# Bloch equations

Equation (A1.6.64) describes the relaxation to equilibrium of a two-level system in tenns of a vector equation. It is the analogue of tire Bloch equation, originally developed for magnetic resonance, in the optical regime and hence is called the optical Bloch equation. [c.234]

Comparing this with the Bloch equation establishes a correspondence between t and /p/j. Putting t = ip/j, one finds [c.455]

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads [c.456]

The solution to this is a Gaussian function, which spreads out in time. Hence the solution to the Bloch equation for a free particle is also a Gaussian [c.457]

The phenomenological Bloch equations assume the magnetization component along Bq (the longitudinal magnetization M ) to relax exponentially to its equilibrium value, Mq. The time constant for the process is called the spin-lattice or longitudinal relaxation time, and is denoted The magnetization components perpendicular to (the transverse, magnetization, are also assumed to relax in an exponential maimer to their equilibrium value of zero. The time constant for this process is called the spin-spm or transverse relaxation time and is denoted T. The inverse of a relaxation time is called a relaxation rate. [c.1500]

The spin-spin relaxation time, T, defined in the Bloch equations, is simply related to the width Av 2 Lorentzian line at the half-height T. Thus, it is in principle possible to detennine by measuring [c.1509]

The classical description of magnetic resonance suffices for understanding the most important concepts of magnetic resonance imaging. The description is based upon the Bloch equation, which, in the absence of relaxation, may be written as [c.1520]

The components of the Bloch equation are hence reduced to [c.1521]

B2.4.2.2 THE BLOCH EQUATIONS APPROACH [c.2094]

With this definition, the Bloch equations can be written as in equation (B2.4.4)). [c.2095]

This apparently artificial way of re-writing the Bloch equations is important, since this fonn applies to all exchanging systems—coupled or uncoupled—m the frequency domain. The description starts with the equilibrium z magnetizations. These are affected by all the NMR interactions chemical shifts, relaxation and exchange. Finally, the observed signal is detected. This is the standard preparation-evolution-detection paradigm used in multi-dimensional NMR. There may be algebraic and numerical complications in setting up and solving the equations for different systems, but the fonn remains the same for all frequency-domain calculations. [c.2096]

Reeves L W and Shaw K N 1970 Nuclear magnetic resonance studies of multi-site chemical exchange. I. Matrix formulation of the Bloch equations Can. J. Chem. 48 3641-53 [c.2112]

This equation is the Blasius equation. [c.54]

R. Dickman, C. K. Hall. High density Monte Carlo simulations of chain molecules Bulk equation of state and density profile near walls. J Chem Phys 59 3168-3174, 1988. [c.627]

Equations (4-58), the Bloch equations, result dM, [c.163]

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. [c.168]

The formation bulk density (p ) can be read directly from the density log (see Figure 5.51) and the matrix density (p J and fluid density (p,) found in tables, assuming we have already identified lithology and fluid content from other measurements. The equation can be rearranged for porosity ((])) as follows [c.146]

Equation III-34 is the same as would apply to the case of two bulk phases separated by a membrane under tension y. [c.59]

This rule is approximately obeyed by a large number of systems, although there are many exceptions see Refs. 15-18. The rule can be understood in terms of a simple physical picture. There should be an adsorbed film of substance B on the surface of liquid A. If we regard this film to be thick enough to have the properties of bulk liquid B, then 7a(B) is effectively the interfacial tension of a duplex surface and should be equal to 7ab + VB(A)- Equation IV-6 then follows. See also Refs. 14 and 18. [c.107]

L. The liquid-expanded, L phase is a two-dimensionally isotropic arrangement of amphiphiles. This is in the smectic A class of liquidlike in-plane structure. There is a continuing debate on how best to formulate an equation of state of the liquid-expanded monolayer. Such monolayers are fluid and coherent, yet the average intermolecular distance is much greater than for bulk liquids. A typical bulk liquid is perhaps 10% less dense than its corresponding solid state. [c.133]

We begin by discussing the methods for estimating the solid-liquid interfacial properties via thermodynamic measurements on bulk systems. We then discuss the contact angle on a uniform perfect solid surface where the Young equation is applicable. The observation that most surfaces are neither smooth nor uniform is addressed in Section X-5, where contact angle hysteresis is covered. Then we describe briefly the techniques used to measure contact angles and some results of these measurements. Section X-7 is taken up with a discussion of theories of contact angles and some practical empirical relationships. [c.347]

The preceding treatment relates primarily to flocculation rates, while the irreversible aging of emulsions involves the coalescence of droplets, the prelude to which is the thinning of the liquid film separating the droplets. Similar theories were developed by Spielman [54] and by Honig and co-workers [55], which added hydrodynamic considerations to basic DLVO theory. A successful experimental test of these equations was made by Bernstein and co-workers [56] (see also Ref. 57). Coalescence leads eventually to separation of bulk oil phase, and a practical measure of emulsion stability is the rate of increase of the volume of this phase, V, as a function of time. A useful equation is [c.512]

