NMR Signal Properties & Data Processing

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COURSE#1022: Biochemical Applications of NMR Spectroscopy

http://www.bioc.aecom.yu.edu/labs/girvlab/nmr/course

/

NMR Signal Properties & Data Processing

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Reading

Selected Readings (any of the following):

• Evans, pp 2-13

• Teng, Chapter 1, pp. 12-17

• Cavanagh, Fairbrother, Palmer, & Skelton (1st Ed), pp. 14-16 and 100-126

• Derome, pp 9-29

• Sanders and Hunter, 10-55

• Keeler, chapters 4 and 5

Online Sources

Content from the following online sources were used for some slides in this

presentation:

• http://tonga.usip.edu/gmoyna/NMR_lectures

• http://www.oci.unizh.ch/group.pages/zerbe/notes.html

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Overview of Basic Principles

Quantum Mechanics picture:

• some nuclei pocess spin which is quantized

• in a magnetic field, B

_o

, the spins have discrete energy states and populate

the energy levels according to Boltzmann distribution

• transitions between energy levels are stimulated by photons with frequency:

 = B

_o

or  = B

_o

/2

where  and  is the Larmor frequency in radians/s and Hz, respectively

• The quantum-mechanical selection rule states that only transitions with

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Overview of Basic Principles

Classical Mechanics picture (vector model):

• When sample is placed in a magnetic field, nuclear spins align with or against

the field and a net magnetization, M

_o

, is created within the sample.

• the bulk magnetization precess around the magnetic field axis at its Larmor

frequency

• an “RF pulse” is sent to probe coil which creates a magnetic field (B

₁

field)

along some axis in transverse plane. The B

₁

field places a torque on M

_o

and

tips the nuclear magnetization from z-axis into transverse plane (according to

right-hand rule).

• shut off RF and detect magnetization in transverse plane - magnetization will

oscillate at its Larmor frequency, , and return back to z-axis due to relaxation.

This gives rise to an FID – a time dependent voltage induced in probe coil

• Fourier transform (FT) of FID gives Larmor frequency of nuclei in sample.



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The NMR Spectrum

The NMR spectrum is “information rich” because it consists of a signal for each detected atom in the molecule and each signal has the following properties:

• position or “chemical shift” (related to its larmor frequency) for each type nuclei which is sensitive to its chemical environment (CH proton will be different than OH proton)

• intensity which is proportional to # of nuclei of that type (for example, a methyl CH3 group proton signal would be 3 times as intense as a CH proton signal)

• width which is sensitive to overall size of molecule and the dynamics or motion of the individual atoms

• splittings or “couplings” which is sensitive to # of bonded nuclei

So the signals within an NMR spectrum depend on the surrounding environment (i.e. structure) and dynamics of each dectected nucleus in the molecule

1_{H NMR spectrum of an organic compound}

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The RF Pulse

To detect the magnetization, we must tip it into the transverse plane by exciting the

spins with a radiofrequency that corresponds to its Larmor frequency. An NMR

sample may contain many different magnetization components, each with its own

Larmor frequency so we must excite the sample with many different

radiofrequencies at the same time. An RF pulse (also called the B

₁

field) is a

poly-chromatic source of radiofrequency and it covers a broad band of

frequencies . The covered band width is proportional to the inverse of the pulse

duration. Short (hard) pulses are required for uniform excitation of large

bandwidths (non-selective excitation). Long (soft) pulses lead to selective

excitation. Hard pulses have lengths of 5-20 μs and can excite over a wide range

of frequencies (>10 kHz), whereas soft pulses may last for 1-100 ms and excite

only a small range of frequencies (<100 Hz):

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Effect of RF Pulse on Equilibrium Magnetization

Spins respond to an RF pulse in such a way as to cause the net magnetization

vector to rotate about the direction of the applied B

₁

field. In the laboratory frame

the equilibrium magnetization spirals down around the Z axis to the XY plane. In

the rotating frame (to be discussed) a pulse about the Y’ axis will cause the

magnetization to tip down toward X’.

