1. Primarily,
Chapter 3 Claridge
Also, Chapter 2 Claridge,

3. Continuous Wave NMR
Why is CW NMR almost never used except as an instructional tool?
Signal-to-noise ratio and speed of data acquisition
NMR is insensitive.
Any noise- thermal, electrical, etc. is a problem for an NMR spectrum.
Signal averaging.
Acquire 2 spectra instead of one. (so nt = 2).
-Signal doubles, noise increases as sq. root

4. Continuous Wave NMR E = h Et ~ h
How long does it take to acquire a decent spectrum in CW mode?
For 1 Hz resolution, the energy difference to resolve is given by:
E = h
If  = 1 Hz, then E = h.
The Heisenberg uncertainty principle states that:
Et ~ h
so if E = h, then t = 1 (second).
On a 500 MHz NMR, the range of frequency required to sweep would be ~10 ppm or 5000 Hz, so in CW mode, 5000 seconds (~80 minutes) would be required per scan.

5. Acquire all frequencies at the same time
Faster CW NMR?
How can we reduce the time required for data acquisition?
1) Acquire each frequency for less time
2) Don't cover such a large frequency range
3) Use a lower field instrument, so the same ppm range is less frequency
4) Don't acquire more than one scan or
Acquire all frequencies at the same time

6. ht ~ h Faster NMR? How can we acquire all frequencies at once?
Apply a pulse of radiation
-except this pulse must excite the whole frequency range with a single pulse
E = handEt ~ h, thus:
ht ~ h
Thus for a 5000 Hz wide spectrum,
-> the time of the pulse must be ~0.2 ms.
An RF pulse for ~0.2 ms will excite a 5000 Hz spectral width at one time!

7. Free Induction Decay How do we observe the signal?
-> we measure the response for a period of time on the order of 1 second.
The response is referred to as the Free Induction Decay or FID.
CW NMR = amplitude vs. frequency
pulse NMR = amplitude vs. time
Time and frequency are inversely related, to go from time domain to frequency domain requires a Fourier transform,
-> so pulse NMR becomes know as Fourier Transform NMR (or FT NMR).

8. How does FT NMR Work?
Convince yourself that these two representations are the same:
In real NMR, there will obviously be more than one signal.
In the time domain, all of the resonances act together and the oscillation of the wave will be a combination of their frequencies and amplitudes.
Waves have a frequency, an amplitude and a phase

9. Fourier Transformation
Thus, to extract the frequency domain information from the time domain, we need this relation:
f() = - f(t)eitdt
f() = frequency domain spectrum
f(t) = time domain spectrum
Algorithms on modern computers can handle this calculation efficiently.

10. Practical Implementation of Pulse NMR
In real NMR, signals decay over time exponentially. Thus, the FID decays to zero amplitude over time exponentially. The resulting line shape is Lorentzian.

12. Digital Sampling Digital sampling is how often and how long data
points are acquired.
How long:
The length of time that we sample
over is the acquisition time.
Acquisition time = at

13. Digital Sampling How often:
The rate of sampling depends upon the total spectral range (spectral width = sw).
If the frequency range is N Hz, then the sampling rate must be 1/2N seconds.
This is known as the Nyquist frequency.
Spectral width is correlated to the number of points and the acquisition time
acquisition time = number of points/(2*spectral width)
Thus, at = np/(2*sw)
[note Claridge uses TD, which is the parameter on Bruker instruments]

14. What happens if you do not sample fast enough?
Thus, the signals are detected, but they are detected at the wrong frequency. This is referred to as folding or aliasing.

15. Folded Resonances
Folded or Aliased (Wrapped) resonances are outside the spectral width, but still observed at incorrect frequency.

16. Not only are real signals folded into the spectrum, but noise will be as well.
Spectrometers have analog and digital filters.
Filter bandwidth = computer controlled filter = fb

17. Bandpass Filters Large Spectral Width/Large Filter Width
Small Spectral Width/Large Filter Width
Small Spectral Width/Small Filter Width

18. Dynamic Range and Receiver Gain
The analog-to-digital converter (ADC) limits both the frequency range by digital sampling rate (correlated to spectral width) and the signal amplitudes accepted.
The ADC converts the electrical NMR signal into a binary number correlated to the amplitude of the signal.
The dynamic range is defined as the ratio of the largest and smallest detectable values.
The dynamic range is correlated to the amount of bits that the digitizer can handle- typical digitizers have 16 bits for a range of +/- 32K.
Thus, the dynamic range is 32767:1.
The dynamic range is fixed for a spectrometer, but the user still controls the minimum amplitude detectable, gain or receiver gain; Varian parameter is gain.

19. Dynamic Range and Receiver Gain
Receiver gain too high
Receiver gain low or small dynamic range:

20. Oversampling
Oversampling is the sampling of the data at a much higher rate than necessary than according to the Nyquist theorem.
Eliminate digitization noise.
Digitization noise arises from creating a digital signal from the analog signal.

