A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

Abstract A microscopic diffusional theory for the motion of a curved antiphase boundary is presented. The interfacial velocity is found to be linearly proportional to the mean curvature of the boundary, but unlike earlier theories the constant of proportionality does not include the specific surface free energy, yet the diffusional dissipation of free energy is shown to be equal to the reduction in total boundary free energy. The theory is incorporated into a model for antiphase domain coarsening. Experimental measurements of domain coarsening kinetics in Fe-Al alloys were made over a temperature range where the specific surface free energy was varied by more than two orders of magnitude. The results are consistent with the theory; in particular, the domain coarsening kinetics do not have the temperature dependence of the specific surface free energy.

A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening

Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys

Electron diffraction and microscopy are among the most important techniques for studying the structures of solids. This review aims to give a comprehensive introduction to the basic principles of the scattering of fast electrons and to highlight selected applications of importance. It begins by discussing electron scattering by single atoms and describes how single atoms may be imaged. The geometry of diffraction from perfect single crystals is then considered and a simple kinematical theory which yields approximate values for diffracted intensities is given. It is shown how simple principles have been used to image monatomic steps on the surface of crystals. More accurate methods of calculating diffracted intensities are given and particular attention is paid to describing concepts often found to be difficult, for example the dispersion surface. Principles of various recent electron scattering techniques are outlined including high-voltage electron microscopy, scanning electron microscopy, convergent-beam electron diffraction and the critical voltage effect. Applications described range from measuring bonding electron charge densities to the imaging of dislocations. Finally some recent theoretical developments on the problem of imaging imperfect crystals at atomic resolution are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/42/11/002/pdf

The scattering of fast electrons by crystals

— A microscopic theory for curved antiphase domain wall motion in ordered structures leads to a prediction that velocity is proportional to mean curvature. Unlike previous models, the velocity is not proportional to domain wall free energy. Experimental results on domain growth in ordered FeAl alloys over a range of temperatures, times and compositions, are consistent with the theory. In the vicinity of the critical temperature where domain wall free energy tends to zero, domain growth is not slowed. Introduction. — Antiphase domain boundaries are a non-equilibrium feature of real ordered alloys. Since there is a positive excess surface free energy associated with these boundaries, they move by diffusion in such a way that the total area of the boundaries is reduced. A long-standing phenomenological theory of interface motion [1-3] states that interfacial velocity is proportional to the thermodynamic driving force, the proportionality constant being a positive quantity called a mobility. The thermodynamic driving force in this theory is the product of the local mean curvature of the boundary and the excess surface free energy per unit area of boundary. A feature of higher-order transitions in ordering alloys is that the antiphase boundary energy approaches zero as the critical temperature is approached [4, 5]. Thus, the phenomenological theory of interface motion would seem to indicate that as the critical temperature for ordering is approached, antiphase domain growth should become very sluggish. In this paper, a new expression for interfacial velocity will be derived using a microscopic model. The excess surface free energy per unit area does not appear in this new formulation, and hence, the behavior of systems near critical points is more clearly represented. Application of the new result to the problem of macroscopic antiphase domain growth leads to a simple experimental test of the theory. Microscopic theory. — The kinetic equations of continuous ordering reflect the fact that the order parameter r\\ is not a conserved quantity. If the free energy is not at a minimum with respect to a local variation in r\\ there is an immediate change in r\\ given by dr\\ SF 5 ? = K ^ (1) where dFjStj is the variational derivative of the free energy of the system with respect to a local change in r\\, and a is a positive kinetic coefficient. In a non-uniform system [5] bF\\br\\ = dFjdri 2 K VTJ (2) where dF/drj is the rate of change in free energy for uniformly ordered systems and 2 K Vr\\ is the gradient energy contribution in a system in which r\\ is varying spatially. We obtain for the time dependent problem

A Microscopic Theory for Domain Wall Motion and Its Experimental Verification in Fe‐Al Alloy Domain Growth Kinetics

