How to Draw Hcp Planes

Trunk Centered Cubic Lattices

Basic Properties of Crystals

Prasanta K. Misra , in Physics of Condensed Thing, 2012

i.two.4 Trunk-Centered Cubic (bcc) Lattice

The torso-centered cubic (bcc) lattice (Figure one.4b) tin can be obtained by adding a 2nd lattice point at the centre of each cubic prison cell of a simple cubic lattice. Thus, the unit prison cell of each bcc lattice can exist considered as two interpenetrating uncomplicated cubic primitive lattices. In fact, there are two alternate ways of considering a bcc lattice, either with a uncomplicated cubic lattice formed from the corner points with a lattice bespeak at the cube middle, or with the unproblematic cubic lattice formed from the lattice points at the center and the corner points located at the center of the new cubic lattice. In either case, each one of the eight lattice points at the corner of a cubic jail cell is shared by eight adjacent cubic cells, while the lattice point at the center of the cubic prison cell exclusively belongs to that prison cell. Therefore, the bcc lattice can be considered equally a unit cubic cell with two lattice points per cell. The number of nearest neighbors of each lattice bespeak is eight. Alternately, 1 tin can state that the coordination number is viii.

Yet, the archaic cell of a bcc lattice can likewise be easily obtained. In fact, there are a diverseness of ways in which the archaic vectors of the bcc lattice tin be described. The most symmetric set of archaic vectors is given as follows:

(1.2) $a_1=\frac{a}{\mathrm{ii}}\left(\hat{y}+\hat{z}-\hat{x}\right),a_2=\frac{a}{2}\left(\hat{z}+\widehat{\mathrm{ten}}-\hat{y}\right),a_{\mathrm{iii}}=\frac{a}{2}\left(\hat{x}+y-\hat{z}\right),$

where $a$ is the lattice constant (the side of the unit cubic prison cell), and $\hat{x},\hat{y},\text{and}\hat{z}$ are orthonormal vectors. It is of import to note that a ₁, a _ii, and a ₃ are not orthogonal vectors. The parallelepiped drawn with these three vectors (shown in Effigy 1.5) is the archaic cell of the bcc lattice. The 8 corners of this primitive cell have eight lattice points, each shared past eight primitive cells.

Effigy i.5. Symmetric set of primitive vectors for the bcc Bravais lattice.

Information technology can be shown that the volume of the primitive Bravais cell is (Problem one.i)

(1.3) $V=a_1·(a_2\times a_3)=\frac{a^3}{2}.$

Because the volume of the unit cubic cell is $a^3$ , and each unit of measurement jail cell has two lattice points, the primitive cell of the bcc lattice is one-half of the volume of the unit prison cell. However, non ane of the lattice points uniquely belongs to any archaic cell shown in Figure 1.5. The lattice constants of bcc lattices at low temperatures are shown in Tabular array ane.ane. To be able to specify the primitive cell around each lattice point, i has to draw the Wigner–Seitz cell of the bcc lattice.

Tabular array 1.one. Lattice Constants of bcc Lattices at Low Temperatures

Chemical element	Lattice Abiding (A °)
Barium	5.02
Chromium	2.88 (Cr as well has fcc and hcp phases)
Cesium	6.05
Europium	4.61
Iron	2.87 (Iron also has fcc phase)
Potassium	v.23
Lithium	3.50
Molybdenum	3.xv (Mo also has fcc stage)
Sodium	iv.29
Niobidium	iii.30
Rubidium	5.59
Tantalum	3.31
Thallium	3.88 (T1 likewise has fcc and hcp phases)
Uranium	3.47
Vanadium	3.02
Tungsten	3.xvi

Source: R. W. G. Wyckoff, Crystal Structures, vol. i (J. Wiley, 1963).

It tin exist easily shown that the Wigner–Seitz cell of a trunk-centered cubic Bravais lattice is a truncated octahedron (see Figure 1.6). The octahedron has iv square faces and four hexagonal faces. The square faces bisect the lines joining the central betoken of a cubic cell to the central points of the half-dozen neighboring cubic cells. The hexagonal faces bisect the lines joining the central bespeak of a cubic cell to the eight corner points of the same cubic cell. The lattice betoken at the center of the bcc lattice is also at the center of this octahedron. Any indicate in the space inside the octahedron (except for points on the mutual surface at 2 or more Wigner–Seitz cells) is closer to this central lattice bespeak than any other cardinal lattice signal.

Figure 1.6. Wigner–Seitz jail cell of a bcc Bravais lattice.

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Residue Stresses and Baloney in Welds

K. Masubuchi , in Encyclopedia of Materials: Science and Technology, 2005

five.1 Breakable Fracture

Since low-carbon steels usually used in many structural applications have body-centered-cubic (b.c.c.) lattice structures, they become brittle when they are exposed to depression temperatures. In fact, engineers have experienced brittle fractures of steel structures since they became widely used around 1850. Brittle fractures of steel structures have go a very serious problem for the post-obit reasons:

(i)

a welded structure does not accept riveted joints that can interrupt the extension of a brittle crack;

(2)

welds may take diverse defects including cracks, slag inclusion, etc.; and

(iii)

loftier tensile residual stresses normally be in welds, as described here.

The nearly extensive and well-known brittle fractures of welded structures are those that occurred in welded ships built in the U.s.a. during World War II. Extensive studies have been conducted on the effects of residue stresses on breakable fractures of welded structures, especially steel structures. When a sharp notch is located in areas where high tensile residual stresses exist, unstable fracture may occur in the weld even when it is subjected to tensile loading considerably below the yield strength of the fabric.

