1. 2. Theory of NMR 1. The Classical mechanical approach to NMR
2. The Quantum Mechanical approach to NMR
3. I = ½ Nuclei
4. The Magnetic Field
5. Boltzmann Distribution
6. Sensitivity of NMR
7. The Macroscopic Magnetization Vector
8. The NMR Pulse
9. Laboratory vs. Rotating Frame
10. The Bloch Equations
11. What is a Free Induction Decay (FID)?
12. What is Fourier Transform?

2. Nuclear Magnetic Resonance
Nuclei absorb & emit energy at their resonance frequencies.
This method observes the nucleus.
A magnetic field is essential for the phenomenon to be observed.
2. Theory of NMR (Dayrit)

3. A Bird’s Eye View The nucleus and its magnetic moment
Through bond coupling B A C
Through space coupling
2. Theory of NMR (Dayrit)

4. Electromagnetic excitation and detection
Introduction
The test substance is placed in a glass tube in the center of a homogeneous magnetic field. A coil is wrapped around the tube. A radio-frequency field (electromagnetic radiation) is imposed on the coil which excites the nuclei of the sample.
RF energy can be absorbed or emitted at specific frequencies. This is detected and outputted as the NMR spectrum.
The basic NMR set-up
Magnetic
pole
Magnetic pole
Sample
Electromagnetic excitation and detection
2. Theory of NMR (Dayrit)

5. What is the physical basis for the NMR signal?
Nuclei without a magnetic field:
Random orientation of nuclear spins
Nuclei in the presence of a magnetic field: Alignment of nuclear spins along or against the direction of magnetic field.
Magnetic pole
Magnetic pole
2. Theory of NMR (Dayrit)

6. There are two main approaches to NMR theory: Classical Mechanics:
based on mass and charge of nucleus
represents the nucleus as a bar magnet
useful as a pictorial model but has limited theoretical utility
Quantum Mechanics:
starts with quantized description of a nuclear spin, I
provides a better basis for understanding the NMR phenomenon
necessary to understand the more complex NMR pulse sequences, in particular 2-D NMR
2. Theory of NMR (Dayrit)

7. 1. Classical Mechanical approach to NMR
Classical picture. In the presence of an external magnetic field, B0 (z-axis), the nucleus precesses at an angle, , around B0. Such a precessing object possesses an angular momentum, P. If the object has a charge, this gives produces a magnetic moment, . The magnetogyric ratio, , is a constant which is characteristic of the nucleus which arises from the ratio of the magnetic moment, , and the angular momentum, P:
 = / P
2. Theory of NMR (Dayrit)

8. Limitations of the Classical mechanics approach:
1. Classical Mechanical approach to NMR
Limitations of the Classical mechanics approach:
In principle, Classical mechanics (CM) predicts that all nuclei should be observable by NMR. This is not so.
In CM, the only property that distinguishes the various nuclei is their nuclear mass. Classical mechanics predicts that the Larmor frequencies should be inversely proportional to the nuclear mass. This is not so.
CM does not provide a theoretical basis for the phenomena of spin-spin coupling and quadrupolar nuclei.
CM cannot predict transition energies.
CM cannot be used to design NMR pulse experiments.
2. Theory of NMR (Dayrit)

9. 2. Quantum Mechanical approach to NMR
Each atomic nucleus has a nuclear magnetic moment , m, which arises from the spin of the protons and neutrons in the nucleus. The nuclear magnetic moment varies from isotope to isotope of an element.
According to the shell model, protons and neutrons form pairs of opposite total angular momentum. Thus, the magnetic moment of a nucleus with even numbers of both protons and neutrons is zero, while that of a nucleus with an odd number of protons and even number of neutrons (or vice versa) depends on the unpaired proton (or neutron). However, the nuclear magnetic moment is only partly predicted by simple versions of the shell model..
2. Theory of NMR (Dayrit)

10. Table. Spin quantum numbers, I, of selected nuclei.
2. Quantum Mechanical approach to NMR
Each nucleus has a spin quantum number, I, which is determined by its nucleus (number of protons and neutrons).
Table. Spin quantum numbers, I, of selected nuclei.
2. Theory of NMR (Dayrit)

