where H and S are the enthalpy and the entropy, respectively, and T is the absolute temperature. The enthalpy H = U – PV where U is the internal energy, P is the pressure, and V is the volume. Gibbs’ free energy is also known as free enthalpy. Gibbs’ phase rule/law Interrelation between the number of components, C, the number of phases, P, and the number of degrees of freedom, F, in some equilibrium thermodynamic system: F = C – P + 2 in the case of varying temperature and pressure. If the pressure is constant, F = C – P + 1 In a binary system (i.e., at C = 2) at a constant pressure, the independent variables can be the temperature and the concentration of one of the components. If P = 1, then F = 2, which means that, in a single-phase field of the corresponding phase diagram, both the temperature and the com - position of the phase can be changed independently as long as the system remains single-phased. If P = 2, then F = 1, which means that, in a two- phase field, only one variable can be changed independently. In the case of the arbitrarily chosen temperature, the compositions of the phases are fixed, and in the case of the arbitrarily chosen composition of one of the phases at a given temperature, the composition of the other phase is fixed. Finally, if P = 3, then F = 0, i.e., in a three-phase field (in binary systems, it is represented by a horizontal line contacting three single-phase fields), the compositions of all the phases concerned are fixed and the phase equilibrium can take place at a constant temperature only. This is the reason why all the three-phase reactions in binary systems are termed invariant. In ternary systems (C = 3), invariants are reactions with four participating phases; whereas in a reaction with three participating phases, the compo - sitions vary in the course of the reaction, and the reaction develops in a temperature range. See, e.g., eutectic and peritectic reactions. Gibbs–Thomson equation Description of an alteration in chemical potential, µ, induced by a curved interface (free surface, phase boundary, or grain boundary): ∆µ = 2σ/ρ where σ and ρ are the interfacial energy and the radius of interface curvature, respectively. Chemical potential reduces when the interface migrates toward its center of curvature. The same effect reveals itself in the dependence of the solubility limit on the interface curvature: © 2003 by CRC Press LLC where H and S are the enthalpy and the entropy, respectively, and T is the absolute temperature. The enthalpy H = U – PV where U is the internal energy, P is the pressure, and V is the volume. Gibbs’ free energy is also known as free enthalpy. Gibbs’ phase rule/law Interrelation between the number of components, C, the number of phases, P, and the number of degrees of freedom, F, in some equilibrium thermodynamic system: F = C – P + 2 in the case of varying temperature and pressure. If the pressure is constant, F = C – P + 1 In a binary system (i.e., at C = 2) at a constant pressure, the independent variables can be the temperature and the concentration of one of the components. If P = 1, then F = 2, which means that, in a single-phase field of the corresponding phase diagram, both the temperature and the com - position of the phase can be changed independently as long as the system remains single-phased. If P = 2, then F = 1, which means that, in a two- phase field, only one variable can be changed independently. In the case of the arbitrarily chosen temperature, the compositions of the phases are fixed, and in the case of the arbitrarily chosen composition of one of the phases at a given temperature, the composition of the other phase is fixed. Finally, if P = 3, then F = 0, i.e., in a three-phase field (in binary systems, it is represented by a horizontal line contacting three single-phase fields), the compositions of all the phases concerned are fixed and the phase equilibrium can take place at a constant temperature only. This is the reason why all the three-phase reactions in binary systems are termed invariant. In ternary systems (C = 3), invariants are reactions with four participating phases; whereas in a reaction with three participating phases, the compo - sitions vary in the course of the reaction, and the reaction develops in a temperature range. See, e.g., eutectic and peritectic reactions. Gibbs–Thomson equation Description of an alteration in chemical potential, µ, induced by a curved interface (free surface, phase boundary, or grain boundary): ∆µ = 2σ/ρ where σ and ρ are the interfacial energy and the radius of interface curvature, respectively. Chemical potential reduces when the interface migrates toward its center of curvature. The same effect reveals itself in the dependence of the solubility limit on the interface