Nuclear magnetic resonance (MRI Imaging and NMR Spectroscopy Principles)

Nuclear magnetic resonance

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Bruker 700 MHz. Nuclear Magnetic Resonance (NMR) spectrometer

Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a magnetic field absorb and re-emit electromagnetic radiation. This energy is at a specific resonance frequency which depends on the strength of the magnetic field and the magnetic properties of the isotope of the atoms; in practical applications, the frequency is similar to VHF and UHF television broadcasts (60–1000 MHz). NMR allows the observation of specific quantum mechanical magnetic properties of the atomic nucleus. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals, and non-crystalline materials through NMR spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).

All isotopes that contain an odd number of protons and/or of neutrons (see Isotope) have an intrinsic magnetic moment and angular momentum, in other words a nonzero spin, while all nuclides with even numbers of both have a total spin of zero. The most commonly studied nuclei are 1H and 13C, although nuclei from isotopes of many other elements (e.g. 2H, 6Li, 10B, 11B, 14N, 15N, 17O, 19F, 23Na, 29Si, 31P, 35Cl, 113Cd, 129Xe, 195Pt) have been studied by high-field NMR spectroscopy as well.

A key feature of NMR is that the resonance frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonance frequencies of the sample's nuclei depend on where in the field they are located. Since the resolution of the imaging technique depends on the magnitude of magnetic field gradient, many efforts are made to develop increased field strength, often using superconductors. The effectiveness of NMR can also be improved using hyperpolarization, and/or using two-dimensional, three-dimensional and higher-dimensional multi-frequency techniques.

The principle of NMR usually involves two sequential steps:

• The alignment (polarization) of the magnetic nuclear spins in an applied, constant magnetic field H₀.
• The perturbation of this alignment of the nuclear spins by employing an electro-magnetic, usually radio frequency (RF) pulse. The required perturbing frequency is dependent upon the static magnetic field (H₀) and the nuclei of observation.

The two fields are usually chosen to be perpendicular to each other as this maximizes the NMR signal strength. The resulting response by the total magnetization (M) of the nuclear spins is the phenomenon that is exploited in NMR spectroscopy and magnetic resonance imaging. Both use intense applied magnetic fields (H₀) in order to achieve dispersion and very high stability to deliver spectral resolution, the details of which are described by chemical shifts, the Zeeman effect, and Knight shifts (in metals).

NMR phenomena are also utilized in low-field NMR, NMR spectroscopy and MRI in the Earth's magnetic field (referred to as Earth's field NMR), and in several types of magnetometers.

History

Nuclear magnetic resonance was first described and measured in molecular beams by Isidor Rabi in 1938,^[1] by extending the Stern–Gerlach experiment, and in 1944, Rabi was awarded the Nobel Prize in physics for this work.^[2] In 1946, Felix Bloch and Edward Mills Purcell expanded the technique for use on liquids and solids, for which they shared the Nobel Prize in Physics in 1952.^[3][4]

Purcell had worked on the development of radar during World War II at the Massachusetts Institute of Technology's Radiation Laboratory. His work during that project on the production and detection of radio frequency power and on the absorption of such RF power by matter laid the foundation for Rabi's discovery of NMR.

Rabi, Bloch, and Purcell observed that magnetic nuclei, like 1H and 31P, could absorb RF energy when placed in a magnetic field and when the RF was of a frequency specific to the identity of the nuclei. When this absorption occurs, the nucleus is described as being in resonance. Different atomic nuclei within a molecule resonate at different (radio) frequencies for the same magnetic field strength. The observation of such magnetic resonance frequencies of the nuclei present in a molecule allows any trained user to discover essential chemical and structural information about the molecule.

The development of NMR as a technique in analytical chemistry and biochemistry parallels the development of electromagnetic technology and advanced electronics and their introduction into civilian use.

Theory of nuclear magnetic resonance

Nuclear spin and magnets

All nucleons, that is neutrons and protons, composing any atomic nucleus, have the intrinsic quantum property of spin. The overall spin of the nucleus is determined by the spin quantum number S. If the number of both the protons and neutrons in a given nuclide are even then S = 0, i.e. there is no overall spin. Then, just as electrons pair up in atomic orbitals, so do even numbers of protons or even numbers of neutrons (which are also spin-¹⁄₂ particles and hence fermions) pair up giving zero overall spin.

However, a proton and neutron will have lower energy when their spins are parallel, not anti-parallel, since this parallel spin alignment does not infringe upon the Pauli Exclusion Principle, but instead it has to do with the quark structure of these two nucleons. Therefore, the spin ground state for the deuteron (the deuterium nucleus, or the ²H isotope of hydrogen)—that has only a proton and a neutron—corresponds to a spin value of 1, not of zero. The single, isolated deuteron therefore exhibits an NMR absorption spectrum characteristic of a quadrupolar nucleus of spin 1, which in the "rigid" state at very low temperatures is a characteristic ('Pake') doublet, (not a singlet as for a single, isolated ¹H, or any other isolated fermion or dipolar nucleus of spin 1/2). 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 1/2. Therefore, the character of the tritium nucleus is again magnetic dipolar, not quadrupolar—like its non-radioactive deuteron cousin—and the tritium nucleus total spin value is again 1/2, just like for the simpler, abundant hydrogen isotope, ¹H nucleus (the proton). The NMR absorption (radio) frequency for tritium is however slightly higher than that of ¹H because the tritium nucleus has a slightly higher gyromagnetic ratio than ¹H. In many other cases of non-radioactive nuclei, the overall spin is also non-zero. For example, the 27Al nucleus has an overall spin value S = ⁵⁄₂.

A non-zero spin is thus always associated with a non-zero magnetic moment (μ) via the relation μ = γS, where γ is the gyromagnetic ratio. It is this magnetic moment that allows the observation of NMR absorption spectra caused by transitions between nuclear spin levels. Most nuclides (with some rare exceptions) that have both even numbers of protons and even numbers of neutrons, also have zero nuclear magnetic moments, and they also have zero magnetic dipole and quadrupole moments. Hence, such nuclides do not exhibit any NMR absorption spectra. Thus, 18O is an example of a nuclide that has no NMR absorption, whereas 13C, 31P, 35Cl and 37Cl are nuclides that do exhibit NMR absorption spectra. The last two nuclei are quadrupolar nuclei whereas the preceding two nuclei (13C and 31P) are dipolar ones.

Electron spin resonance (ESR) is a related technique in which transitions between electronic spin levels are detected rather than nuclear ones. 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. ESR has much higher sensitivity than NMR does.