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The Landau Theory of Phase Transitions: Full general Concept and Its Microscopic Relation to Mean Field Theory

Jurgen Honig , Józef Spałek , in A Primer to the Theory of Disquisitional Phenomena, 2018

Problem 4.one: A Unproblematic Physical Instance of a Continuous Stage Transition: Order–disorder Transformation

The metallic blend CuZn contains exactly a fifty–fifty limerick within the body centered cubic lattice (β-brass). At high temperatures ( $T>T_c$ ) the A ≡ Cu and B ≡ Zn atoms are arranged in a random way on this bcc lattice (cf. Fig. 4.half-dozenA). At $T=0$ , the alloy is completely ordered, as shown schematically in Fig. four.half dozenB. The positions of A and B tin be interchanged, but within a single domain of macroscopic size, but 1 of them gets selected. This is what is meant by spontaneous breakup of discrete A↔B interchange symmetry.

Effigy four.six. Schematic representation of the CuZn (AB) blend on the bcc lattice in the disordered phase (A) and in the fully ordered land (B). For details, see the principal text.

The order parameter tin be defined in a natural manner as follows: Suppose $N_{AA}$ is the number of A atoms in A (corner) positions, whereas $N_{AB}$ is those in B (middle) positions. In the aforementioned manner, one tin define $N_{BB}$ and $N_{BA}$ , respectively. We can define the caste of order as

(4.96) $\eta =\frac{{\mathrm{Northward}}_{AA}-{\mathrm{Northward}}_{AB}}{N_{AA}+{\mathrm{North}}_{AB}}=\frac{N_{BB}-{\mathrm{Northward}}_{BA}}{N_{BB}+{\mathrm{North}}_{BA}}$

For $N_{AB}=N_{BA}=0$ , nosotros have complete order, $\eta =1$ . On the reverse, for ${\mathrm{Northward}}_{AA}={\mathrm{North}}_{AB}=N_{BA}={\mathrm{Northward}}_{AB}$ , we take $\eta =0$ , i.east., in this case, a continuous phase transition point. This situation is reminiscent of that for a binary mixture, which starts separating at $T=T_c$ .

The question centers on the microscopic cause of the separation into ordered AB blend, and why it takes the class of a stage transition. We recognize that the natural pair attraction $V_{AB}<0$ must exist stronger than the sum of $5_{AA}$ and $5_{BB}$ , where $i,j\equiv 〈i,j〉$ , and ${\mathrm{northward}}_{iA,B}$ are the respective numbers $n_{iA,B}=0,1$ of atoms at site i. So the total energy of the system is

(4.97) $\mathrm{East}=\sum _{〈i,j〉}(V_{AB}{\mathrm{due\; north}}_{iA}{\mathrm{due\; north}}_{jB}+V_{AA}{\mathrm{northward}}_{iA}{\mathrm{north}}_{jA}+V_{BB}{n}_{iB}{n}_{jB}).$

Now, introducing the spin variables

(4.98) $n_{iA,B}=\frac{1}{\mathrm{ii}}+S_i^z,$

where $S_i^z=\pm \frac{\mathrm{ane}}{2}$ for A or B cantlet occupancy, respectively, and similarly

(4.99) $n_{jB,A}=\frac{\mathrm{ane}}{\mathrm{two}}-S_j^z,$

we have, upward to a constant (with the status $n_A+{\mathrm{due\; north}}_B=1$ ),

(4.100) $E=\sum _{〈i,j〉}(-{\mathrm{Five}}_{AB}{\mathrm{South}}_i^z{S}_j^z+{\mathrm{Five}}_{AA}{\mathrm{Due\; south}}_i^z{S}_j^z+V_{BB}{\mathrm{South}}_i^z{S}_j^z).$

Hence, the organisation is described by the Ising model, with the effective substitution integral $J\equiv +(5_{AB}-V_{AA}-{\mathrm{Five}}_{BB})$ , with an antiferromagnetic sign $J<0$ , unlike in Chapters ii and 3, i.e.,

(four.101) $\mathrm{East}=|J|\sum _{〈i,j〉}{S}_i^z{S}_j^z.$

Thus, the ordering in this spin representation is a checkerboard design, equally shown in Fig. ii.2, and equally can be shown explicitly, by subdividing the whole lattice of N sites into two sublattices, one with all spins up and the other with spins down. The solution of Eq. (4.101) is non simple; nevertheless, we can can generalize the mean-field solution to the example of two sublattices.

Of import note

The separation into the two sublattices breaks non only the equivalence of occupancies between A and B, but simultaneously breaks the translational symmetry of the lattice: If the lattice parameter is a above $T_c$ then it is 2a below $T_c$ . We see that the stage transition may alter the symmetry in the system in a nonobvious manner. See Fig. 4.6.

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On the theory of phase transitions of the second order

East.M. LIFSHITZ , in Perspectives in Theoretical Physics, 1992

3 Summary

In the appended tables all the possible superstructures (in face up-centred and body-centred cubic lattices and hexagonal close-packed ones) for which phase transitions of the second order are possible are enumerated.†

Graph I shows the space group of an ordered lattice. The numbers in graph Three show how many times the simple cell is increased when the lattice is ordered.

Graph Ii shows the types of Bravais lattices, P, F, I announce uncomplicated, confront-centred and body-centred lattices respectively, H and R are the hexagonal and rhombohedral lattices. Graph Iv shows how the cell of the Bravais lattice of ordered crystals is obtained from the initial disordered lattice. The simple lattice P is determined by the coordinates of the ends of three edges of its prison cell; in lattice I coordinates of the center of the prison cell are added to these 3 indices, and in the F lattice − the coordinates of the centres of iii of its faces. In these notations the original cell of the disordered lattice of the types A1 and A2 must be correspondingly written equally (100, 010, 001) $\left(\frac{1}{\mathrm{ii}}\frac{\mathrm{i}}{2}0,\frac{1}{\mathrm{two}}0\frac{1}{\mathrm{ii}},0\frac{1}{2}\frac{\mathrm{ane}}{2}\right)$ and (100, 010, 001), $\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ . In the hexagonal lattice H the centrality z is chosen along the tiptop of the cell, and the axes x and y forth the sides of its rhombic base of operations which intersect at an angle of 120°; the cell of such a lattice we will also determine by the ends of its three edges and then that the original lattice of solution of the type A3 is written as (100, 010, 001) (the cell is chosen in such a manner that the coordinates of the atoms it contains are $\left(\frac{1}{3}\frac{2}{3}\frac{\mathrm{one}}{4}\right),\left(\frac{\mathrm{two}}{3}\frac{1}{\mathrm{iii}}\frac{3}{4}\right)$ ). In the lattice R three axes are directed along three edges of the rhombohedral cell. The indices in the graph Four decide the Bravais lattice of an ordered crystal, the coordinates existence given in units of the lengths of the edges of the original cell; this establishes connection between the Bravais lattices of the disordered and ordered alloys.

