Those among you who are accustomed to regarding religion merely as a disease of the mind no doubt also entertain the idea that it is an evil that is easier to tolerate, indeed, perhaps to subdue, so long as only isolated individuals here and there are afflicted. But you think the common danger soars to its highest and everything is lost as soon as an all-too-close community should exist among several such unfortunates. In the former case, you think one can weaken the paroxysms through purposeful treatment, as it were, through a diet that counteracts the inflammation and through healthy air, and at least render the characteristic contagious element harmless where it is not fully conquered. But in the latter case you believe one must abandon every hope of deliverance. You fear the evil will become far more devastating and accompanied by the most dangerous symptoms, if each individual's disease is fostered and intensified by too close proximity to the others. You say the whole atmosphere will be poisoned by a few, even the soundest bodies will be contaminated, all channels in which the process of life is to take place destroyed, all fluids broken down, and, seized by the same feverish delirium, whole generations and peoples irretrievably affected. For this reason, your opposition to the church, to every event aimed at the communication of religion, is still greater than your opposition to religion itself; for this reason, priests, as the pillars and the really active members of such institutions, are for you the most despised of persons.

It may be an unexpected undertaking, and you might rightly be surprised that someone can demand from just those persons who have raised themselves above the herd, and are saturated by the wisdom of the century, a hearing for a subject so completely neglected by them. I confess that I do not know how to indicate anything that presages a fortunate outcome for me, not even the one of winning your approval of my efforts, much less the one of communicating my meaning and enthusiasm to you. From time immemorial faith has not been everyone's affair, for at all times only a few have understood something of religion while millions have variously played with its trappings with which it has willingly let itself be draped out of condescension.

Especially now, the life of cultivated persons is removed from everything that would in the least way resemble religion. I know that you worship the deity in holy silence just as little as you visit the forsaken temples, that in your tasteful dwellings there are no other household gods than the maxims of the sages and the songs of the poets, and that humanity and fatherland, art and science (for you imagine yourselves capable of all of this) have taken possession of your minds so completely that no room is left over for the eternal and holy being that for you lies beyond the world, and that you have no feelings for and with it.

What I myself have freely confessed as deeply lodged in the character of religion, the endeavor to make proselytes out of unbelievers, is nevertheless not what impels me to speak to you now about the formation of humans toward this sublime capacity and about its conditions. Religion knows no other means to that final goal than that it expresses and imparts itself freely.

When religion stirs with all its own power, when in the flow of this movement it sweeps along with it every faculty of one's mind into its service, it thus also expects to penetrate to the innermost being of every individual who breathes its atmosphere. Every homogeneous particle will be touched and, seized by the same vibration, the awaiting ear of a seeker will rejoice upon attaining consciousness of its existence through an answering, kindred tone. Only in this way, through the natural expressions of its own life, does religion wish to arouse what is similar. Where this is not successful, it proudly disdains every external attraction, every violent procedure, calmed in the conviction that the hour is not yet present in which something congenial to itself could stir. This unsuccessful outcome is not new to me. How often have I struck up the music of my religion in order to move those present, beginning with soft individual tones and longingly progressing with youthful impetuosity to the fullest harmony of religious feelings?

On Religion in its cultural milieu

The work of Friedrich Daniel Ernst Schleiermacher (1768–1834), inaugurated by this book, brilliantly reflects the tensions between the religious thought of the Enlightenment and Romanticism. When On Religion: Speeches to its Cultured Despisers was published in 1799, its author was an all but unknown cleric and member of the German Romantic circle. At the time of his death Schleiermacher was the most distinguished theologian of Protestant Germany, the author of a modern post-Enlightenment system of theology that ranks with Calvin and Aquinas in the history of Christian thought. On Religion is the premier expression of an understanding of religion as rooted in immediate pre-reflexive feeling and intuition, and only secondarily at the level of intellectual cognition or in moral systems and deeds. This classic theory of religion arose from the Romantics' intense critique of Kant's moral and religious philosophy in the repressive political atmosphere of a Prussia that feared the social upheavals of the French revolution. A many-faceted work, On Religion belongs to modern intellectual history, to German studies, to philosophy, religious studies, and theology.

Ultimately Schleiermacher's fame derives from his systematic interpretation of Christian theology, The Christian Faith (Glaubenslehre [1821–2, 1830–1]), whose relationship to On Religion is often disputed. Yet Schleiermacher never renounced his early book and considered his revisions (1806, 1821) to be more stylistic than substantive.

You know how the aged Simonides, through repeated and prolonged hesitation, reduced to silence the person who had bothered him with the question, “What are the gods after all?” I should like to begin with a similar hesitation about the far greater and more comprehensive question, “What is religion?”