Values of spin angular momentum

The angular momentum associated with nuclear spin is quantized. This means both that the magnitude of angular momentum is quantized (i.e. S can only take on a restricted range of values), and also that the orientation of the associated angular momentum is quantized. The associated quantum number 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 2S + 1 angular momentum states.
The z-component of the angular momentum vector (S) is therefore S_z = mħ, where ħ is the reduced Planck constant. The z-component of the magnetic moment is simply:

Spin behavior in a magnetic field

Splitting of nuclei spin states in an external magnetic field

An intuitive model. Nuclei behave like they had own magnetic moments (spin magnetic moments). By itself, there is no energetic difference for any particular orientation (only one energy state, on the left), but in external magnetic field there is a high-energy state and a low-energy state depending on the relative orientations of the magnet to the external field, and the orientation of the magnetic moment can precess relative to it. The external field can be supplied by a large magnet and also by other nuclei in the vicinity.

Consider nuclei which have a spin of one-half, like 1H, 13C or 19F. The nucleus has two possible spin states: m = ¹⁄₂ or m = −¹⁄₂ (also referred to as spin-up and spin-down, or sometimes α and β spin states, respectively). These states are degenerate, that is they have the same energy. Hence the number of atoms in these two states will be approximately equal at thermal equilibrium.

If a nucleus is placed in a magnetic field, however, the interaction between the nuclear magnetic moment and the external magnetic field mean the two states no longer have the same energy. The energy of a magnetic moment μ when in a magnetic field B₀ is given by:

Usually the z axis is chosen to be along B₀, and the above expression reduces to:

or alternatively:

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 ¹⁄₂ as being aligned either with or against the magnetic field. If γ is positive (true for most isotopes) then m = ¹⁄₂ is the lower energy state.

The energy difference between the two states is:

and this difference results in a small population bias toward the lower energy state.

Magnetic resonance by nuclei

Resonant absorption by nuclear spins will occur only when electromagnetic radiation of the correct frequency (e.g., equaling the Larmor precession rate) is being applied to match the energy difference between the nuclear spin levels in a constant magnetic field of the appropriate strength. The energy of an absorbed photon is then E = hν₀, where ν₀ is the resonance radiofrequency that has to match (that is, it has to be equal to the Larmor precession frequency ν_L of the nuclear magnetization in the constant magnetic field B₀). Hence, a magnetic resonance absorption will only occur when ΔE = hν₀, which is when ν₀ = γB₀/(2π). Such magnetic resonance frequencies typically correspond to the radio frequency (or RF) range of the electromagnetic spectrum for magnetic fields up to roughly 20 T. It is this magnetic resonant absorption which is detected in NMR.^{[citation needed]}

Nuclear shielding

It might appear from the above that all nuclei of the same nuclide (and hence the same γ) would resonate at the same frequency. This is not the case. The most important perturbation of the NMR frequency for applications of NMR is the "shielding" effect of the surrounding shells of electrons.^[5] Electrons, similar to the nucleus, are also charged and rotate with a spin to produce a magnetic field opposite to the magnetic field produced by the nucleus. In general, this electronic shielding reduces the magnetic field at the nucleus (which is what determines the NMR frequency).

As a result the energy gap is reduced, and 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 orbital, 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 values close to the average chemical shifts. This is unnecessary in conventional NMR investigations of molecules, since rapid "molecular tumbling" averages out the chemical shift anisotropy (CSA). In this case, the term "average" chemical shift (ACS) is used.

Relaxation

The process called population relaxation refers to nuclei that return to the thermodynamic state 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. Once the nuclear spin population is 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 (returning the net magnetization vector to a non-precessing field) and stop producing a signal. This is called T₂ or transverse relaxation. Because of the difference in the actual relaxation mechanisms involved (for example, inter-molecular vs. intra-molecular 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 which is the actually observed decay time of the observed NMR signal, or free induction decay, (to 1/e of the initial amplitude immediately after the resonant RF pulse)-- also depends on the static magnetic field inhomogeneity, which is 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 time is inversely related to the width of the NMR signal in frequency units. 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.

A Hahn echo decay experiment can be used to measure the dephasing time, as shown in the animation below. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence which is not refocused by the pulse. In simple cases, an exponential decay is measured which is described by the time.

NMR spectroscopy

900 MHz, 21.2 T NMR Magnet at HWB-NMR, Birmingham, UK

NMR spectroscopy is one of the principal techniques used to obtain physical, chemical, electronic and structural information about molecules due to either the chemical shift, Zeeman effect, or the Knight shift effect, or a combination of both, on the resonant frequencies of the nuclei present in the sample. It is a powerful technique that can provide detailed information on the topology, dynamics and three-dimensional structure of molecules in solution and the solid state. Thus, structural and dynamic information is obtainable (with or without "magic angle" spinning (MAS)) from NMR studies of quadrupolar nuclei (that is, those nuclei with spin S > ¹⁄₂) even in the presence of magnetic "dipole-dipole" interaction broadening (or simply, dipolar broadening) which is always much smaller than the quadrupolar interaction strength because it is a magnetic vs. an electric interaction effect.

Additional structural and chemical information may be obtained by performing double-quantum NMR experiments for quadrupolar nuclei such as 2H. Also, nuclear magnetic resonance is one of the techniques that has been used to design quantum automata, and also build elementary quantum computers.^[6][7]

Continuous wave (CW) spectroscopy

In its first few decades, nuclear magnetic resonance spectrometers used a technique known as continuous-wave spectroscopy (CW spectroscopy). Although NMR spectra could be, and have been, obtained using a fixed magnetic field and sweeping the frequency of the electromagnetic radiation, this more typically involved using a fixed frequency source and varying the current (and hence magnetic field) in an electromagnet to observe the resonant absorption signals. This is the origin of the counterintuitive, but still common, "high field" and "low field" terminology for low frequency and high frequency regions respectively of the NMR spectrum.

CW spectroscopy is inefficient in comparison with Fourier analysis techniques (see below) since it probes the NMR response at individual frequencies in succession. Since the NMR signal is intrinsically weak, the observed spectrum suffers from a poor signal-to-noise ratio. This can be mitigated by signal averaging i.e. adding the spectra from repeated measurements. While the NMR signal is constant between scans and so adds linearly, the random noise adds more slowly – proportional to the square-root of the number of spectra (see random walk). Hence the overall signal-to-noise ratio increases as the square-root of the number of spectra measured.

