Basic principles of NMR NMR signal origin, properties, detection, and processing

NTDR, 2012

Nils NybergNPR, Department of Drug Design and Pharmacology

Basic principles of NMRNMR signal origin, properties, detection, and processing

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Outline

1000 – 1045

• Establishing current knowledge level• Nuclear Magnetic Resonance phenomenon• Vector model, in and out of the rotating frame

1045 – 1100

• Short break1100 – 1130

• The phase of pulses and signals• Effect of different chemical shifts in the vector model• Effect of homonuclear coupling in the vector model• The spin-echo sequence (homonuclear case)• The spin-echo sequence (heteronuclear case)

1130 – 1200

• Spin-echo exercise1215 – 1315

• Lunch

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Outline

1215 – 1315

• Lunch1315 – 1415

• Signal processing• Window functions• Fourier transform• Real and imaginary parts• Phasing• Topspin starter

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Establishing current knowledge level

Instrument and terms• Magnet• Probes• Amplifiers• Receiver• ADC• Gradients• Temperature control• Lock• Shimming

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Establishing current knowledge level

Parameters• Chemical shifts• Integrals• Phases• Coupling constants• Line widths

• life time of signals, shimming, exchange, dynamics

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Nuclear Magnetic Resonance phenomenon

Nuclear: concerns the nuclei of atoms.

Magnetic: uses the magnetic properties of the nuclei.

Resonance: physics term describing oscillations.

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Resonance

A system prefers some frequencies over others…

A small energy input at the right frequency will give large oscillations…

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The magnetic properties of atomic nuclei

Atoms has a spin quantum number, I, and a magnetic quantum number, m = 2×I +1.

The magnetic quantum number = the number of different energy levels when the atom is placed in an external magnetic field.

Spin I = 0: 12C, 16OSpin I = ½: 1H, 13C, 15N, 19F, 31P, 77SeSpin I = 1: 2H, 14NSpin I = 1½:33S, 35Cl, 37Cl

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Chemical shifts

The energy for a spin ½ nuclei can take two different levels in a magnetic field.

The population of the two states is almost equal. A small surplus in the low energy α spin state and slightly fewer atoms in the higher β spin state.

Stronger magnetic field = larger energy differences between the states.

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Chemical shifts

A magnet provides the static field (B0) in the NMR instrument.

The rest of the molecule provides a ’local magnetic field’, which is dependent on structure.

Chemical shifts

The chemical shifts are expressed on a frequency scale (by convention plotted in reverse direction).

To make spectra comparable between instruments, the frequencies are expressed in parts per million [ppm] relative to a reference frequency.

Early instruments with electromagnets worked by slowly change the magnetic field. Hence the terms ‘Downfield’ and ‘Upfield’.

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• Less shielded• More

deshielded• Downfield• Higher

frequency

• More shielded• Less

deshielded• Upfield• Lower

frequency

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Vector model (a statistical abstraction…)

Unordered collection of½-spin nuclei, with a magnetic moment (μ).

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Vector model

Unordered collection of½-spin nuclei, with a magnetic moment (μ).

In an external magnetic field, the magnetic moment starts to precess…

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Vector model

Unordered collection of½-spin nuclei, with a magnetic moment (μ).

In an external magnetic field, the magnetic moment starts to precess…

…and aligns, at an angle of 54.7°, with the external field…

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Vector model

Unordered collection of½-spin nuclei, with a magnetic moment (μ).

In an external magnetic field, the magnetic moment starts to precess…

…and aligns, at an angle of 54.7°, with the external field…

…either up (along the field, slightly lower energy) or down (opposite the field, slightly higher energy) according to the Boltzmann distribution.

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Boltzmann distribution

The distribution of spins in a-state relative those in the b-state is described by the Boltzmann distribution.

The number of spins in each state is almost equal. There is a small surplus in the lower state.