TABLE one. Curie points in superstructures of face-centred cubic lattice

I	II	III	IV	V	Half dozen	Seven
${\text{D}}_{3d}^5$	R	ii	( $\frac{1}{2}\frac{1}{2}\mathrm{i},$ )	AB	1A (000) (a), 1B $\left(\frac{1}{2}\frac{\mathrm{ane}}{2}\frac{1}{2}\right)$ (b)	L-11
${\text{O}}_h^{\mathrm{v}}$	F	eight	(200, )(100, )	ABC6	4A (000) (a),4B $\left(\frac{1}{\mathrm{two}}\frac{1}{2}\frac{1}{\mathrm{ii}}\right)$ (b), 24C ( $0\frac{1}{4}\frac{1}{\mathrm{four}},$ ) ( $0\frac{1}{\mathrm{iv}}\frac{3}{4},$ ) 4A (a), 28B(b) (d)
				AB7
${\text{O}}_h^{\mathrm{vii}}$	F	8	(200, )(110, )	AB	16A (000) ( $0\frac{1}{\mathrm{iv}}\frac{1}{4},$ ) (c), 16B $\left(\frac{\mathrm{i}}{2}\frac{1}{\mathrm{two}}\frac{1}{2}\right)$ $\left(\frac{1}{2}\frac{3}{\mathrm{iv}}\frac{3}{4}\right)$ (d)	L-xiii
${\text{D}}_{4h}^{17}$	I	4	(100,010,002)	ABC2	2A (000) (a), 2B $\left(00\frac{1}{\mathrm{two}}\right)$ (b), 4C $\left(0\frac{1}{2}\frac{1}{4}\right)$ $\left(\frac{1}{\mathrm{two}}0\frac{\mathrm{ane}}{\mathrm{four}}\right)$ (d) 2A (a), 6B(b) (d)
			$\left(\frac{1}{2}\frac{1}{2}1\right)$	AB3
${\text{D}}_{\mathrm{four}h}^{\mathrm{nineteen}}$	I	4	(100,010,002)	AB	4 (000) $\left(0\frac{1}{2}\frac{1}{\mathrm{iv}}\right)$ (a), 4 $\left(00\frac{\mathrm{one}}{\mathrm{two}}\right)$ $\left(0\frac{1}{2}\frac{3}{4}\right)$ (b)
			$\left(\frac{1}{\mathrm{ii}}\frac{\mathrm{i}}{\mathrm{two}}1\right)$
${\text{O}}_h^1$	P	32	(200, )	ABC3 Dthree E12 F12	1A (000) (a), 1B $\left(\frac{1}{2}\frac{1}{\mathrm{two}}\frac{1}{2}\right)$ (b), 3C ( $0\frac{1}{2}\frac{\mathrm{i}}{\mathrm{ii}}$ ) (c), 3D ( $\frac{\mathrm{i}}{\mathrm{ii}}00,$ ) (b), 12E ( $0\frac{1}{4}\frac{\mathrm{one}}{4},$ ) ( $0\frac{\mathrm{iii}}{\mathrm{four}}\frac{3}{\mathrm{iv}},$ ) ( $0\frac{\mathrm{ane}}{4}\frac{3}{\mathrm{iv}},$ ) (i), 12F (( $\frac{1}{2}\frac{\mathrm{ane}}{\mathrm{four}}\frac{1}{4},$ ) (( $\frac{1}{2}\frac{\mathrm{iii}}{\mathrm{iv}}\frac{1}{4},$ ) ( $\frac{1}{2}\frac{1}{4}\frac{3}{\mathrm{iv}},$ ( $\frac{\mathrm{one}}{\mathrm{ii}}\frac{\mathrm{three}}{4}\frac{\mathrm{iii}}{\mathrm{four}},$ ) (j)
${\text{O}}_h^3$	P	32	(200, )	Athree B3 Cfour Dvi	8C (000) ( $0\frac{1}{\mathrm{ii}}\frac{1}{\mathrm{two}},$ ) $\left(\frac{\mathrm{one}}{\mathrm{ii}}\frac{1}{2}\frac{1}{\mathrm{ii}}\right)$ ( $00\frac{1}{2},$ ) (e), 12D ( $\frac{\mathrm{three}}{4}\frac{\mathrm{three}}{4}0,$ ) ( $\frac{\mathrm{three}}{4}\frac{3}{4}\frac{\mathrm{one}}{\mathrm{ii}},$ ) ( $\frac{\mathrm{i}}{\mathrm{iv}}\frac{1}{4}\frac{\mathrm{ane}}{2},$ ) ( $\frac{1}{4}\frac{\mathrm{i}}{4}0,$ ) (f), 6A ( $0\frac{3}{4}\frac{1}{2},$ ) ( $\frac{1}{2}\frac{3}{4}\frac{1}{4},$ ) (c) 6B ( $0\frac{\mathrm{ane}}{\mathrm{iv}}\frac{3}{4},$ ) ( $\frac{\mathrm{i}}{2}\frac{3}{4}\frac{1}{4},$ ) (d)
Ovi, O7	P	32	(200, )	ABCthree Diii	4A (000) ( $\frac{1}{2}\frac{1}{\mathrm{four}}\frac{\mathrm{iii}}{4},$ ) (a), 4B $\left(\frac{1}{\mathrm{two}}\frac{1}{\mathrm{two}}\frac{\mathrm{one}}{2}\right)$ ( $0\frac{\mathrm{three}}{4}\frac{\mathrm{i}}{\mathrm{iv}},$ ) (b), 12C ( $\frac{1}{4}\frac{\mathrm{ane}}{4}0,$ ) ( $\frac{\mathrm{i}}{2}\frac{1}{2}0,$ ) ( $\frac{3}{\mathrm{iv}}\frac{3}{\mathrm{four}}0,$ ) ( $\frac{\mathrm{i}}{\mathrm{two}}\frac{3}{4}\frac{1}{\mathrm{iv}},$ ) (d), 12D ( $\frac{1}{4}\frac{1}{\mathrm{four}}\frac{1}{2},$ ) ( $00\frac{1}{2},$ ) ( $\frac{\mathrm{i}}{\mathrm{two}}\frac{3}{4}\frac{3}{4},$ ) ( $\frac{1}{\mathrm{four}}\frac{3}{4}0,$ ) (d)