Naturally, this would not be with the intention of keeping silent and leaving you in embarrassment as he did, but that, kept waiting in impatient expectation, you might for a time steadily direct your gaze to the point we seek, while completely excluding all other thoughts. It is, after all, the first requirement of those who only conjure common spirits that onlookers, who want to see their manifestations and be initiated into their secrets, prepare themselves through abstinence from earthly things and through holy silence; then, without distracting themselves by the sight of other objects, they look with undivided attention at the place where the vision is to show itself. How much more will I be permitted to insist on a similar obedience, since I am to call forth a rare spirit that does not deign to appear in any oft-seen familiar guise, a spirit you will have to observe attentively a long time in order to recognize it and understand its significant features.

It is beyond all doubt that a human being, engaged in the intuition of the universe, must be an object of esteem and veneration for all of you. No one who is still capable of understanding something of that condition can, upon considering it, withhold these feelings. You may despise anyone whose mind is readily and completely filled up with petty things, but you will try in vain to scorn the person who absorbs the greatest in himself and sustains himself by it.

You may love or hate each individual according to whether he moves with you or toward you on the limited path of activity and culture. But even that fairest feeling among those who are rooted in equality will no longer remain in you in relation to someone who is raised as far above you as the contemplator of the universe stands above anyone who is not in the same condition. Your wisest people say you must honor, even against your will, the virtuous individual who according to the laws of moral nature tries to determine the finite in conformity with infinite demands. But even if it were possible for you to find virtue itself somewhat ludicrous in the contrast of finite forces with an infinite undertaking, you would still be unable to deny esteem and reverence to that person whose senses are open to the universe.

Two important binary systems

Copper-Zinc (‘Brass’) [2.1]

Fig. 6.1 shows the phase diagram, which is characterized by a series of five peritectic reactions from the copper to the zinc-rich side. At 73 at.% zinc the δ-phase undergoes eutectoid decomposition into γ and ε on cooling.

The α-phase is close packed cubic, i.e. fee like copper; the β-phase is body centered cubic (bcc). Below 468 to 454 °C an ordered arrangement β′ of Cu and Zn atoms (present in the ratio of approximately 1:1) forms on the bcc lattice, as shown in fig. 6.2. It is interesting that the β field widens with increasing temperature above the β′β transition temperature and that consequently the solubility of Zn and Cu is reduced, contrary to (5-22). The gain in entropy on disordering apparently stabilizes β, i.e. lowers the F β parabola with increasing temperature by more than F α and F γ are lowered. The γ-phase is complex cubic with 52 atoms per unit cell. It can be imagined as consisting of 3 × 3 × 3 unit cells of the bcc β′-brass from which 2 atoms have been removed and the remainder slightly displaced. δ also has a very large and complicated cubic unit cell, ε and η, on the other hand, are hexagonal structures, described as close-packed hexagonal or cph.

Superlattices

We have seen in section 6.1 that the ordered β′-brass structure is produced from the disordered β structure when the corners of the bcc unit cell are occupied by one atomic species and the cell centre positions by the other. The crystallographic nomenclature for β is A2 and for β′ is B2 (for the lattice type nomenclature A, B,…, D, L,…, see [7.11]).

An ordered distribution of the alloying components over the lattice sites, i.e. the formation of sublattices, results in a larger unit cell or superlattice. Often additional X-ray (superlattice) reflections are observed if the waves reflected from parallel A atom and B atom planes do not cancel each other completely. (The structure factor {fA + fB exp[− iπ(h + k + l]} see (2-7) with un = 0, is no longer zero for odd values of (h + k + l), if fA ≠ fB , see [2.2].)

Other superlattices based on A 2 are observed in Fe3Al(D03), fig. 7.1, and the Heusler alloy, Cu2MnAl(L21), which is employed as an iron-free ferromagnet. (Al on the sublattice X, Mn on the sublattice Y in fig. 7.1.) The superlattice Cu3Au I (L12), fig. 7.2, and the tetragonal superlattice CuAu I (L10), fig. 7.3, are based on the disordered fee lattice A1. The superlattices Mg3Cd (D019) with an axial ratio for the ordered cell of 0.8038 is formed on a cph A3 lattice.

Since it is primarily on account of their mechanical properties that metals are of such technological interest, this chapter is concerned with one of the vital themes of this book. Only by alloying do metals achieve sufficient strength, i.e. a sufficiently high yield stress, to be able to withstand the stresses applied in practice. Unlike the hardening due to deformation discussed in chapter 12 or that produced by the martensitic transformation (section 13.7) the hardening we are concerned with here is produced by the addition of alloying elements. As described in chapter 11, plastic deformation is brought about by dislocations. Hardening thus implies restriction of dislocation movement due to interaction of dislocations with the solute. The solute can be present in a variety of forms: as dissolved atoms, as precipitate particles, as ordered regions in a disordered matrix, as particles of a second phase, or as a constituent of a phase mixture. In all these examples hardening is achieved by making the material heterogeneous. We shall discuss these various alloy-heterogeneities one after the other.

Solid solution hardening

In a solid solution the heterogeneity is on an atomic scale. We assume a very dilute solution of B and A in which the dislocation interacts with individual solute atoms. This interaction involves various mechanisms, which are the first problem to be discussed in the following. In reality the dislocation interacts simultaneously with many solute atoms, regarded here (at low temperatures) as immobile whereas the dislocation is mobile.

