1 / 34

PH0101 UNIT 4 LECTURE 3

PH0101 UNIT 4 LECTURE 3. CRYSTAL SYMMETRY CENTRE OF SYMMETRY PLANE OF SYMMETRY AXES OF SYMMETRY ABSENCE OF 5 FOLD SYMMETRY ROTOINVERSION AXES SCREW AXES GLIDE PLANE. CRYSTAL SYMMETRY. Crystals have inherent symmetry. The definite ordered arrangement of the faces

zach
Download Presentation

PH0101 UNIT 4 LECTURE 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PH0101 UNIT 4 LECTURE 3 • CRYSTAL SYMMETRY • CENTRE OF SYMMETRY • PLANE OF SYMMETRY • AXES OF SYMMETRY • ABSENCE OF 5 FOLD SYMMETRY • ROTOINVERSION AXES • SCREW AXES • GLIDE PLANE PH 0101 UNIT 4 LECTURE 3

  2. CRYSTAL SYMMETRY • Crystals have inherent symmetry. • The definite ordered arrangement of the faces • and edges of a crystal is known as `crystal • symmetry’. • It is a powerful tool for the study of the internal • structure of crystals. • Crystals possess different symmetries or • symmetry elements. PH 0101 UNIT 4 LECTURE 3

  3. CRYSTAL SYMMETRY What is a symmetry operation ? • A `symmetry operation’ is one, that leaves • the crystal and its environment invariant. • It is an operation performed on an object or pattern • which brings it to a position which is absolutely • indistinguishable from the old position. PH 0101 UNIT 4 LECTURE 3

  4. CRYSTAL SYMMETRY • The seven crystal systems are characterised by • three symmetry elements. They are • Centre of symmetry • Planes of symmetry • Axes of symmetry. PH 0101 UNIT 4 LECTURE 3

  5. CENTRE OF SYMMETRY • It is a point such that any line drawn through it will • meet the surface of the crystal at equal distances • on either side. • Since centre lies at equal distances from various • symmetrical positions it is also known as `centre • of inversions’. • It is equivalent to reflection through a point. PH 0101 UNIT 4 LECTURE 3

  6. CENTRE OF SYMMETRY • A Crystal may possess a number of planes or • axes of symmetry but it can have only one centre • of symmetry. • For an unit cell of cubic lattice, the point at the • body centre represents’ the `centre of • symmetry’ and it is shown in the figure. PH 0101 UNIT 4 LECTURE 3

  7. CENTRE OF SYMMETRY PH 0101 UNIT 4 LECTURE 3

  8. PLANE OF SYMMETRY • A crystal is said to have a plane of symmetry, when • it is divided by an imaginary plane into two halves, • such that one is the mirror image of the other. • In the case of a cube, there are three planes of • symmetry parallel to the faces of the cube and six • diagonal planes of symmetry PH 0101 UNIT 4 LECTURE 3

  9. PLANE OF SYMMETRY PH 0101 UNIT 4 LECTURE 3

  10. AXIS OF SYMMETRY • This is an axis passing through the crystal such that if • the crystal is rotated around it through some angle, • the crystal remains invariant. • The axis is called `n-fold, axis’ if the angle of rotation • is . • If equivalent configuration occurs after rotation of • 180º, 120º and 90º, the axes of rotation are known as • two-fold, three-fold and four-fold axes of symmetry • respectively. PH 0101 UNIT 4 LECTURE 3

  11. AXIS OF SYMMETRY • If equivalent configuration occurs after rotation of 180º, • 120º and 90º, the axes of rotation are known as two- • fold, three-fold and four-fold axes of symmetry . • If a cube is rotated through 90º, about an axis normal to • one of its faces at its mid point, it brings the cube into • self coincident position. • Hence during one complete rotation about this axis, i.e., • through 360º, at four positions the cube is coincident • with its original position.Such an axis is called four-fold • axes of symmetry or tetrad axis. PH 0101 UNIT 4 LECTURE 3