curvature: © 2003 by CRC Press LLC H habit Shape of a precipitate or a grain, e.g., a plate-like habit, a dendritic habit, etc. habit plane In martensitic transformation, a plane in the parent phase lattice retaining its position and remaining undistorted during the transformation. In precipitation, a lattice plane of the parent phase parallel to the flat interfacial facets of precipitates. Hall–Petch equation Relationship describing an interconnection between the flow stress (or the lower yield stress in materials with the yield point phenomenon) σ, and the mean grain size, : where σ i i s the friction stress (it equals the flow stress in a coarse-grained material), k is a coefficient characterizing the grain-boundary strengthen- ing, and the exponent m = 1/2. This effect of grain size may be connected with pile-ups at grain boundaries triggering dislocation sources in the adjacent grains. An increase of the grain size results in a larger number of dislocations in the pile-ups and, thus, in the onset of slip in the neigh - boring grains at a lower stress level. If the obstacles to the dislocation glide motion are not grain boundaries, but subboundaries or twin bound- aries, the exponent m is between 1/2 and 1. In this case, is either a mean subgrain size or an average distance between the twin boundaries. In nanocrystalline materials, k = 0. hardenability Ability to form martensite on steel quenching; it can be enhanced by alloying. hardening [treatment] See quenching. hardness Resistance to the penetration of an object into the sample surface layer; in hardness tests, the object is called indenter. In metallic alloys, hardness is proportional to the yield stress. Harper–Dorn creep Steady-state creep at low stresses and temperatures ≥ 0.6 T m that evolves due to the dislocation glide motion controlled by climb. Dislocation density during Harper–Dorn creep does not increase, and the creep rate is described by a power law with the exponent n = 1 (see power D σσ i kD m– += D © 2003 by CRC Press LLC I ideal orientation See texture component. immersion objective/lens In optical microscopes, an objective with a numerical aperture A N >1.0 (up to ∼1.30). It works with a special medium between the lens and an object whose refraction index exceeds 1.0. imperfect dislocation See partial dislocation. impurity Incidentally present substance or chemical element, unlike alloying element. In semiconductors, impurity frequently means the same as dopant. impurity cloud See Cottrell and Suzuki atmospheres. impurity drag Inhibition of grain boundary migration by equilibrium grain- boundary segregations. Since the segregations decrease the grain-bound - ary energy, they reduce the capillary driving force, thus causing a drag force. At the same time, impurity drag is most often used in the sense that the segregated impurity reduces the effective mobility of grain bound - aries because the diffusivity of impurity atoms differs from that of the host atoms. Impurity drag is also called solute drag. incoherent interface Phase boundary in which there is no coincidence of the lattice points of neighboring lattices, in contrast to coherent or partially coherent interfaces. incoherent precipitate/particle Second phase precipitate whose interface with the matrix phase is incoherent. Incoherent precipitates have little to no orientation relationship with the matrix. incoherent twin boundary Twin boundary whose plane does not coincide with the twinning plane (see twin). A boundary of this kind is always joined to either a coherent twin boundary or the boundaries of the twinned grain. The energy and mobility of an incoherent twin boundary are rather close to those of general high-angle grain boundaries, in contrast to a coherent twin boundary. incubation period In materials science, the time duration (at a constant temper- ature) necessary for the first stable nuclei of a new phase to occur. The incubation period found experimentally is often greater than the true incubation period, due to an insufficient sensitivity of investigation tech - niques used. Incubation period is sometimes called induction period. indirect replica See replica. © 2003 by CRC Press LLC . between the number of components, C, the number of phases, P, and the number of degrees of freedom, F, in some equilibrium thermodynamic system: F = C – P + 2 in the case of varying temperature. independently. In the case of the arbitrarily chosen temperature, the compositions of the phases are fixed, and in the case of the arbitrarily chosen composition of one of the phases at a given. between the number of components, C, the number of phases, P, and the number of degrees of freedom, F, in some equilibrium thermodynamic system: F = C – P + 2 in the case of varying temperature