The rotation angle (tip angle) depends on the length of time the field is on,



,

and its magnitude B

₁

:

 =  B

₁



Apply pulse along one axis (eg. y-axis) which tips I

_z

through flip angle a towards

x-axis (according to Right-hand rule):

z

x

y

z

x

y



I

_y

using right-hand rule:

I_z I_zcos + I_xsin for I_yRF pulse I_z I_zcos - I_ysin for I_x RF pulse

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Some Useful Pulses

• The most commonly used pulse is the 90 degree pulse (



/2), because it puts as

much magnetization as possible in the <xy> plane (more signal can be detected by

the instrument):

• Also important is the 180 degree pulse (



) pulse, which has the effect of inverting

the populations of the spin system (to be discussed later):

• With control of the spectrometer we can basically obtain any pulse width we want

and flip angle we want.

z x

M

_xy y z x y

M

_o

 / 2

z x

-M

_o y z x y

M

_o



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SIDE NOTE: only ½ of the B

₁

field interacts with the magnetization. The B

₁

field is linearly polarized along the axis of RF pulse. This B

₁

field can be

decomposed into two circularly rotating fields rotating in opposite

directions along the z-axis:

Only the field rotating in the same sense as the magnetic moment will

interact with the magnetization and tip it; the counter-rotating field does not

influence the magnetization and can be ignored.

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The Time Domain NMR Signal – Free Precession

An NMR sample may contain many different magnetization components, each

with its own Larmor frequency. These magnetization components are associated

with the nuclear spin configurations joined by an allowed transition line in the

energy level diagram. Based on the number of allowed absorptions due to

chemical shifts and spin-spin couplings of the different nuclei in a molecule, an

NMR spectrum may contain many different frequency lines. After the RF pulse,

each magnetization component will rotate in the transverse plane according to its

own Larmor frequency (



):

y x y x y x y x



This is a rotation around z-axis with angle given by t:

I

_x



I

_x

cost + I

_y

sint

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The RF Transmitter and Receiver Coil

An RF coil creates the B

₁

field which rotates the net magnetization during an RF

pulse (excitation). It also detects the transverse magnetization as it precesses in

the XY plane (detection). Most RF coils on NMR spectrometers are of the saddle

coil design and act as the transmitter of the B

₁

field and receiver of RF energy

from the sample. You may find one or more RF coils in a probe depending on how

many different types of nuclei are manipulated during a given experiment. Each of

these RF coils must resonate, that is they must efficiently store energy, at the

Larmor frequency of the nucleus being examined with the NMR spectrometer. All

NMR coils are composed of inductive elements and capacitive elements and the

resonant frequency,



, of an RF coil is determined by the inductance (L) and

capacitance (C) of the circuit:

TUNING and MATCHING PROBE (to be discussed in Lab) involves changing “L” and “C” of coil using knobs until desired frequency is obtained.

For example: detecting 1H will require a certain L & C that is different than what is used to detect 13C

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As the magnetization rotates in the transverse plane, it generates a fluctuating

magnetic field which induces a time dependent voltage (the FID) in the receiver

coil – the time dependence is determined by the Larmor frequency. If the

magnetization starts along the x-axis (eg. if a 90 degree pulse is applied along

the y-axis), then the time dependence would look as follows:

t t

I_x I_y

view along x-axis view along y-axis

We will see later that both the x and y components of the magnetization are

detected using a process called quadrature detection. It is convenient to

express I

_x

and I

_y

as “real” and “imaginary” parts of I, the total magnetization:

I(t) = Icost + iIsint = Iexp(it)

The detected time domain signal is complex with real and imaginary parts

corresponding to x and y components of the signal.

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Return to Equilibrium – Relaxation

The magnetization does not precess infinitely in the transverse plane but returns

back to the equilibrium state by a process called relaxation. We’ll see the physics

that govern relaxation in a later lecture. Two different time-constants describe this

behavior:

• the re-establishment of the equilibrium α/β state distribution (T

₁

)

• dephasing of the transverse component (destruction of the coherent state, T

₂

).

