**UFRJ**

# Apostila cristalografia (ing)

(Parte **1** de 8)

1 Pattern | 1 |

1.1 Pattern: An Introduction | 1 |

1.2 Summary | 7 |

References | 7 |

2 Lattices | 9 |

2.1 Plane Lattice | 9 |

2.2 Space Lattice | 12 |

2.3 Lattice Planes and Miller Indices | 12 |

2.4 Lattice Directions | 13 |

2.5 Summary | 14 |

References | 14 |

3 Symmetry in Lattices | 15 |

3.1 Symmetry Operations in Plane Lattices | 15 |

3.1.1 Rotational Symmetry | 15 |

3.1.2 Mirror Plane of Symmetry | 16 |

3.1.3 Centre of Symmetry | 19 |

3.2 Symmetry Operation in Space Lattices | 19 |

3.2.1 Rotation Inversion (Rotary Inversion) Symmetry | 19 |

3.3 Summary | 21 |

References | 21 |

4 Crystal Symmetry (Crystal Pattern): I | 23 |

4.1 Macroscopic Symmetry Elements | 23 |

in Hermann–Mauguin Notations | 24 |

4.3 Crystal Systems | 27 |

4.4 Bravais Lattices | 28 |

4.5 Summary | 3 |

Contents 4.2 Thirty-Two Point Groups of Symmetries References .................................................. 3

5 Crystal Symmetry (Crystal Pattern): I | 35 |

5.1 Microscopic Symmetry Elements in Crystals | 35 |

5.2 Space Groups | 40 |

5.3 Constitution of Space Groups | 42 |

5.4 Summary | 42 |

References | 42 |

6 Crystals and X-Ray | 43 |

6.1 Production and Properties of X-Ray | 43 |

6.2 Laue Equations | 45 |

6.3 Bragg’s Law | 47 |

6.4 Reciprocal Lattice | 47 |

6.5 Geometry of X-Ray Diﬀraction: Use of Reciprocal Lattice | 50 |

Systems | 53 |

6.7 Weighted Reciprocal Lattice | 5 |

6.8 Summary | 56 |

References | 56 |

X Contents 6.6 The Interplanar Distance (d-Spacing) of Diﬀerent Crystal

X-Ray Diﬀraction Techniques | 57 |

7.1 Experimental Techniques for Single Crystal | 57 |

7.1.1 Laue Camera and Laue Pattern | 57 |

7.1.2 Rotation/Oscillation Camera and the Applications | 60 |

7.1.3 Weissenberg Camera and Moving Film Technique | 64 |

7.1.4 de Jong–Boumann and Precession Camera | 68 |

7.2 Experimental Techniques for Polycrystals | 70 |

The Plan View | 72 |

7.4 Indexing of the Debye–Scherrer Pattern | 73 |

7.4.1 Cubic Systems | 73 |

7.4.2 Tetragonal System | 73 |

7.4.3 Hexagonal System | 74 |

7.4.4 Orthorhombic System | 74 |

7.4.5 Monoclinic and Triclinic Systems | 75 |

7.5 Summary | 75 |

References | 75 |

8 Determination of Space Group and Crystal Structure | 7 |

from Moving Film | 7 |

8.1.1 Weissenberg Photographs | 7 |

8.2 Determination of Crystal Structure | 79 |

7 Experimental Methods for Structure Analysis: 7.3 Debye–Scherrer Cylindrical Powder Camera: 8.1 Determination of Space Group from Data Obtained 8.2.1 Trial-and-Error Method . ........................... 79