Table 2. Curie points in superstructures of torso-centred cubic lattice

I	II	III	4	V	VI	7
${\text{O}}_h^1$	P	2	(100, ↻)	AB	$1\text{A}\left(000\right)\left(\text{a}\right),1\text{B}\left(\frac{1}{2}\frac{\mathrm{one}}{\mathrm{ii}}\frac{\mathrm{i}}{2}\right)\left(\text{b}\right)$	50-20
${\text{O}}_h^{\mathrm{five}}$	F	4	$\left(200,\circlearrowright \right)\left(100,\circlearrowright \right)$	ABC2	$\begin{array}{c}4\text{A}\left(000\right)\left(\text{a}\right),4\text{B}\left(\frac{1}{2}\frac{\mathrm{one}}{\mathrm{four}}\frac{\mathrm{i}}{2}\right)\left(\text{b}\right),\\ 8\text{C}\left(\frac{\mathrm{i}}{4}\frac{1}{\mathrm{four}}\frac{1}{4}\right)\left(\frac{3}{4}\frac{3}{4}\frac{3}{\mathrm{iv}}\right)\left(\text{c}\right)\end{array}$	L-21
				AB3	4A (a), 12B (b) (c)	Practisethree
${\text{O}}_4^{\mathrm{seven}}$	F	4	$\left(200,\circlearrowright \right)\left(100,\circlearrowright \right)$	AB	$\begin{array}{l}8\text{A}\left(000\right)\left(\frac{\mathrm{i}}{4}\frac{\mathrm{ane}}{\mathrm{four}}\frac{1}{4}\right)\left(\text{a}\right),\\ \mathrm{eight}\text{B}\left(\frac{1}{\mathrm{ii}}\frac{1}{2}\frac{\mathrm{i}}{2}\right)\left(\frac{3}{4}\frac{3}{\mathrm{iv}}\frac{3}{4}\right)\left(\text{b}\right)\end{array}$	B-32

Tabular array 3. Curie points in superstructures of hexagonal shut-packed lattice

I	II	III	Iv	V	Half dozen
${\text{D}}_{3h}^1$	H	1	(100, ↻)	AB	$1\text{A}\left(\frac{1}{3}\frac{\mathrm{ii}}{\mathrm{iii}}\frac{\mathrm{ane}}{\mathrm{four}}\right)\left(\text{a}\right),1\text{B}\left(\frac{2}{\mathrm{three}}\frac{1}{\mathrm{three}}\frac{\mathrm{three}}{4}\right)\left(\text{b}\right)$
${\text{D}}_{2h}^5$	P	2	( $110,1\overline{\mathrm{one}}0,001$ )	AB	$\begin{array}{l}\mathrm{two}\text{A}\left(\frac{1}{2}\frac{\mathrm{i}}{6}\frac{1}{4}\right)\left(\frac{\mathrm{ane}}{\mathrm{two}}\frac{5}{6}\frac{3}{4}\right)\left(\text{f}\right),\\ 2\text{B}\left(0\frac{2}{3}\frac{1}{4}\right)\left(0\frac{\mathrm{ane}}{3}\frac{3}{4}\right)\left(\text{b}\right)\end{array}$
${\text{D}}_{\mathrm{three}h}^1$	H	four	(200, 020, 001)	ABC3D3	$\begin{array}{l}\mathrm{one}\text{A}\left(\frac{1}{6}\frac{1}{3}\frac{1}{\mathrm{iv}}\right)\left(\text{a}\right),1\text{B}\left(\frac{5}{6}\frac{2}{\mathrm{three}}\frac{3}{\mathrm{iv}}\right)\left(\text{f}\right)\\ \mathrm{three}\text{C}\left(\frac{1}{3}\frac{1}{6}\frac{\mathrm{ane}}{\mathrm{iv}}\right)\left(\frac{5}{6}\frac{1}{6}\frac{3}{\mathrm{four}}\right)\left(\frac{\mathrm{one}}{3}\frac{2}{\mathrm{iii}}\frac{3}{4}\right)\left(\text{k}\right)\\ \mathrm{three}\text{D}\left(\frac{\mathrm{ii}}{3}\frac{\mathrm{ane}}{3}\frac{\mathrm{one}}{\mathrm{iv}}\right)\left(\frac{1}{6}\frac{\mathrm{v}}{\mathrm{six}}\frac{\mathrm{ane}}{\mathrm{iv}}\right)\left(\frac{\mathrm{two}}{\mathrm{three}}\frac{5}{6}\frac{1}{4}\right)\left(\text{j}\right)\end{array}$

Graph V gives the full general formula for the composition of ordered alloys. If the given order corresponds to an alloy of more two metals it might be possible to fill the non-equivalent sites with atoms of the same kind which leads to an ordered blend with a smaller number of components. The respective compositions are given only in the case of ternary alloys. In more complicated cases nosotros get a good many different possible compositions and we do not give them in the table. They are all easy to notice by filling the different sites in the cell with similar atoms in such a way, withal, that the symmetry of the lattice obtained would not be higher than in the general case when the not-equivalent places are filled with different atoms.

