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NMR Spectroscopy Short Course 2013

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1. A Brief Introduction to NMR Technique Development

Nuclear Magnetic Resonance spectroscopy has been developed from experiments performed to accurately measure nuclear magnetogyric ratio sixty-five years ago. The technique depends on the fact that some atomic nuclei possess a nonzero spin angular momentum. A spinning charge generates magnetic field associated with its angular momentum. This phenomenon has long been known in molecular beams and has yielded a great deal of information on nuclear properties. Two independent groups in 1945, Purcell et al. at Harvard and Bloch et al. at Stanford reported the first observation of nuclear magnetic resonance in bulk matter. They were jointly awarded the Nobel Prize for physics in 1952 for this discovery.  In 1949 and 1950, Pake noted that nuclei of the same species absorbed energy at different frequencies. In 1951, Arnold’s discovery of three magnetically nonequivalent protons in ethyl alcohol paved the way for NMR to become a powerful tool for chemists. The importance of NMR spectroscopy is paramount in the fields of organic, inorganic and analytical chemistry for the investigation of molecular structure and dynamics. New developments have been applied to biochemistry, materials and medicine research. As a natural consequence of continuous development both in NMR instrumentation and methodology, more and more scientists will employ.

 

Over the last decade we have witnessed a substantial increase in the range and power of NMR experiments, which allow chemists to gain an order of magnitude more information than that provided by standard or traditional experiments. Three developments were necessary for this revolutionary change. First, spectrometer hardware including fast computing and networking had to become very reliable. Second, the software had to become easy to use and fast enough to control all experimental parameters by a keyboard and a mouse.   Third, the superconducting magnet had to reach the highest field ever. The 900MHz NMR instrument is available now. 

 

Continuing efforts have been made to develop methods to obtain more information from NMR measurements such as COSY, NOESY, ROESY, TOCSY, HETCOR, J-Resolved Spectroscopy, INADEQUATE, HMQC, Multiquantum Filter COSY etc. for liquids and CRAMPS, CP/MAS, TOSS, DOR, REDOR etc. for solids. One recent advance is Pulsed Field Gradient NMR. This technique can measure molecular diffusivities in a variety of samples such as liquids, solids and polymers. It can also be used to select specific coherence pathways and provide the NMR spectroscopists with a powerful method to improve the efficiency of multidimensional techniques and to obtain new information.

 

Nowadays, NMR probably is the most important technique for structure elucidation, material characterization and studying molecular motion.  As practicing chemists who are not NMR spectroscopists begin to consider using these NMR techniques in their work, they are almost immediately confronted by a series of questions. In this book those questions have been collected and organized along with appropriate answers.

 

 

2. Basic Theory of NMR

 

A. Magnetization of Nuclei in Magnetic Field

 

All nuclei carry a charge. In some nuclei this charge spins around the nuclear axis.  This spin generates a magnetic dipole along the axis. The spin angular momentum is decried in terms of the spin quantum number I.  If the sum of protons and neutrons is even, the spin quantum number, I will be 0, 1, 2, etc. For example, the 2H nucleus has one proton and one neutron. I is 1. If the sum of protons and neutrons is odd, I will be 1/2, 3/2 ...etc. For example, 13C nucleus has six protons and seven neutrons. I is ½. If both protons and neutrons are even numbers, I will be zero and then it is NMR insensitive.

 

 
 

 

 

 

 

 

Figure 2 - 1.  A spin with nonzero spin angular momentum m.  

There are a large number of nuclei, such as 1H, 13C, and 31P, they have a nonzero spin angular momentum, I ¹ 0, then Ih/2p ¹ 0.  The Zeeman Hamiltonian for a spin with quantum number I in a magnetic filed is:

 

 

                                           (2 – 1)

 

 

Where g is the magnetogyric ratio, a characteristic of the nucleus and it could be either positive or negative and B0 is the magnetic filed chosen by convention to be the z axis of the laboratory coordinate. For a spin I under the influence of a fixed magnetic field, the energy levels split into (2I + 1) sublevels, which are represented in Figure 1-2. The energy difference between neighboring levels can be expressed by:

 

 

                           (2 – 2)

 

Where mI takes values ±I, ±(I-1), ...... ±1/2 or zero depending on whether I is a half-odd integer or an integer.