21. Oversampling at a rate of N reduces digitization noise by N1/2.
Oversampling is generally done with values of 16 or
32 times the spectral width.
There is no effect on thermal noise.
This is particularly useful with low settings of receiver
gain.

22. Oversampling is usually combined with digital signal processing (dsp) and digital filters to allow the digitizer to work at such a high rate and give sharp cutoff of frequencies outside the spectral width.

23. Fourier Transformation
f() = - f(t)eitdt
With NMR, we are detecting a real quantity: amplitude as a function of time,
Fourier transformation has real and imaginary parts.
Thus, converting the exponential into the trig identity:
eit = cos(t) + isin(t)
thus the transform looks like:
real part: f() = - f(t) cos(t)dt
imaginary part: f() = - f(t) sin(t)dt
The real and imaginary parts contribute to the spectrum,
the real part is absorptive, the imaginary is dispersive 90˚ out of phase

24. Phase of the Wave Two waves out of phase from each other:
As the phase in the time domain
changes (in this case in 10˚
increments), the line becomes
more dispersive in the frequency
domain.
To fix the dispersive signal
the spectrum is phased.
aph = autophase command
rp = right phase correct parameter
lp = left phase correct parameter

25. FID is a wave which has
1) Amplitude- correlated to signal-to-noise, dynamic
range and gain
2) Frequency- correlated to chemical shift and spectral width
3) Decay time- correlated to acquisition time, number of points,
and relaxation
4) Phase- correlated to phase corrects

26. Digital Resolution at = 65 seconds at = 1 second
Digital Resolution is the interval in frequency between data points.
Digital Resolution = 1/at where at = acquisition time.
To properly define a line, multiple points per line are required.
at = 65 seconds
at = 1 second

27. Zero Fill No Zero-filling Zero-filled spectrum fn = 8*np
Acquiring for 65 seconds is not very practical for achieving high resolution.
If the FID has already decayed to zero amplitude, the signal to the computer will just be all zeroes, so instead of wasting time acquiring, we can just have the computer add zero amplitude signals to the end of the FID. This process is called zero-filling.
No Zero-filling
fn = Fourier Number
the amount of points
that are Fourier transformed
If fn > np, then zero amplitude
signals are added to the end
of the FID
If fn < np, then data points acquired
beyond that are not transformed
Zero-filled spectrum
fn = 8*np

28. Truncation Full FID: Truncated FID (acquisition time, at, too short):
Truncation occurs when the acquisition time is too short so that the FID has not fully decayed. When the acquisition time (at) is too short, the end of the fid is a step function
from signal to nothing.
Full FID:
Truncated FID (acquisition time, at, too short):
Expansion of last real data point of FID acquired:

29. Truncation Wiggles Truncation of the FID creates a Step Function
the Fourier transformation of a step function is a sinc function, (sin x)/x:

30. Apodization
The easy way to eliminate the wiggles in the spectrum is by acquiring longer, but that is not always possible.
Thus, to eliminate the wiggles in the spectrum with data processing, the FID can be multiplied by a mathematical function that smoothly takes the FID to zero, rather than the harsh step function.
Without apodization:
With apodization:

31. Sensitivity Enhancement with Apodization
Exponential Mulitply/Line Broadening (em or lb):
By multiplying by an exponentially decaying function, we can lessen the significance of the right end of the FID on the spectrum; emphasizing the left end of the FID leads to improvement in signal-to-noise at the expense of resolution.
Line-broadened
No Line-broadening

32. Resolution Enhancement with Apodization
If weighting the left side of the FID leads to signal-to-noise improvement, then weighting the right side of the FID leads to resolution enhancement. The ideal way for resolution enhancement is to convert a Lorentzian line into a Gaussian line as Gaussian lines are narrower at the base (gaussian function = gf).
Resolution-enhanced weighting function applied
No weighting function applied

33. Linear Prediction dN = a1dN-1 + a2dN-2 + a3dN-3 + ...
Linear prediction is the adding of discreet non-zero points to the end of the FID (forward linear prediction) to account for the points not acquired.
The value of a data point (dN) can be estimated by the equation:
dN = a1dN-1 + a2dN-2 + a3dN
Advantages of linear prediction over zero-filling:
1) There is greater improvement in resolution in comparison with zero-filling
2) The truncation wiggles are removed without line-broadening apodization.
Disadvantages of linear prediction:
1) Requires good signal-to-noise for accurate prediction of points.
2) The amount of points used for the prediction must be >> that the number of lines in the FID

34. 1D Linear Prediction Full FID,
No apodization, zero-fill or linear prediction
Truncated FID with zero-fill, No apodization or linear prediction
Apodized, zero-filled FID
Linear-predicted FID

35. Linear-Prediction in 2D
No Zero-fill/linear prediction Zero-filled Linear Predicted

36. Backwards Linear Prediction
Phase errors and baseline errors are often a problem in spectra because there are real time delays between the end of a pulse and the turning on of the receiver and the digitizer. Backward linear prediction predicts the first points of the FID by taking the first "good" points and using those points to predict what the first points should be.
Fourier transform, no backward linear prediction
Fourier transform, with backward linear prediction

37. Instrument demo