Liquid Crystal Displays 2 Symmetry Smaller, lighter, with no radiation problems. Found on portables and notebooks, and starting to appear and more on desktops ASymmetry: Point Symmetry rather than emitted. Use of super-twisted crystals pace Symmetry have improved the viewing angle, and response rates 832 Point Groups of Crystals are improving all the time (necessary for tracking .Unit Cell, 7 Crystal Systems, Lattice Planes cursor accurately) Miller indices .Lattices and 14 Bravias Types of Lattices ●230 Space Group Crystal Symmetry Symmetry in Nature, Art and Math Symmetry is one idea by which man through the ages has tried to comprehend and create order Mathematics of Symmetry beauty and perfection. - Hermann Weyl Crystals Symmetry Physical Properties caused by Symmetry 米" Eiffel tower in Paris. France is a wonderful example of symmetry Macroscopic Symmetry Elements Point Symmetry Elements) Mirror Plane Symmetry mirror plane symmetry arise E Point symmetry elements operate to change the a°。o。°° when one half of an object is orientation of structural motifs A point symmetry operation does not alter at °。。o。 the mirror image of the other least one point that it operates on Symmetry Elements and Symmetry Operations 1. Mirror Planes -Reflection or Mirror 2. Center of Symmetry 3. Rotation axis -Rotate Can be folded in half 4. Rotoinversion axis -Rotate and inverse Seen externally with animals
6 Smaller, lighter, with no radiation problems. Found on portables and notebooks, and starting to appear more and more on desktops. Less tiring than c.r.t. (Cathode -ray tube) displays, and reduce eye-strain, due to reflected nature of light rather than emitted. Use of super -twisted crystals have improved the viewing angle, and response rates are improving all the time (necessary for tracking cursor accurately). Liquid Crystal Displays 2 Symmetry Symmetry: Point Symmetry Space Symmetry 32 Point Groups of Crystals Unit Cell, 7 Crystal Systems, Lattice Planes, Miller indices Lattices and 14 Bravias Types of Lattices 230 Space Groups Crystal Symmetry ß Mathematics of Symmetry ß Crystal’s Symmetry ß Physical Properties caused by Symmetry Symmetry in Nature, Art and Math Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection. ¾ Hermann Weyl Eiffel tower in Paris, France is a wonderful example of symmetry Macroscopic Symmetry Elements (Point Symmetry Elements) Point symmetry elements operate to change the orientation of structural motifs A point symmetry operation does not alter at least one point that it operates on Symmetry Elements and Symmetry Operations: 1. Mirror Planes — — Reflection or Mirror 2. Center of Symmetry — — Inverse 3. Rotation Axis — — Rotate 4. Rotoinversion Axis — — Rotate and inverse Mirror Plane Symmetry Mirror plane symmetry arises when one half of an object is the mirror image of the other half •Can be folded in half •Seen externally with animals s s
Mirror Plane Symmetry Symmetry Operation Reflect sThis molecule has two mirror planes flips all points in the ci .One is horizontal, in the plane of the asymmetric unit over a paper and bisects the H-C-H bond line which is called the MOther is vertical, perpendicular to mirror and thereby the plane of the paper and bisects the changes the handedness of CkC· Cl bonds WY any figures in the asymmetric unit. The points along the mirror has reflectional symmetry if an line are all invariant lane can divide the crvstal into halves points under a reflection. hich is the mirror image of the other Rotational Symmetry Symmetry Operation Rotation turns all the points in the o Rotated about a point symmetric unit around one o Allows chirality o In crystals limited to 1,2,3,4,and6 he handedness of figures in rotations nly invariant point Symmetry Axis of Rotation Rotational Symmetry We say a crystal has a symmetry coincidence upon rotation axis of rotation when we can turn it the pattern looks exactly the same. nfold axis of rotational Think of the center of a pizza. If it the same size and have the same then the pizza could be turned and you couldn't tell the difference element for which the operation is a rotation of 3607n retry. The below has rotational symmetry of graphite 60 degrees
