightly byfollowing with two R pulses with reverse phase.

)(234213243324213243324213243324213243

phasereverseR−

NMR IN BIOMEDICINE, VOL. 10, 372–380 (1997)

C) Signal detected in NMR: FID

signal right after the pulse

... and now we must let the system evolve

transform from rotating back to lab frame

Spectrometers are equipped with a phase sensitive detector – so detect:

Quadrature detection

• In the old days the frequency of B1 (carrier) was somewhere higher than all other frequencies. This was done to avoid having frequencies faster (or slower) than the carrier, so the computer always knew the sign of the frequencies in the FID.

• Two problems: 1) noise, which is always there, is not sampled properly and its aliased into our spectrum. 2) in order to excite lines far from the carrier, we need very good pulses, which is never the case.

carrier

carrier

Source: http://tonga.usip.edu/gmoyna/NMR_lectures/NMR_lectures.html

Quadrature detection

• The phase of the Faster signals is opposite to that of the Slower signals, andthe computer is then able to sort this out.

ω (B1)

S

F

S

F

PH = 0

PH

= 9

0P

H =

0P

H =

90

F

F

S

S

Source: http://tonga.usip.edu/gmoyna/NMR_lectures/NMR_lectures.html

But we don’t measure the time-domain signal, but rather the frequency domain signal, so:

Some important points about the data:

1) Sampling rate

•The Nyquist Theorem says that we have to sample at least twice as fast than the fastest (higher frequency) signal

• If we sample twice as fast as the frequency, i.e. where the dots are then we have sufficient sampling

• If, on the other hand, we sample at half the speed at ,we end up with a digitized signal in the computer at 1/2 of the real frequency. These peaks will fold over with the wrong phase in our spectrum. This is called aliasing.

SR = 1 / (2 * SW)

Source: http://tonga.usip.edu/gmoyna/NMR_lectures/NMR_lectures.html

2) Data processing:

Signal + noise… Noise…

F(t) = 1 * e - ( LB * t ) - or - F(t) = 1 * e - ( t / τ )

FT FT

LB = -1.0 HzLB = 5.0 Hz

Other useful window functions

• Gaussian/Lorentzian

• Hanning

• Cosine/Sine

F(t) = e - ( t * LB + σ2 t2 / 2 )

F(t) = 0.5 + 0.5 * cos( π t / tmax )

F(t) = cos( π t / tmax )