Metallurgists employ a number of experimental methods not normally encountered by the solid state physicist. Several of these will be described and critically assessed in the following chapter because the data they can provide will be drawn upon in subsequent chapters. Naturally many experimental techniques are standard for both solid state physicists and metallurgists, for example X-ray methods for the determination of crystal structure, lattice parameters and crystallographic orientation [2.1], [2.2], based on the Bragg relationship for constructive interference of X-rays scattered by the atoms in the lattice. Similarly, both physicists, and metallurgists make use of measurements of electrical conductivity, Hall voltage, macroscopic density and its variation with temperature (thermal expansion), although the metallurgist usually measures the latter uni-dimensionally in a dilatometer. Nor do measurements of elastic moduli, specific heat or magnetic susceptibility introduce the physicist to any new methods. Results of these and related investigations will be referred to without any detailed description of the procedure. Relevant techniques are summarized in [2.3], [2.4], [2.5].

Microscopy of surfaces

Apart from their crystal structure, metals possess a microstructure, in that they consist of differently oriented grains or differently constituted phases. The observation and if possible quantitative description of this microstructure is the aim of metallography, which employs optical, electron, ion and X-ray microscopical methods, some of which are described in sections 2.2 and 2.4.

Measurement of the equilibrium vacancy concentration

In section 8.2.1 the vacancy was recognized as being the predominant means of diffusion in metals. Vacancies are present as part of a structure in thermal equilibrium; the equilibrium concentration c v(T) at a temperature T can be calculated from (3-1). Vacancies can be detected directly but tediously by counting the occupied lattice sites at a quenched metal tip using a field ion microscope [10.5]. In principle, it is also possible to demonstrate their existence by any physical property which varies proportional to c v, for example the electrical resistivity. Nevertheless, since c v decreases exponentially with decreasing temperature from 10−4 at the melting point, high-temperature measurements are necessary to study vacancies in equilibrium. The electrical resistance of metals at these temperatures is, however, largely determined by lattice vibrations and scarcely influenced by lattice defects. There are very few remaining methods, only two according to A. Seeger [10.1], which are sufficiently sensitive to vacancies at high temperatures to permit their effects to be separated from those of the ‘background’. These are (differential) thermal expansion and the lifetime of positrons.

Definitions

In previous chapters we have discussed processes which introduce lattice defects into metals: deformation generates dislocations, irradiation can cause displacement cascades, martensitic and diffusion-controlled transformations produce grain boundaries, etc. These processes change the microstructure creating a state of higher free energy. Recrystallization is the formation of a new microstructure with a lower free energy by a reaction in the solid state similar to the formation of the microstructure by the crystallization of a melt (chapter 4). In a typical recrystallization experiment, a heavily deformed metal is annealed at a temperature higher than one-half its melting point. This removes many of the lattice defects introduced by deformation and a new arrangement of grain boundaries is established. As was stated in chapter 3, the microstructure is, by definition, not in thermodynamic equilibrium. The grain boundaries remaining after recrystallization constitute a metastable arrangement separating grains with a certain orientation distribution. In other words the metal exhibits a characteristic recrystallization texture. Such textures have a vital influence on many of the physical properties of technological materials, e.g. magnetization losses in transformer sheet.

Recrystallization, the generation of a system of new grains, is fundamentally different from recovery which always precedes it and in which the lattice defects within a given arrangement of high-angle grain boundaries either anneal out or rearrange themselves. The recovery stages I to IV described in section 10.4 in which point defects created by irradiation, deformation or quenching anneal out are thus recovery in the true sense. Stage V is the recrystallization of deformed material.

In the preceding chapter we have already employed the concept of the dislocation, which is an essential constituent of the metallic microstructure. So far we have been able to refer to a chapter in Kittel [1.1] in which dislocations were introduced to explain plastic deformation and were described in terms of geometry and elasticity theory. Since the following chapters develop concepts based on the properties of these lattice defects, the present chapter presents the necessary elements of dislocation theory and verifies them by experimental results of a fundamental character. There is no lack of more advanced texts on dislocations, see for example [11.1] to [11.5].

Topological properties of dislocations

Definition

As was stated in [1.1], dislocations are needed to explain why, in principle, metallic crystals are very easy to deform. The fundamental process in plastic deformation consists of shear on a definite crystallo-graphic plane (slip plane) by a unit translation vector b (Burgers vector) in a definite lattice direction (slip direction), fig. 11.1. It can be explained in terms of two possible intermediate steps, the edge dislocation and the screw dislocation (figs. 11.1(b) and (c)). In the first case fig. 11.1(b), slip has started on the right-hand side of the crystal but has not yet reached the left-hand side. In between, therefore, there is a distorted region (edge dislocation) with five half-planes above the slip plane and four half-planes below it (or three above two) which extends from the front to the back of the crystal at right angles to b.