The T

₂

value characterizes the exponential decay of the signal in the receiver coil

– the shorter T

₂

is, the faster the signal decays:

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The precessing spins return to the z-axis. Instead of moving circularly in the

transverse plane, they follow a spiral trajectory until they have reached their

initial position aligned with the z-axis and the time dependence of total

magnetization becomes:

I(t) = Iexp(it)exp(-t/T

₂

) = Iexp(it - t/T

₂

)

FIGURE left: Trajectory of the magnetization, right: individual x,y,z component

governed by T₂

governed by T₁ governed by T₂

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Signal Detection – The Carrier Frequency

In order to be stored in a computer, the analog signal that comes from the coil must be converted into a number which is done by the analog-to-digital converter (ADC) and is known as signal digitization or signal sampling. The resulting FID is a time-domain signal with numbers which represent signal intensity vs. time [f(t)]. However, before digitization, the signal is mixed with a reference frequency (called the carrier frequency) to produce a

frequency which is the difference between the two – the resultant frequency is an audio

frequency which is easier to detect and handle than a radio frequency. In the rotating

frame of reference, a frame of reference that rotates at the carrier frequency, _ref relative to the laboratory frame, the vector would appear to rotate at a frequency equal to the

difference between and the Larmor frequency (_o) and the rotation frame frequency (_ref):

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The +/- Frequency Convention

Transverse magnetization vectors rotating faster than the rotating frame of

reference are said to be rotating at a positive frequency relative to the rotating

frame (+). Vectors rotating slower than the rotating frame are said to be

rotating at a negative frequency relative to the rotating frame (-).



_

= -10hz



_

= +10hz

0 Hz 0 Hz -10 Hz +10 Hz

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Quadrature Detection

Can we discriminate between frequencies which only differ in their sign; the

answer is yes, but only provided we use a method known as quadrature

detection. The figure below illustrates the problem. The left side shows a vector

precessing with a positive offset frequency (in the rotating frame); the x

component is a cosine wave and the y component is a sine wave. The right side

shows the evolution of a vector with a negative frequency. The x component

remains the same but the y component has changed sign when compared with

evolution at a positive frequency. In order to distinguish between positive and

negative frequencies we need to know both the x and y components of the

magnetization and thus detect two RF signals that are 90 degrees out of phase.

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Signal Digitization

In order to be stored in a computer, the analog signal that comes from the coil must

be converted into a number which is done by the analog-to-digital converter

(ADC) and is known as signal digitization or signal sampling. The resulting FID

is a time-domain signal with numbers that represent signal intensity vs. time [f(t)].

The signal is sampled stroboscopically and neighboring data points are separated

by the dwell time (DW).

Left:Rotating spin with its position at certain time intervals 1-6 are marked. Right: Corresponding signal in the receiver coil.

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Spectral Sweep Width and the Nyquist Theorem

The Nyquist theorem says that to reproduce an analog signal of frequency f, the sampling rate of the ADC is required to be at least 2f. Since the highest frequency of the spectrum is SW (in Hz), the dwell time (in s) must be set to:

DW = 1/(2SW)

When quadrature detection is being used, the dwell time becomes: DW = 1/SW

because we can discriminate between positive and negative frequencies. So, for example, if the desired SW=10000 Hz, then the DW must be set to 100s.

NOTE: signals that have a frequency larger than SW, will appear folded or aliased in the spectrum and their position will be incorrect (Figure 3).

Figure 1: Left: Sampled data points. Right: Black: Sampled data points. Red: Missing sampling point for a frequency of twice the nyquist value..