 

 

 

Figure 2 - 2. The energy level splits into (2I +1) levels under the influence of a magnetic field.

 

Zeeman energy levels are displaced by a constant value, ghBo/2p, which generally can be expressed in frequency unit and is called the Larmor frequency of the isotope in the field of Bo.  This resonance frequency is found to vary in direct proportion to the applied field, thus the larger the magnetic field, the higher the resonance frequency. For proton we can represent this effect as in Figure 2-3.

 

 

 

 

Figure 2 - 3. The energy difference between two adjacent levels depends on the strength of applied magnetic field B0 (Tesla or Gauss.  1.00 T = 10,000 G).

 

For an ensemble of nuclear spins I, the (2I + 1) allowed energy levels are populated in thermal equilibrium in accordance with the Boltzmann distribution. For a spin I=1/2, the ratio of the number of spins in the higher energy state (b) compared to the lower energy state (a) is given by:

 

 

                                   (2 – 3)

 

 

Where e =ghB0/2pT. In principle, a bulk magnetization, M, is directly proportion to the net population difference between energy levels:

 

 

           (2 – 4)

 

 

Where m=ghI/2p. The value of the magnetization, M, can be shown to determine the signal intensity. From the equation it is shown that the concentration of nuclei in the sample, the strength of the magnet field B0, the magnetogyric ratio of the nuclei under the observation are directly proportion to the NMR signal intensity.  However, increase sample temperature T will reduce the NMR signal intensity.

 

B.    The Larmor Frequency

 

A typical magnetic field strength used for NMR is 9.395 Tesla. For proton and carbon, the resonance frequencies can be calculated by:

 

wo= gBo                                                                                 (2 – 5)

 

    

 

             

 

Where g is the magnetogyric ratio, and B0 is the strength of the magnetic field.

If a magnetic field B1 (typical strength 7.34 ´ 10-4 T) is placed along the X’ axis, a 90 degree pulse width can be calculated by:

 

 

 
 

 

 

 

 

 

 Figure 2 – 4. A  90 degree flip of a spin under magnetic field B1 along the X axis.

 

 

C. Spin-Lattice and Spin-Spin Relaxation

 

When a sample is inserted into the magnetic field B0, the Boltzmann distribution of spins occurs between the energy levels. The equilibrium is established by means of specific relaxation process and gives rise to a small excess of nuclei in the lower state. We can apply an oscillating field, B1 perpendicular to the B0 axis, to manipulate this spin system. After B1 is removed, there are two different mechanisms that allow spins return to equilibrium of the longitudinal and transverse components. The spin-lattice relaxation is a process whereby non-radiative energy transfer takes place from “excited” spins to the surrounding of the molecules.  These relaxation processes can be described by the Bloch equations:

 

                         (2 – 6)

 

                                                (2 – 7)

 

                                                  (2 – 8)

 

Where T1 is the spin-lattice relaxation time and T2 is the spin-spin relaxation time. The magnitude of T1 and T2 is related to the relaxation efficiency that is a property of the molecule. T1 and T2 are also related to the structure and mobility of the molecule.

 

D. Chemical Shift

 

When an atom is placed in a magnetic field, its electrons circulate about the direction of the applied magnetic field. This circulation causes a small magnetic field at the nucleus that opposes the externally applied field.

 
 

 

 

 

 

 

 

 Figure 2 – 5. A   water molecule is placed in a magnetic filed. Its electrons cause a small magnetic field that opposes the applied filed.

 

The magnetic field at the nucleus (the effective field) is therefore generally less than the applied field by a fraction. 