11. 2. Quantum Mechanical approach to NMR
(Jacobsen, 2007)
I = ½
Spin ½ nuclei have two quantum states that can be visualized as having the spin axis pointing “up” or “down”. In the absence of an external magnetic field, these two states have the same energy. In the presence of an external magnetic field, B0, and at thermal equilibrium a little less than half of a large population of nuclei will be in the higher state (b) and a little more than one half will be in the lower state (a).
2. Theory of NMR (Dayrit)

12. Periodic Table of Isotopes:
Spin Quantum Numbers, I
2. Theory of NMR (Dayrit)

13. The nuclear spin quantum number, I, has (2I + 1) values
2. Quantum Mechanical approach to NMR
The nuclear spin quantum number, I, has (2I + 1) values
The magnetic spin quantum numbers, mI:
mI = +I, (I - 1), ...-I
Each mI value represents a specific spin state.
The number of mI values represents the number of spin states available to a particular nucleus.
2. Theory of NMR (Dayrit)

14. 2. Quantum Mechanical approach to NMR
Table. I, mI and the number of spin states available to a particular nucleus: I mI
Number of mI values*
Examples 1
12C, 16O, 32S
+½, -½ 2
1H, 13C, 15N, 19F, 29Si, 31P
+1, 0, -1 3 2H
3/2
+3/2, +½, -½, -3/2 4
7Li, 11B, 23Na, 19F, 33S
* 2I + 1
Nuclei with I = 0 have no nuclear magnetic properties; they are not observable by NMR (e.g., 12C, 16O, 32S).
2. Theory of NMR (Dayrit)

15. Angular momentum, Pm = I (h/2)
2. Quantum Mechanical approach to NMR
The quantum mechanical description of NMR tells us that the maximum observable component of the angular momentum is:
Angular momentum, Pm = I (h/2)
(J-sec/rad)
m =  I (h/2)
The magnetic moment, m, is quantized along (2I + 1) orientations with respect to the static magnetic field:
(rad/T-sec)·(J-sec/rad) = J/T
where , the magnetogyric ratio, is the proportionality factor between the magnetic moment and the angular momentum.
Note: magnetic moment, m =  · Pm
2. Theory of NMR (Dayrit)

16. Table. Magnetogyric ratios, , of common nuclei.
1 rad = 2p, or approximately
2. Theory of NMR (Dayrit)

17. 3. I = ½ systems For I = ½ systems, there are two energy levels:
mI = +½ and -½.
The energy separation between these two levels is given by:
E = h 0
where o =  B0 / 2.
Therefore:
E =  (h/2) B0
and
E = 2B0
for I=½, m = ½  (h / 2)
Note that according to Quantum Mechanics:
a magnetic field is needed to observe the NMR phenomenon
E  B0
2. Theory of NMR (Dayrit)

18. Some common I = ½ nuclei: 1H, 13C, 19F, 31P
3. I = ½ systems
Splitting of a I = ½ nucleus in a magnetic field.
Energy level diagram for I=½. There are two mI levels: ½. These are separated by an energy of: E = 2B0 = (h/2)  B0. (Note that E is directly proportional to  and B0.)
By convention, the upper energy level is designated -½ or , while the lower energy level is +½ or .
Some common I = ½ nuclei:
1H, 13C, 19F, 31P
2. Theory of NMR (Dayrit)

19. 3. I = ½ systems
What is the energy difference between the two spin states of 1H at different magnetic fields?
E = (h/2)  B0
where h = x J-s
 = x 107 rad/T-s
B0 (T) 0 MHz (1H) E (x J)
2. Theory of NMR (Dayrit)

20. What is the energy difference between the transitions of 1H and 13C ?
3. I = ½ systems
What is the energy difference between the transitions of 1H and 13C ?
E = (h/2)  B0
Let B0 = 9.4 T
where h = x J-s
 (x 107 rad/T-s) E (x J) E (J/mol)
1H:
13C:
Note: E of 1H is about 4 times that of 13C.
2. Theory of NMR (Dayrit)