Figure 2: Basic parameter for specifying the acquired spectral region

Figure 3: Illustration of the concept of folding. In spectrum (a) the peak (shown in grey) is at a higher frequency than the maximum set by the Nyquist condition. In

practice, such a peak would appear in the position

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The Fourier Transformation

The analog signal that comes from the coil is converted into a number by the analog-to-digital converter (ADC). The resulting FID is a time-domain signal with numbers which represent signal intensity vs. time [f(t)]. Each time point is complex – it has two values which represent the I_x and I_y components of magnetization that are the result of quadrature detection. This data is converted into a spectrum that represents signal intensity vs. frequency [f(ω)] by use of the

Fourier Transformation (FT). The Fourier theorem states that every periodic function may be

decomposed into a series of sine- and cosine functions of different frequencies and the Fourier transform is defined by the integral:

The complex Fourier transform can be decomposed into a cosine transform and a sine

transform that yield the real and imaginary parts of the signals, respectively and are stored in

separate memory locations. The cosines give absorptive lineshapes while sines give

dispersive lineshapes:

A(ω) is an absorption mode

lorentzian lineshape centred at ω = Ω

D(ω) is a dispersion mode lorentzian lineshape centred at ω = Ω

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The discrete FT is implemented in form of the very fast Cooley-Tukey algorithm.

The consequence of using this algorithm is, that the number of points the spectrum

has must be a power of 2 (2

n

_).

So will often see (where SI is # of points in FID to collect):

SI=8192

SI=16384

SI=32768

SI=65536, etc.

Discrete Fourier Transformation

Note that in FT-NMR the signal is sampled as discrete points. Hence, the

discrete FT has to be utilized, which transforms a N-point time series

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This figure summarizes the FT of some important functions (the so called Fourier pairs):

(A) The FT of a cosine-wave gives two delta functions at

the appropriate frequency.

(B) The FT of a decaying exponential gives a lorentzian

function with a characteristic shape.

(C) The FT of a properly shimmed sample containing a

single frequency gives two signals at the appropriate frequency with lorentzian lineshape. In fact the FT of (C) can be thought of a convolution of (A) and (B).

(D) The FT of a step function gives a sinc function The

sinc function has characteristic wiggles outside the central frequency band.

(E) A FID that has not decayed properly can be thought

of as a convolution of a step function with an exponential which is again convoluted with a cosine function. The FT gives a signal at the appropriate frequency but containing the wiggles arising from the sinc function.

(F) A gaussian function yields a gaussian function after

FT.

Results of Different Functions after

Fourier Transformation in Frequency Space

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Phasing

Purely absorptive signals have a much narrower base than a dispersive signal, so that

differentiation of peaks very close to each other is easier if all the signals in the spectrum have purely absorptive line shape. The phase of the signals after Fourier Transformation depends on the first point of the FID. If the oscillation of the signal of interest is purely

cosine-modulated, the signal will have pure absorptive line shape. If the signal starts as a

pure sine-modulated oscillation, the line shape will be purely dispersive (see Figure 1). Ideally, collection of a purely cosine-modulated signal would occur if one turned on the receiver immediately after the pulse and start digitizing the signal. However, in a real

experiment the signal has to travel through cables before it is digitized which causes a delay and there is also a protection delay (called DE on Bruker instruments), to wait for the pulse

ring-down (see Figure 2) – otherwise, the detector would be overloaded by the RF pulse.

Therefore, a phase correction is needed to yield purely absorptive signals.

Figure 2: A) Ideal FID. B) Introduction of the pre-scan delay to enable pulse power ring-down.

Figure 1: The relative phase of the

magnetization with respect to the receiver coil determines the phase of the signal after

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Phasing, cont.

There are two different types of phase-correction. The zero-order phase correction (PH0) applies the phase change to all signals in the same way. The first order phase

correction (PH1) applies a phase change, whose amount increases linearly with the

distance to the reference signal (see figure below).

Top: The zero order phase correction changes the signal phase for all signals by the same amount. Lower: For the first order phase correction the applied correction depends on the frequency

difference to the reference signal (marked by an arrow).