 

 

B = Bo (1-s                                                                                      (2 – 9)

 

 

So the Larmor frequency of the nucleus under observation is smaller. w=gB. We use TMS as an chemical standard, its frequency under the filed refer to wo=gBo, or fref. In an NMR spectrum, each nucleus has a characteristic frequency or chemical shift.  It is defined as:

 

 

                                                    (2 – 10)

 

The chemical shift is a finger printer of a nucleus in the molecule. It relates to nucleus’s environment and relative position in the molecule. For the proton NMR, 1ppm is equal to 500 Hz under the static filed of 11.74 Tesla (500MHz NMR instrument). However, under the static filed of 4.70 Tesla (200 MHz NMR instrument), 1ppm is equal to 200Hz.

 

 

 

Figure 2 - 6. Two spins coupled each other with a coupling constant J. The chemical shift d, 1ppm is equal to 60, 200 and 500 Hz respect to the static field of 60, 200 and 500 MHz instruments.  The J is a constant in different field, however, Chemical shift between peaks d (in Hz) is increased as field strength increasing, so the two pair of peaks will be resolved in a spectrum acquired in a high field while overlapped in lower filed.

 

Chemical shifts arise from the simultaneous interaction of a nucleus with an electron and the electron with the applied static magnetic field. It is practically impossible to calculate a chemical shift value from the screening factor due to the complexity of the mechanisms that give rise to it. 

 

E. Spin-Spin Coupling (Scalar Coupling or J-Coupling)

 

Spin-Spin coupling is the coupling of spins through the bonding electrons. It results in the multiple peaks observed in the NMR spectra. The distance (in Hz) between the multiple peaks (JH-H or JH-X) provides important molecular structure information.

 

If two protons are magnetically inequivalent, there are two peaks in the spectrum for each proton. If these two protons are scalar coupled, then the other senses the spin states of one nucleus.  Since proton (I=1/2) has two energy levels (+1/2, -1/2), the coupled proton will be splited to two lines relative to the two energy states.   If one of the nuclei has a spin of one (I=1), then the nucleus to which it is coupled become split into three lines because the nucleus has three energy levels (+1, 0, -1).  A good example is CDCl3, the carbon spectrum will have a triplet with equivalent intensity since deuterium has spin of one (I=1).

 

 

 

Figure 2 – 7, (A). The proton a and proton b are not coupled. (B). The proton a and proton b are coupled. (C). The carbon is coupled with D (I=1) and the carbon spectrum will be a triplet.

 

 

F. Dipole-Dipole Coupling

 

Dipole-Dipole coupling is the coupling of spins through the space. They are not necessary bonded. The Dipole-Dipole interaction is an important source of relaxation effect but not necessary broaden the lines in liquids. In solids, however, it is the dominant source of line broadening. If there are two spins, I and S, an approximate dipolar Hamiltonian can be written as:

 

                                (2 – 11)

 

Where q is the angle between the internuclear vector and the applied field, r is the distance between two nuclei.  In liquid, due to the random motion of molecules, q is a random value. The average value of (1-3cos2q) is zero for all possible directions. In solids, the (1-3cos2q) is not zero since molecule can not move freely. Hd-d is the major line-broadening factor. In order to narrowing the line, the CP/MAS (Cross Polarization/ Magic Angle Spinning) probe is designed so that the angle between sample tube and field is 54.7 degree. The term (1-3cos2q) in the equation will be zero when q is equal to 54.70 (Refer to Magic Angle)

 

G. Cross Polarization (CP)

 

The presence of strong dipole coupling between rare spin (such as 13C) and abundant spin (such as 1H) in solids or the presence of scalar J coupling (JC-H) in liquids can be used to enhance the sensitivity of the rare spin observation under an appropriate conditions.

 

Cross Polarization or Polarization Transfer is very important technique to observe chemical shift correlation between two different nuclei, to observe very insensitive nuclei coupled to proton, such as 15N.  The key to this type of experiment is that the signal of the nucleus that we observe in t2 in somehow modulated by the chemical shift of one or more other nuclei through the polarization transfer.  

