Electron paramagnetic resonance
Electron paramagnetic resonance or electron spin resonance spectroscopy is a method for studying materials with unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance, but it is electron spins that are excited instead of the spins of atomic nuclei. EPR spectroscopy is useful for studying metal complexes or organic radicals. EPR was first observed in Kazan State University by Soviet physicist Yevgeny Zavoisky in 1944, was developed independently at the same time by Brebis Bleaney at the University of Oxford; every electron has a magnetic moment and spin quantum number s = 1 2, with magnetic components m s = + 1 2 and m s = − 1 2. In the presence of an external magnetic field with strength B 0, the electron's magnetic moment aligns itself either parallel or antiparallel to the field, each alignment having a specific energy due to the Zeeman effect: E = m s g e μ B B 0, where g e is the electron's so-called g-factor, g e = 2.0023 for the free electron, μ B is the Bohr magneton.
Therefore, the separation between the lower and the upper state is Δ E = g e μ B B 0 for unpaired free electrons. This equation implies that the splitting of the energy levels is directly proportional to the magnetic field's strength, as shown in the diagram below. An unpaired electron can move between the two energy levels by either absorbing or emitting a photon of energy h ν such that the resonance condition, h ν = Δ E, is obeyed; this leads to the fundamental equation of EPR spectroscopy: h ν = g e μ B B 0. Experimentally, this equation permits a large combination of frequency and magnetic field values, but the great majority of EPR measurements are made with microwaves in the 9000–10000 MHz region, with fields corresponding to about 3500 G. Furthermore, EPR spectra can be generated by either varying the photon frequency incident on a sample while holding the magnetic field constant or doing the reverse. In practice, it is the frequency, kept fixed. A collection of paramagnetic centers, such as free radicals, is exposed to microwaves at a fixed frequency.
By increasing an external magnetic field, the gap between the m s = + 1 2 and m s = − 1 2 energy states is widened until it matches the energy of the microwaves, as represented by the double arrow in the diagram above. At this point the unpaired electrons can move between their two spin states. Since there are more electrons in the lower state, due to the Maxwell–Boltzmann distribution, there is a net absorption of energy, it is this absorption, monitored and converted into a spectrum; the upper spectrum below is the simulated absorption for a system of free electrons in a varying magnetic field. The lower spectrum is the first derivative of the absorption spectrum; the latter is the most common way to publish continuous wave EPR spectra. For the microwave frequency of 9388.2 MHz, the predicted resonance occurs at a magnetic field of about B 0 = h ν / g e μ B = 0.3350 teslas = 3350 gausses. Because of electron-nuclear mass differences, the magnetic moment of an electron is larger than the corresponding quantity for any nucleus, so that a much higher electromagnetic frequency is needed to bring about a spin resonance with an electron than with a nucleus, at identical magnetic field strengths.
For example, for the field of 3350 G shown at the right, spin resonance occurs near 9388.2 MHz for an electron compared to only about 14.3 MHz for 1H nuclei. (For NMR spectroscopy, the corresponding resonance equation is h ν = g N μ N B 0 where g N and
Microwaves are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter. Different sources define different frequency ranges as microwaves. A more common definition in radio engineering is the range between 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are referred to by their IEEE radar band designations: S, C, X, Ku, K, or Ka band, or by similar NATO or EU designations; the prefix micro- in microwave is not meant to suggest a wavelength in the micrometer range. Rather, it indicates that microwaves are "small", compared to the radio waves used prior to microwave technology; the boundaries between far infrared, terahertz radiation and ultra-high-frequency radio waves are arbitrary and are used variously between different fields of study. Microwaves travel by line-of-sight. At the high end of the band they are absorbed by gases in the atmosphere, limiting practical communication distances to around a kilometer.
Microwaves are used in modern technology, for example in point-to-point communication links, wireless networks, microwave radio relay networks, radar and spacecraft communication, medical diathermy and cancer treatment, remote sensing, radio astronomy, particle accelerators, industrial heating, collision avoidance systems, garage door openers and keyless entry systems, for cooking food in microwave ovens. Microwaves occupy a place in the electromagnetic spectrum with frequency above ordinary radio waves, below infrared light: In descriptions of the electromagnetic spectrum, some sources classify microwaves as radio waves, a subset of the radio wave band; this is an arbitrary distinction. Microwaves travel by line-of-sight paths. Although at the low end of the band they can pass through building walls enough for useful reception rights of way cleared to the first Fresnel zone are required. Therefore, on the surface of the Earth, microwave communication links are limited by the visual horizon to about 30–40 miles.