In practice what happens is that we Fourier transform the FID and look at the spectrum. If the signals look “out of phase” and have some dispersive appearance, we then adjust the phase until the spectrum appears to be in pure absorption mode – usually this adjustment is made by turning a knob or by a “click and drag” operation with the mouse or most modern spectrometers can do this automatically. The whole process is called phasing the

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Resolution of the Spectrum

The spectral resolution is a measure of how well we can resolve or separate signals within a spectrum. The spectral resolution that can be obtained from a spectrum is determined by two factors:

• the natural line width of the signal at half-height (LW_1/2) which is related to the transverse relaxation time or T₂:

_1/2 or LW_1/2 (Hz) = 1/T₂

Fast transverse relaxation may also be caused because the B_o magnetic field is inhomogenous (inhomogenous broadening or T₂*_{) – recall that as B}

o changes, 

changes so that a range of B_o’s within a sample will cause a range of ’s.

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Resolution of the Spectrum, cont.

• the number of points which are used to digitize the signal. The digital resolution (in Hz/point) of the FID collected can be calculated as:

Digital resolution = SW (Hz) / TD = 1/AQ (s)

where TD is the number of complex points acquired and AQ is the acquisition time for each FID. It is often useful to append zeros to the FID to add additional resolution (called

zero-filling). Adding zeros once (that means making SI = 2*TD where SI is the number of complex

points processed) gives a significant improvement in resolution but adding more zeros gives cosmetic effects but does not add more information.

Effect of increasing number of data points, TD, of FID .

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Sensitivity Enhancement of the Spectrum by Apodization

The resolution is determined by the transverse relaxation time T₂ of the spins. A long transverse relaxation time leads to a slowly decaying FID and fast relaxation leads to a quickly decaying FID (Figure A). Therefore, the resolution is determined by the amount of signal which remains towards the end of the FID. The sensitivity (signal-to-noise, S/N) of a spectrum is determined by the amount of signal at the beginning of the FID and the

amount of noise collected throughout the FID. Once the signal has decayed, only noise is being sampled by the spectrometer. If too many data points (TD) are sampled, the spectrum will become excessively noisy (Figure B). There are a number of so-called window or

apodization functions that can be multiplied with the FID in order to enhance the beginning

or end of the FID to improve either S/N or resolution, respectively (Figure C). Most

manipulations that improve resolution lead to a loss in S/N and vice versa. The goal is to

achieve a balance between resolution and sensitivity.

Figure A

Figure B

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Window Functions

(A) raw spectrum after FT.

(B) Multiplication with exponential,LB=5.

(C) as (B), but LB=1.

(D) sinebell.

(E) 45 degree shifted sine-bell.

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Window Functions, cont.

Exponential multiplication: The line broadening constant LB can be positive (sensitivity enhancement) or negative (resolution enhancement). A lorentzian line of width W will have a width of W + LB Hz after

apodization. It can be shown that the best SNR is obtained by applying a weighting function which matches the linewidth in the original spectrum – such a weighting function is called a matched filter. So, for

example, if the linewidth is 2 Hz in the original spectrum, applying an additional line broadening of 2 Hz will give the optimum S/N ratio:

Lorentz-to-Gauss transformation: This a combination of exponential and gaussian multiplication. The factor LB determines the broadening, whereas GB determines the center of the maximum of the gaussian curve. When GB is set 0.33, the maximum of the function occurs after 1/3 of the acquisition time. By this multiplication, the Lorentzian lineshape is converted into a gaussian lineshape, which has a narrower base

Sine-Bell apodization: Sine-bell apodization is frequently used in 2D NMR processing. A pure sinebell corresponds to the first half-lobe of a sine-wave and multiplies the FID such that it brought to zero towards the end. However, such a function also sets the first data points to zero and hence leads to severe loss of S/N and introduces negative wiggles at the bases of the signals in addition to a strong improvement in resolution. Shifting the sine-bell by 90 degrees (a cosine-bell) means that the function used starts of the maximum of the first lobe and extends to zero towards the end of the FID. This is usually used for FIDs that have maximum signal in the first data points (NOESY, HSQC). A pure sine-bell is used for such experiment in which the FID does not have maximum signal in the first points (such as COSY) or for improving

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Different window functions:

A) exponential multiplication with LB=1,3,5Hz.

