Bloch’s equation

Before developing the theory when interactions are present, it is convenient to derive Bloch s equation for the density matrix. 

Bloch s Equation for the Density Matrix.—Bloch s equation will be derived here both for the equilibrium ensemble density matrix of Eq. (8-204), and for the equilibrium grand ensemble density matrix of Eq. (8-219). 

This is Bloch s equation. We note that this equation is essentially of the same form as Schrbdinger s equation, with playing the role of wave function, and ]8 playing the role of time, j8 (i]E)t, and the operator H — E representing deviation of the hamiltonian from the ensemble average. 

Combining Eqs. (8-231) and (8-232) we then have Bloch s equation again ... 

Density matrix and ensembles, 465 Bloch s equation, 475 Derogatory form, 73... 

The frequency ranges for maximum and minimum excitation can be exchanged, when the phases of successive pulses are alternating (Fig. 6a, version I vs. II). Since experiments were carried out at 1.5 T, theoretical calculations were based on intervals of 2.32 ms between consecutive pulses. Transverse magnetization in Fig. 6b was calculated by a program simulating Bloch s equations. 

The most expensive parts of a conventional NSE instrument are the main solenoids providing the precession field. A closer look at Bloch s equation of motion for the spins (Eq. 2.11) shows that in a coordinate system that rotates with the precession frequency around Bg the spin is stationary, the coordinate system rotation is equivalent to the addition of - to all magnetic fields. By this means the large precession field inside the main coils may be transformed to zero - zero field spin-echo). The flippers are viewed as elements rotating... 

The repeated tt - t - 7t/2 rf pulse trains are used in the method. The Ti follows the Equation (20.1) derived from Bloch s equation ... 

Reaction rates were obtained by total line-shape analysis of the spectra, the equations used for the simulation being Bloch s equations modified for exchange phenomena, The experimental and theoretical spectra for the Mg-NIPA system are shown in Figure 1. 

CALCULATION OF THE MODULATION SIGNAL BY SOLVING BLOCH S EQUATIONS IN THE PRESENCE OF AMPLITUDE MODULATION... 

MATRIX TECHNIQUE TO SOLVE BLOCH S EQUATIONS IN A ROTATING FRAME USING FOURIER-SERIES EXPANSION... 

In order to solve Bloch s equations using the matrix technique one introduces expansions (12) and (13) into Bloch equations (5) and (6), respectively. Using the relation cos f2t=%(e +e ) and comparing the coefficients of e " on the two... 

Misra SK. 2005. Microwave amplitude modulation technique to measure spin-lattice relaxation times solution of Bloch s equations by a matrix technique. Appl Magn Reson 28 55-67. 

Faced with a broad range of prospective spin-lattice relaxation times, the investigator needs two types of spectrometers, a situation that is further complicated if multi-frequency measmements are required. Furthermore, the phenomenological descriptions of measurements made by cw and transient spectrometers differ, as they correspond to separate solutions to Bloch s equations. This chapter describes refinements of both instrumental and theoretical/computational techniques that facilitate the measurement of spin-lattice relaxation times. 

Bloch s phenomenological equations were solved for the case of slow passage through the resonance, by which it is meant that the rate of varying field Ho is small compared to the time required to acquire a spectroscopic data point (Hq is assumed to remain constant over this period). In the laboratory frame of reference (x,y,z), magnetic moments fx are subjected to time-dependent field H = lH cos at) + ]H sm at)) + m , and so the evolution of vector product HxM) in Bloch s equations gives rise to first-order differential equations... 

From Bloch s equations and their solution there follows several experimental i.e., resonant) methods of measuring T. For example, it is a common practice to plot both the peak-to-peak amplitude and the peak-to-peak linewidth of the derivative EPR signal as a function of the square root of the microwave power VP (c/ Poole, 1967). The square root of the microwave power is proportional to H, and so one has an experimental handle on the term that appears in the denomina-... 

Bloch s equations can be expressed in matrix form as follows ... 

The effect of amplitude modulation on magnetization is determined by substituting the modified expression for Hi into Bloch s equations for the rotating field ... 

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