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Sherwood Park (Richmond, VA)
Sherwood Park (Richmond, VA)
	Coordinates: 37°34′30″N 77°27′18″W / 37.575°N 77.455°W
Government
• Body	Sherwood Park Community Association

Sherwood Park (Richmond, VA)
Not what I want here
	Meditation labyrinth in the Westwood tract of Sherwood Park
	v; t; e;

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37°32′N 77°28′W / 37.533°N 77.467°W / 37.533; -77.467

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math examples How I did NOE article

A series of experiments are carried out with increasing mixing times, and the increase in NOE enhancement is followed. The closest protons will show the most rapid build-up rates of the NOE. While rf irradiation can only induce single-quantum transitions (due to so-called quantum mechanical selection rules) giving rise to observable spectral lines, dipolar relaxation may take place through any of the pathways. The dipolar mechanism is the only common relaxation mechanism that can cause transitions in which more than one spin flips. Specifically, the dipolar relaxation mechanism gives rise to transitions between the αα and ββ states (W₂) and between the αβ and the βα states (W₀).

Expressed in terms of their bulk NMR magnetizations, the experimentally observed steady-state NOE for nucleus I when the resonance of nucleus S is saturated ( $M_{S}=0$ ) is defined by the expression:

\eta _{I}^{S}=\left({\frac {M_{I}^{S}-M_{0I}}{M_{0I}}}\right)

where $M_{0I}$ is the magnetization (resonance intensity) of nucleus $I$ at thermal equilibrium. An analytical expression for the NOE can be obtained by considering all the relaxation pathways and applying the Solomon equations to obtain

\eta _{I}^{S}={\frac {M_{I}^{S}-M_{0I}}{M_{0I}}}={\frac {\gamma _{S}}{\gamma _{I}}}{\frac {\sigma _{IS}}{\rho _{I}}}={\frac {\gamma _{S}}{\gamma _{I}}}\left({\frac {W_{2}-W_{0}}{2W_{1}^{I}+W_{0}+W_{2}}}\right)

where

\rho _{I}=2W_{1}^{I}+W_{0}+W_{2}

and

\sigma _{IS}=W_{2}-W_{0}

.

$\rho _{I}$ is the total longitudinal dipolar relaxation rate ( $1/T_{1}$ ) of spin I due to the presence of spin s, $\sigma _{IS}$ is referred to as the cross-relaxation rate, and $\gamma _{I}$ and $\gamma _{S}$ are the magnetogyric ratios characteristic of the $I$ and $S$ nuclei, respectively.

Saturation of the degenerate W₁^S transitions disturbs the equilibrium populations so that P_αα = P_αβ and P_βα = P_ββ. The system's relaxation pathways, however, remain active and act to re-establish an equilibrium, except that the W₁^S transitions are irrelevant because the population differences across these transitions are fixed by the RF irradiation while the population difference between the W_I transitions does not change from their equilibrium values. This means that if only the single quantum transitions were active as relaxation pathways, saturating the $S$ resonance would not affect the intensity of the $I$ resonance. Therefore to observe an NOE on the resonance intensity of I, the contribution of $W_{0}$ and $W_{2}$ must be important. These pathways, known as cross-relaxation pathways, only make a significant contribution to the spin-lattice relaxation when the relaxation is dominated by dipole-dipole or scalar coupling interactions, but the scalar interaction is rarely important and is assumed to be negligible. In the homonuclear case where $\gamma _{I}=\gamma _{S}$ , if $W_{2}$ is the dominant relaxation pathway, then saturating $S$ increases the intensity of the $I$ resonance and the NOE is positive, whereas if $W_{0}$ is the dominant relaxation pathway, saturating $S$ decreases the intensity of the $I$ resonance and the NOE is negative.

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vinculum

T

Deuterium NMR

Deuterium NMR is NMR spectroscopy of deuterium (²H or D), an isotope of hydrogen. ^[1]

Deuterium is an isotope with spin = 1, unlike hydrogen which is spin = 1/2. Deuterium NMR has a range of chemical shift similar to proton NMR but with poor resolution. It may be used to verify the effectiveness of deuteration: a deuterated compound will show a peak in deuterium NMR but not proton NMR.

