Lecture 2 Flashcards
crystal
any hmogenous solid with long-range, three-dimensional internal order (symmetry)
crystal space
can only observe at atomic level
reciprocal space
observe diffraction patterns from crystal structures with x-ray laser beam
same intensities
group of atoms being repeated
long-range
location of beam, symmetry of crystal structure
symmetry element
rotation element - point, line, or plane about which symmetry operation is being performed
symmetry operation
a real or imagined movement of a body about a symmetry element, such that after movement, every point on the object is coincident with an equivalent point
proper rotation
movement about a line, no inter-conversion of handedness occurs
-positive = proper
-matrix describing a 2-fold rotation
-determinant is +1
improper rotation
rotation about a line followed by inversion
-a change of handedness occurs
-negative = improper
-matrix describing an inversion center
-determinant is -1
point group
a set of symmetry elements in which one or more points remain fixed under all symmetry operations (at least one point doesn’t move)
-stereogram
-dot
-outer norm
-inner norm
-make a 3d drawing out of 2d paper
-normal of a plane that connects to the south plane
-facing directly towards you which is perpendicular
-going towards the center
certain types of rotation axes can fill all space
1-fold
2-fold
3-fold
4-fold
6-fold
NOT 5-fold
types of uniaxial point groups
uniaxial- proper rotation (1,2,3,4,6)
uniaxial - improper rotation (right, left, right, left)
uniaxial - coincident rotations (proper and improper together)
types of polyaxial point groups
polyaxial - proper rotations
polyaxial - combinations of proper and improper rotations (improper axes must come in pairs)
polyaxial - coincident rotations (all three axes must be coincident)
which way do rotations go?
counterclockwise - right hand rule
improper symmetry elements
center of symmetry = inversion center
roto-inversion = rotate, then invert
fractional c coordinates
fraction is the equipoint is along a particular axis
x is fraction along ? axis
y is fraction along ? axis
z is frcation along ? axis
a-axis
b-axis
c-axis
equivalent positions for a 4-fold proper rotation (90 degrees)
(x, y, z)
(-y, x, z)
(-x, -y, z)
(y, -x, z)
equivalent positions for a 3-fold proper rotation (120 degrees)
(x, y)
(-y, x-y)
(-x+y, -x)
seven crystal systems
triclinic
monoclinic
orthorhombic
trigonal
tetragonal
hexagonal
cubic
rotation axes for the seven crystal systems
1
2
222
3, 322
4, 422
6, 622
332, 432
folds on rotation axis
-1-fold rotation axis
-1 2-fold rotation axis
-3 independent mutually perpendicular 2-fold axes
-1 unique 3-fold rotation axis
-1 unique 4-fold rotation axis
-1 unique 6-fold rotation axis
-4 intersecting 3-fold axes
symmetry element: rotation axis HM notation:
x
symmetry element: rotoinversion axis HM notation:
x bar
symmetry element: rotation w/ perpendicular mirror plane HM notation:
x/m
symmetry element: rotation w/ perpendicular 2-fold axis HM notation:
x2
symmetry element: rotation w/ parallel mirror plane HM notation:
xm
symmetry element: Rotation with perpendicular and parallel mirror planes HM notation:
x bar / m * m
uniaxial point groups tetragonal crystal system
P, I, P/I
polyaxial point groups tetragonal crystal system
PPP, PII, IPI, IIP, P/I P/I P/I
polar axis
an axis that has distinguishable ends