21. What is the energy difference between the transitions of 1H and 13C ?
3. I = ½ systems
What is the energy difference between the transitions of 1H and 13C ?
n0 = 400 MHz
n0 = 100 MHz
E = (h/2)  B0
2. Theory of NMR (Dayrit)

22. 4. The Magnetic Field
A magnetic field is necessary to be able to observe the NMR phenomenon.
Electromagnetic properties are inherent in matter because of their associated electronic properties and magnetic moments.
A nuclear spin with I = ½ is like a microscopic magnet whose orientation is arbitrary until it is placed in a magnetic field. The magnetic field splits it according to its magnetic energy levels. This is called the Zeeman effect.
2. Theory of NMR (Dayrit)

23. 4. The Magnetic Field
Effect of magnetic field on Larmor frequency. Splitting of the magnetic energy levels of 1H as a function of 0 (in MHz) vs. magnetic field strength, B0 (in tesla, T). Chemists generally use the 1H Larmor frequency when describing the magnetic field strength.
Relationship between magnetic field, B0 (tesla) and the 1H Larmor frequency (MHz).
Relationship between magnetic field, B0 (tesla), and the 1H and 13C Larmor frequencies (MHz).
2. Theory of NMR (Dayrit)

24. 4. The Magnetic Field
The NMR phenomenon (nuclear precession) is observable only in the presence of an external magnetic field, B0.
The natural frequency of the precession of a spin is called the Larmor frequency, 0:
0 =  B0 / 2
sec-1 = (rad T-1 s-1) (T)
where 0 is Larmor frequency, in Hz*
 is the magnetogyric ratio, in (107 radians/Tesla-sec)
B0 is the magnetic field, in Tesla
* The Larmor frequency, designated 0, has units in Hz:
0 = 2 0
* This is also alternatively given in units of (radians/sec):
0 =  B0
2. Theory of NMR (Dayrit)

25. Table. Larmor frequencies at B0 = 2.35 T and 9.4 T.
4. The Magnetic Field
Table. Larmor frequencies at B0 = 2.35 T and 9.4 T.
2. Theory of NMR (Dayrit)

26. Spin Quantum Numbers, I, and Larmor Frequencies, n0 (at 2.35 T)
NMR Periodic Table
Spin Quantum Numbers, I, and Larmor Frequencies, n0 (at 2.35 T)
2. Theory of NMR (Dayrit)

27. 4. The Magnetic Field
2. Theory of NMR (Dayrit)

28. N / N = exp (-h  B0 / 2 kB T )
5. Boltzmann Distribution
I = ½
In the absence of an external magnetic field (B0 = 0), the nuclear spin energy levels are degenerate (E = 0). When B0>0, the nuclei are split between the nuclear spin energy levels:  and .
Using the Boltzmann equation, the population distribution is given as:
N / N = exp (-E / kB T )
where kB = x J/K
T is the absolute temperature in Kelvin
Since E = (h/2)  B0, the Boltzmann equation can be rewritten as:
N / N = exp (-h  B0 / 2 kB T )
2. Theory of NMR (Dayrit)

29. At T = 300 K, the Boltzmann equation gives:
5. Boltzmann Distribution
Let us use the Boltzmann equation to calculate the population distribution for: 1H at 60 MHz (1.41 T), T = 300 K:
-E = -(6.626 x J s / 2) (26.75 x 107 rad / T s) (1.41 T)
-E = x J per 1H nucleus
At T = 300 K, the Boltzmann equation gives:
N / N = exp{(-2.50 x J) / [(1.381 x J / K) (300 K)]}
N / N =
For 1 million nuclei, 499,985 will be in N and 500,015 will be in N (diff. = 30 spins).
To solve: 1. N / N =
N = ( ) N
2. N + N = 1
2. Theory of NMR (Dayrit)

30. At T = 300 K, the Boltzmann equation gives:
5. Boltzmann Distribution
Let us compare the population difference for the following: 1H at 400 MHz (9.4 T), T = 300 and 400 K:
-E = -(6.626 x J s / 2) (26.75 x 107 rad / T s) (9.4 T)
-E = x J
At T = 300 K, the Boltzmann equation gives:
N / N = exp{(-1.67 x J) / [(1.381 x J / K) (300 K)]}
N / N =
For 1 million nuclei, 499,900 will be in N and 500,100 will be in N (diff. = 200 spins).
At T = 400 K, the Boltzmann equation gives:
N / N =
For 1 million nuclei, 499,924 will be in N and 500,076 will be in N (diff. = 152 spins).
2. Theory of NMR (Dayrit)

