Nuclear spins and magnets All nucleons, that is
neutrons and
protons, composing any atomic
nucleus, have the intrinsic quantum property of
spin, an intrinsic
angular momentum analogous to the classical angular momentum of a spinning sphere. The overall spin of the nucleus is determined by the
spin quantum number S. If the numbers of both the protons and neutrons in a given
nuclide are even then , i.e. there is no overall spin. Then, just as electrons pair up in nondegenerate
atomic orbitals, so do even numbers of protons or even numbers of neutrons (both of which are also
spin- particles and hence
fermions), giving zero overall spin. However, an unpaired proton and unpaired neutron will have a lower energy when their spins are parallel, not anti-parallel. This parallel spin alignment of distinguishable particles does not violate the
Pauli exclusion principle. The lowering of energy for parallel spins has to do with the
quark structure of these two nucleons. As a result, the spin
ground state for the deuteron (the nucleus of
deuterium, the 2H isotope of hydrogen), which has only a proton and a neutron, corresponds to a spin value of
1,
not of zero. On the other hand, because of the Pauli exclusion principle, the
tritium isotope of hydrogen must have a pair of anti-parallel spin neutrons (of total spin zero for the neutron spin-pair), plus a proton of spin . Therefore, the tritium total nuclear spin value is again , just like the simpler, abundant hydrogen isotope, 1H nucleus (the
proton). The NMR absorption frequency for tritium is also similar to that of 1H. In many other cases of
non-radioactive nuclei, the overall spin is also non-zero and may have a contribution from the
orbital angular momentum of the unpaired nucleon. For example, the nucleus has an overall spin value . A non-zero spin \vec{S} is associated with a non-zero
magnetic dipole moment, \vec{\mu} , via the relation \vec{\mu} = \gamma \vec{S} where
γ is the
gyromagnetic ratio. Classically, this corresponds to the proportionality between the angular momentum and the magnetic dipole moment of a spinning charged sphere, both of which are vectors parallel to the rotation axis whose length increases proportional to the spinning frequency. It is the magnetic moment and its interaction with magnetic fields that allows the observation of NMR signal associated with transitions between nuclear spin levels during resonant RF irradiation or caused by Larmor precession of the average magnetic moment after resonant irradiation. Nuclides with even numbers of both protons and neutrons have zero
nuclear magnetic dipole moment and hence do not exhibit NMR signal. For instance, is an example of a nuclide that produces no NMR signal, whereas , , and are nuclides that do exhibit NMR spectra. The last two nuclei have spin
S > and are therefore quadrupolar nuclei.
Electron spin resonance (ESR) is a related technique in which transitions between
electronic rather than nuclear spin levels are detected. The basic principles are similar but the instrumentation, data analysis, and detailed theory are significantly different. Moreover, there is a much smaller number of molecules and materials with unpaired electron spins that exhibit ESR (or
electron paramagnetic resonance (EPR)) absorption than those that have NMR absorption spectra. On the other hand, ESR has much higher signal per spin than NMR does.
Values of spin angular momentum Nuclear
spin is an intrinsic
angular momentum that is quantized. This means that the magnitude of this angular momentum is quantized (i.e.
S can only take on a restricted range of values), and also that the x, y, and z-components of the angular momentum are quantized, being restricted to integer or half-integer multiples of
ħ, the reduced
Planck constant. The integer or half-integer quantum number associated with the spin component along the z-axis or the applied magnetic field is known as the
magnetic quantum number,
m, and can take values from +
S to −
S, in integer steps. Hence for any given nucleus, there are a total of angular momentum states. The
z-component of the angular momentum vector ( \vec{S} ) is therefore . The
z-component of the magnetic moment is simply: \mu_z = \gamma S_z = \gamma m\hbar.