Fourier transform spectroscopy

Most applications of NMR involve full NMR spectra, that is, the intensity of the NMR signal as a function of frequency. Early attempts to acquire the NMR spectrum more efficiently than simple CW methods involved illuminating the target simultaneously with more than one frequency. A revolution in NMR occurred when short pulses of radio-frequency radiation began to be used—centered at the middle of the NMR spectrum. In simple terms, a short pulse of a given "carrier" frequency "contains" a range of frequencies centered about the carrier frequency, with the range of excitation (bandwidth) being inversely proportional to the pulse duration, i.e. the Fourier transform of a short pulse contains contributions from all the frequencies in the neighborhood of the principal frequency. The restricted range of the NMR frequencies made it relatively easy to use short (millisecond to microsecond) radio frequency pulses to excite the entire NMR spectrum.^{[citation needed]}

Applying such a pulse to a set of nuclear spins simultaneously excites all the single-quantum NMR transitions. In terms of the net magnetization vector, this corresponds to tilting the magnetization vector away from its equilibrium position (aligned along the external magnetic field). The out-of-equilibrium magnetization vector precesses about the external magnetic field vector at the NMR frequency of the spins. This oscillating magnetization vector induces a current in a nearby pickup coil, creating an electrical signal oscillating at the NMR frequency. This signal is known as the free induction decay (FID), and it contains the vector sum of the NMR responses from all the excited spins. In order to obtain the frequency-domain NMR spectrum (NMR absorption intensity vs. NMR frequency) this time-domain signal (intensity vs. time) must be Fourier transformed. Fortunately the development of Fourier Transform NMR coincided with the development of digital computers and the digital Fast Fourier Transform. Fourier methods can be applied to many types of spectroscopy.

Richard R. Ernst was one of the pioneers of pulse NMR, and he won a Nobel Prize in chemistry in 1991 for his work on Fourier Transform NMR and his development of multi-dimensional NMR (see below).

Multi-dimensional NMR Spectroscopy

The use of pulses of different shapes, frequencies and durations in specifically designed patterns or pulse sequences allows the spectroscopist to extract many different types of information about the molecule. Multi-dimensional nuclear magnetic resonance spectroscopy is a kind of FT NMR in which there are at least two pulses and, as the experiment is repeated, the pulse sequence is systematically varied. In multidimensional nuclear magnetic resonance there will be a sequence of pulses and, at least, one variable time period. In three dimensions, two time sequences will be varied. In four dimensions, three will be varied.

There are many such experiments. In one, these time intervals allow (amongst other things) magnetization transfer between nuclei and, therefore, the detection of the kinds of nuclear-nuclear interactions that allowed for the magnetization transfer. Interactions that can be detected are usually classified into two kinds. There are through-bond interactions and through-space interactions, the latter usually being a consequence of the nuclear Overhauser effect. Experiments of the nuclear Overhauser variety may be employed to establish distances between atoms, as for example by 2D-FT NMR of molecules in solution.

Although the fundamental concept of 2D-FT NMR was proposed by Jean Jeener from the Free University of Brussels at an International Conference, this idea was largely developed by Richard Ernst who won the 1991 Nobel prize in Chemistry for his work in FT NMR, including multi-dimensional FT NMR, and especially 2D-FT NMR of small molecules.^[8] Multi-dimensional FT NMR experiments were then further developed into powerful methodologies for studying biomolecules in solution, in particular for the determination of the structure of biopolymers such as proteins or even small nucleic acids.^[9]

In 2002 Kurt Wüthrich shared the Nobel Prize in Chemistry (with John Bennett Fenn and Koichi Tanaka) for his work with protein FT NMR in solution.

Solid-state NMR spectroscopy

This technique complements X-ray crystallography in that it is frequently applicable to molecules in a liquid or liquid crystal phase, whereas crystallography, as the name implies, is performed on molecules in a solid phase. Though nuclear magnetic resonance is used to study solids, extensive atomic-level molecular structural detail is especially challenging to obtain in the solid state. There is little signal averaging by thermal motion in the solid state, where most molecules can only undergo restricted vibrations and rotations at room temperature, each in a slightly different electronic environment, therefore exhibiting a different NMR absorption peak. Such a variation in the electronic environment of the resonating nuclei results in a blurring of the observed spectra—which is often only a broad Gaussian band for non-quadrupolar spins in a solid- thus making the interpretation of such "dipolar" and "chemical shift anisotropy" (CSA) broadened spectra either very difficult or impossible.
Professor Raymond Andrew at Nottingham University in the UK pioneered the development of high-resolution solid-state nuclear magnetic resonance. He was the first to report the introduction of the MAS (magic angle sample spinning; MASS) technique that allowed him to achieve spectral resolution in solids sufficient to distinguish between chemical groups with either different chemical shifts or distinct Knight shifts. In MASS, the sample is spun at several kilohertz around an axis that makes the so-called magic angle θ_m (which is ~54.74°, where cos²θ_m = 1/3) with respect to the direction of the static magnetic field B₀; as a result of such magic angle sample spinning, the chemical shift anisotropy bands are averaged to their corresponding average (isotropic) chemical shift values. The above expression involving cos²θ_m has its origin in a calculation that predicts the magnetic dipolar interaction effects to cancel out for the specific value of θ_m called the magic angle. One notes that correct alignment of the sample rotation axis as close as possible to θ_m is essential for cancelling out the dipolar interactions whose strength for angles sufficiently far from θ_m is usually greater than ~10 kHz for C-H bonds in solids, for example, and it is thus greater than their CSA values.

There are different angles for the sample spinning relative to the applied field for the averaging of quadrupole interactions and paramagnetic interactions, correspondingly ~30.6° and ~70.1°
A concept developed by Sven Hartmann and Erwin Hahn was utilized in transferring magnetization from protons to less sensitive nuclei (popularly known as cross-polarization) by M.G. Gibby, Alex Pines and John S. Waugh. Then, Jake Schaefer and Ed Stejskal demonstrated also the powerful use of cross-polarization under MASS conditions which is now routinely employed to detect low-abundance and low-sensitivity nuclei.

Sensitivity

Because the intensity of nuclear magnetic resonance signals and, hence, the sensitivity of the technique depends on the strength of the magnetic field the technique has also advanced over the decades with the development of more powerful magnets. Advances made in audio-visual technology have also improved the signal-generation and processing capabilities of newer instruments.

As noted above, the sensitivity of nuclear magnetic resonance signals is also dependent on the presence of a magnetically susceptible nuclide and, therefore, either on the natural abundance of such nuclides or on the ability of the experimentalist to artificially enrich the molecules, under study, with such nuclides. The most abundant naturally occurring isotopes of hydrogen and phosphorus (for example) are both magnetically susceptible and readily useful for nuclear magnetic resonance spectroscopy. In contrast, carbon and nitrogen have useful isotopes but which occur only in very low natural abundance.