Microwaves are absorbed by moisture in the atmosphere, the attenuation increases with frequency, becoming a significant factor at the high end of the band. Beginning at about 40 GHz, atmospheric gases begin to absorb microwaves, so above this frequency microwave transmission is limited to a few kilometers. A spectral band structure causes absorption peaks at specific frequencies. Above 100 GHz, the absorption of electromagnetic radiation by Earth's atmosphere is so great that it is in effect opaque, until the atmosphere becomes transparent again in the so-called infrared and optical window frequency ranges. In a microwave beam directed at an angle into the sky, a small amount of the power will be randomly scattered as the beam passes through the troposphere. A sensitive receiver beyond the horizon with a high gain antenna focused on that area of the troposphere can pick up the signal; this technique has been used at frequencies between 0.45 and 5 GHz in tropospheric scatter communication systems to communicate beyond the horizon, at distances up to 300 km.
The short wavelengths of microwaves allow omnidirectional antennas for portable devices to be made small, from 1 to 20 centimeters long, so microwave frequencies are used for wireless devices such as cell phones, cordless phones, wireless LANs access for laptops, Bluetooth earphones. Antennas used include short whip antennas, rubber ducky antennas, sleeve dipoles, patch antennas, the printed circuit inverted F antenna used in cell phones, their short wavelength allows narrow beams of microwaves to be produced by conveniently small high gain antennas from a half meter to 5 meters in diameter. Therefore, beams of microwaves are used for point-to-point communication links, for radar. An advantage of narrow beams is that they don't interfere with nearby equipment using the same frequency, allowing frequency reuse by nearby transmitters. Parabolic antennas are the most used directive antennas at microwave frequencies, but horn antennas, slot antennas and dielectric lens antennas are used. Flat microstrip antennas are being used in consumer devices.
Another directive antenna practical at microwave frequencies is the phased array, a computer-controlled array of antennas which produces a beam which can be electronically steered in different directions. At microwave frequencies, the transmission lines which are used to carry lower frequency radio waves to and from antennas, such as coaxial cable and parallel wire lines, have excessive power losses, so when low attenuation is required microwaves are carried by metal pipes called waveguides. Due to the high cost and maintenance requirements of waveguide runs, in many microwave antennas the output stage of the transmitter or the RF front end of the receiver is located at the antenna; the term microwave has a more technical meaning in electromagnetics and circuit theory. Apparatus and techniques may
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which nuclei in a strong static magnetic field are perturbed by a weak oscillating magnetic field and respond by producing an electromagnetic signal with a frequency characteristic of the magnetic field at the nucleus. This process occurs near resonance, when the oscillation frequency matches the intrinsic frequency of the nuclei, which depends on the strength of the static magnetic field, the chemical environment, the magnetic properties of the isotope involved. NMR results from specific magnetic properties of certain atomic nuclei. Nuclear magnetic resonance spectroscopy is used to determine the structure of organic molecules in solution and study molecular physics, crystals as well as non-crystalline materials. NMR is routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging. All isotopes that contain an odd number of protons and/or neutrons have an intrinsic nuclear magnetic moment and angular momentum, in other words a nonzero nuclear spin, while all nuclides with numbers of both have a total spin of zero.
The most used nuclei are 1H and 13C, although isotopes of many other elements can be studied by high-field NMR spectroscopy as well. A key feature of NMR is that the resonance frequency of a particular simple substance is directly proportional to the strength of the applied magnetic field, it is this feature, exploited in imaging techniques. Since the resolution of the imaging technique depends on the magnitude of the magnetic field gradient, many efforts are made to develop increased gradient field strength; the principle of NMR involves three sequential steps: The alignment of the magnetic nuclear spins in an applied, constant magnetic field B0. The perturbation of this alignment of the nuclear spins by a weak oscillating magnetic field referred to as a radio-frequency pulse; the oscillation frequency required for significant perturbation is dependent upon the static magnetic field and the nuclei of observation. The detection of the NMR signal during or after the RF pulse, due to the voltage induced in a detection coil by precession of the nuclear spins around B0.