31. At T = 300 K, the Boltzmann equation gives:
5. Boltzmann Distribution
Now, let us use the Boltzmann equation to calculate the populations for: 13C at 9.4 T, T = 300 K:
-E = -(6.626 x J s / 2) (6.72 x 107 rad / T s) (9.4 T)
-E = x J per 13C nucleus
At T = 300 K, the Boltzmann equation gives:
N / N = exp{(-4.19 x J) / [(1.381 x J / K) (300 K)]}
N / N =
For 1 million nuclei, 499,975 will be in N and 500,025 will be in N (diff. = 50 spins).
2. Theory of NMR (Dayrit)

32. Magnetogyric ratio, g: NMR is more sensitive towards high g nuclei.
5. Boltzmann Distribution
To summarize, the Boltzmann equation predicts the following very important rules regarding the sensitivity of the NMR method:
Magnetogyric ratio, g: NMR is more sensitive towards high g nuclei.
Magnetic field, B0: Higher B0 results in higher sensitivity.
Temperature, K: Higher T leads to lower sensitivity.
2. Theory of NMR (Dayrit)

33. 6. Sensitivity of NMR
The sensitivity of a spectroscopic technique depends on the strength of the absorption signal. The strength of the absorption signal, in turn, depends on the population distribution in the energy levels.
The larger the energy difference between the ground and excited states, the larger is the difference in population. This population difference determines the sensitivity.
Because UV-visible spectroscopy involves electronic energy levels which have relatively large energy separations, it is the most sensitive among the spectroscopic techniques.
Sensitivity: UV-visible > Infrared > NMR
2. Theory of NMR (Dayrit)

34. 2. Theory of NMR (Dayrit)

35. Technique Amount of sample needed Sensitivity UV-visible ~mg/mL ppm
6. Sensitivity of NMR
Table. Comparison of measurement limts for UV-visible, infrared and NMR.
Technique Amount of sample needed Sensitivity
UV-visible ~mg/mL ppm
Infrared ~1 mg %
NMR ~1-10 mg/mL %
2. Theory of NMR (Dayrit)

36. Table. Relative sensitivity of some common I = ½ nuclei.
6. Sensitivity of NMR
Table. Relative sensitivity of some common I = ½ nuclei.
2. Theory of NMR (Dayrit)

37. NMR sensitivity  [(I+1) / (I2)] (5/2) (A%) (Bo3/2) (K-1)
6. Sensitivity of NMR
In sum, there are five factors which affect NMR sensitivity:
the spin quantum number, I;
the magnetogyric ratio of the nucleus, ;
the isotopic abundance, A%;
the strength of the magnetic field, Bo;
the temperature, K.
Theoretical parameters
Experimental parameters
NMR sensitivity  [(I+1) / (I2)] (5/2) (A%) (Bo3/2) (K-1)
2. Theory of NMR (Dayrit)

38. 6. Sensitivity of NMR
Larmor frequencies of various nuclei at 9.4 T and their relative intensities (where 13C = 1). 
13C
2. Theory of NMR (Dayrit)

39. Periodic Table of Isotopes and their NMR Parameters
2. Theory of NMR (Dayrit)

40. 7. The Macroscopic Magnetization Vector
The Boltzmann equation states that the population of spins will be distributed between the quantized lower and upper states. To represent the behavior of large populations, it is more convenient to use a macroscopic magnetization vector, M. While each nucleus has quantized behavior, the entire population of spins exhibits classical (continuous) behavior.
M1 = 2/3 M2
2. Theory of NMR (Dayrit)