Other limitations on sensitivity arise from the quantum-mechanical nature of the phenomenon. For quantum states separated by energy equivalent to radio frequencies, thermal energy from the environment causes the populations of the states to be close to equal. Since incoming radiation is equally likely to cause stimulated emission (a transition from the upper to the lower state) as absorption, the NMR effect depends on an excess of nuclei in the lower states. Several factors can reduce sensitivity, including

• Increasing temperature, which evens out the population of states. Conversely, low temperature NMR can sometimes yield better results than room-temperature NMR, providing the sample remains liquid.
• Saturation of the sample with energy applied at the resonant radiofrequency. This manifests in both CW and pulsed NMR; in the first case (CW) this happens by using too much continuous power that keeps the upper spin levels completely populated; in the second case (pulsed), each pulse (that is at least a 90° pulse) leaves the sample saturated, and four to five times the (longitudinal) relaxation time (5 T₁) must pass before the next pulse or pulse sequence can be applied. For single pulse experiments, shorter RF pulses that tip the magnetization by less than 90° can be used, which loses some intensity of the signal, but allows for shorter recycle delays. The optimum there is called an Ernst angle, after the Nobel laureate. Especially in solid state NMR, or in samples with very few nuclei with spins > 0, (diamond with the natural 1% of Carbon-13 is especially troublesome here) the longitudinal relaxation times can be on the range of hours, while for proton-NMR they are more on the range of one second.
• Non-magnetic effects, such as electric-quadrupole coupling of spin-1 and spin-³⁄₂ nuclei with their local environment, which broaden and weaken absorption peaks. 14N, an abundant spin-1 nucleus, is difficult to study for this reason. High resolution NMR instead probes molecules using the rarer 15N isotope, which has spin-¹⁄₂.

Isotopes

Many isotopes of chemical elements can be used for NMR analysis.^[10]

Commonly used nuclei:

• 1H, the most commonly used spin ½ nucleus in NMR investigation, has been studied using many forms of NMR. Hydrogen is highly abundant, especially in biological systems. It is the nucleus most sensitive to NMR signal (apart from 3H which is not commonly used due to its instability and radioactivity). Proton NMR produces narrow chemical shift with sharp signals. Fast acquisition of quantitative results (peak integrals in stoichiometric ratio) is possible due to short relaxation time. The 1H signal has been the sole diagnostic nucleus used for clinical magnetic resonance imaging.
• 2H, a spin 1 nucleus commonly utilized as signal-free medium in the form of deuterated solvents during proton NMR, to avoid signal interference from hydrogen-containing solvents in measurement of 1H solutes. Also used in determining the behavior of lipids in lipid membranes and other solids or liquid crystals as it is a relatively non-perturbing label which can selectively replace 1H. Alternatively, 2H can be detected in media specially labeled with 2H. Deuterium resonance is commonly used in high-resolution NMR spectroscopy to monitor drifts in the magnetic field strength (lock) and to improve the homogeneity of the external magnetic field.
• 3He, is very sensitive to NMR. There is a very low percentage in natural helium, and subsequently has to be purified from 4He. It is used mainly in studies of endohedral fullerenes, where its chemical inertness is beneficial to ascertaining the structure of the entrapping fullerene.
• 11B, more sensitive than 10B, yields sharper signals. Quartz tubes must be used as borosilicate glass interferes with measurement.
• 13C spin-1/2, is widely used, despite its relative paucity in naturally occurring carbon (approximately 1%). It is stable to nuclear decay. Since there is a low percentage in natural carbon, spectrum acquisition on samples which have not been experimentally enriched in 13C takes a long time. Frequently used for labeling of compounds in synthetic and metabolic studies. Has low sensitivity and wide chemical shift, yields sharp signals. Low percentage makes it useful by preventing spin-spin couplings and makes the spectrum appear less crowded. Slow relaxation means that spectra are not integrable unless long acquisition times are used.
• 14N, spin-1, medium sensitivity nucleus with wide chemical shift. Its large quadrupole moment interferes in acquisition of high resolution spectra, limiting usefulness to smaller molecules and functional groups with a high degree of symmetry such as the headgroups of lipids.
• 15N, spin-1/2, relatively commonly used. Can be used for labeling compounds. Nucleus very insensitive but yields sharp signals. Low percentage in natural nitrogen together with low sensitivity requires high concentrations or expensive isotope enrichment.
• 17O, spin-5/2, low sensitivity and very low natural abundance (0.037%), wide chemical shifts range (up to 2000 ppm). Quadrupole moment causing a line broadening. Used in metabolic and biochemical studies in studies of chemical equilibria.
• 19F, spin-1/2, relatively commonly measured. Sensitive, yields sharp signals, has wide chemical shift.
• 31P, spin-1/2, 100% of natural phosphorus. Medium sensitivity, wide chemical shifts range, yields sharp lines. Spectra tend to have a moderate amount of noise. Used in biochemical studies and in coordination chemistry where phosphorus containing ligands are involved.
• 35Cl and 37Cl, broad signal. 35Cl significantly more sensitive, preferred over 37Cl despite its slightly broader signal. Organic chlorides yield very broad signals, its use is limited to inorganic and ionic chlorides and very small organic molecules.
• 43Ca, used in biochemistry to study calcium binding to DNA, proteins, etc. Moderately sensitive, very low natural abundance.
• 195Pt, used in studies of catalysts and complexes.

Other nuclei (usually used in the studies of their complexes and chemical binding, or to detect presence of the element):

• 6Li, 7Li
• 9Be
• 19F
• 21Ne
• 23Na
• 25Mg
• 27Al
• 29Si
• 31P
• 33S
• 39K, 40K, 41K
• 45Sc
• 47Ti, 49Ti
• 50V, 51V
• 53Cr
• 55Mn
• 57Fe
• 59Co
• 61Ni
• 63Cu, 65Cu
• 67Zn
• 69Ga, 71Ga
• 73Ge
• 75As
• 77Se
• 81Br
• 87Rb
• 87Sr
• 95Mo
• 109Ag
• 113Cd
• 119Sn
• 125Te
• 127I
• 133Cs
• 135Ba, 137Ba
• 139La
• 183W
• 199Hg

Applications

Medicine

Medical MRI

The application of nuclear magnetic resonance best known to the general public is magnetic resonance imaging for medical diagnosis and magnetic resonance microscopy in research settings, however, it is also widely used in chemical studies, notably in NMR spectroscopy such as proton NMR, carbon-13 NMR, deuterium NMR and phosphorus-31 NMR. Biochemical information can also be obtained from living tissue (e.g. human brain tumors) with the technique known as in vivo magnetic resonance spectroscopy or chemical shift NMR Microscopy.

These studies are possible because nuclei are surrounded by orbiting electrons, which are charged particles that generate small, local magnetic fields that add to or subtract from the external magnetic field, and so will partially shield the nuclei. The amount of shielding depends on the exact local environment. For example, a hydrogen bonded to an oxygen will be shielded differently from a hydrogen bonded to a carbon atom. In addition, two hydrogen nuclei can interact via a process known as spin-spin coupling, if they are on the same molecule, which will split the lines of the spectra in a recognizable way.

As one of the two major spectroscopic techniques used in metabolomics, NMR is used to generate metabolic fingerprints from biological fluids to obtain information about disease states or toxic insults.

Chemistry

By studying the peaks of nuclear magnetic resonance spectra, chemists can determine the structure of many compounds. It can be a very selective technique, distinguishing among many atoms within a molecule or collection of molecules of the same type but which differ only in terms of their local chemical environment. NMR spectroscopy is used to unambiguously identify known and novel compounds, and as such, is usually required by scientific journals for identity confirmation of synthesized new compounds. See the articles on carbon-13 NMR and proton NMR for detailed discussions.