In graph VI the positions of the atoms on an ordered lattice are given (the coordinates existence given in the new, i.e. ordered jail cell). The letters (a), (b), … are the notations of these places in the International Tables [6].

Finally graph Vii gives types of an ordered lattice according to Strukturbericht (if the lattice is found to exist one of the known types).

In Figure ii the but three superstructures of a body-centred cubic lattice for which Curie points are possible, are shown equally an case. Transitions of whatever other ordered alloy in the disordered state are possible only with an absorption of a latent heat.

Some of the types of order enumerated in the tabular array are already known experimentally, others have non yet been observed. Then, for face-centred matted alloys the phase transitions of the 2nd club are possible for ordered types of the rhombohedral (50-11) and cubic (L-13) modifications of CuPt. For body-centred solid solutions the Curie points are found to be possible for ordered alloys of the type of the β-brass (50-20), Heusler alloys Cu₂AlMn (L-21) and of the type of Fe₃Al. On the other hand, the Curie points are found to be impossible for ordered alloys of the blazon CuAu (50-x) and Cu₃Au (L-12). In the known superstructures of the hexagonal close-packed lattice at that place is not a unmarried example where the being of the Curie indicate is possible.

The experimental data concerning the phase transitions of the second guild are scanty (cf., eastward.chiliad., the survey [8] †). Phase transitions of the second order are characterised by the absence of latent heat and the presence of a aperture in the rut capacity. Both are established for β-contumely in which the presence of the Curie point can actually be expected. A large discontinuity in the heat capacity is attributed to the alloy Cu₃Au [9]. Although latent oestrus has been observed here such a big aperture in the rut capacity could only show that nosotros are in the neighbourhood of Curie betoken (i.e. most the point in which the line of the ordinary phase transitions goes over into a line of Curie points). Inasmuch every bit the Curie betoken is found to be thermodynamically impossible in alloys of the blazon of Cu₃Au the data concerning the large discontinuity in the estrus capacity in this case seem doubtful. This also refers to the indications of the lack of latent rut at the point of transition of the alloy Cu₃Pd [10].

The interrelation of the present piece of work with the theories advanced past Bragg and Williams [11], Bethe [12] and Peierls [thirteen] must exist mentioned. Bragg and Williams come to the conclusion that in alloys of composition AB the transition from an ordered to a disordered country takes place at the Curie point, and in alloys AB₃ at the point of phase transition of the beginning social club. Meanwhile the results of the thermodynamic investigation in the nowadays work evidence that the possible presence of the Curie point is non determined by the ratio between the 'correct' and 'incorrect' sites for the atoms in the lattice, and is fundamentally continued with the properties of the symmetry of the crystal. In particular, the Curie point is possible in by no ways all the alloys of composition AB, e.k. it is non possible in such a unproblematic case as the alloy of the type of CuAu. Bethe investigated the special case of the ordered blend of the β-contumely type for which the Curie indicate is possible thermodynamically. Peierls investigated the ordered alloy of the Cu₃Au type and found that the Curie point was non possible which also does not contradict the thermodynamic theory. We may, nevertheless, think that these coincidences are purely accidental and would not occur in more complicated cases.

In this way those far reaching simplifying suppositions and neglections which are made in these theories of ordering can even atomic number 82 to a qualitative inapplicability of the theory which can become contradicting with thermodynamics.

In conclusion I wish to express my sincere thanks to Prof. 50. Landau for valuable discussions and his constant interest in my work.

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Computational Thermodynamics: Application to Nuclear Materials☆

Christine Guéneau , ... Bo Sundman , in Comprehensive Nuclear Materials (2nd Edition), 2020

1.26.two.6.2.3 Interstitial solutions

In steels the carbon and nitrogen atoms are much smaller than the metallic atoms and in the bcc lattice, they occupy the octahedral interstitial sites and there are 3 times every bit many sites in that sublattice. The default elective in an interstitial sublattice is the vacancy denoted Va.

For a binary A-B arrangement with interstitial B using a two sublattice CEF model for bcc with (A)_ane(B,Va)₃, the terms in (5) will be:

(39) ${}^{\mathrm{srf}}K_G=y_{2,\mathrm{Va}}{}^{°}\mathrm{Chiliad}_{\mathrm{A}:\mathrm{Va}}+y_{\mathrm{ii},\mathrm{B}}{}^{°}G_{\mathrm{A}:\mathrm{B}}$

(40) ${}^{\mathrm{cfg}}S_{\mathrm{Grand}}=-3R\left(y_{2,\mathrm{Va}}\ln \left(y_{\mathrm{ii},\mathrm{Va}}\right)+y_{2,\mathrm{B}}\ln \left(y_{\mathrm{ii},\mathrm{B}}\right)\right)$

(41) ${}^{\mathrm{Eastward}}\mathrm{1000}_K=y_{2,\mathrm{B}}{y}_{2,\mathrm{Va}}{L}_{B,Va}$

(42) ${}^{\mathrm{phy}}G_M={}^{\mathrm{mgn}}\mathrm{One\; thousand}_M$

where $y_{2,\mathrm{B}}$ is the constituent fraction of B in the second (interstitial) sublattice. At that place is a factor three in the configurational entropy because in that location are 3 times equally many interstitial sites equally sites for A. In bcc A there is an important ferro-magnetic transition.

Substitutional elements like Cr, Ni, Ti dissolve on the same sublattice as A whereas small atoms such equally C and N occupy the interstitial sublattice as shown in Fig. 5. The mole fractions of the elements are calculated from the general Eq. (15):

Fig. 5. Sublattice models for fcc (left) and bcc (correct) solutions with vacancies on the second interstitial sublattice.