Contents XI

and Patterson Function | 80 |

8.3 Summary | 83 |

References | 84 |

9 The World of Symmetry | 85 |

9.1 Symmetry in Living Bodies | 86 |

9.2 Symmetry in Patterns, Snow Flakes, and Gems | 89 |

9.3 Symmetry in Architecture | 91 |

9.4 Symmetry in Fundamental Particles | 91 |

9.5 Symmetry (Invariance) of Physical Laws | 95 |

9.6 Summary | 98 |

References | 9 |

10 Asymmetry in Otherwise Symmetrical Matter | 101 |

10.1 Single Crystals, Poly Crystals, and Asymmetry–Symmetry | 101 |

of Matter | 106 |

10.3 A Symmetry in Asymmetry I: Liquid Crystalline Phase | 1 |

10.3.1 Optical Study of Liquid Crystals | 114 |

(Electro-Optical Eﬀect) | 116 |

10.4 Symmetry Down to the Bottom: The Nanostructures | 116 |

10.5 Summary | 126 |

References | 126 |

Epilogue | 127 |

8.2.2 The Electron Density Equation 10.2 A Symmetry in Asymmetry I: Quasi Crystalline State 10.3.2 Eﬀect of Electric Field on Nematic Liquid Crystal

and Aperiodic Structures | 129 |

References | 135 |

B Solved Problems | 137 |

Further Reading | 145 |

A An Outline of the Diﬀraction Theory of Periodic Index .......................................................... 147

1.1 Pattern: An Introduction

Let us begin with the question: What is a pattern? The answer to this question is as much objective as it may be subjective. From the days unknown, the human race have started studying and appreciating the regular periodic features like movement of stars, moon, sun, the beautiful arrangement of petals in ﬂowers, the shining faces of gems, and also the beautiful wings of a butterﬂy. They have constructed many architectural marvels like tombs, churches, pyramids, and forts having symmetries, which still attract tourists. The regularities observed in nature either in the worlds of plants, animals, or inanimate objects are patterns and people get startled by observing them and thrilled by inspecting them. That may be the beginning of the study of pattern. Every music or song has two aspects: the tal or the tune and the verse of the song. The composer composes the tune on the verse of the song made by the lyric. This composition must satisfy certain harmonic conditions and the listener of the song has the right to appreciate a song or reject it. This judgment is more subjective than objective as to some listener some songs are very pleasing and appreciating where as to others its appeal may not be that much deep rooted. The appeal of a song sometimes also changes with time; new tunes come in the front and the old one are either rejected or forgotten. A pattern has much similarity with the sense of order, harmony that a song brings forward to us. It is also composed of two aspects: The ﬁrst one like the verse of the song is the motif or the object which is the constituent of the pattern which can be compared with the song. The other is the order of the arrangement of the motif, which has similarity with the tune or the music. The most common example of a geometric pattern is a printed cloth or a wall paper. In a printed cloth the motif can be any object, it can be any geometrical ﬁgure, a leaf or a bud or anything. But the conventional sense of pattern can only be created if these motifs are either all similar, regular, or may be even a combination of more than one type. Now about the arrangement of these motifs in two-dimensional

♣♣♣♣♣ Fig. 1.1. Same motif, a perfect pattern

♥ | •∗♠♦ |

♦ | ♠♠∗♥ |

♠∗ ♠∗ ♥ Fig. 1.2. Regular arrangement of random motifs, not a pattern

Fig. 1.3. The motifs are diﬀerent but they bear a constant regularity in their arrangement, and so it constitute a pattern printed cloths must also satisfy an order of their arrangement both in their translation or position and after orientation [1].

In Figs.1.1 and 1.2, two arrangements of motifs are created such that in one all the motifs are similar and in the other, they are diﬀerent. Now the question is, which one of these two appears more soothing to our eye? There lies perhaps the subjectivity of the problem. But if we do not only restrict ourselves to the more common geometric senses, then also the ﬁrst one (Fig.1.1) appears deﬁnitely more soothing to the eye and so can be accepted as a pattern and whereas the latter is not at least as in Fig.1.1. In Fig.1.3 motifs are not all same row-wise but they are same column-wise.