By studying T₂ information, a chemist can determine the identity of a compound by comparing the observed nuclear precession frequencies to known frequencies. Further structural data can be elucidated by observing spin-spin coupling, a process by which the precession frequency of a nucleus can be influenced by the magnetization transfer from nearby chemically bound nuclei. Spin-spin coupling is observed in NMR of hydrogen-1 (1H NMR), since its natural abundance is nearly 100%; isotope enrichment is required for most other elements.

Because the nuclear magnetic resonance timescale is rather slow, compared to other spectroscopic methods, changing the temperature of a T₂*experiment can also give information about fast reactions, such as the Cope rearrangement or about structural dynamics, such as ring-flipping in cyclohexane. At low enough temperatures, a distinction can be made between the axial and equatorial hydrogens in cyclohexane.

An example of nuclear magnetic resonance being used in the determination of a structure is that of buckminsterfullerene (often called "buckyballs", composition C₆₀). This now famous form of carbon has 60 carbon atoms forming a sphere. The carbon atoms are all in identical environments and so should see the same internal H field. Unfortunately, buckminsterfullerene contains no hydrogen and so 13C nuclear magnetic resonance has to be used. 13C spectra require longer acquisition times since carbon-13 is not the common isotope of carbon (unlike hydrogen, where 1H is the common isotope). However, in 1990 the spectrum was obtained by R. Taylor and co-workers at the University of Sussex and was found to contain a single peak, confirming the unusual structure of buckminsterfullerene.^[11]

Purity determination (w/w NMR)

NMR is primarily used for structural determination, however it can also be used for purity determination, providing that the structure and molecular weight of the compound is known. This technique requires the use of an internal standard of a known purity. Typically this standard will have a high molecular weight to facilitate accurate weighing, but relatively few protons so as to give a clear peak for later integration e.g. 1,2,3,4-tetrachloro-5-nitrobenzene. Accurately weighed portions of both the standard and sample are combined and analysed by NMR. Suitable peaks are selected for both compounds and the purity of the sample determined via the following equation.

Where:

: Weight of internal standard

: Weight of sample

: The integrated area of the peak selected for comparison in the standard, corrected for the number of protons in that functional group

: The integrated area of the peak selected for comparison in the sample, corrected for the number of protons in that functional group

: Molecular weight of standard

: Molecular weight of sample

: Purity of internal standard

Non-destructive testing

Nuclear magnetic resonance is extremely useful for analyzing samples non-destructively. Radio waves and static magnetic fields easily penetrate many types of matter and anything that is not inherently ferromagnetic. For example, various expensive biological samples, such as nucleic acids, including RNA and DNA, or proteins, can be studied using nuclear magnetic resonance for weeks or months before using destructive biochemical experiments. This also makes nuclear magnetic resonance a good choice for analyzing dangerous samples.

Acquisition of dynamic information

In addition to providing static information on molecules by determining their 3D structures in solution, one of the remarkable advantages of NMR over X-ray crystallography is that it can be used to obtain important dynamic information.

Data acquisition in the petroleum industry

Another use for nuclear magnetic resonance is data acquisition in the petroleum industry for petroleum and natural gas exploration and recovery. A borehole is drilled into rock and sedimentary strata into which nuclear magnetic resonance logging equipment is lowered. Nuclear magnetic resonance analysis of these boreholes is used to measure rock porosity, estimate permeability from pore size distribution and identify pore fluids (water, oil and gas). These instruments are typically low field NMR spectrometers.

Flow probes for NMR spectroscopy

Recently, real-time applications of NMR in liquid media have been developed using specifically designed flow probes (flow cell assemblies) which can replace standard tube probes. This has enabled techniques that can incorporate the use of high performance liquid chromatography (HPLC) or other continuous flow sample introduction devices.^[12]

Process control

NMR has now entered the arena of real-time process control and process optimization in oil refineries and petrochemical plants. Two different types of NMR analysis are utilized to provide real time analysis of feeds and products in order to control and optimize unit operations. Time-domain NMR (TD-NMR) spectrometers operating at low field (2–20 MHz for 1H) yield free induction decay data that can be used to determine absolute hydrogen content values, rheological information, and component composition. These spectrometers are used in mining, polymer production, cosmetics and food manufacturing as well as coal analysis. High resolution FT-NMR spectrometers operating in the 60 MHz range with shielded permanent magnet systems yield high resolution 1H NMR spectra of refinery and petrochemical streams. The variation observed in these spectra with changing physical and chemical properties is modeled using chemometrics to yield predictions on unknown samples.
The prediction results are provided to control systems via analogue or digital outputs from the spectrometer.

Earth's field NMR

In the Earth's magnetic field, NMR frequencies are in the audio frequency range, or the very low frequency and ultra low frequency bands of the radio frequency spectrum. Earth's field NMR (EFNMR) is typically stimulated by applying a relatively strong dc magnetic field pulse to the sample and, after the end of the pulse, analyzing the resulting low frequency alternating magnetic field that occurs in the Earth's magnetic field due to free induction decay (FID). These effects are exploited in some types of magnetometers, EFNMR spectrometers, and MRI imagers. Their inexpensive portable nature makes these instruments valuable for field use and for teaching the principles of NMR and MRI.
An important feature of EFNMR spectrometry compared with high-field NMR is that some aspects of molecular structure can be observed more clearly at low fields and low frequencies, whereas other aspects observable at high fields are not observable at low fields. This is because:

• Electron-mediated heteronuclear J-couplings (spin-spin couplings) are field independent, producing clusters of two or more frequencies separated by several Hz, which are more easily observed in a fundamental resonance of about 2 kHz. "Indeed it appears that enhanced resolution is possible due to the long spin relaxation times and high field homogeneity which prevail in EFNMR."^[13]
• Chemical shifts of several ppm are clearly separated in high field NMR spectra, but have separations of only a few millihertz at proton EFNMR frequencies, so are usually lost in noise etc.

Zero Field NMR

In Zero Field NMR all magnetic fields are shielded such that magnetic fields below nT (nano-Tesla) are achieved and the nuclear precession frequencies of all nuclei are close to zero and indistinguishable. Under those circumstances the observed spectra are no-longer dictated by chemical shifts but primarily by J-coupling interactions which are independent of the external magnetic field.
Since inductive detection schemes are not sensitive at very low frequencies, on the order of the J-couplings (typically between 0 and 1000 Hz), alternative detection schemes are used. Specifically, sensitive magnetometers turn out to be good detectors for Zero Field NMR. A zero magnetic field environment does not provide any polarization hence it is the combination of zero-field NMR with hyperpolarization schemes that makes zero field NMR attractive.

Quantum computing

NMR quantum computing uses the spin states of molecules as qubits. NMR differs from other implementations of quantum computers in that it uses an ensemble of systems, in this case molecules.