Spin energy in a magnetic field s (
spin magnetic moments). By itself, there is no energetic difference for any particular orientation of the nuclear magnet (only one energy state, on the left), but in an external magnetic field there is a high-energy state and a low-energy state depending on the relative orientation of the magnet to the external field, and in thermal equilibrium, the low-energy orientation is preferred. The average orientation of the magnetic moment will
precess around the field. The external field can be supplied by a large magnet and also by electrons and other nuclei in the vicinity. Consider nuclei with a spin of one-half, like , or . Each nucleus has two linearly independent spin states, with
m = or
m = − (also referred to as spin-up and spin-down, or sometimes α and β spin states, respectively) for the z-component of spin. In the absence of a magnetic field, these states are degenerate; that is, they have the same energy. Hence the number of nuclei in these two states will be essentially equal at
thermal equilibrium. If a nucleus with spin is placed in a magnetic field, however, the two states no longer have the same energy as a result of the interaction between the nuclear magnetic dipole moment and the external magnetic field. The
energy of a magnetic dipole moment \vec{\mu} in a magnetic field
B0 is given by: E = -\vec{\mu} \cdot \mathbf{B}_0 = -\mu_x B_{0x}-\mu_y B_{0y} - \mu_z B_{0z} . Usually the
z-axis is chosen to be along
B0, and the above expression reduces to: E = -\mu_\mathrm{z} B_0 \, , or alternatively: E = -\gamma m\hbar B_0 \, . As a result, the different nuclear spin states have different energies in a non-zero magnetic field. In less formal language, we can talk about the two spin states of a spin as being
aligned either with or against the magnetic field. If
γ is positive (true for most isotopes used in NMR) then ("spin up") is the lower energy state. The energy difference between the two states is: \Delta{E} = \gamma \hbar B_0 \, , and this results in a small population bias favoring the lower energy state in thermal equilibrium. With more spins pointing up than down, a net spin magnetization along the magnetic field
B0 results.
Precession of the spin magnetization A central concept in NMR is the precession of the spin magnetization around the magnetic field at the nucleus, with the angular frequency \omega = -\gamma B where \omega = 2 \pi \nu relates to the oscillation frequency \nu and
B is the magnitude of the field. This means that the spin magnetization, which is proportional to the sum of the spin vectors of nuclei in magnetically equivalent sites (the
expectation value of the spin vector in quantum mechanics), moves on a cone around the
B field. This is analogous to the precessional motion of the axis of a tilted spinning top around the gravitational field. In quantum mechanics, \omega is the
Bohr frequency Magnetic resonance and radio-frequency pulses A perturbation of nuclear spin orientations from equilibrium will occur only when an oscillating magnetic field is applied whose frequency
νrf sufficiently closely matches the
Larmor precession frequency
νL of the nuclear magnetization. The populations of the spin-up and -down energy levels then undergo
Rabi oscillations, The stronger the oscillating field, the faster the Rabi oscillations or the precession around the effective field in the rotating frame. After a certain time on the order of 2–1000 microseconds, a resonant RF pulse flips the spin magnetization to the transverse plane, i.e. it makes an angle of 90° with the constant magnetic field
B0 ("90° pulse"), while after a twice longer time, the initial magnetization has been inverted ("180° pulse"). It is the transverse magnetization generated by a resonant oscillating field which is usually detected in NMR, during application of the relatively weak RF field in old-fashioned continuous-wave NMR, or after the relatively strong RF pulse in modern pulsed NMR.
Chemical shielding It might appear from the above that all nuclei of the same nuclide (and hence the same
γ) would resonate at exactly the same frequency but this is not the case. The most important perturbation of the NMR frequency for applications of NMR is the "shielding" effect of the shells of electrons surrounding the nucleus. Electrons, similar to the nucleus, are also charged and rotate with a spin to produce a magnetic field opposite to the applied magnetic field. In general, this electronic shielding reduces the magnetic field
at the nucleus (which is what determines the NMR frequency). As a result, the frequency required to achieve resonance is also reduced. This shift in the NMR frequency due to the electronic molecular orbital coupling to the external magnetic field is called
chemical shift, and it explains why NMR is able to
probe the chemical structure of molecules, which depends on the electron density distribution in the corresponding molecular orbitals. If a nucleus in a specific chemical group is shielded to a higher degree by a higher electron density of its surrounding molecular orbitals, then its NMR frequency will be shifted "upfield" (that is, a lower chemical shift), whereas if it is less shielded by such surrounding electron density, then its NMR frequency will be shifted "downfield" (that is, a higher chemical shift). Unless the local
symmetry of such molecular orbitals is very high (leading to "isotropic" shift), the
shielding effect will depend on the orientation of the molecule with respect to the external field (
B0). In
solid-state NMR spectroscopy,
magic angle spinning is required to average out this orientation dependence in order to obtain frequency values at the average or isotropic chemical shifts. This is unnecessary in conventional NMR investigations of molecules in solution, since rapid "molecular tumbling" averages out the
chemical shift anisotropy (CSA). In this case, the "average" chemical shift (ACS) or isotropic chemical shift is often simply referred to as the chemical shift.