(43) $x_{\mathrm{A}}=\frac{1}{1+\mathrm{iii}{y}_{2,\mathrm{B}}}$

(44) ${\mathrm{ten}}_{\mathrm{B}}=\frac{3y_{2,\mathrm{B}}}{1+3y_{2,\mathrm{B}}}$

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A Tribute to F.R.N. Nabarro

V. Vitek , Five. Paidar , in Dislocations in Solids, 2008

ten.three Molecular crystals

In all the previous discussions of metallic and ceramic materials the edifice units of the crystal structures considered are individual atoms. However, in that location is a large family of crystalline materials, chosen molecular crystals, in which the building units are molecules that tin can be rather circuitous. Even so, such crystals may exhibit mechanical beliefs alike to that of metals or ceramics and dislocations over again mediate plastic deformation. The cores of these dislocations may exist both planar and not-planar and affect significantly the plastic properties.

One of the first studies revealing a possible dislocation core effects was investigation of plastic deformation of hexamine by Dipersio and Escaig [133]. This fabric crystallizes in the BCC lattice and the building units are molecules with the molecular formula (CH₂)₆Northward₄ [533,534] with Van der Waals intermolecular forces. The observed skid planes are {110} or {112} [535,536] and the slip direction is 〈111〉, just as in BCC metals. Dipersio and Escaig [133] observed expanding dislocation loops and the striking characteristic was that when reaching the 〈111〉 spiral orientation the dislocation stopped moving and became apparently sessile. Hence, it is likely that in the screw orientation the core of the screw dislocation becomes not-planar, following the aforementioned spreading every bit in BCC metals (see Section 3.ii). Withal, since no atomistic written report of the dislocation cores in hexamine has notwithstanding been made this core construction remains a speculation. However, such calculations are feasible equally the description of interatomic and intermolecular forces is well developed ¹⁶ and was employed in a successful molecular dynamics study of incommensurate structures of hexamine [537].

A complete molecular statics written report of dislocation core construction and dislocation glide was performed by Ide et al. [538,539] for crystalline anthracene, post-obit a limited study of dislocations in naphthalene [540]. Anthracene crystallizes in base centered monoclinic construction (space group $P{\mathrm{two}}_1/a$ ) and its molecule consists of 3 interconnected benzene rings with the molecular formula C₁₄H₁₀ [541]. In anthracene crystals the observed slip systems are (001)[010] and (001)[110] [542,543]. In atomistic calculations of Ide et al. [538,539] no intramolecular displacements were allowed. The individual molecules were taken every bit rigid only molecular rotations and translations were immune. The interactions of atoms belonging to different molecules were described by the Buckingham potential developed in [544]. The Peierls stresses of the dislocations studied were determined by directly awarding of external shear stresses in the same way as described in Section iii.3, past gradually increasing the practical stress until the dislocation started to motion.

The edge dislocation studied was the [010] dislocation on the (001) slip aeroplane. Its core was plant to be planar and spread into the slip plane [538]. In contrast, the core of the [010] screw dislocation was found to be spread spatially [539]. However, the most interesting result is that the Peierls stress of the edge dislocation is nigh ${10}^{\mathrm{-three}}\mu$ , where μ is the shear modulus in the (001) plane in the [010] direction, while the Peierls stress of the screw dislocation is near $2\times {10}^{\mathrm{-ane}}\mu$ [539]. This large Peierls of screw dislocations is obviously the consequence of their non-planar cores and thus the [010] screw dislocations command the plastic deformations of anthracene crystals at low temperature, similarly equally exercise the $\text{one/two}〈111〉$ screw dislocations in BCC metals.

Recently an atomistic study of dislocations in cyclotrimethylene trinitramine (RDX) single crystals has been made in connection with large-scale molecular dynamics simulations of daze loading [545]. This is an energetic material with xx ane atoms per molecule (C₃H₆N₆O₆) that crystallizes in orthorhombic structure (space grouping Pbca) with unit of measurement cell containing viii molecules [546]. Atomic interactions in this material were described by potentials for nitramines developed by Smith and co-workers [547,548]. Similarly as in the study of hexamine [537], intramolecular interactions of atoms are characterized by elastic springs and interactions of atoms belonging to dissimilar molecules are of the Van der Waals type. Long-range electrostatic interactions were included explicitly. Perfect dislocations possess 〈100〉 type Burgers vectors and glide on (001), (021) and $(02\overline{1})$ planes. In order to assess the possibility of stacking faults and dislocation dissociation the [010] cross-section of the (001) γ-surface was calculated. No local minima were found at zilch external pressure but when the pressure of one   GPa was applied perpendicular to the plane of the fault, a minimum corresponding to the displacement almost 0.16[010] occurred. Thus a metastable stacking fault may exist on (001) planes but only at a loftier pressure.

This finding explains an unusual change in the machinery of dislocation mediated plasticity in daze loaded (111)-oriented single crystals that has been plant in both experiments and molecular dynamics simulations [545]. At shock pressures below 1.78   GPa plastic deformation is attributed to the heterogeneous nucleation and glide of perfect dislocations with the 〈100〉 type Burgers vector. However, to a higher place this pressure, when the resolved stress normal to (001) is larger than ane GPa, partial dislocation loops are nucleated homogeneously on (001) planes and their Burgers vector is 0.16[010]. The stacking faults generated by the partials form obstacles for the glide of perfect dislocations on other skid planes. Moreover, metastable stacking faults may exist stabilized by the large pressure on other glide planes and the cores of total dislocations change, possibly becoming spread to more than one airplane. A more detailed study of such phenomena would be very interesting but information technology is at present excessively demanding on calculating power.