Relaxation experiments were among the earliest applications of time-domain high-resolution NMR spectroscopy, invented more than 30 years ago by Ernst and Anderson [23]. The progress of the experimental methodology has been enonnoiis and only some basic ideas of the experiment design will be presented here. This section is divided into three subsections. The first one deals with Bloch equation-type experiments, measuring and T2 when such quantities can be defined, i.e. when the relaxation is monoexponential. As a slightly oversimplified rule of thumb, we can say that this happens in the case of isolated spins. The two subsections to follow cover miiltiple-spm effects. [c.1506]

The Bloch equation is simplified, and the experiment more readily understood, by transfonnation into a frame of reference rotating at the frequency ciDq=X Bq about die z-axis whereupon [c.1521]

A more detailed description of the action of an arbitrary pulse on a sample in a gradient can be obtained from a solution of the Bloch equations, either numerically or using advanced analytic teclmiques. In general this is complicated smce the effective field in the rotating frame is composed not only of the spatially varying gradient field in the /-direction but also the transverse excitation field which, in general, varies with time. What follows is therefore an approximate treatment which nonetheless provides a surprisingly accurate description of many of the more connnonly used slice selection pulses [4]. [c.1522]

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. [c.1616]

In liquids, the Bloch equations (single-dephasing time scale) picture [175] used above to describe the fomiation of echo signals is apparently inadequate. It is now known that electronic dephasing occurs over a distribution of time scales, so a single time constant T2 is insufficient to describe all of the line-broadening dynamics [177]. The two-pulse echo method described above only is sensitive to the fastest of processes in organic molecules in solution, the two-pulse echoes typically decay on the 20 fs or shorter time scale [176]. This is the time scale usually assigned to homogeneous Ime broadening. The slower electronic dephasing processes that contribute to inliomogeneous line broadening, involving solvent-induced fluctuations or radiationless decay between imcorrelated states, extend over the 10 fs to 100 ps (or longer) time scales in liquids and proteins [77, 78, 177]. [c.1986]

In chemical exchange, tire two exchanging sites, A and B, will have different Lannor frequencies, and cOg. Assuming equal populations in the two sites, and the rate of exchange to be k, the two coupled Bloch equations for the two sites are given by equation (B2.4.5)). [c.2095]

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. [c.2099]

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. [c.2104]

For turbulent flow in smooth tubes, the Blasius equation gives the friction facdor accurately for a wide range of Reynolds numbers. [c.636]

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. [c.73]

TRIFOU is a general code for solving Maxwell s equations. It has been developed by EDF/R D for ten years. TRIFOU is able to treat different electromagnetic problems such as magnetostatic, magnetodynamic (transient or harmonic regime), electrostatic and micro wave. It is able to treat the thin-skin regime by reducing mesh step size in the test block depth direction. [c.140]

TRIFOU is a combined Finite Elements/Boundary Integral formulation code. The BIM formulation in vacuum is suitable for NDT simulation where the probe moves in the air around the test block. The FEM formulation needs more calculation time, but tetrahedral elements enable a large variety of specimens and defect geometries to be modelled. TRIFOU uses a formulation of Maxwell Equations using magnetic field vector h, where h is decomposed as h = hs + hr (hj source field, and hr reaction field). [c.141]

A more elaborate treatment of ester hydrolysis was attempted by Davies and Rideal [308] in the case of the alkaline hydrolysis of monolayers of monocetyl-succinate ions. The point in mind was that since the interface was charged, the local concentration of hydroxide ions would not be the same as in the bulk substrate. The surface region was treated as a bulk phase 10 A thick and, using the Donnan equation, actual concentrations of ester and hydroxide ions were calculated, along with an estimate of their activity coefficients. Similarly, the Donnan effect of added sodium chloride on the hydrolysis rates was measured and compared with the theoretical estimate. The computed concentrations in the surface region were rather high (1-3 M), and since the region is definitely not isotropic because of orientation effects, this type of approach would seem to be semiempirical in nature. On the other hand, there was quite evidently an electrostatic exclusion of hydroxide ions from the charged monocetyl-succinate film, which could be predicted approximately by the Donnan relationship. [c.154]

The dififiision time gives the same general picture. The bulk self-diffusion coefficient of copper is 10"" cm /sec at 725°C [12] the Einstein equation [c.258]

As with the BET equation, a number of modihcations of Eqs. XVII-77 or XVn-79 have been proposed, for example Ref. 71. FHH-type equations go to inhnite him thickness (i.e., bulk liquid), as P - F and this cannot be the case if the liquid does not wet the solid, and Adamson [72] proposed [c.628]

See pages that mention the term

**Bloch equations**:

**[c.455] [c.708] [c.708] [c.1501] [c.2458] [c.672] [c.160] [c.132] [c.271] [c.123] [c.89] [c.377]**

Chemical kinetics the study of reaction rates in solution (1990) -- [ c.160 , c.163 ]