Deuterium NMR spectra are especially informative in the solid state because of its relatively small quadrupole moment in comparison with those of bigger quadrupolar nuclei such as chlorine-35, for example.^{[citation needed]} One example is the use of deuterium NMR to study lipid membrane phase behavior.

References

^ Mantsch, Henry H.; Saitô, Hazime; Smith, Ian C. P. (1977). "Deuterium magnetic resonance, applications in chemistry, physics and biology". Progress in Nuclear Magnetic Resonance Spectroscopy. 11 (4): 211–272. doi:10.1016/0079-6565(77)80010-1. ISSN 0079-6565.

Articles about solid-state deuterium NMR

NMR Relaxation

old lede

In magnetic resonance imaging (MRI) and nuclear magnetic resonance spectroscopy (NMR), the term relaxation describes how signals change with time. In general signals deteriorate with time, becoming weaker and broader. The deterioration reflects the fact that the NMR signal, which results from nuclear magnetization, arises from the over-population of an excited state. Relaxation is the conversion of this non-equilibrium population to a normal population. In other words, relaxation describes how quickly spins "forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.^[1]

The energy gap between the spin-up and spin-down states in NMR is really quite small by atomic emission standards — at 1.5T it is only about 2 x 10−7 eV (electron-volts). By comparison, visible light photons have energies of about 2 eV, or 10 million times higher. There is thus a considerable "advantage" for a high-energy light photon to be emitted by phosphorescence, but relatively little "motivation" for an already low energy nuclear spin to switch states spontaneously.

Most energy emission in NMR must be induced through a direct interaction of a nucleus with its external environment This interaction may be through the electrical or magnetic fields generated by other nuclei, electrons, or molecules

another lede alteration

In MRI and NMR spectroscopy, nuclear spin polarization (magnetization) forms inside a homogeneous magnetic field where the magnetic moments of the nuclei in the sample precess about the direction of the applied field at a characteristic frequency called the Larmor frequency. At thermal equilibrium, this polarization is not detectable because the phase of the spins is random and does not result in a net polarization orthogonal to the magnetic field. During an intense RF pulse, the spin polarization rotates with the magnetic component of the RF field. Following the pulse, any of the resultant transverse polarization that remains orthogonal to the field can induce a signal in an RF coil or detector, which can be observed when amplified by an RF receiver. The RF pulse causes the population of spin-states to be perturbed from their thermal equilibrium value. The return of the longitudinal component of the magnetization to its equilibrium value is termed spin-lattice relaxation while the loss of phase-coherence of the spins is termed spin-spin relaxation, which is manifest as an observed free induction decay (FID).

Altered Lede

In MRI and NMR spectroscopy, an observable nuclear spin polarization (magnetization) is created by an RF pulse or a train of pulses applied to a sample in a homogeneous magnetic field at the resonance (Larmor) frequency of the nuclei. At thermal equilibrium, nuclear spins precess randomly about the direction of the applied field but become abruptly phase coherent when any of the resultant polarization is created orthogonal to the field. This transverse magnetization can induce a signal in an RF coil that can be detected and amplified by an RF receiver. The RF pulses cause the population of spin-states to be perturbed from their thermal equilibrium value. The return of the longitudinal component of the magnetization to its equilibrium value is termed spin-lattice relaxation while the loss of phase-coherence of the spins is termed spin-spin relaxation, which is manifest as an observed free induction decay (FID).

For spin=½ nucleic (such as ¹H), the polarization due to spins oriented with the field N_- relative to the spins oriented against the field N₊ is given by the Boltzmann distribution:

{\frac {N_{+}}{N_{-}}}=e^{\frac {\Delta E}{kT}}

where ΔE is the energy level difference between the two populations of spins, k is the Boltzmann constant, and T is the sample temperature. At room temperature, the number of spins in the lower energy level, N−, slightly outnumbers the number in the upper level, N+. The energy gap between the spin-up and spin-down states in NMR is minute by atomic emission standards at magnetic fields conventionally used in MRI and NMR spectroscopy. Energy emission in NMR must be induced through a direct interaction of a nucleus with its external environment rather than by spontaneous emission. This interaction may be through the electrical or magnetic fields generated by other nuclei, electrons, or molecules. Spontaneous emission of energy is a radiative process involving the release of a photon and typified by phenomena such as fluorescence and phosphorescence. As stated by Abragam, the probability per unit time of the nuclear spin-1/2 transition from the + into the - state through spontaneous emission of a photon is a negligible phenomenon.^[2]^[3] Rather, the return to equilibrium is a much slower thermal process induced by the fluctuating local magnetic fields due to molecular or electron (free radical) rotational motions that return the excess energy in the form of heat to the surroundings.

Rather, relaxation of nuclear spins requires a microscopic mechanism for a nucleus to change orientation with respect to the applied magnetic field and/or interchange energy with the surroundings (the "lattice"). The return to thermal equilibrium is a much slower thermal process than spontaneous emission that is induced by the fluctuating local magnetic fields due to molecular or electron (e.g., free radicals or paramagnetic ions) rotational motions that return the excess energy in the form of heat to the surroundings. Molecular tumbling can then modulate various orientation-dependent spin-interactions called "relaxation mechanisms".

old mechanism text: blue font example

Relaxation of nuclear spins requires a microscopic mechanism for a nucleus to change orientation with respect to the applied magnetic field and/or interchange energy with the surroundings (called the lattice). The most common mechanism is the magnetic dipole-dipole interaction between the magnetic moment of a nucleus and the magnetic moment of another nucleus or other entity (electron, atom, ion, molecule). This interaction depends on the distance between the pair of dipoles (spins) but also on their orientation relative to the external magnetic field. Several other relaxation mechanisms also exist. The chemical shift anisotropy (CSA) relaxation mechanism arises whenever the electronic environment around the nucleus is non spherical, the magnitude of the electronic shielding of the nucleus will then be dependent on the molecular orientation relative to the (fixed) external magnetic field. The spin rotation (SR) relaxation mechanism arises from an interaction between the nuclear spin and a coupling to the overall molecular rotational angular momentum. Nuclei with spin I ≥ 1 will have not only a nuclear dipole but a quadrupole. The nuclear quadrupole has an interaction with the electric field gradient at the nucleus which is again orientation dependent as with the other mechanisms described above, leading to the so-called quadrupolar relaxation mechanism. Molecular reorientation or tumbling can then modulate these orientation-dependent spin interaction energies. According to quantum mechanics, time-dependent interaction energies cause transitions between the nuclear spin states which result in nuclear spin relaxation. The application of time-dependent perturbation theory in quantum mechanics shows that the relaxation rates (and times) depend on spectral density functions that are the Fourier transforms of the autocorrelation function of the fluctuating magnetic dipole interactions. The form of the spectral density functions depend on the physical system, but a simple approximation called the BPP theory is widely used. Another relaxation mechanism is the electrostatic interaction between a nucleus with an electric quadrupole moment and the electric field gradient that exists at the nuclear site due to surrounding charges. Thermal motion of a nucleus can result in fluctuating electrostatic interaction energies. These fluctuations produce transitions between the nuclear spin states in a similar manner to the magnetic dipole-dipole interaction.

new mechanism text

Relaxation of nuclear spin polarization requires a microscopic mechanism for nuclei to interchange energy with their surroundings. In principle, any fluctuating magnetic field that has frequency components at the Larmor or resonance frequency of the polarized spins can induce spin transitions that will return the system to thermal equilibrium. Such fluctuating fields are provided by the weak magnetic fields induced by the tumbling or translational motions of other nearby dipolar nuclei, paramagnetic ions, or free radicals.