Calculate how many spins in total you need to get one extra spin in the low energy state![1H, 600 MHz, 298 K]

273.15) (TcK in eTemperatur T

)(s Hz Frequency,

constant)(Planck Js 106.6

constant) (BoltzmannJ/K 104.1

logarithm) (natural 718.2

)exp(

1-

34

23

)(

h

k

hE

e

eTk

E

N

N Tk

E

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Boltzmann distribution

One spin extra in the low energy state![1H, 600 MHz, 298 K]

Nβ = 12 922

Nα = 12 923

Σ = 25 845

12922

)1(

1

))exp(1(

1

1))exp(1(

1)exp(

)exp(1

)exp(1

)exp(

)298104.1

10600106.6(

23

634

e

N

Tkh

N

Tk

hN

NTk

hN

NTk

hN

Tk

h

N

N

Tk

E

N

N

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Vector model

The ordered collection of spins can be handled from a common origin.

The Boltzmann distribution of up- and down-spins, makes a net magnetic vector along the external field (green).

An external magnetic field (radio frequency pulse, B1) perpendicular to the first (B0) have two effects:

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Vector model

The ordered collection of spins can be handled from a common origin.

The Boltzmann distribution of up- and down-spins, makes a net magnetic vector along the external field (green).

An external magnetic field (radio frequency pulse, B1) perpendicular to the first (B0) have two effects:

• Creation of phase coherence (‘bunching of spins’)

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Vector model

The ordered collection of spins can be handled from a common origin.

The Boltzmann distribution of up- and down-spins, makes a net magnetic vector along the external field (green).

An external magnetic field (radio frequency pulse, B1) perpendicular to the first (B0) have two effects:

• Creation of phase coherence (‘bunching of spins’)

• Switch from up- to down-spin (or down- to up- !)

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Vector model

The resultant magnetic vector is spinning at the precession frequency, which is the same as the frequency of the external magnetic field.

The spinning magnetic vector induces a current in the detector coil around the sample. The alternating current is recorded.

The detector senses the absolute length of the magnetic vector in the horizontal plane (XY-plane).

• Cosine curve along y-axis.• Sine curve along x-axis.

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Vector model

The resultant magnetic vector is spinning at the precession frequency, which is the same as the frequency of the external magnetic field.

The ‘rotating frame’ reference is used to simplify the model.

The coordinate system is spun at the same speed as the vectors the vectors appear as fixed.

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Relaxation

T1-relaxation• Exponential recovery of

magnetization along B0-axis• Back to equilibrium populations

of up- and down-spin

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Relaxation

T1-relaxation• Exponential recovery of

magnetization along B0-axis• Back to equilibrium populations

of up- and down-spin

T2-relaxation• Gradual ‘fanning’ out of

individual magnetic vector.• emission-absorption

among spins (changes phase)

• bad homogeneity of magnetic field

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Relaxation

T1-relaxation• Exponential recovery of

magnetization along B0-axis• Back to equilibrium populations

of up- and down-spin

T2-relaxation• Gradual ‘fanning’ out of

individual magnetic vector.• emission-absorption

among spins (changes phase)

• bad homogeneity of magnetic field

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Pulsed experiments

The basic 1D-FT NMR experiment• Pulse (μseconds)

• Broadband (covers a wide range of frequencies)

• Acquisition (seconds)• Records all frequencies

within a preset frequency width

• Relaxation delay (seconds)• To return the

magnetization vector close to equilibrium

• Repeat and add results• signals increases linearly

with n, while the noise partly cancels out and increases with n½.

nN

Sn

n

N

S

11 pB

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Phase of pulses and signals

Basic 1D NMR-experiment: With a 90°-pulse along the x-axis

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Phase of pulses and signals

Basic 1D NMR-experiment: With a 90°-pulse along the y-axis

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Phase of pulses and signals

The phase of the pulse gives the phase of the signal…

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Phase of pulses and signals

X

Y

X

Y

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Phase of pulses and signals

X

Y

X

Y

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Different chemical shifts in the vector model

Two signals with different chemical shifts rotates with different speed in the vector model• Interpreted as two different frequencies in the spectrum

X

Y

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Different chemical shifts in the vector model

Two signals with different chemical shifts rotates with different speed in the vector model• Interpreted as two different frequencies in the spectrum

X

Y

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Different chemical shifts in the vector model

One of the signals right on the carrier frequency• The other resonance will have a different speed

X

Y

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Different chemical shifts in the vector model

One of the signals right on the carrier frequency• The other resonance will have a different speed

X

Y

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Coupling in the vector model

A doublet with two signals• The same effect as two different chemical shifts, but

usually depicted with the carrier frequency in the middle of the doublet.