After an RF pulse, precession occurs with the nuclei's intrinsic Larmor frequency and, in itself, does not involve transitions between spin states or energy levels. The two magnetic fields are chosen to be perpendicular to each other as this maximizes the NMR signal strength; the frequencies of the time-signal response by the total magnetization of the nuclear spins are analyzed in NMR spectroscopy and magnetic resonance imaging. Both use applied magnetic fields of great strength produced by large currents in superconducting coils, in order to achieve dispersion of response frequencies and of high homogeneity and stability in order to deliver spectral resolution, the details of which are described by chemical shifts, the Zeeman effect, Knight shifts; the information provided by NMR can be increased using hyperpolarization, and/or using two-dimensional, three-dimensional and higher-dimensional techniques. NMR phenomena are utilized in low-field NMR, NMR spectroscopy and MRI in the Earth's magnetic field, in several types of magnetometers.
Nuclear magnetic resonance was first described and measured in molecular beams by Isidor Rabi in 1938, by extending the Stern–Gerlach experiment, in 1944, Rabi was awarded the Nobel Prize in Physics for this work. 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. Yevgeny Zavoisky observed nuclear magnetic resonance in 1941, well before Felix Bloch and Edward Mills Purcell, but dismissed the results as not reproducible. Russell H. Varian filed the "Method and means for correlating nuclear properties of atoms and magnetic fields", U. S. Patent 2,561,490 on July 24, 1951. Varian Associates developed the first NMR unit called NMR HR-30 in 1952. 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 his discovery of NMR in bulk matter.
Rabi 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 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. All nucleons, 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 b
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an atom or molecule making a transition from a high energy state to a lower energy state. The photon energy of the emitted photon is equal to the energy difference between the two states. There are many possible electron transitions for each atom, each transition has a specific energy difference; this collection of different transitions, leading to different radiated wavelengths, make up an emission spectrum. Each element's emission spectrum is unique. Therefore, spectroscopy can be used to identify the elements in matter of unknown composition; the emission spectra of molecules can be used in chemical analysis of substances. In physics, emission is the process by which a higher energy quantum mechanical state of a particle becomes converted to a lower one through the emission of a photon, resulting in the production of light; the frequency of light emitted is a function of the energy of the transition.
Since energy must be conserved, the energy difference between the two states equals the energy carried off by the photon. The energy states of the transitions can lead to emissions over a large range of frequencies. For example, visible light is emitted by the coupling of electronic states in molecules. On the other hand, nuclear shell transitions can emit high energy gamma rays, while nuclear spin transitions emit low energy radio waves; the emittance of an object quantifies. This may be related to other properties of the object through the Stefan–Boltzmann law. For most substances, the amount of emission varies with the temperature and the spectroscopic composition of the object, leading to the appearance of color temperature and emission lines. Precise measurements at many wavelengths allow the identification of a substance via emission spectroscopy. Emission of radiation is described using semi-classical quantum mechanics: the particle's energy levels and spacings are determined from quantum mechanics, light is treated as an oscillating electric field that can drive a transition if it is in resonance with the system's natural frequency.
The quantum mechanics problem is treated using time-dependent perturbation theory and leads to the general result known as Fermi's golden rule. The description has been superseded by quantum electrodynamics, although the semi-classical version continues to be more useful in most practical computations; when the electrons in the atom are excited, for example by being heated, the additional energy pushes the electrons to higher energy orbitals. When the electrons fall back down and leave the excited state, energy is re-emitted in the form of a photon; the wavelength of the photon is determined by the difference in energy between the two states. These emitted photons form the element's spectrum; the fact that only certain colors appear in an element's atomic emission spectrum means that only certain frequencies of light are emitted. Each of these frequencies are related to energy by the formula: E photon = h ν,where E photon is the energy of the photon, ν is its frequency, h is Planck's constant.
This concludes. The principle of the atomic emission spectrum explains the varied colors in neon signs, as well as chemical flame test results; the frequencies of light that an atom can emit are dependent on states. When excited, an electron moves to orbital; when the electron falls back to its ground level the light is emitted. The above picture shows the visible light emission spectrum for hydrogen. If only a single atom of hydrogen were present only a single wavelength would be observed at a given instant. Several of the possible emissions are observed because the sample contains many hydrogen atoms that are in different initial energy states and reach different final energy states; these different combinations lead to simultaneous emissions at different wavelengths. As well as the electronic transitions discussed above, the energy of a molecule can change via rotational and vibronic transitions; these energy transitions lead to spaced groups of many different spectral lines, known as spectral bands.