  12. AXIS OF SYMMETRY • If n=1, the crystal has to be rotated through an angle = • 360º, about an axis to achieve self coincidence. Such an • axis is called an `identity axis’. Each crystal possesses an • infinite number of such axes. • If n=2, the crystal has to be rotated through an angle = • 180º about an axis to achieve self coincidence. Such an • axis is called a `diad axis’.Since there are 12 such edges in • a cube, the number of diad axes is six. PH 0101 UNIT 4 LECTURE 3

  13. AXIS OF SYMMETRY • If n=3, the crystal has to be rotated through an • angle = 120º about an axis to achieve self • coincidence. Such an axis is called is `triad • axis’. In a cube, the axis passing through a • solid diagonal acts as a triad axis. Since there • are 4 solid diagonals in a cube, the number of • triad axis is four. • If n=4, for every 90º rotation, coincidence is • achieved and the axis is termed `tetrad axis’. • It is discussed already that a cube has `three’ • tetrad axes. PH 0101 UNIT 4 LECTURE 3

  14. AXIS OF SYMMETRY • If n=6, the corresponding angle of rotation is • 60º and the axis of rotation is called a hexad • axis. A cubic crystal does not possess any • hexad axis. • Crystalline solids do not show 5-fold axis of • symmetry or any other symmetry axis higher • than `six’, Identical repetition of an unit can take • place only when we consider 1,2,3,4 and 6 fold • axes. PH 0101 UNIT 4 LECTURE 3

  15. SYMMETRICAL AXES OF CUBE PH 0101 UNIT 4 LECTURE 3

  16. SYMMETRICAL ELEMENTS OF CUBE (a) Centre of symmetry 1 (b) Planes of symmetry 9 (Straight planes -3,Diagonal planes -6) (c) Diad axes 6 (d) Triad axes 4 (e) Tetrad axes 3 ---- Total number of symmetry elements = 23 ---- Thus the total number of symmetry elements of a cubic structure is 23. PH 0101 UNIT 4 LECTURE 3

  17. ABSENCE OF 5 FOLD SYMMETRY • We have seen earlier that the crystalline solids show only • 1,2,3,4 and 6-fold axes of symmetry and not 5-fold axis of • symmetry or symmetry axis higher than 6. • The reason is that, a crystal is a one in which the atoms or • molecules are internally arranged in a very regular and • periodic fashion in a three dimensional pattern, and • identical repetition of an unit cell can take place only • when we consider 1,2,3,4 and 6-fold axes. PH 0101 UNIT 4 LECTURE 3

  18. MATHEMATICAL VERIFICATION • Let us consider a lattice P Q R S as shown in figure   P Q R S a a a PH 0101 UNIT 4 LECTURE 3

  19. MATHEMATICAL VERIFICATION • Let this lattice has n-fold axis of symmetry and the • lattice parameter be equal to ‘a’. • Let us rotate the vectors Q P and R S through an • angle  = , in the clockwise and anti clockwise • directions respectively. • After rotation the ends of the vectors be at x and y. • Since the lattice PQRS has n-fold axis of symmetry, • the points x and y should be the lattice points. PH 0101 UNIT 4 LECTURE 3

  20. MATHEMATICAL VERIFICATION • Further the line xy should be parallel to the line PQRS. • Therefore the distance xy must equal to some integral • multiple of the lattice parameter ‘a’ say, m a. • i.e., xy = a + 2a cos  = ma (1) • Here, m = 0, 1, 2, 3, .................. • From equation (1), • 2a cos  = m a – a PH 0101 UNIT 4 LECTURE 3

  21. MATHEMATICAL VERIFICATION i.e., 2a cos  = a (m - 1) (or) cos  = (2) Here, N = 0, 1, 2, 3, ..... since (m-1) is also an integer, say N. We can determine the values of  which are allowed in a lattice by solving the equation (2) for all values of N. PH 0101 UNIT 4 LECTURE 3