Magnetometers

Various magnetometers use NMR effects to measure magnetic fields, including proton precession magnetometers (PPM) (also known as proton magnetometers), and Overhauser magnetometers. See also Earth's field NMR.

Near-infrared spectroscopy (Also Needed to Understand Greenhouse Gases)

Near-infrared spectroscopy

From Wikipedia, the free encyclopedia

Near-IR absorption spectrum of dichloromethane showing complicated overlapping overtones of mid IR absorption features.

Near-infrared spectroscopy (NIRS) is a spectroscopic method that uses the near-infrared region of the electromagnetic spectrum (from about 800 nm to 2500 nm). Typical applications include pharmaceutical, medical diagnostics (including blood sugar and pulse oximetry), food and agrochemical quality control, and combustion research, as well as research in functional neuroimaging, sports medicine & science, elite sports training, ergonomics, rehabilitation, neonatal research, brain computer interface, urology (bladder contraction), and neurology (neurovascular coupling).

Theory

Near-infrared spectroscopy is based on molecular overtone and combination vibrations. Such transitions are forbidden by the selection rules of quantum mechanics. As a result, the molar absorptivity in the near-IR region is typically quite small. One advantage is that NIR can typically penetrate much farther into a sample than mid infrared radiation. Near-infrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little or no sample preparation.

The molecular overtone and combination bands seen in the near-IR are typically very broad, leading to complex spectra; it can be difficult to assign specific features to specific chemical components. Multivariate (multiple variables) calibration techniques (e.g., principal components analysis, partial least squares, or artificial neural networks) are often employed to extract the desired chemical information. Careful development of a set of calibration samples and application of multivariate calibration techniques is essential for near-infrared analytical methods.^[1]

History

Near-infrared spectrum of liquid ethanol.

The discovery of near-infrared energy is ascribed to William Herschel in the 19th century, but the first industrial application began in the 1950s. In the first applications, NIRS was used only as an add-on unit to other optical devices that used other wavelengths such as ultraviolet (UV), visible (Vis), or mid-infrared (MIR) spectrometers. In the 1980s, a single-unit, stand-alone NIRS system was made available, but the application of NIRS was focused more on chemical analysis. With the introduction of light-fiber optics in the mid-1980s and the monochromator-detector developments in early-1990s, NIRS became a more powerful tool for scientific research.

This optical method can be used in a number of fields of science including physics, physiology, or medicine. It is only in the last few decades that NIRS began to be used as a medical tool for monitoring patients.

Instrumentation

Instrumentation for near-IR (NIR) spectroscopy is similar to instruments for the UV-visible and mid-IR ranges. There is a source, a detector, and a dispersive element (such as a prism, or, more commonly, a diffraction grating) to allow the intensity at different wavelengths to be recorded. Fourier transform NIR instruments using an interferometer are also common, especially for wavelengths above ~1000 nm. Depending on the sample, the spectrum can be measured in either reflection or transmission.

Common incandescent or quartz halogen light bulbs are most often used as broadband sources of near-infrared radiation for analytical applications. Light-emitting diodes (LEDs) are also used; they offer greater lifetime and spectral stability and reduced power requirements.^[2]

The type of detector used depends primarily on the range of wavelengths to be measured.
Silicon-based CCDs are suitable for the shorter end of the NIR range, but are not sufficiently sensitive over most of the range (over 1000 nm). InGaAs and PbS devices are more suitable though less sensitive than CCDs. In certain diode array (DA) NIRS instruments, both silicon-based and InGaAs detectors are employed in the same instrument. Such instruments can record both UV-visible and NIR spectra 'simultaneously'.

Instruments intended for chemical imaging in the NIR may use a 2D array detector with an acousto-optic tunable filter. Multiple images may be recorded sequentially at different narrow wavelength bands.^[3]

Many commercial instruments for UV/vis spectroscopy are capable of recording spectra in the NIR range (to perhaps ~900 nm). In the same way, the range of some mid-IR instruments may extend into the NIR. In these instruments, the detector used for the NIR wavelengths is often the same detector used for the instrument's "main" range of interest.

Applications

Typical applications of NIR spectroscopy include the analysis of foodstuffs, pharmaceuticals, combustion products, and a major branch of astronomical spectroscopy.

Astronomical spectroscopy

Near-infrared spectroscopy is used in astronomy for studying the atmospheres of cool stars where molecules can form. The vibrational and rotational signatures of molecules such as titanium oxide, cyanide, and carbon monoxide can be seen in this wavelength range and can give a clue towards the star's spectral type. It is also used for studying molecules in other astronomical contexts, such as in molecular clouds where new stars are formed. The astronomical phenomenon known as reddening means that near-infrared wavelengths are less affected by dust in the interstellar medium, such that regions inaccessible by optical spectroscopy can be studied in the near-infrared. Since dust and gas are strongly associated, these dusty regions are exactly those where infrared spectroscopy is most useful. The near-infrared spectra of very young stars provide important information about their ages and masses, which is important for understanding star formation in general.

Agriculture

Near-infrared spectroscopy is widely applied in agriculture for determining the quality of forages, grains, and grain products, oilseeds, coffee, tea, spices, fruits, vegetables, sugarcane, beverages, fats, and oils, dairy products, eggs, meat, and other agricultural products. It is widely used to quantify the composition of agricultural products because it meets the criteria of being accurate, reliable, rapid, non-destructive, and inexpensive.^[4]

Remote monitoring

Techniques have been developed for NIR spectroscopic imaging. Hyperspectral imaging has been applied for a wide range of uses, including the remote investigation of plants and soils. Data can be collected from instruments on airplanes or satellites to assess ground cover and soil chemistry.

Materials Science

Techniques have been developed for NIR spectroscopy of microscopic sample areas for film thickness measurements, research into the optical characteristics of nanoparticles and optical coatings for the telecommunications industry.

Medical uses

The primary application of NIRS to the human body uses the fact that the transmission and absorption of NIR light in human body tissues contains information about hemoglobin concentration changes. When a specific area of the brain is activated, the localized blood volume in that area changes quickly. Optical imaging can measure the location and activity of specific regions of the brain by continuously monitoring blood hemoglobin levels through the determination of optical absorption coefficients.

NIRS can be used for non-invasive assessment of brain function through the intact skull in human subjects by detecting changes in blood hemoglobin concentrations associated with neural activity, e.g., in branches of Cognitive psychology as a partial replacement for fMRI techniques.^[5] NIRS can be used on infants, and NIRS is much more portable than fMRI machines, even wireless instrumentation is available, which enables investigations in freely moving subjects.^[6][7] However, NIRS cannot fully replace fMRI because it can only be used to scan cortical tissue, where fMRI can be used to measure activation throughout the brain. Special public domain statistical toolboxes for analysis of stand alone and combined NIRS/MRI measurement have been developed^[8] (NIRS-SPM).