 

 

H. Nuclear Overhauser Effect (NOE)

 

A change in the integrated NMR absorption intensity of a spin when the NMR absorption of another spin is saturated is know as the Nuclear Overhauser Effect (NOE). It depends on the observing field and mobility in solution of the molecule under study. The NOE is a very important tool for determination of the distance between spins.

 

The NOE is characterized by an enhancement factor:

 

 

                                                                        (2 – 12)

 

 

Where I0 is the intensity of a peak without irradiation of the other spin, and I with irradiation. The maximum NOE in liquid is:

 

 

                                                                                          (2 – 13)             

 

 

Where gi is the magnetogyric ratio of irradiating nucleus; go is that of observed nucleus. For homonuclear, the maximum NOE is 0.5. For heteronuclear, the NOE depends on the value of g and its sign. One very important application of NOE is enhancement of S/N when the go is low. In some case, even if the observing nucleus is not directly connected to protons, it could help to develop a potential intermolecular NOE enhancement by dissolving the compound in a protonated solvent, rather than a pure deuterated solvent.

 

 

Table  of  Magnetogyric Ratio

Nucleus

Natural Abundance

(%)

Magnetogyric Ratio g

(107 rad T-1 S-1)

Larmor Frequency

(MHz)

1H

99.985

26.753

100.000,000

13C

1.108

6.7283

25.145,004

15N

0.37

-2.7126

10.136,767

19F

100.0

25.1815

94.094,003

29Si

4.70

-5.319

19.867,184

31P

100.0

10.8394

40.480,747

183W

14.28

1.1283

4.151,888

2H

0.015

4.1064

15.351

17O

0.037

-3.6266

13.557

27Al

100

6.9762

26.077

51V

99.76

7.0492

26.350

95Mo

15.72

1.7514

6.547

I. Magnetic Field Strength and Transmitter Frequency

 

The nature action of a nuclear spin in a magnetic field, whether a static field (The magnetic field generated by a DC current in a coil) or an oscillatory field (The magnetic field generated by an AC current in a coil) is that of precession, likes a top. The frequency of precession depends on the strength of the field. For example, a TMS proton signal will be at the frequency of 100,000,000 Hz in a static field (2.34874 Tesla), a water proton will be at the frequency of 100,000,480 Hz (100,000,000 + 4.80ppm ´ 100Hz). If the instrument has a 100,000,000 Hz transmitter, then the TMS signal will be at the frequency of 0.0 Hz, i.e. on resonance (at 0.0 ppm) and the water signal has a frequency of 480Hz.  The difference between two signals is 4.8 ppm. If the static magnetic field is 2.3380 Tesla, transmitter frequency is set to 99.6 MHz. If we still set reference to TMS as 0.0 ppm, the water signal will still 4.8 ppm away from TMS, but at this filed 1 ppm is equal to 99.6 HZ, rather 100 Hz. At this point we know it is important when a internal reference is selected. In most case, the residual amount of CHCl3 in CDCl3 is enough as reference. Sometimes, however the signals may overlap, it is hard to determine the reference peak. A good practice is inert a external reference by using coaxial tube.  

 

 

J.     Laboratory Frame and Rotating Frame

 

A laboratory Frame is refer to XYZ axis. In convention, the spin precesses along the Z axis with Larmor Frequency in a magnetic field. Rotating Frame is an imaginary frame refer to X’Y’Z’ axis which precesses as the same frequency as an observe spin. In the other words, the spin in the rotating frame is stationary in a fixed magnetic field. When a B1 along the X axis is applied to the spin, the spin will rotate around the X’ axis. So the spin manipulation is much simplified comparing with in the laboratory frame.    

 

 
 

 

 

 

 

 

 

 

Figure 2 - 8. (A) Spins process in a static magnetic field under the laboratory frame; (B) Spins process in a static magnetic field under rotating frame; (C) Spins rotate in an oscillatory field along the X’; (D) Spins rotate in an oscillatory field along the Y’.  The w1 is the frequency depending on the strength of the filed of B1.

 

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