• J = Coupling constant in Hz

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Spin-echoes in pulse sequences

Chemical shifts are refocused

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Spin-echoes in pulse sequences

Chemical shifts are refocused

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Spin-echoes in pulse sequences

Chemical shifts are refocused

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Spin-echoes in pulse sequences

Couplings evolve (if both of the coupled nuclei are inverted)

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Spin-echoes in pulse sequences

Couplings evolve (if both of the coupled nuclei

are inverted)

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Spin-echoes in pulse sequences

Couplings evolve (if both of the coupled nuclei are inverted)

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Spin-echo example

Explain the appearance of the normal 1H spectrum of the hypothetical molecule.

12C 13C

Hb Ha

3JH,H=10 Hz

1JC,H=145 Hz

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Spin-echo exercise I12C 13C

Hb Ha

3JH,H=10 Hz

1JC,H=145 Hz

Explain the appearance of the spin-echo spectrum…• Use vector model• What delay was used around the 180-degree pulse?

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Spin-echo exercise II

Explain the appearance of the spin-echo spectrum with simultaneous 180-pulses at both proton and carbon…• Use vector model• What delay was used around the 180-degree pulse?

12C 13C

Hb Ha

3JH,H=10 Hz

1JC,H=145 Hz

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Spin-echo exercise I12C 13C

Hb Ha

3JH,H=10 Hz

1JC,H=145 Hz

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Spin-echo exercise II12C 13C

Hb Ha

3JH,H=10 Hz

1JC,H=145 Hz

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LUNCH

The lunch is served in the cafeteria in building 22

1215-1315

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Outline

1215 – 1315

• Lunch1315 – 1415

• Signal processing• Window functions• Fourier transform• Real and imaginary parts• Phasing• Topspin starter

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Acquisition time

The acquisition time is usually ~100 ms – 10 sec depending of type of experiment.

The best theoretical resolution in the spectrum is the inverse of the acquisition time (ta).

ta = 10 seconds Δν= 0.1 Hz

ta. = 0.1 seconds Δν= 10 Hz

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Acquisition time

The line width is determined by the acquisition time and the relaxation!

Fast relaxation => the signal fades out fast => broad lines• long acquisition time will in this case only increase the

noise

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Spectral width, sampling rate & dwell time

Dwell time = Time between sampling pointsSampling rate = Number of data points per secondSampling rate = Total no. of data points /

acquisition timeDwell time = (Sampling rate)-1

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Spectral width, sampling rate & dwell time

Dwell time = Time between sampling pointsSampling rate = Number of data points per secondSampling rate = Total no. of data points /

acquisition timeDwell time = (Sampling rate)-1

Faster sampling larger spectral width (sw)Spectral width = ½ × Sampling rate (according to

Nyquist)

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Experimental

What is the acquisition time (ta) for the 1D NMR experiment described in this article?

SW = 7.2 kHz Sampling rate = 2 × 7.2 kHz = 14400 HzTD = 32k = 32 × 1024 = 32768 data pointsAcquisition time; ta = 32768 / 14400 ≈ 2.3 seconds

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Sweep width, dwell time and sampling rate

The sampling rate must be high enough to determine the frequency of the signal (at least twice per period).

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Sweep width, dwell time and sampling rate

The sampling rate must be high enough to determine the frequency of the signal (at least twice per period).

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Sweep width, dwell time and sampling rate

The sampling rate must be high enough to determine the frequency of the signal (at least twice per period).