The application in functional mapping of the human cortex is called diffuse optical tomography (DOT), near-infrared imaging (NIRI) or functional NIRS (fNIR). The term diffuse optical tomography is used for three-dimensional NIRS. The terms NIRS, NIRI, and DOT are often used interchangeably, but they have some distinctions. The most important difference between NIRS and DOT/NIRI is that DOT/NIRI is used mainly to detect changes in optical properties of tissue simultaneously from multiple measurement points and display the results in the form of a map or image over a specific area, whereas NIRS provides quantitative data in absolute terms on up to a few specific points. The latter is also used to investigate other tissues such as, e.g., muscle,^[9] breast and tumors.^[10] NIRS can be used to quantify blood flow, blood volume, oxygen consumption, reoxygenation rates and muscle recovery time in muscle.^[9]

By employing several wavelengths and time resolved (frequency or time domain) and/or spatially resolved methods blood flow, volume and absolute tissue saturation ( or Tissue Saturation Index (TSI)) can be quantified.^[11] Applications of oximetry by NIRS methods include neuroscience, ergonomics, rehabilitation, brain computer interface, urology, the detection of illnesses that affect the blood circulation (e.g., peripheral vascular disease), the detection and assessment of breast tumors, and the optimization of training in sports medicine.

The use of NIRS in conjunction with a bolus injection of indocyanine green (ICG) has been used to measure cerebral blood flow^[12][13] and cerebral metabolic rate of oxygen consumption (CMRO2).^[14] It has also been shown that CMRO2 can be calculated with combined NIRS/MRI measurements.^[15]

NIRS is starting to be used in pediatric critical care, to help deal with cardiac surgery post-op. Indeed, NIRS is able to measure venous oxygen saturation (SVO2), which is determined by the cardiac output, as well as other parameters (FiO2, hemoglobin, oxygen uptake). Therefore, following the NIRS gives critical care physicians a notion of the cardiac output. NIRS is liked by patients, because it is non-invasive, is painless, and uses non-ionizing radiation.

Optical Coherence Tomography (OCT) is another NIR medical imaging technique capable of 3D imaging with high resolution on par with low-power microscopy. Using optical coherence to measure photon pathlength allows OCT to build images of live tissue and clear examinations of tissue morphology. Due to technique differences OCT is limited to imaging 1–2 mm below tissue surfaces, but despite this limitation OCT has become an established medical imaging technique especially for imaging of the retina and anterior segments of the eye.

The instrumental development of NIRS/NIRI/DOT/OCT has proceeded tremendously during the last years and, in particular, in terms of quantification, imaging and miniaturization.^[11]

Particle measurement

NIR is often used in particle sizing in a range of different fields, including studying pharmaceutical and agricultural powders.

Industrial uses

As opposed to NIRS used in optical topography, general NIRS used in chemical assays does not provide imaging by mapping. For example, a clinical carbon dioxide analyzer requires reference techniques and calibration routines to be able to get accurate CO₂ content change. In this case, calibration is performed by adjusting the zero control of the sample being tested after purposefully supplying 0% CO₂ or another known amount of CO₂ in the sample. Normal compressed gas from distributors contains about 95% O₂ and 5% CO₂, which can also be used to adjust %CO₂ meter reading to be exactly 5% at initial calibration.^[16]

Molecular vibration (Needed to Understand Greenhouse Gases)

Molecular vibration

From Wikipedia, the free encyclopedia

 

A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 10¹² to approximately 10¹⁴ Hz.

In general, a molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 such modes, as rotation about its molecular axis cannot be observed.^[1] A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E = hν (where h is Planck's constant). A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, as the potential energy of the molecule is more like a Morse potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation (vibronic transition), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state.
Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C-H or C-C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule.
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule,
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups.
• Out-of-plane: a change in the angle between any one of the C-H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

Vibrations of a Methylene group (-CH₂-) in a molecule for illustration

The atoms in a CH₂ group, commonly found in organic compounds, can vibrate in six different ways: symmetric and asymmetric stretching, scissoring, rocking, wagging and twisting as shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates.^[2] The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by

where are the internal coordinates for stretching of each of the four C-H bonds.

Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto.^[3]

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The advantage of working in normal modes is that they diagonalize the matrix governing the molecular vibrations, so each normal mode is an independent molecular vibration, associated with its own spectrum of quantum mechanical states. If the molecule possesses symmetries, it will belong to a point group, and the normal modes will "transform as" an irreducible representation under that group. The normal modes can then be qualitatively determined by applying group theory and projecting the irreducible representation onto the cartesian coordinates. For
example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch.

• symmetric stretching: the sum of the two C-O stretching coordinates; the two C-O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C-O stretching coordinates; one C-O bond length increases while the other decreases. Q = q₁ - q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

1. principally C-H stretching with a little C-N stretching; Q₁ = q₁ + a q₂ (a << 1)
2. principally C-N stretching with a little C-H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.^[4]

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere.^[5]

By Newton’s second law of motion this force is also equal to a reduced mass, μ, times acceleration.

Since this is one and the same force the ordinary differential equation follows.

The solution to this equation of simple harmonic motion is

A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as

The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,ν_i are obtained from the eigenvalues,λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.^[4] F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.^[6]

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

,

where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.^[7][8]

The difference in energy when n (or v) changes by 1 is therefore equal to , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band.

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate.^[9] The intensity of Raman bands depends on polarizability.

The Russo-Ukrainian War

August 29, 2014 

Original link:  http://20committee.com/2014/08/29/the-russo-ukrainian-war/#ixzz3BsOi4qqI

 

This week Vladimir Putin’s war on Ukraine became overt for all the world to see. Since February, Moscow waged a semi-covert campaign that I term Special War, with the initial aim of taking Crimea. This succeeded almost bloodlessly thanks to confusion in Kyiv. Over the past six months, inspired by Crimean success, Russian strategy has focused on creating and preserving Kremlin-controlled pseudo-states, the so-called Donetsk and Luhansk “People’s Republics,” which are in fact subsidiaries of Russian intelligence.

This, however, is a far more ambitious goal than the Crimean operation, and resistance has mounted. In recent weeks, Ukrainian efforts to retake territory around Donetsk and Luhansk in what Kyiv calls the anti-terrorist operation (ATO) have gained momentum, and this week Moscow sent troops across the border more or less openly since the alternative is the defeat and collapse of its proxies in southeast Ukraine. That Putin will not allow, and it’s difficult to see how he could, after months of stoking fiery Russian nationalism over the Ukraine issue, with casual talk of “Nazis” ruling in Kyiv ready to inflict “genocide” on ethnic Russians in Donetsk and Luhansk.