Relaxation Mechanisms

Nucelar Spin Relaxation Mechanisms
Mechanism	b_loc	Correlation Time	Comments
Dipole-Dipole, nuclear-nuclear	R1C2	reorientation/translational	r1c4
Dipole-Dipole, electron-nuclear	R1C2	reorientation/translational	r1c4
Spin Rotation	R2C2	angular momentum	e1c4
Chemical Shift Anisotropy	R2C2	reorientation	e1c4
Scalar Coupling	R2C2	R2C3	e1c4
Quadrupolar	R2C2	reorientation	e1c4

Dipole-Dipole

Scalar J

Chemical shift anisotropy

Quadrupolar

Spin rotation

Relaxation of nuclear spins requires a microscopic mechanism for a nucleus to change orientation with respect to the applied magnetic field and/or interchange energy with the surroundings (called the lattice). Molecular reorientation or tumbling can then modulate these orientation-dependent spin interaction energies. The most commonly encountered mechanism is the magnetic dipole-dipole interaction between the magnetic moment of a nucleus and the magnetic moment of another nucleus or other entity (electron, atom, ion, molecule). This interaction depends on the distance between the pair of dipoles (spins) but also on their orientation relative to the external magnetic field. Several other relaxation mechanisms also exist. The chemical shift anisotropy (CSA) relaxation mechanism arises whenever the electronic environment around the nucleus is non spherical, the magnitude of the electronic shielding of the nucleus will then be dependent on the molecular orientation relative to the (fixed) external magnetic field. The spin rotation (SR) relaxation mechanism arises from an interaction between the nuclear spin and a coupling to the overall molecular rotational angular momentum. Nuclei with spin I ≥ 1 will have not only a nuclear dipole but a quadrupole. The nuclear quadrupole has an interaction with the electric field gradient at the nucleus which is again orientation dependent as with the other mechanisms described above, leading to the so-called quadrupolar relaxation mechanism.

Molecular reorientation or tumbling can then modulate these orientation-dependent spin interaction energies. According to quantum mechanics, time-dependent interaction energies cause transitions between the nuclear spin states which result in nuclear spin relaxation. The application of time-dependent perturbation theory in quantum mechanics shows that the relaxation rates (and times) depend on spectral density functions that are the Fourier transforms of the autocorrelation function of the fluctuating magnetic dipole interactions.^[4] The form of the spectral density functions depend on the physical system, but a simple approximation called the BPP theory is widely used.

Another relaxation mechanism is the electrostatic interaction between a nucleus with an electric quadrupole moment and the electric field gradient that exists at the nuclear site due to surrounding charges. Thermal motion of a nucleus can result in fluctuating electrostatic interaction energies. These fluctuations produce transitions between the nuclear spin states in a similar manner to the magnetic dipole-dipole interaction.

Ends and Odds

Susskind, Leonard (2008). The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Little, Brown and Company. ISBN 978-0316016407.

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Ref List

^ Friebolin, H., "Basic One- and Two- Dimensional NMR Spectroscopy, 4th ed.," VCH: Weinheim, 2008. ISBN 978-3-527-31233-7
^ Abragam, A. (1961). "VII Thermal Relaxation in Liquids and Gases". Principles of Nuclear Magnetism. Oxford University Press. p. 264. ISBN 019852014X.}
^ Hoult, D.I.; Bahkar, B. (1998). "NMR Signal Reception: Virtual Photons and Coherent Spontaneous Emission". Concepts in Magnetic Resonance. 9 (5): 277–297. doi:10.1002/(SICI)1099-0534(1997)9:5<277::AID-CMR1>3.0.CO;2-W.
^ A. Abragam "Principles of Nuclear Magnetism" (Oxford University Press, 1961)

NOE Ref

^[1] Kulhman et al. ^[2]

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I think our house was one of the first five or ten along Gloucester and Loxley. This was a development called Sherwood Park. I don't know whether the neighborhood has hung on to that name or not. In any case, no neighborhood ever had a lovelier group of people. In the twenty-five years we lived there we never had a bad neighbor. ... We had a coal-burning furnace. The coal bin was in the southeast corner of the cellar. We also had an old-fashioned ice box. A couple of times a week the ice man brought an enormous block of ice which he carried on a rubber mantle that went across his back. He got a grip on the ice with a huge pair of tongs. The arrival of our first electric refrigerator---in the late 30s, I suppose---was quite an event.

Tom Wolfe, https://www.architecturaldigest.com/story/tom-wolfes-sweet-memories-of-his-childhood-home-will-make-you-cry

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