41. 1. B0 is set along the z-axis.
7. The Macroscopic Magnetization Vector
From nuclear spin vectors to macroscopic magnetization vectors: the convention:
1. B0 is set along the z-axis.
2. Net spin population is represented by length of the macroscopic magnetization vector, M.
3. Continuous orientation of M.
4. Equilibrium position (ground state) is +z direction; vector is written as M0. +z +y +x M
(from: Lambert & Mazzola, 2003)
2. Theory of NMR (Dayrit)

42. 7. The Macroscopic Magnetization Vector
5. Phase of M vector is described by two angles from z-axis and x-axis.
6. Na and Nb are the populations of the lower and upper spin states, respectively.
7. This is not an energy diagram. This is used to represent the orientation of the net macroscopic magnetization vector during the course of the NMR experiment. +z +y +x M Nb Na
2. Theory of NMR (Dayrit)

43. 7. The Macroscopic Magnetization Vector
In a large collection of nuclei (I=½) at equilibrium, each nuclear vector will be rotating (precessing) around the z-axis at the same frequency, 0, but will be doing so randomly. Therefore, they will simply trace a uniform cone around the z-axis, which is represented by M0.
m2 = +½ +z +y +x M Nb Na
m2 = -½
2. Theory of NMR (Dayrit)

44. The net (x,y) component of M gives the phase coherence of the spins. M
7. The Macroscopic Magnetization Vector
The macroscopic magnetization vector, M, is the net vector representing all the spins.
M can take any angle although the individual spins can have only quantized orientations of ½.
The z-axis is called the longitudinal axis and the (x,y)- plane is called the transverse plane.
The net (x,y) component of M gives the phase coherence of the spins.
m2 = +½ M Nb Na
m2 = -½
2. Theory of NMR (Dayrit)

45. 8. The NMR pulse
Spectroscopic methods are based on the absorption or emission of a specific frequency of electromagnetic energy by the sample. In the case of NMR, the energy is in the radio frequency (RF) range and the absorbing species are the nuclei.
Pulse FT-NMR utilizes short bursts of RF energy, designated as B1, in the form of pulses to excite a broad range of frequencies. B1 can be considered as a second field.
2. Theory of NMR (Dayrit)

46. Increase in number of spins in higher spin energy level
8. The NMR pulse
At equilibrium, M is aligned along B0 (the z-axis), and its projections along the y-x plane, My and Mx = 0.
By convention, the RF pulse, B1, is implemented along the +x-axis.
NMR pulse B1
Increase in number of spins in higher spin energy level
2. Theory of NMR (Dayrit)

47. 8. The NMR pulse
A pulse at a right angle to B0 acts as a twisting force on M which causes M to tip away from its equilibrium position into the y'-axis.
A /2 (90o) pulse angle tips M into the x-y plane. The pulse causes a fraction of the spins to precess in-phase. This produces phase coherence.
Phase coherence
90 NMR pulse B1
2. Theory of NMR (Dayrit)

48. E . t  h (h . ) t  h   (1 / t)
8. The NMR pulse
According to the Heisenberg Uncertainty Principle:
E . t  h
where t is the pulse width (PW or p) usually ~ 10-6 s
h is Planck’s constant = x J-s
Expanding, we get:
(h . ) t  h
  (1 / t)
 refers to the range of the frequencies which is affected by the pulse, t. A short pulse excites a wide range. A long continuous pulse which excites a narrow range is the same as CW NMR irradiation.
2. Theory of NMR (Dayrit)

49. 8. The NMR pulse
In a pulsed NMR experiment, the nuclei are irradiated with an intense B1 field for the duration of the pulse. This causes M to precess at an angle, q, away from B0. During this time, the primary field present is B1, and the precession frequency becomes: w = g B1.
2. Theory of NMR (Dayrit)

50. 8. The NMR pulse
The angle or rotation, q, depends on the duration the B1 field is applied, tp. Since w = g B1, it follows that:
q = g B1 tp
2. Theory of NMR (Dayrit)

51. 15 ms 7 x 104 typical pulse width 1 s 1 used for double resonance and
8. The NMR pulse
Some typical values:
  (1 / t)
t  (Hz) Comments
1 ms 106
15 ms 7 x 104 typical pulse width
1 s 1 used for double resonance and
continuous wave (CW) experiments
2. Theory of NMR (Dayrit)

