### Basic principles of NMR NMR signal origin, properties, detection

```Basic principles of NMR
NMR signal origin, properties, detection, and processing
Nils Nyberg
NPR, Department of Drug Design and Pharmacology
NTDR, 2014
Outline
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1045 –
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1100 –
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1130 –
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1215 –
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1045
Establishing current knowledge level
Nuclear Magnetic Resonance phenomenon
Vector model, in and out of the rotating frame
1100
Short break
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)
1200
Spin-echo exercise
1315
Lunch
Outline
1215 –
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1315 –
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1315
Lunch
1415
Signal processing
Window functions
Fourier transform
Real and imaginary parts
Phasing
Topspin starter
Establishing current knowledge level
Build (sketch) a NMR-instrument!
•Magnet
•Probes
•Amplifiers
•Temperature control
•Lock
•Shimming
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Establishing current knowledge level
Draw a spectrum!
•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
Spin
Spin
Spin
I
I
I
I
=
=
=
=
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0:
½:
1:
1½:
12C, 16O
1H, 13C, 15N, 19F, 31P, 77Se
2H, 14N
33S, 35Cl, 37Cl
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.
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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’.
•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 of
½-spin nuclei, with a magnetic
moment (μ).
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Vector model
Unordered collection of
½-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 of
½-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 of
½-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 bstate is described by the
Boltzmann distribution.
Na
 exp(
Nb
E
k T
)e
(
E
k T
)
e  2 . 718 (natural logarithm)
 E  h 
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]
k  1 . 4  10
 23
h  6 . 6  10
 34
J/K (Boltzmann
constant)
Js (Planck constant)
  Frequency, Hz (s )
-1
T  Temperatur e in K (Tc  273.15)
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Boltzmann distribution
One spin extra in the low
energy state!
[1H, 600 MHz, 298 K]
Na
Nb
Nb 1
Nβ
Nα
Σ
=
=
=
12 922
12 923
25 845
 exp(
 exp(
Nb
N b  1  exp(
N b  exp(
h 
k T
N b (1  exp(
E
k T
h 
k T
h 
k T
)
)
) Nb
)  N b  1
h 
k T
))   1
1
Nb 
(1  exp(
h 
k T
))
1
Nb 
(1  e
(
6 . 6 10
34
1 . 4 10
 600 10
 23
 298
 12922
6
)
)
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Vector model
The ordered collection of spins can
be handled from a common origin.
The Boltzmann distribution of upand down-spins, makes a net
magnetic vector along the
external field (green).
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 upand down-spins, makes a net
magnetic vector along the
external field (green).
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 upand down-spins, makes a net
magnetic vector along the
external field (green).
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
•
individual magnetic vector.
• emission-absorption
among spins (changes
phase)
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
•
individual magnetic vector.
• emission-absorption
among spins (changes
phase)
magnetic field
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Pulsed experiments
The basic 1D-FT NMR experiment
•
•
•
•
Pulse (μseconds)
range of frequencies)
Acquisition (seconds)
• Records all frequencies
within a preset frequency
width
Relaxation delay (seconds)
• To return the
magnetization vector close
to equilibrium
• signals increases linearly
with n, while the noise
partly cancels out and
increases with n½.
S

N
S
N
n
n

n
    B1  p1
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
Y
X
Y
X
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Phase of pulses and signals
Y
X
Y
X
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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
Y
X
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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
Y
X
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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
Y
X
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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
Y
X
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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 (Hz = rounds per seconds)
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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.
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Spin-echo exercise I
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?
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Spin-echo exercise I
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Spin-echo exercise II
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LUNCH
The lunch is served in the
cafeteria in building 22
1215-1315
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Outline
1215 –
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1315 –
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1315
Lunch
1415
Signal processing
Window functions
Fourier transform
Real and imaginary parts
Phasing
Topspin starter
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!
• long acquisition time will in this case only increase the
noise
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Spectral width, sampling rate & dwell time
Dwell time
Sampling rate
Sampling rate
Dwell time
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=
=
=
=
Time between sampling points
Number of data points per second
Total no. of data points / acquisition time
(Sampling rate)-1
Spectral width, sampling rate & dwell time
Dwell time
Sampling rate
Sampling rate
Dwell time
=
=
=
=
Time between sampling points
Number of data points per second
Total no. of data points / acquisition 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
SW = 7.2 kHz  Sampling rate = 2 × 7.2 kHz = 14400 Hz
TD = 32k = 32 × 1024 = 32768 data points
Acquisition 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
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
100
M Z  99 . 3 %
60
z
M (%)
80
40
M Z  M 0  (1  e
20
0
0
1
2
3
4
Time (t/T1)
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5
6
 t / T1
7
)
8
Relaxation delay
• …but what if the recovery is slow and the repetition time
too fast?
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Relaxation delay
• …but what if the recovery is slow and the repetition time
too fast? Use a small flip angle!
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Relaxation delay
• …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!
For high resolution 1H spectra (aq ≈3×T1)
For accurate quantitative measurements!
For carbons with long T1’s
cos( a e )  e
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 t r / T1
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
• 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 -1.0 Hz, GB = 0.25
• 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
• 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.
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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 order
phase
correction
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Phasing
1:st order
phase
correction
Freq. dep.
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Phasing
0:th order
+
1:st order
phase
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 spectrum
2.) Adjust PHC1 on signals in the other end…
Consider the relative phase (phase errors) of signals…
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Topspin in D1
User name:
upnmr
nmr2013!
Folder hierarchy:
<Dir>/data/<user>/nmr/<experiment name>
<Dir>/<experiment name>
<Dir> = C:/data/ntdr2014/nmr/
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Topspin basics
1.Prepare data directory
• Make a directory named ‘data’ in C:\
• Make a directory named ‘NTDR2014’ in C:\data
• Make a directory named ‘nmr’ in the ‘NTDR2014’-folder
• http://drug.ku.dk/research/npr/nmr/ntdr2014/