The microstructural changes during the α→ FeAl, FeAl → Fe3Al, and α→ Fe3Al transitions were studied by transmission electron microscopy. The ordering of ferromagnetic α was observed to occur in a classical manner by the nucleation and growth of particles of the FeAl or Fe3Al type phases. However, the ordering of paramagnetic α to FeAl and paramagnetic FeAl to Fe3Al occurred by a mechanism which showed many of the characteristics expected of a second or higher degree transition. These included critical point fluctuations in the degree of long-range order which also appeared to be greater in the vicinity of antiphase domain boundaries. The FeAl phase was also observed to partially disorder in the initial stages of the FeAl → α + FeAl transition and this effect appears to account for the anomalous magnetic behavior of some FeAl alloys.

The electron metallography of ordering reactions in Fe-Al alloys

Some consequences of Bloch's theorem on the diffraction of fast electrons in solids are considered. It is shown that the absorption coefficients of the various crystal wavefields excited in a crystal by the incident electron are periodic functions of crystal orientation. As a result it is shown that the two-beam theory, when applied to the calculation of the absorption coefficients, results in a poor approximation. Recently several authors have shown that the higher-order even reflections, such as the 222 reflection in Al or the 400 reflection in MgO, become vanishingly weak at certain critical values of the incident beam potential. From an investigation of the symmetry properties of the crystal wavefields a simple physical picture of this effect is obtained, and it is shown that at the critical voltage a reversal of the crystal wavefield symmetry occurs.



Es werden einige Konsequenzen des Blochschen Theorems uber die Beugung schneller Elektronen in Festkorpern betrachtet. Es wird gezeigt, das die Absorptionskoeffizienten der verschiedenen im Kristall durch die einfallenden Elektronen angeregten Kristallwellenfelder periodische Funktionen der Kristallorientierung sind. Als Ergebnis wird gezeigt, das die Zweistrahltheorie, wenn sie auf die Berechnung des Absorptionskoeffizienten angewendet wird, nur eine schlechte Naherung darstellt. Kurzlich haben verschiedene Autoren gezeigt, das die geraden Reflexionen hoherer Ordnung, wie die 222-Reflexion in al oder die 400-Reflexion in MgO, bei gewissen kristischen Werten des Potentials des einfallenden Strahles verschwindend schwach werden. Aus einer Untersuchung der Symmetrieeigenschaften des Kristallwellenfeldes wird ein einfaches physikalisches Bild dieses Effekts erhalten, und es wird gezeigt, das bei der kristischen Spannung eine Umkehrung der Kristallfeldsymmetrie erfolgt.

Consequences of Bloch's Theorem on the Dynamical Theory of Electron Diffraction Contrast

We have performed Mossbauer experiments on FeMn carbides having the cementite structure (Fe1−xMnx)3C, where x ranges from 0 to 0.6, and on a Hagg carbide with the Pd5B2 structure, (Fe1.1Mn3.9)C2, over the temperature range from 12 to 500 °K. Above the Curie temperature, the spectra exhibit a quadrupole interaction ranging from about 0.20 to 0.28 mm/s and the contributions from Fe atoms in the two types of cementite lattice sites can be identified. It is found that Mn atoms have a slight preference for “general” over “special” lattice sites in cementite. The specimens containing 0 to 20% Mn all become ferromagnetic at low enough temperatures, and the effective field decreases with Mn concentration very rapidly, extrapolating to 0 at about 34% Mn. Below Tc, the quadrupole splitting decreases drastically to less than 0.05 mm/s. Most of the results seem to be consistent with a donor theory in which C donates both of its 2p electrons to the metal d-band, giving the Fe in cementite an average electron configuration of approximately 3d7.66 4 s1.