52. 9. Laboratory vs. Rotating Frame
After the RF pulse, B1, M will be tipped along the (x,y)-plane and will rotate at the Larmor frequency, say 400 MHz. Analysis of a rapidly moving vector is very cumbersome. For convenience, it is preferred to use the rotating frame of reference. When using the rotating frame, the axes are designated with primes (’), i.e., x’ and y’; the z-axis is sometimes designated as z’.
2. Theory of NMR (Dayrit)

53. Let us observe a sample with a single nucleus, such as the 13C in CHCl3, and that our NMR pulse is set exactly on-resonance of the 13C peak.
In the rotating frame, the frame rotates around the z-axis at the Larmor frequency. Peaks which are on-resonance are co-incident with the y’-axis and appear stationary.
In the fixed laboratory frame of reference, M revolves around the z-axis at the Larmor frequency.
2. Theory of NMR (Dayrit)

54. 9. Laboratory vs. Rotating Frame
If we set our pulse off-resonance,  (say 10 Hz), from the frequency position of the 13C peak of chloroform, then M will precess at frequency of  (10 Hz) relative to o in the rotating frame.
In the rotating frame therefore, the vectors move depending on its frequency relative to the center of the NMR pulse. It can be at higher frequency or lower frequency (~10s to 1000s of Hz). The rotating frame simplifies the vector description by removing the effect of the Larmor frequency which is in MHz.
2. Theory of NMR (Dayrit)

55. 10. The Bloch Equations
In 1946, Felix Bloch proposed that the population of nuclear spins could be represented by the macroscopic magnetization vector, M0, which showed the net orientation and magnitude of magnetization of the entire population of spins.
At equilibrium, M0 will align with the orientation of the external magnetic field, B0.
2. Theory of NMR (Dayrit)

56. 10. The Bloch Equations
The magnetization can be tipped away from its equilibrium value by the application of a second field, such an RF pulse.
This tipped magnetization will then precess about the external magnetic field at a well-defined frequency; this precession would generate a discrete energy emission, also in the RF range, which can be detected by a RF coil positioned at right angle to the external magnetic field.
Detector
2. Theory of NMR (Dayrit)

57. 10. The Bloch Equations
Immediately after a 90o pulse, M is aligned along the y’-axis and Mz = 0. M will relax back to its equilibrium position with a characteristic duration of time. This return of the z-component of the magnetization to its equilibrium position, Mz  M0, is known as the longitudinal relaxation and the length of time (in seconds) this takes is the longitudinal relaxation time, T1.
The relaxation (or the return) of the z component towards its equilibrium value, M0, is given by:
dMz / dt = -( Mz - M0 ) / T1
2. Theory of NMR (Dayrit)

58. 10. The Bloch Equations
Immediately following the 90o pulse, M is tipped into the y’-axis and the FID has its maximum value. If the pulse is on-resonance, M will stay on the y’-axis, but its projection on this axis will decrease as M relaxes to its equilibrium position along the z’-axis. The loss of magnetization on the (x’,y’)-plane, Mx, My  0, is called transverse relaxation, T2:
dMy / dt = - My / T2
dMx / dt = - Mx / T2
2. Theory of NMR (Dayrit)

59. dMz / dt = -( Mz - M0 ) / T1 dMy / dt = - My / T2 dMx / dt = - Mx / T2
10. The Bloch Equations Mz
dMz / dt = -( Mz - M0 ) / T1
dMy / dt = - My / T2
Mx, My
dMx / dt = - Mx / T2
T2 is affected both by the motion of M in the (x’, y’)-plane, as well as the return of Mz to M0 (that is, T1).
That is, the longitudinal relaxation time, T1, is the upper limit of the relaxation process. Other processes can shorten the transverse relaxation time, T2.
Therefore:
T1  T2
2. Theory of NMR (Dayrit)

