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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

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PH0101 UNIT 4 LECTURE 3

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  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

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