Mosbaueruntersuchungen wurden im Temperaturintervall 12 bis 500 °K an FeMn Karbiden mit der Zementitstruktur (Fe1−x Mnx)3C, mit 0 ≦ x ≦ 0,6, und an einem Hagg-Karbid mit der Pd5B2-Struktur (Fe1,1Mn3,9)C2 durchgefuhrt. Oberhalb der Curietemperatur zeigen die Spektren eine Quadrupolwechselwirkung zwischen 0,20 und 0,28 mm/s, und die Beitrage von den Fe-Atomen in den beiden Typen der Zementit-Gitterplatze konnen identifiziert werden. Es wird gefunden, das die Mn-Atome die „allgemeinen” vor den „speziellen” Gitterplatzen in Zementit leicht bevorzugen. Die Proben, die 0 bis 20% Mn enthalten, werden bei genugend niedrigen Temperaturen alle ferromagnetisch und das effektive Feld nimmt mit der Mn-Konzentration sehr schnell ab (extrapoliert auf null bei rund 34% Mn). Unterhalb Tc nimmt die Quadrupolaufspaltung stark auf weniger als 0,05 mm/s ab. Die meisten Ergebnisse scheinen mit der Donatortheorie, in der C seine beiden 2p-Elektronen in das Metall-d-Band liefert, im Einklang zu sein und ergeben fur Fe in Zementit eine mittlere Elektronenkonfiguration von annahernd 3d7,664s1.

Mössbauer Study of the FeMn Carbides (Fe1−xMnx)3C and (Fe1.1Mn3.9)C2

Transmission electron metallography has been used to resolve the microstructural changes which occur during the D03⇄L20 transition in an Fe26.5at%Al alloy. The micrographs show evidence for a) long-range-order fluctuations near the critical temperature, b) local lowering of a critical temperature in the neighborhood of domain boundaries, and c) absence of a stable two-phase field separating the D03 and L20 phases. From these results it is concluded that the subject transition is not a classical (nucleation and growth) phase change.



Transmissions-Elektronenmetallographie wurde zur Auflosung mikrostruktureller Anderungen, die wahrend des D03⇄L20−Ubergangs in einer Fe26,5At%Al-Legierung auftreten, verwendet. Die Mikrographien zeigen a) Fernordnungsfluktuationen in der Nahe der kritischen Temperatur, b) lokale Erniedrigung einer kritischen Temperatur in der Nachbarschaft von Domanengrenzen und c) die Abwesenheit eines stabilen Zwei-Phasenfeldes, das die D03− und L20−Phasen trennt. Aus diesen Ergebnissen wird geschlossen, das der untersuchte Ubergang keine klassische (Keimbildungs- und Wachstums-) Phasenanderung ist.

Electron Metallography of a Non-Classical Order—Disorder Transition

The sensitivity of electron microscope image intensity profiles to the form of the absorption and extinction parameters has been analyzed for many-beam diffraction conditions which predominate at high voltage for most metals. The results show that the profiles are most sensitive to the extinction parameter which must be known to better than 5% if good agreement with experiment is to be obtained. It appears that the form of the absorption parameters has little influence on the intensity distributions of n-beam profiles, especially for medium atomic number elements such as copper.



Die Empfindlichkeit der Intensitatsprofile von Elektronenmikroskopbildern hinsichtlich der Form der Absorptions- und Extinktionsparameter wird fur die Bedingungen der Vielstrahlbeugung, die fur die meisten Metalle bei hohen Spannungen vorherrschend ist, untersucht. Die Ergebnisse zeigen, das die Profile am empfindlichsten von dem Extinktionsparameter abhangen, der bis auf 5% bekannt sein mus, um gute Ubereinstimmung mit dem Experiment zu erhalten. Es scheint, das die Form der Absorptionsparameter nur geringen Einflus auf die Intensitatsverteilung von n-Strahlprofilen hat, speziell fur Elemente mit mittlerer Atomnummer wie Kupfer.

R. M. Fisher

Papers

The electron metallography of ordering reactions in Fe-Al alloys

Consequences of Bloch's Theorem on the Dynamical Theory of Electron Diffraction Contrast

Mössbauer Study of the FeMn Carbides (Fe1−xMnx)3C and (Fe1.1Mn3.9)C2

Electron Metallography of a Non-Classical Order—Disorder Transition

Sensitivity of Many‐Beam Electron Microscope Images to the Form of Absorption and Extinction Parameters