1.1 Pattern: An Introduction 3

Note: The scheme of repetition or the mode of arrangement of motifs is the same in all ﬁgures. The only diﬀerence is that in Fig.1.1 the motifs are similar and in Figs.1.2 and 1.3 they are diﬀerent. The pattern in Fig.1.1 is more regular than that in Fig.1.2 and also in Fig.1.3, but Fig.1.3 is more “patternlike” than Fig.1.2.

Conclusion: To constitute a pattern, the motifs are either to be same or should be regularly arranged in the same scheme of repetition even if they are different. The scheme of repetition comprises of position and orientation of the motifs.

Now, if the motifs are identical but the mode of their arrangement, that is, the scheme of their repetition is changed then the pattern will also be changed. This is shown in Figs.1.4 and 1.5.

Note: In both Figs.1.4 and 1.5, the patterns are created but they look diﬀerent because their schemes of repetition are diﬀerent. Again in Fig.1.6 though there is regularity in arrangement, it looks less pattern-like than Fig.1.7 as the color of the motifs is random in Fig.1.6, whereas the motifs are systematically colored in Fig.1.7 and so it appears more pattern-like. In Fig.1.8 each motif are similar but randomly oriented but positioned in regular order, and in Fig.1.9 both are maintained and so it is a regular geometric pattern. Figure 1.10 shows another pattern though motifs are oriented but in symmetrical right-hand screw order. This can be understood in more elaborate

♣ ♣ ♣ ♣ ♣ Fig. 1.4. A pattern with one scheme of repetition

♣♣ | ♣♣♣♣ |

Fig. 1.5. A pattern with diﬀerent scheme of repetition

Fig. 1.7. Regular arrangement of symmetrically coloured motifs

◄▼ ◄ ► ► Fig. 1.8. Same motifs, symmetrically placed but randomly oriented

◄◄ ◄ ◄ ◄ Fig. 1.9. Same motifs symmetrically placed and oriented

◄► ► ◄ Fig. 1.10. Motifs symmetrically placed and also oriented

Fig. 1.1. Motifs are triangles and each is placed in perfect symmetrical position to constitute a pattern

Fig. 1.12. Motifs are triangles and are same as Fig.1.1 and they are placed in perfect symmetrical positions but are randomly oriented about their positions and so it loses the characteristics of the pattern way from Figs.1.1 and 1.12, where in the motifs are triangles and they are diﬀerently placed so far as their orientation is concerned but in symmetrical positions.

Conclusion: The change in scheme of repetition either in position or orientation changes the patterns and even loses the sense of pattern if there is no regularity in their orientation, though the motifs remain same and they are placed at equal intervals.

Therefore, a pattern must possess two characteristics, the regular motifs stationed at sites that obey a certain scheme of repetition [1,2]. Any change of motif or the change of the repetition scheme both in their positions and orientation changes the pattern. So to study a pattern these two aspects are to be looked into and studied. It is more convenient to start with the study of diﬀerent possible schemes of repetition that exist in their position rather than with the orientation symmetry, if there is any, in the motifs. The former class of study may be categorized as Macroscopic whereas the latter may be called as Microscopic. To begin with the investigation on scheme of repetition, it should be appreciated that the regularity of the scheme of repetition including both macro and micro is the essence of a perfect pattern [3]. This regularity generates a sense of symmetry, which is later described as Macroscopic and Microscopic. It is better to start with the symmetry present in the scheme of repetition, that is, with the symmetry operations.

Note: Two bodies or conﬁguration of bodies (i.e., Pattern) may be called symmetrical if and only if they are indistinguishable in all respect. The symmetry operations are those operations which when performed on a pattern, the pattern returns back to its state of self-coincidence or invariance.

Thus the total identiﬁcation of the symmetry operations that can be performed to bring a pattern in to its self coincidence, gives the knowledge of the symmetry and the scheme of repetition present in the pattern.

This should be appreciated that the entire knowledge of symmetry or scheme of repetition present in the pattern though include the sense of motifs present, it would be much beneﬁcial at least to start with if we consider the symmetry present within the sites of the motifs only (Macroscopic), and when these are known, then the symmetry operations including the motifs (Microscopic) can be considered. Therefore, let us start with the symmetry present within the sites where the motifs are to be placed to generate the patterns.