There is no doubt that hundreds of Russian armored vehicles and thousands of troops are operating in southeast Ukraine now. Dead Russian paratroopers are coming home for burial and NATO has shown satellite imagery that leaves no doubt that the Russo-Ukrainian War, which began in the winter, has become a full-fledged conflict this summer. As I write, the city of Mariupol on the coast of the Sea of Azov is preparing to defend itself against an expected Russian onslaught this weekend. If Mariupol falls, a land corridor to Crimea will open up and the war will likely grow wider, fast. Certainly Russian tanks provocatively flying the flag of Novorossiya, which was the Tsarist-era name for south and east Ukraine — a term that Putin himself has picked up recently — gives a clear indication of what the Kremlin wants.

The next few days will be decisive in determining if Russia’s war against Ukraine remains limited or expands significantly into a major conflict that will imperil European security in a manner not witnessed in decades. The course that Putin has plotted is described ably in an article today in Novaya Gazeta, the last Kremlin-unfriendly serious newspaper in Russia, by Pavel Felgenhauer, a noted Russian defense commentator. “We are still a half step from full-scale war,” he states, explaining why:

War will happen if the current alignment does not achieve the strategic goals that Moscow is setting itself. The strategic goal, as Putin has been saying since April, is a stable ceasefire. In order to achieve it, it is necessary to achieve a military balance on the battlefield: To rout the Ukrainian forces, throw them back from Donetsk and Luhansk, and consolidate the territory that the insurgents are controlling. Donetsk People’s Republic representatives have repeatedly stated that they want the complete withdrawal of the Ukrainian troops from the territory of Donetsk and Luhansk.

To date, Moscow has shown restraint, Felgenhauer notes, committing only a few thousand Russian troops to battle in Ukraine, rather than the tens of thousands it could deploy. But that may not last:
The main battle now will obviously take place around and within Mariupol. Unless the Ukrainians are driven back, a real war will begin. Furthermore, there will be an air war on all of Ukraine’s territory. Then tens of thousands of Russian military will intervene. They will try to achieve air superiority and throw the Ukrainians out. In an extended version, perhaps, this will not only be from the territories of Donetsk and Luhansk.

Time, that trickiest of strategic concerns, is not on Putin’s side any longer, as Felgenhauer observes accurately, between weather and the Russian military’s conscript cycle:

There is not much time left. Fall is approaching. The short hours of daylight and low clouds will complicate the matters for the air force. It will have difficulty supporting ground troops — pilots in Su-25 ground-attack planes need to see the targets on the ground. In addition, starting 1 October, it is necessary to conduct a new draft and begin the demobilization of those conscripts who are stationed on the border as part of the artillery battalion groups. It is specifically for these reasons that the question must be resolved now.

We will know in a few days, then, if Putin has achieved his relatively limited military aims in eastern Ukraine. If he does not manage a quick win, there is every reason to think Ukraine and Russia will become embroiled in a protracted war for which neither Moscow nor Kyiv is ready.

Despite the impressive tenacity shown by Ukrainian volunteer units in the ATO, the overall condition of the country’s military remains lamentable, thanks to a generation of political and financial neglect after independence from the USSR in 1991. Moreover, too many Ukrainian senior officers retain Soviet-era habits of sloth, drunkenness and thievery, which has led to protests this week by citizens angry at military corruption and poor support for the men who are fighting and dying in the southeast. While the courage of Ukrainian troops is not in question, the competence of the military system certainly is.

There is ample reason to doubt the staying power of Ukrainian forces against a genuine Russian onslaught in the southeast. How badly things are going with the Ukrainian military in the field was laid bare in a recent interview with Serhiy Chervonopyskyy in the Kyiv daily Obozrevatel. A highly decorated veteran of the Soviet war in Afghanistan in the 1980s, Chervonopyskyy heads the country’s Afghan War veterans’ association and has observed the situation around Donetsk and Luhansk. He’s not impressed:

Many military leaders show a criminal lack of professionalism.  Which, as always, is compensated for by the heroism of the ordinary soldiers.  Afghanistan veterans are fighting in Donbas, working as instructors, delivering humanitarian aid, freeing captives, living in the battle zone.  In the [Afghan veteran's association] we receive a lot of information, particularly from experienced men who know more about war than just what you see in films. We can make an objective assessment of the situation.

Chervonopyskyy minces no words, finding fault with nearly everything about Ukraine’s military, save the soldiers themselves, citing poor logistics, outdated weaponry, abysmal staff work, plus a dysfunctional medical system that does not care for the many wounded properly. The Russians seem to know when the Ukrainians are coming, not only due to numerous Russians spies, but because Ukrainian troops, officers too, use their mobile phones constantly in the combat zone, creating a bonanza for Russian military intelligence, which is listening in. His verdict is harsh: “in recent years the army in our country has been systematically destroyed. Unlike Russia’s army.”

Chervonopyskyy leaves no doubt that Ukraine’s military needs root-and-branch reform that is nearly impossible to achieve while it is at war. Too many officers engage in profiteering while soldiers die without necessary supplies, including ammunition. Repeated offers from Afghan war veterans to assist the war effort have been rebuffed in Kyiv:

Our generals, colonels, and other commanders, of whom there are too many, are very frightened of appearing incompetent in front of us, as we understand military matters well.  They are more frightened of this than of losing soldiers or suffering a defeat.  Again it is a question of professionalism.

As long as this lamentable situation continues, it is unrealistic to expect the Ukrainian military to successfully defend their country against attacks by some of the best units in the Russian army, the demonstrated heroism of Kyiv’s soldiers notwithstanding. Ukraine is fighting for its life now, and the utmost seriousness is required to prevent defeat at the hands of Putin’s soldiers and proxies.

But we must not find fault only with Ukraine. It is far from encouraging that Western leaders, including the White House, will not use the word “invasion” in connection with Russian moves. Such institutionalized escapism in the West does not discourage Russian aggression, rather it encourages more of it. Putin is playing va banque now, his two options are a quick win in the southeast of Ukraine or a protracted conflict: backing down is not an option in the Kremlin anymore, and only naive Westerners think it is.

Sanctions will have no short-term impact on Russian behavior at this point. Vaunted Western “soft power” has been run over by Russian tanks. The decision for war has been made in Moscow, and it will be prosecuted until Putin achieves his objectives or the cost — rising numbers of Russian dead — becomes politically prohibitive. Putin knows that the Russian public, heady after the nearly bloodless conquest of Crimea, has no stomach for a costly war of choice with Ukraine, their “Slavic brothers.”

If the West wants to prevent more Russian aggression and save Ukraine from further Kremlin depradations, it must offer Kyiv armaments, logistics, training, and above all intelligence support without delay. Nothing else will cause Moscow to back down. Only by arming and enabling Ukraine’s military can the West make the cost of Putin’s war prohibitive for Russia. Ukraine’s defense ministry and armed forces require major Western aid to transform its underperforming military from bad Soviet habits to real fighting capability, but that is a long-term enterprise. Right now, Kyiv needs direct military aid. If NATO does not provide it, a wider war for Ukraine becomes more likely by the day, with grave consequences for the European peace that NATO has preserved, at great expense, for sixty-five years.