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Tungsten as a Plasma-Facing Material☆

Gerald. Pintsuk , Akira. Hasegawa , in Comprehensive Nuclear Materials (2d Edition), 2020

six.02.4.3.2 Mechanical properties

Data in the literature on mechanical properties of neutron-irradiated tungsten are very limited. ^265,269,270 However, in combination with experimental results obtained for other refractory metals, it has been shown that in metals with a bcc lattice construction, irradiation hardening causes a steep increase in yield stress and a subtract in ductility. ¹²⁶ Consequently, the fabric fails past brittle cleavage fracture as soon as the yield stress exceeds the cleavage forcefulness. Therefore, the increase of the DBTT depends on the neutron fluence, the neutron spectrum (volition be addressed by the International Fusion Materials Irradiation Facility, IFMIF), and the irradiation temperature. Irradiation hardening behavior of neutron irradiated Tungsten was already mentioned before. The tensile behavior and fracture style are reported using textile testing reactor HFIR, in which nuclear transmutation of thermal neutron was extremely high above 1 dpa. ²⁷¹ Fig. 17 is a map against the irradiation/examination temperature and irradiation dose of single crystal tungsten. The figure depicts transition of the fracture mode from fully ductile or uniform plasticity for the unirradiated material to the ductile beliefs involving a steep yield drop, then to embrittlement. Although the high temperature exam data are bachelor but up to 0.5dpa (6.5% Re+Os) at which point the embrittlement is accompanied by very big hardening, a weakening /embrittlement is anticipated to follow at college doses exceeding 1dpa(11.5% Re+Os), where irradiated specimens did non survive handling in the hot cell.

Fig. 17. Tensile beliefs of neutron-irradiated unmarried-crystal tungsten mapped against fluence (or concentration of transmuted Re+Bone) and irradiation/test temperature.

Reproduced from Katoh, Y., et al., 2019. J. Nucl. Mater. 520, 193–207.

The radiation hardening in bcc alloys at low temperatures (<0.3 T _chiliad) occurs even for doses as low every bit ~0.xv–0.6 dpa (irradiation of plasma facing materials for ITER and DEMO, PARIDE campaigns ²⁴⁹ ), which corresponds to the expected ITER conditions. Therefore, operation of tungsten at temperatures >chiliad°C would be preferred as total or at least partial recovery of defect-induced material degradation is achieved by annealing at 1200°C. ²⁶⁵ This implies that the well-nigh-surface part of a Due west component will retain its ductility, which has a beneficial effect on the crack resistance at the plasma loaded surface. However, such temperatures are not feasible at the interface to the heat sink where tungsten will be in contact with copper (ITER) or steel (DEMO), which are limited to significantly lower operational temperatures. Hence, amend understanding of the irradiation effects on tungsten at temperatures between 700 and 1000°C is needed, peculiarly related to reactor application in DEMO. ^125,126,272 In add-on to the influencing factors on the DBTT mentioned above, that is, neutron fluence, neutron spectrum, and irradiation temperature, the material'due south limerick also plays an of import role. While the addition of Re has a beneficial effect on the material's ductility in the nonirradiated state, nether neutron irradiation it results in more rapid and astringent embrittlement than it is observed for pure Westward. ²⁷⁰ Similarly, less mechanical strength and an increased loss of ductility compared to pure W is institute for particle- strengthened W alloys (east.g., W–ane% La₂O₃) when tested upwards to 700°C. The but exception among all explored tungsten alloys might be mechanically assimilated W–TiC that showed no irradiation hardening as measured past Vickers hardness at 600°C. ⁹⁴ Finally, the mechanical properties are influenced by neutron-induced He-generation and the transmutation of tungsten. While He generation in Due west is, compared to Cfc and Be, very small (~0.vii appm He per dpa) and its influence on the mechanical properties of Westward negligible, ^80,90,256 the transmutation of Due west into Re and subsequently Os significantly alters the material structure and its properties. The generation of pregnant amounts of ternary a and subsequently σ-phases results in extreme cloth embrittlement and will cause shrinkage. In combination with thermally induced strains, this might produce high tensile stresses causing the extremely brittle fabric to extensively cleft and perhaps even crumble to pulverisation. ^twoscore

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Processing ultrafine-grained and nanostructured materials

Farid Z. Utyashev , ... Alexander P. Zhilyaev , in Superplasticity and Grain Boundaries in Ultrafine-Grained Materials (Second Edition), 2021

3.iii.ii.3 Forging of titanium alloys

The basis of the structure of titanium alloys is solid solutions of α and/or β titanium. The beginning one is a low-temperature modification of alloyed titanium with hexagonal crystal lattice, and the second one is a high-temperature phase with body-centered cubic lattice. Alloying elements in titanium alloys are classified in relation to how the temperature of polymorphic transformation (T _p.t.) of an alloy changes with their introduction into titanium. If the introduced components are more than soluble in the α-phase and increase the temperature of the polymorphic transformation, and then they are chosen α-stabilizers. Those are Al and O ₂ . The elements that reduce Т _p.t. are called β-stabilizers, they include 5, Nb, Ta, Mo, and H. Neutral hardeners are also used, which have little upshot on the modify in Т _p.t., for example, Zr, Hf, Sn.

College oestrus-resistant backdrop are shown by alloys with high content of α titanium solid solution doped with elements that increase temperature Т_p.t. and recrystallization temperature, which, in addition to Al, besides include Sn, Si, C. Therefore pseudo-α alloys and two-phase martensitic (α   +   β)-alloys with low content of β-stabilizers are mainly used to produce GTE components. The chemic limerick of a number of heat-resistant titanium alloys is given in Table 3.4.

Table 3.iv. Chemical limerick of heat-resistant Ti alloys.

Class	Main alloying elements (wt%)
	Al	Sn	Zr	Mo	Si	Others
VT9	5.8–7.0	–	0.8–2.5	2.8–3.viii	0.2–0.35
VT25	6.2–7.2	0.viii–2.v	3.v–four.5	one.5–2.v	0.ane–0.25	Due west (0.five–1.5)
VT25U	6.0–7.0	1.0–2.5	3.0–four.5	three.v–four.5	0.1–0.25	W(0.four–1.5)
VT18U	6.2–7.3	two.0–3.0	3.five–four.5	0.4–1.0	–	Nb (0.v–ane.5)
Ti 6242	5.5–6.five	1.8–2.2	3.half-dozen–four.four	one.2–2.2
IMI550	3.5–4.5	1.5–2.5		three.5–4.four	0.iv–0.6	Atomic number 26—0.12
Ti64	5.five–6.75					Fe(0.25–0.4) Five(3.5–four.5)

Estrus-resistant titanium alloys also contain a small-scale corporeality of chemic compounds (ТiAl, Ti ₅ Si, TiC, TiB), strengthening α- and β-phases.

