Nuclear spin is represented by an operator ˆI associated with the spin quantum
number I. The value of I corresponds to a spin angular momentum which is given by ~ ˆI. Nuclei with a certain value of spin angular momentum can be visualized
as tiny bar magnets with a magnetic dipole moment [87–89]
ˆ
µ = γ~ ˆI (2.1)
where γ is the gyromagnetic ratio of a nucleus. The z component of the magnetic moment is given by
ˆ
µz = γ~ ˆIz. (2.2)
In the absence of an external magnetic field, all magnetic moments in a sample are randomly oriented. This results in a net zero magnetic moment. When an external magnetic field is applied, the interaction between the magnetic field and the dipole moment is governed by the Hamiltonian [87–90]
ˆ
H = −ˆµ · ~B. (2.3)
When the applied magnetic field is in the z direction, this Hamiltonian is becomes
ˆ
Hz = −γ~ ˆIZB0. (2.4)
The energy eigenvalues corresponding to the above Hamiltonian can be found by using the Schrodinger equation [87–89]
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Hz|Ψi = −γ~B0Iz|Ψi
= −γ~B0m |Ψi
(2.6)
where m is the magnetic spin quantum number. Therefore, the energy of the magnetic interaction is given as
Em = −γ~m (2.7)
and it depends on the spin state of the nucleus. For a spin 1
2 system, m has two values ±1
2 . The energy corresponding to each value of m are
Em=1/2 = −1 2~γB0. (2.8) Em=−1/2 = 1 2~γB0. (2.9) The m = 1
2 state is lower in energy and corresponds to the parallel orientation of the dipole moment with respect to the external magnetic field. This energy state is represented by α. The m = −1
2 state is higher in energy and represents the anti-parallel orientation of the magnetic moment with the applied magnetic field. This energy state is represented by β. Hence, in the presence of an external magnetic field, the degenerate nuclear spin states split into different energy levels according to their alignment with the direction of field. This is called the Zeeman splitting [88–91] and is schematically shown in Figure 2.1 for a spin 1
2 system. At thermal equilibrium, the relative populations of the states α and β are given by the Boltzmann distribution. According to this distribution, the α state is more populated than the β state [89, 92]. Therefore, in the presence of an external magnetic field, the alignment of magnetic moments in the direction of the field gives rise to the net magnetic moment. Averaged over an entire sample, this net magnetic moment is called the equilibrium magnetization. This is schematically
Chapter 2. Biophysical methods to study the interactions of AMPs with model
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Figure 2.1: The depiction of the Zeeman splitting for spin 1
2 system. In the presence of the external magnetic field, the degenerate energy states split into two non-degenerate states. The splitting between the two energy states depends on the strength of the external magnetic field.
shown in Figure 2.2. Once the equilibrium magnetization is established, it does
Figure 2.2: In the absence of the external magnetic field, the magnetic moments of the individual spins are randomly oriented. On the other hand, if a magnetic field is applied in the z direction, then there is a slight energy preference for magnetic moments in the direction of the field. This results in a net magnetization in the direction of the field.
not change with time unless perturbed by some torque. This external torque can be provided by an oscillating magnetic field produced by a radio frequency (RF) pulse. Once the magnetization vector moves away from the z axis, it rotates about the direction of the magnetic field. This is called Larmor precession and
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the corresponding frequency is called the Larmor frequency [87–90,92]
ω0 = −γB0 (2.10)
where γ is the gyromagnetic ratio. The values of γ for common NMR-active nuclei are given in Table 2.1. The gyromagnetic ratio of a nucleus can be positive or neg- ative and determines the direction of precession. For example, a 1H has a positive value of γ and hence precesses clockwise in the presence of the external magnetic field, while 15N has negative a gyromagnetic ratio that results in counterclockwise precession. Nuclei Spin γ(M HzT−1) ω0(M Hz) 1 H 12 42.576 400.0 2H 1 6.536 61.4 15N 1 2 -4.316 40.5 31 P 12 17.235 162.0 13 C 12 10.705 94.7
Table 2.1: The gyromagnetic ratio and corresponding Larmor frequency of various nuclei. The Larmor frequency is calculated in a magnetic field strength of 9.4 T.
2.1.1.2 Free induction decay (FID) signal and NMR spectrum
In an NMR experiment, the magnetization is manipulated via RF pulses. For this purpose, a coil is mounted around the sample. The axis of the coil is aligned in the xy plane and the external magnetic field is along the z axis. The RF pulse generates an oscillating magnetic field in the coil which flips the equilibrium magnetization vector away from the z axis. This is illustrated in Figure 2.3 (a). The precessing magnetization vector produces an oscillating current in the coil that can be amplified and detected as a time-domain signal. This signal is termed
Chapter 2. Biophysical methods to study the interactions of AMPs with model
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the Free Induction Decay (FID) signal. The NMR spectrum is acquired by Fourier transformation of the FID signal which gives the spectrum in the frequency domain [87, 90, 92]. An FID signal and the corresponding NMR spectrum are shown schematically in Figure 2.3 (b).
(a)
(b)
Figure 2.3: (a) The schematic representation of the precessional motion of the magnetization vector about the z axis. A coil positioned along the x axis is used to apply the RF pulse and to detect the FID signal.(b) The time domain FID signal and the corresponding Fourier transformed NMR spectrum. This figure is inspired from “Understanding NMR spectroscopy” by James Keeler [92].
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