Spin states, magnetic fields, and electromagnetic radiation

In the presence of an applied external magnetic field, ¹H and ¹³C (also ¹⁹F and ³¹P) nuclei exist in two nuclear spin states of different energy. Slightly more than half of the nuclei exist in the lower energy state called alpha () than in the higher energy state called beta (). The external magnetic field B_o, which is homogeneous, is created with a large magnet, commonly a super-conducting solenoid. The energy difference E between the spin states is proportional to the strength of B_o. A typical B_o field is 7.05 T (Tesla) or 70,500 Gauss. NMR spectroscopy records transitions between these spin states induced by a radio frequency electromagnetic field called the B₁ field. With an applied magnetic field of 7.05 T, the frequency of the B₁ field is about 300 MHz (megahertz, million cycles per second) for ¹H NMR or 75 MHz for ¹³C NMR. Correspondingly, if the applied field is 11.75 T, the frequency is 500 MHz for ¹H NMR or 125 MHz for ¹³C NMR. Instruments operating at 500 MHz give better sensitivity and resolution than those operating at 300 MHz. The relationship between field and frequency is shown mathematically by Equation 1 and graphically in Figure 1.

Equation 1 introduces the important term B_e. In a spherically symmetric s orbital, the B_o field (the external magnetic field) causes the electrons in the cloud surrounding the nucleus to circulate through the orbital (Figure 2). This circulation of electrons produces B_e: an induced magnetic field opposed to the B_o field. Thus, the B_e field slightly shields the nucleus from the applied magnetic field B_o and the value (B_o–B_e) represents the actual magnetic field experienced at the nucleus. The B_e field produced by electrons in p and d orbitals are more complex (discussed further in Proton Chemical Shift).

Figure 2 shows that the B_e field slightly shields the nucleus from the applied magnetic field B_o and the value (B_o-B_e) represents the actual magnetic field experienced at the nucleus.

If we divide through by h, equation 1 simplifies to

or the frequency of a nuclear spin transition is directly proportional to the magnetic field experienced at the nucleus. This nuclear spin transition frequency is also called the resonance or precession frequency. The proportionality constant is four times larger for the ¹H nucleus than it is for the ¹³C nucleus. This is why the resonance frequency () is 300 MHz for ¹H and only 75 MHz for ¹³C.

Almost equal population of the and nuclear spin states occurs because the energy difference between the spin states is quite small, 3x10^-5 kcal/mol at 300 MHz, and the higher energy state is heavily populated by available thermal energy. The actual population ratio is given by the Boltzmann equation that relates the population ratio to E and the absolute temperature. In contrast, higher energy vibrational states in IR spectroscopy are not significantly populated by available thermal energy because the energy difference between vibrational states is much larger, 5.8 kcal/mol at 2000 cm^-1. An excess population of the spin state is required in order to observe an NMR signal, and the slight excess of spins over spins at 7.05 T is more than sufficient. The excess is higher at higher B_o fields (11.75T/500 MHz versus 7.05 T/300 MHz) because E is higher. Correspondingly, a 500 MHz instrument has higher sensitivity than a 300 MHz instrument.

Next section: Chemical Shift

Copyright information: Original content © University of Colorado, Boulder, Chemistry and Biochemistry Department, 2011. The information on these pages is available for academic use without restriction.

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