The solid-solution and intermetallic hardening of titanium alloys are weaker as compared to oestrus-resistant nickel alloys. This along with a relatively depression homological temperature of polymorphic transformation of Ti alloys explains their lower strength and rut resistance. Therefore parts made of titanium alloys, different nickel-based alloys, are operated at lower temperatures. However, high specific force of titanium alloys that makes them capable of withstanding loftier loads at moderate temperatures (~   400°С), makes them unchallenged in comparison with other materials in the manufacturing of shipping engine rotor parts.

When comparing the technological backdrop of heat-resistant nickel and titanium alloys, some preference should be given to the latter. Titanium alloys have a wider temperature range for hot deformation in the single-phase expanse ~   300–350°C. All the same, information technology is rather difficult to use loftier ductility and depression deformation resistance of these alloys in the single-phase β area due to their sensitivity to strain heterogeneity and high grain growth charge per unit, which leads to formation of an inhomogeneous coarse construction and, equally a consequence, low mechanical properties. The formation of an inhomogeneous coarse-grained structure in the β-region is facilitated by low thermal conductivity. The low thermal conductivity leads to nonuniformity of the temperature field in the deformation zone. High chemic activity of these alloys that results in saturation with atmospheric gases and a tendency to deformation localization too contribute to formation of an inhomogeneous coarse-grained structure. Therefore the processing of these alloys in the unmarried-stage surface area is rarely used as a final shape-forming operation.

Despite the relatively high ductile properties and depression flow stress in the two-phase (α   +   β) expanse, titanium alloys, similar nickel alloys, vest to the class of hard-to-work materials, which is primarily due to the nonuniform distribution of strain due to the peculiarities of their fibroid-grained structure.

The structure of 2-phase titanium alloys externally resembles the structure of plate perlite in steels. In coarse-grained titanium alloys, lamellar morphology is inherent in the α-phase, which precipitates during polymorphic transformation from the β-phase. Office of the β-phase remains in the grade of thin layers between α-stage plates, Fig. 3.48.

Fig. 3.48. Microstructure of coarse-grained VT8 alloy.

The α-phase plates in the localized zones of intensive shear larn a globular shape, which leads to further deformation localization in these zones. To create the conditions for transformation of a lamellar structure into a globular one in the unabridged volume of deformable billets, these alloys are processed with a regulated amount of strain and with a frequent change in the deformation direction. Such processing leads to multiple and long-term cycles of manufacturing of semifinished items and products thereof.

During common cold plastic deformation of two-stage titanium alloys, bundles of α-phase plates are separated in the transverse directions by shear lines. Parallel plates in bundles are deformed as a whole unit, the semicoherent interface α/β and the bcc β-phase do non interfere with the transfer of deformation between them. A block structure is formed in the volume of grains. However, due to the significant nonuniformity of cold deformation of titanium alloys, their depression ductility and loftier flow stresses, this method of structure preparation is not used in practice. Large cold deformation of titanium alloys is impossible without the awarding of high hydrostatic pressure; therefore, it is accumulated only by torsion of thin disks in Bridgman anvils. However, as already noted, this method is not suitable for producing bulk semifinished items for structural purposes.

The construction refinement in big-sized workpieces of titanium alloys is achieved as a event of hot deformation in the α   +   β region. The main stages of the grain refinement procedure in this instance are schematically presented in Fig. 3.49.

Fig. 3.49. Scheme of transformation of a plate structure into a globular i [37].

At the kickoff stage of deformation, the external load P leads to the division of plates in the transverse direction by coarse shear bands, thereby forming steps and grooves on the interfacial longitudinal surface of plates. Equally a issue of these violations in the interfacial boundaries, α/β plates are distorted, lose their coherency, and their surface energy grows. Thus a thermodynamic stimulus to spheroidization of plates takes place. A block structure with low-angle misorientations of transverse boundaries is formed in the shear bands in the α-phase.

In titanium alloys, interlayers of the β-phase are softer than plates of the α-phase, and therefore, the plastic flow is more localized in this phase, which is also facilitated past its bcc lattice. As a outcome of the action of harder α-phase plates on the soft β-stage, the stress-strain land in the latter becomes similar to the pinch scheme. Under the action of compressive loads, oppositely directed flows arise: vacancies and Al atoms, which are the most mobile alloying chemical element in titanium alloys. Vacancies motility to the longitudinal boundaries of plates, and Al atoms motion to the transverse boundaries. The outflow of atoms is most intense from spots where the local stresses are college, in particular, from the joints of transverse boundaries of the α-phase with the β-phase subgrains. This occurs due to depletion of the α-phase by the stabilizing element Al and polymorphic transformation of office of the α-phase into the β-stage. Every bit the deformation develops, the volumes of material undergoing this transformation increase and expand the depth and thickness of the grooves in the transverse management in the α-plates, upwardly to their complete separation into separate parts. In improver to the polymorphic transformation, the process of plate partitioning is facilitated by interaction of lattice dislocations with depression-bending boundaries in the α-phase and interphase α/β. The enhancement of misorientations and energy of an interphase boundary with loftier angles stimulates the evolution of spheroidization of plates and the formation of a globular structure.

Dynamic recrystallization in the β-stage makes a significant contribution to misorientation enhancement in the longitudinal interphase boundaries. At a later stage of deformation, active participation in the completion of globularization is conditioned past the activity of the mechanism of grain-boundary sliding along grain boundaries, which have gained high misorientation angles.

Thus the transformation of a lamellar structure into a globular one is a complex process caused past the motion of defects (dislocations, vacancies, boundaries), redistribution of alloying elements, polymorphic phase transformation, dynamic recrystallization, and GBS. The intensity and abyss of this process should depend on the way and route of deformation of titanium alloys.

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