60. 11. What is a Free Induction Decay (FID)?
As already stated, at equilibrium, the individual 13C nuclei of a sample like chloroform precess at the same frequency, , but are randomly distributed (that is, out-of-phase or incoherent). An RF pulse does two things: first, it increases the population of the upper spin energy level; and second, it introduces phase coherence.
Immediately after the RF pulse is turned off, the spins relax back to its equilibrium condition: the spin population returns to its equilibrium distribution, and the spins lose coherence. During this process, the signal collected is called the free induction decay, FID. The FID is the electrical signal produced during the precession of the nuclei.
2. Theory of NMR (Dayrit)

61. 11. What is a Free Induction Decay (FID)?
By convention the pulse is implemented along the x-axis. This brings M into the y-axis which is the observation axis (detector). It is the y-component of M which is detected as the free induction decay (FID). A vector along +y is read as a positive FID signal, while a vector in the -y direction is negative.
NMR pulse
detector B1
2. Theory of NMR (Dayrit)

62. 11. What is a Free Induction Decay (FID)?
M undergoes a precessing motion around Bo with frequency, 0. Since the detector is aligned along the y-axis, the receiver detects My as the FID signal.
The FID is a time-domain signal: the nucleus precesses at its resonance frequency  over time.
2. Theory of NMR (Dayrit)

63. 11. What is a Free Induction Decay (FID)?
The FID follows an exponential decay which is proportional to [exp (-t / T2)]. It is analogous to the damping of vibrations on a bell, where one type of nucleus is like one type of bell with the same note.
2. Theory of NMR (Dayrit)

64. 12. What is Fourier Transform?
Richard Ernst in the 1960s suggested the use of Fourier transform to analyze the FID and he proceeded to develop numerous techniques which became the basis of a new generation of NMR spectroscopy.
Fourier transform converts the time-domain FID into a frequency domain spectrum.
Given a sample with five different nuclei, the FID in the time domain contains the sum of the responses of all the nuclei acting at the same time. Fourier transform converts each signal to its corresponding frequency characteristic.
2. Theory of NMR (Dayrit)

65. 12. What is Fourier Transform?
FIDs from two separate samples containing nuclei 1 and 2 having different frequencies:
2. Theory of NMR (Dayrit)

66. 12. What is Fourier Transform?
FID showing addition of different frequencies from the two non-interacting nuclei, 1 and 2, mixed in one sample:
2. Theory of NMR (Dayrit)

67. f(n) =  f(t) exp(2pint) dt
12. What is Fourier Transform?
In converting time-domain FID to the familiar NMR frequency spectrum, we need to mathematically separate the various contributions from individual nuclei in the FID to yield the NMR spectrum with the correct combination of frequencies and amplitudes. The famous Fourier transform expression is given by the following equation:
f(n) =  f(t) exp(2pint) dt + -
The left side of the equation represents the NMR frequency spectrum while the right side of the equation is the time-domain FID.
2. Theory of NMR (Dayrit)

68. 12. What is Fourier Transform?
2. Theory of NMR (Dayrit)

69. 13. Summary of basic NMR theory
The theoretical basis of NMR starts with the magnetic spin quantum numbers, I and mI, and the magnetogyric ratio of the nucleus, g.
An external magnetic field, B0, is essential for the observation of the NMR phenomenon.
The Larmor frequency of a nucleus is given by the following equation: 0 = B0 /2.
NMR can be applied to all elements in the Periodic Table. NMR uses a particular isotope for each element with suitable values of I, g, and isotopic abundance.
2. Theory of NMR (Dayrit)

70. 13. Summary of basic NMR theory
The Boltzmann distribution is the population distribution between the upper and lower spin states. This depends on the energy difference between the spin states.
The sensitivity of NMR is determined by I, B0, g, isotopic abundance, and temperature.
The macroscopic magnetization, M, is a representation of a large population of quantized spins. We use the rotating frame of reference to follow M.
The Bloch equations describe the relaxation of nuclei: the longitudinal relaxation, T1, and the transverse relaxation, T2.
2. Theory of NMR (Dayrit)

71. An RF pulse is used to excite a wide range of frequencies.
13. Summary of basic NMR theory
An RF pulse is used to excite a wide range of frequencies.
An FID is produced following an RF pulse on a population of nuclear spins. This is a time-domain signal which is processed by Fourier transformation to convert it to a frequency spectrum (the NMR spectrum).
2. Theory of NMR (Dayrit)