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Relaxation delay

0 1 2 3 4 5 6 7 80

20

40

60

80

100

Mz (

%)

Time (t/T1)

)1( 1/0

TtZ eMM

%3.99ZM

After a pulse: The magnetization returns to equilibrium• Mz increases, Mxy decreases• Exponentially = fast in the beginning, very slowly in the

end• Time constant; T = longitudinal relaxation• Small molecules, 1H: 0.5-5 sec, 13C: 2-60 sec

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Relaxation delay

Pulsed NMR! Add several transients!• …but what if the recovery is slow and the repetition time

too fast?

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Relaxation delay

Pulsed NMR! Add several transients!• …but what if the recovery is slow and the repetition time

too fast? Use a small flip angle!

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Relaxation delay

Pulsed NMR! Add several transients!• …but what if the recovery is slow and the repetition time

too fast? Use a small flip angle! Use the delay to acquire!

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Optimum flip angle

Optimize the sensitivity with the Ernst angle!

1/)cos( Tte

re

For carbons with long T1’s

For high resolution 1H spectra (aq ≈3×T1)

For accurate quantitative measurements!

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Processing of spectra

Fourier transform (time domain -> frequency domain)

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Processing of spectra

Fourier transform (time domain -> frequency domain)

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Processing of spectra

Fourier transform (time domain -> frequency domain)

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Processing of spectra

Fourier transform (time domain -> frequency domain)

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Processing of spectra

Window function: Exponential multiplication• Line broadening 0.3 Hz• Increases apparent T2

• Apodization (‘removal of feet’), end of FID forced to zero.

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Processing of spectra

Window function: Lorentz-Gauss• Line broadening -0.3 Hz, GB = 0.5• Resolution enhancement, trade S/N for better resolved

signals

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Processing of spectra

Window function: Lorentz-Gauss• Line broadening -0.3 Hz, GB = 0.5• Resolution enhancement, trade S/N for better resolved

signals

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Processing of spectra

Window function: Traficante• Line broadening 0.3 Hz• Keep line shape, increase S/N• Real and imaginary multiplied with two different

functions

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Processing of spectra

Window function: Sine• Sine-bell shape, for data with few points• Strong apodization function

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Processing of spectra

Window function: 90 degree shifted sine• Cosine shape• Used in the indirect dimension of 2D-data

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Processing of spectra

Window function: Mixed cosine and sine bell shape• Mixture of sine and cosine shape• Used in the indirect dimension of 2D-data

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Real and imaginary parts

Two phase shifted signals detected simultaneously to separate frequencies on either side of the carrier frequency.• Quadrature detection

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Phasing

Fourier transform => two components• ‘Real’ and ‘imaginary’• Linear combinations => pure absorption + pure

dispersion• The base of the dispersion signal is wide (unwanted

feature)

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Phasing

Good phasing

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Phasing

0:th orderphase correction

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Phasing

1:st orderphase correction

Freq. dep.

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Phasing

0:th order+

1:st orderphase correction

Freq. dep.

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Phasing, tips and tricks

Reset the phase parameters (PHC0 and PHC1) to zero

1.) Adjust PHC0 on one signal in one end of the spectrum2.) Adjust PHC1 on signals in the other end…

Consider the relative phase (phase errors) of signals…

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Topspin in D1

User names: Kursist1 – Kursist16Passwords: Kursist1 – Kursist16

Computer D1-01 = license server (must be started)

Save data on H:/ (net), D:/ (locally) or on memory stick

Download from: http://www.farma.ku.dk/index.php?id=ntdr

<Dir>/data/<user>/nmr/<experiment name>

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Topspin basics

1. Prepare data directory• Make a directory named ‘data’ in H:\• Make a directory named ‘NTDR_2012’ in H:\data• Make a directory named ‘nmr’ in the ‘NTDR_2012’-folder

2. Download 1D-dataset• http://www.farma.ku.dk/index.php?id=ntdr• Download ‘caffeic_acid.zip’ to ‘H:\data\NTDR_2012\

nmr’• Unzip

3. Start Topspin 3.0 (Topspin 3.0.8.b).• Right click in browser pane and select “Add New Data

Dir..”.• Add “H:\”

4. Fourier transform [ft] and phase. Assign signals…

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Topspin basics

They try to be more like an apple…