  22. MATHEMATICAL VERIFICATION • For example, if N = 0, cos  = 0 i.e.,  = 90o •  n = 4. • In a similar way, we can get four more rotation axes • in a lattice, i.e., n = 1, n = 2, n = 3, and n = 6. • Since the allowed values of cos  have the limits –1 • to +1, the solutions of the equation (2) are not • possible for N > 2. • Therefore only 1, 2, 3, 4 and 6 fold symmetry axes • can exist in a lattice. PH 0101 UNIT 4 LECTURE 3

  23. ROTATION AXES ALLOWED IN A LATTICE n= PH 0101 UNIT 4 LECTURE 3

  24. ROTO INVERSION AXES • Rotation inversion axis is a symmetry element which • has a compound operation of a proper rotation and • an inversion. • A crystal structure is said to possess a rotation – • inversion axis if it is brought into self coincidence by • rotation followed by an inversion about a lattice point • through which the rotation axis passes. PH 0101 UNIT 4 LECTURE 3

  25. X 3 2 4 1 X1 ROTO INVERSION AXES PH 0101 UNIT 4 LECTURE 3

  26. ROTO INVERSION AXES • Let us consider an axis xx, normal to the circle passing • through the centre. • Let it operates on a point (1) to rotate it through 90o to the • position (4) followed by inversion to the position (2), this • compound operation is then repeated until the original • position is again reached. PH 0101 UNIT 4 LECTURE 3

  27. ROTO INVERSION AXES • Thus, from position (2), the point is rotated a further 90o • and inverted to the position (3); from position (3), the point • is rotated a further 90o and inverted to a position (4); from • position (4), the point is rotated a further 90o and inverted • to resume position (1). • Thus if we do this compound operation about a point four • times, it will get the original position. This is an example • for 4-fold roto inversion axis. Crystals possess 1,2,3,4 • and 6-fold rotation inversion axes. PH 0101 UNIT 4 LECTURE 3

  28. TRANSLATIONAL SYMMETRY SCREW AXES • This symmetry element has a compound operation of • a proper rotation with a translation parallel to the • rotation axis • This is shown in the figure.In this operation, a rotation • takes place from A to B by an amount of  and it • combines with a translation from B to C by an amount • of T, which is equivalent to a screw motion from A to C. • The symmetry element that corresponds to such a • motion is called a screw axis. PH 0101 UNIT 4 LECTURE 3

  29. C T θ A B TRANSLATIONAL SYMMETRY SCREW AXES PH 0101 UNIT 4 LECTURE 3

  30. TRANSLATIONAL SYMMETRY GLIDE PLANE • This symmetry element also has a compound • operation of a reflection with a translation parallel • to the reflection plane. • Figure shows the operation of a glide plane • If the upper layer of atoms is moved through a • distance of a/2, and then reflected in the plane • mm1, the lower plane of atoms is generated. PH 0101 UNIT 4 LECTURE 3

  31. a m m1 a / 2 TRANSLATIONAL SYMMETRY GLIDE PLANE PH 0101 UNIT 4 LECTURE 3

  32. COMBINATION OF SYMMETRY ELEMENTS • Apart from the different symmetry elements different • combinations of the basic symmetry elements are also • possible. • They give rise to different symmetry points in the • crystal. • The combination of symmetry elements at a point is • called a `point group’. PH 0101 UNIT 4 LECTURE 3

  33. COMBINATION OF SYMMETRY ELEMENTS • In crystals, 32 point groups are possible. • The combination of 32 point groups with 14 • Bravais lattices lead to 230 unique • arrangements of points in space. • They are called as `space groups’. PH 0101 UNIT 4 LECTURE 3

  34. Physics is hopefully simple but Physicists are not PH 0101 UNIT 4 LECTURE 3

More Related