Discontinuous change

A logical consequence of this trend is a quantum w ell laser in which tire active region is reduced furtlier, to less tlian 10 nm. The 2D carrier confinement in tire wells (fonned by tire CB and VB discontinuities) changes many basic semiconductor parameters, in particular tire density of states in tire CB and VB, which is greatly reduced in quantum well lasers. This makes it easier to achieve population inversion and results in a significant reduction in tire tlireshold carrier density. Indeed, quantum well lasers are characterized by tlireshold current densities lower tlian 100 A cm . ... 

The heat capacity can be computed by examining the vibrational motion of the atoms and rotational degrees of freedom. There is a discontinuous change in heat capacity upon melting. Thus, different algorithms are used for solid-and liquid-phase heat capacities. These algorithms assume different amounts of freedom of motion. 

The transition from a ferromagnetic to a paramagnetic state is normally considered to be a classic second-order phase transition that is, there are no discontinuous changes in volume V or entropy S, but there are discontinuous changes in the volumetric thermal expansion compressibility k, and specific heat Cp. The relation among the variables changing at the transition is given by the Ehrenfest relations. 

Blast A transient change in the gas density, pressure, and velocity of the air surrounding an explosion point. The initial change can be either discontinuous or gradual. A discontinuous change is referred to as a shock wave, and a gradual change is known as a pressure wave. 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Passivation looks different when observed under galvanostatic conditions (Fig. 16.2b). The passive state will be attained after a certain time t when an anodic current which is higher than is applied to an active electrode. As the current is fixed by external conditions, the electrode potential at this point undergoes a discontinuous change from E to Ey, where transpassive dissolution of the metal or oxygen evolution starts. The passivation time t will be shorter the higher the value of i. Often, these parameters are interrelated as... 

If one of these quantities experiences a discontinuous change, i.e. if AS 0 or AV 0, then the phase transition is called a first-order transition according to Ehrenfest. It is accompanied by the exchange of conversion enthalpy AH = TAS with the surroundings. 

Backmix flow (BMF) is the flow model for a CSTR, and is described in Section 2.3.1. BMF implies perfect mixing and, hence, uniform fluid properties throughout the vessel. It also implies a continuous distribution of residence times. The stepwise or discontinuous change in properties across the point of entry, and the continuity of property behavior across the exit are illustrated in Figure 2.3. 

Second-order phase transitions are those for which the second derivatives of the chemical potential and of Gibbs free energy exhibit discontinuous changes at the transition temperature. During second-order transitions (at constant pressure), there is no latent heat of the phase change, but there is a discontinuity in heat capacity (i.e., heat capacity is different in the two... 

If the process is second- or higher-order, we will not be able to make a discontinuous change in the slope of the response curve. Consequently we would expect a second-order process to overshoot the setpoint if we forced it to reach the setpoint in one sampling period. The output would oscillate between sampling periods and the manipulated variable would change at each sampling period. This is called rippling and is illustrated in Fig. 20.2c. 

In alloys, structural ordering transformations can proceed with either a continuous or discontinuous change in entropy at the transformation temperature, while in... 

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