B) Lorentz-Gaussian transformation with LB=-5 and GB =0.1, 0.3, 0.5 Hz. C) Sine-bell shifted by 0, π/2, π/4, π/8.

D) Sine-bell squared shifted by 0, π/2, π/4, π/8.

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Illustration of how truncation leads to artifacts (called sinc wiggles) in the spectrum. The FID on the left has been recorded for sufficient time that it has decayed almost the zero; the corresponding spectrum shows the expected lineshape. However, if data recording is stopped before the signal has fully decayed the corresponding spectra show oscillations around the base of the peak.

FID Truncation

If we stop recording the signal before it has fully decayed the FID is said to be “truncated”. As is shown in the figure below, a truncated FID leads to oscillations around the base of the peak; these are usually called sinc wiggles or truncation artifacts – the name arises as the peak shape is related to a sinc function. Clearly these oscillations are undesirable as they may obscure nearby weaker peaks. To remove, one can either increase the acquisition

time or apply a decaying weighting function (line broadening) to the FID so as to force the

signal to go to zero at the end. Unfortunately, the later will have the side effects of broadening the lines and reducing the S/N.

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Sensitivity Enhancement of the Spectrum by Signal Averaging

The signal-to-noise (S/N) of a spectral peak is the ratio of the average height of the peak to the standard deviation of the noise height in the baseline. To correctly interpret an NMR

spectrum, it is usually important to see all signal features clearly which requires a spectrum with sufficient S/N (the weakest signal must at least be above the noise!). The signal to noise ratio may be improved by performing signal averaging. Signal averaging is the

collection and averaging together of multiple spectra. The signals are present in each of the averaged spectra so their contribution to the resultant spectrum add. Noise is random so it does not add, but begins to cancel as the number of spectra averaged increases. The signal-to-noise improvement from signal averaging is proportional to the square root of the number of scans (NS) averaged. S/N  NS1/2

NS

1/2

1 1.00

8 2.83

16 4.00

80 8.94

800 28.28

_{Other ways to enhance S/N:}Spectrum with NS=1 Spectrum with NS=800

• use more sample (double sample saves 4x in time for same S/N) • use higher field, B_o(S/N~B_o3/2₎

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Putting It All Together: The Basic 1D FT-NMR Experiment

;zg ;avance-version ;1D sequence #include <Avance.incl> 1 ze 2 d1 p1 ph1 go=2 ph31 wr #0 exit ph1=0 2 2 0 1 3 3 1 ph31=0 2 2 0 1 3 3 1

;pl1 : f1 channel - power level for pulse (default) ;p1 : f1 channel - high power pulse

;d1 : relaxation delay; 1-5 * T1

Basic 1D pulse Bruker Pulse Sequence:

ref=300.000MHz

=300.0015MHz =300.001MHz =300.000MHz =299.9988MHz

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pulprog zg basic 90 pulse sequence TD 16k number of data points (time

domain size) NS 8 number of scans

DS 0 number of dummy scans SW 6000 Hz sweep width in hertz

O1p 5ppm center of spectrum

D1 1.5 delay parameter in seconds P1 9.5 pulse length in microseconds PL1 -6 pulse power level in terms of dB

SI 8k real spectrum size, si = td/2 (no zero fill)

LB 0.30 exponential weighing factor

The ASED and EDA displays on a Bruker lists the acquisition parameters for a given

experiment

Parameters for a 1D FT-NMR Experiment

typical acquisition parameters for a 1D NMR experiment:

typical processing parameters for a 1D NMR experiment:

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Old School: Continuous Wave NMR

In continuous wave NMR (CW-NMR), the sample is continuously irradiated with a

frequency while the magnetic field is varied and the spectrum is a recording of

which magnetic fields caused the sample to absorb RF (happens when the Larmor

frequency condition is met like in FT-NMR

= B

_o

The main disadvantages of CW-NMR are:

• time consuming – takes 100-1000 times longer to record a scan relative to

FT-NMR

• the data is in frequency domain so time domain manipulations such as filtering,

apodization and linear prediction cannot be readily applied to the data

time

B

_o