Unresolved band spectra may appear as a spectral continuum. Light consists of electromagnetic radiation of different wavelengths. Therefore, when the elements or their compounds are heated either on a flame or by an electric arc they emit energy in the form of light. Analysis of this light, with the help of a spectroscope gives us a discontinuous spectrum. A spectroscope or a spectrometer is an instrument, used for separating the components of light, which have different wavelengths; the spectrum appears in a series of lines called the line spectrum. This line spectrum is called an atomic spectrum; each element has a different atomic spectrum. The production of line spectra by the atoms of an element indicate that an atom can radiate only a certain amount of energy; this leads to the conclusion that bound electrons cannot have just any amount of energy but only a certain amount of energy. The emission spectrum can be used to determine the composition of a material, since it is different for each element of the periodic table.
One example is astronomical spectroscopy: iden
Spectral line shape
Spectral line shape describes the form of a feature, observed in spectroscopy, corresponding to an energy change in an atom, molecule or ion. Ideal line shapes include Lorentzian and Voigt functions, whose parameters are the line position, maximum height and half-width. Actual line shapes are determined principally by Doppler and proximity broadening. For each system the half-width of the shape function varies with temperature and phase. A knowledge of shape function is needed for spectroscopic curve deconvolution. An atomic transition is associated with a specific amount of energy, E. However, when this energy is measured by means of some spectroscopic technique, the line is not infinitely sharp, but has a particular shape. Numerous factors can contribute to the broadening of spectral lines. Broadening can only be mitigated by the use of specialized techniques, such as Lamb dip spectroscopy; the principal sources of broadening are: Lifetime broadening. According to the uncertainty principle the uncertainty in energy, ΔE and the lifetime, Δt, of the excited state are related by Δ E Δ t ⪆ ℏ This determines the minimum possible line width.
As the excited state decays exponentially in time this effect produces a line with Lorentzian shape in terms of frequency. Doppler broadening; this is caused by the fact that the velocity of atoms or molecules relative to the observer follows a Maxwell distribution, so the effect is dependent on temperature. If this were the only effect the line shape would be Gaussian. Pressure broadening. Collisions between atoms or molecules reduce the lifetime of the upper state, Δt, increasing the uncertainty ΔE; this effect depends on the temperature, which affects the rate of collisions. The broadening effect is described by a Lorentzian profile in most cases. Proximity broadening; the presence of other molecules close to the molecule involved affects both line width and line position. It is the dominant process for solids. An extreme example of this effect is the influence of hydrogen bonding on the spectra of protic liquids. Observed spectral line shape and line width are affected by instrumental factors.
The observed line shape is a convolution of the intrinsic line shape with the instrument transfer function. Each of these mechanisms, others, can act in isolation or in combination. If each effect is independent of the other, the observed line profile is a convolution of the line profiles of each mechanism. Thus, a combination of Doppler and pressure broadening effects yields a Voigt profile. A Lorentzian line shape function can be represented as L = 1 1 + x 2, where L signifies a Lorentzian function standardized, for spectroscopic purposes, to a maximum value of 1; the unit of p0, p and w is wavenumber or frequency. The variable x is dimensionless and is zero at p=p0; the Gaussian line shape has the standardized form, G = e − x 2. The subsidiary variable, x, is defined in the same way as for a Lorentzian shape. Both this function and the Lorentzian have a maximum value of 1 at x = 0 and a value of 1/2 at x=±1; the third line shape that has a theoretical basis is the Voigt function, a convolution of a Gaussian and a Lorentzian, V = ∫ − ∞ ∞ G L d x ′, where σ and γ are half-widths.
The computation of a Voigt function and its derivatives are more complicated than a Gaussian or Lorentzian. A spectroscopic peak may be fitted to multiples of the above functions or to sums or products of functions with variable parameters; the above functions are all symmetrical about the position of their maximum. Asymmetric functions have been used. For atoms in the gas phase the principal effects are pressure broadening. Lines are sharp on the scale of measurement so that applications such as atomic absorption spectroscopy and Inductively coupled plasma atomic emission spectroscopy are used for elemental analysis. Atoms have distinct x-ray spectra that are attributable to the excitation of inner shell electrons to excited states; the lines are sharp because the inner electron energies are not sensitive to the atom's environment. This is applied to X-ray fluorescence spectroscopy of solid materials. For molecules in the gas phase, the principal effects are pressure broadening; this applies to rotational spectroscopy, r