These sites of the motifs, which are represented by simple geometrical points, have a special signiﬁcance and they are known as “Lattice.”

Conclusion: The Lattice is the sites of motifs where they can be placed to generate the pattern. It can be two-dimensional regular arrays of points for two-dimensional patterns or it can be three-dimensional arrays of points for three-dimensional patterns. Therefore, the lattice bears the knowledge of the scheme of repetition and when the motifs are placed in the lattice sites the entire pattern takes the shape and changes whenever the orientation of the motifs takes their role to play. If the order of this orientation of the motifs is maintained in some way or other, it retains the pattern characteristics of being geometrically symmetric otherwise not.

References 7 1.2 Summary

1. A pattern has two constituents: one is the motif and the other is the scheme of repetition of the motif. 2. The scheme of repetition of motifs again has two aspects: Positions in the lattice sites (macroscopic) and the orientation (microscopic) in their respective lattice sites. 3. A pattern has two aspects: one is the structure of the motif and its schemes of repetition and the other is its pleasant visual eﬀect. This is the after eﬀect when these two aspects are followed. Therefore, both these two aspects, that is, the symmetry and the visual eﬀect are important and are usually supplementary to each other. 4. When the ﬁrst aspect is not at all or is only partially followed, it is not necessary that the assembly of the motifs will not satisfy the second aspect of it, that is, the visual pleasure and will fail to constitute a pattern. It is then an incidence of an exception in the geometric rule of pattern, that is, asymmetry in symmetry, which is abundantly present in nature.

References

1. N.F. Kenon, Patterns in Crystals (Wiley, New York, 1978) 2. L.S. Dent Glasser, Crystallography and its Applications (Van Nostrand Reinhold

Co. Ltd, New York, 1977) 3. L.V. Azaroﬀ, Introduction to Solids (McGraw-Hill, New York, 1960)

2 Lattices

2.1 Plane Lattice

The two-dimensional inﬁnite array of geometrical points symmetrically arranged in a plane where the diﬀerent motifs may be placed to create the patterns is known as plane lattice. Figures 2.1 and 2.2 are the examples of plane lattices, where the neighborhood of every point is identical. Figures 2.1 and 2.2 are two types of plane lattices, where motifs are to be placed to generate the desired patterns [1].

Note: The plane lattice of a two-dimensional pattern is that array of geometric points which speciﬁes the scheme of repetition that is present in the pattern. There is no diﬀerence between the neighbors of one lattice site from any of its neighborhood. Actually the situation becomes so much identical that if the attention from one lattice point is removed, it then becomes impossible to identify and locate the same lattice point.

Note: The unit translation is the distance shown by arrow between adjacent points along any line in a lattice. This distance taken between any adjacent point is a,a nd d is the distance of separation between any line drawn through the points in the lattice (Figs.2.3 and 2.4).

Thep roducto f a and d, that is, a×d is the total area associated with each lattice point and the inverse, that is, 1/(a×d) is the number of points in each unit area of the lattice. The unit cell of the plane lattice is a parallelogram of two unit translations with lattice points at the corners and is the representative of the lattice, that is, the lattice is constituted by repeating this unit cell in two directions of the plane lattice.

This unit cell is completely speciﬁed by the lengths of the edges known as unit translations and the angle between them. So, the unit cell is the building block of the lattice (Fig. 2.5) [1–3].

Now as this unit cell is the building block of the pattern, it may be thought that there can be inﬁnite number of patterns even for same motifs for diﬀerent

••••• Fig. 2.1. Plane Lattice (Type I)

Fig. 2.2. Plane Lattice (Type I)

Fig. 2.3. Distances between lattice sites and lattice rows (Type I)

Fig. 2.4. Distances between lattice sites and rows (Type I)

2.1 Plane Lattice 1 A

(Parte **1** de 8)