7 This molecule has two mirror planes: One is horizontal, in the plane of the paper and bisects the H-C-H bonds Other is vertical, perpendicular to the plane of the paper and bisects the Cl-C-Cl bonds A crystal has reflectional symmetry if an imaginary plane can divide the crystal into halves, each of which is the mirror image of the other. Mirror Plane Symmetry Symmetry Operation Reflection qflips all points in the asymmetric unit over a line, which is called the mirror and thereby changes the handedness of any figures in the asymmetric unit. The points along the mirror line are all invariant points under a reflection. Rotational Symmetry Rotated about a point Allows chirality In crystals limited to 1,2,3,4, and 6 rotations Symmetry Operation Rotation turns all the points in the asymmetric unit around one point, the center of rotation. A rotation does not change the handedness of figures in the plane. The center of rotation is the only invariant point. Symmetry Axis of Rotation We say a crystal has a symmetry axis of rotation when we can turn it by some degree about a point and the pattern looks exactly the same. Think of the center of a pizza. If it is made so that all the pieces are the same size and have the same ingredients in the same places, then the pizza could be turned and you couldn't tell the difference. This means the pizza has rotational symmetry. The pizza below has rotational symmetry of 60 degrees. Rotational Symmetry coincidence upon rotation about the axis of 360°/n Þ n-fold axis of rotational symmetry graphite O H Symbol for a symmetry element for which the operation is a rotation of 360°/n C2 = 180°, C3=120°, C4 = 90°, C5 = 72°, C6 = 60°, etc
Rotation Axis(Cn) Center of Symmetry In general present if you can draw a straight line from any n fold rotation axis s rotation by(360/n) point, through the center, to an equal distance the other side, and arrive at an identical point Can rotate by120° about the o·Cl bond and the molecule looks identical This is called a rotation ax tation axis, as rotate by 120(= 888 3603)to reach an identical o。。0 Center of No center of I symmetry ats symmetry Symmetry Operation n Axis - Symmetry Inversion Axis of rotary Inversion Rotoinversion Axis(Sn or n):nfold rotation combined with an inversion side of a center of symmetry has a similar point at an equal distance the er of 1=i 2 3=3-fold rotation Mac Symmetry Elements Point groups Electrical resistance Thermal expansion Magnetic susceptibility= Macroscopic symmetry 8 =3-fold rotation Macroscopically measured Translation symmetry mirror plane Combination of mirror, center of symmetry rotational symmetry, center of inversion point groups
8 Can rotate by 120° about the C-Cl bond and the molecule looks identical Þ the H atoms are indistinguishable. This is called a rotation axis Þ in particular, a three fold rotation axis, as rotate by 120° (= 360°/3) to reach an identical configuration Rotation Axis (Cn) In general: n-fold rotation axis = rotation by (360°/n) “present if you can draw a straight line from any point, through the center, to an equal distance the other side, and arrive at an identical point”. Center of symmetry at S No center of symmetry (x,y,z) (-x,-y,-z) Center of Symmetry (Inversion symmetry) i Symmetry Operation Inversion every point on one side of a center of symmetry has a similar point at an equal distance on the opposite side of the center of symmetry. Rotoinversion Axis ¾¾ Symmetry Axis of Rotary Inversion Rotoinversion Axis (Sn or ) : n-fold rotation combined with an inversion. n 1 = i 2 = m 3 = 3-fold rotation + inversion 4 6 =3-fold rotation with perpendicular mirror plane Macroscopic Symmetry Elements: Point Groups Electrical resistance Thermal expansion Magnetic susceptibility Elastic constants Macroscopically measured properties Þ Macroscopic symmetry XÛ Translation symmetry Combination of mirror, center of symmetry, rotational symmetry, center of inversion Þ point groups