Radiation Damping In 1949, Suryan first suggested that the interaction between a
radiofrequency coil and a sample's bulk magnetization could explain why experimental observations of relaxation times differed from theoretical predictions. Building on this idea, Bloembergen and Pound further developed Suryan's hypothesis by mathematically integrating the
Maxwell–Bloch equations, a process through which they introduced the concept of "radiation damping." Radiation damping (RD) in
Nuclear Magnetic Resonance (NMR) is an intrinsic phenomenon observed in many high-field NMR experiments, especially relevant in systems with high concentrations of nuclei like protons or fluorine. RD occurs when transverse bulk magnetization from the sample, following a radio frequency pulse, induces an
electromagnetic field (emf) in the receiver coil of the NMR spectrometer. This generates an oscillating current and a non-linear induced transverse magnetic field which returns the spin system to equilibrium faster than other mechanisms of relaxation. RD can result in line broadening and measurement of a shorter spin-lattice relaxation time (T_{1}). For instance, a sample of water in a 400 MHz NMR spectrometer will have T_{RD} around 20 ms, whereas its T_{1} is hundreds of milliseconds. T_{R D}=\frac{2}{\gamma \mu_0 \eta Q M_0} [1] where \gamma is the
gyromagnetic ratio, \mu_0 is the
magnetic permeability, M_0 is the equilibrium magnetization per unit volume, \eta is the filling factor of the probe which is the ratio of the probe coil volume to the sample volume enclosed, Q=\frac{\omega L}{R} is the quality factor of the probe, \omega, L, and R are the resonance frequency, inductance, and resistance of the coil, respectively. The quantification of line broadening due to radiation damping can be determined by measuring the \Delta v_{\frac{1}{2}} and use equation [2]. T_{R D}^{-1}=\frac{\pi}{0.8384} \Delta v_{\frac{1}{2}} [2] Radiation damping in NMR is influenced significantly by system parameters. It is notably more prominent in systems where the NMR probe possesses a high quality factor (Q) and a high filling factor, resulting in a strong coupling between the probe coil and the sample. The phenomenon is also impacted by the concentration of the nuclei within the sample and their magnetic moments, which can intensify the effects of radiation damping. The strength of the magnetic field is inversely proportional to the lifetime of RD. and Q-factor switches reduce the feedback loop between the sample magnetization and the electromagnetic field induced by the coil and function successfully. Other approaches such as designing selective pulse sequences also effectively manage the fields induced by radiation damping. These approaches aim to control and limit the disruptive effects of radiation damping during NMR experiments and all approaches are successful in eliminating RD to a fairly large extent. Overall, understanding and managing radiation damping is crucial for obtaining high-quality NMR data, especially in modern high-field spectrometers where the effects can be significant due to the increased sensitivity and resolution.
Relaxation The process of population relaxation refers to nuclear spins that return to thermodynamic equilibrium in the magnet. This process is also called
T1, "
spin-lattice" or "longitudinal magnetic" relaxation, where
T1 refers to the mean time for an individual nucleus to return to its thermal equilibrium state of the spins. After the nuclear spin population has relaxed, it can be probed again, since it is in the initial, equilibrium (mixed) state. The
precessing nuclei can also fall out of alignment with each other and gradually stop producing a signal. This is called
T2, "
spin-spin" or
transverse relaxation. Because of the difference in the actual relaxation mechanisms involved (for example, intermolecular versus intramolecular magnetic dipole-dipole interactions),
T1 is usually (except in rare cases) longer than
T2 (that is, slower spin-lattice relaxation, for example because of smaller dipole-dipole interaction effects). In practice, the value of
T2*, which is the actually observed decay time of the observed NMR signal, or
free induction decay (to of the initial amplitude immediately after the resonant RF pulse), also depends on the static magnetic field inhomogeneity, which may be quite significant. (There is also a smaller but significant contribution to the observed FID shortening from the RF inhomogeneity of the resonant pulse). In the corresponding FT-NMR spectrum—meaning the
Fourier transform of the
free induction decay— the width of the NMR signal in frequency units is inversely related to the
T2* time. Thus, a nucleus with a long
T2* relaxation time gives rise to a very sharp NMR peak in the FT-NMR spectrum for a very homogeneous (
"well-shimmed") static magnetic field, whereas nuclei with shorter
T2* values give rise to broad FT-NMR peaks even when the magnet is shimmed well. Both
T1 and
T2 depend on the rate of molecular motions as well as the gyromagnetic ratios of both the resonating and their strongly interacting, next-neighbor nuclei that are not at resonance. decay experiment measuring dephasing time. A
Hahn echo decay experiment can be used to measure the dephasing time, as shown in the animation. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence that is not refocused by the 180° pulse. In simple cases, an
exponential decay is measured which is described by the
T2 time. ==NMR spectroscopy==