god i hate

*Groups are*

* *Closed*
* *In symmetry group, if A & B leave an object looking the same, then performing A and then B will also leave the object looking the same*
* *Contain the identity*
* *In symmetry group, doing nothing to an object will (obviously) keep it looking unchanged*
* *Contain the inverse of every element in the group* 
* *In symmetry group, if the transformation leaves an object looking unchanged, then reversing the transformation will also leave the object looking unchanged*
* *Associative*
* *In symmetry group, since the operation of combining transformations is associative, thus the group is associative*

* *Abelian vs. Non-Abelian* 

* *Abelian - order does not matter (A*B = B*A)* 
* *Non-Abelian - order DOES matter (A*B != B*A)*
* *Ex. 120 degree rotation -> vertical mirror will not produce the same image as vertical mirror -> 120 degree rotation* 

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* *1D lattice* 

* *Infinite row of regularly spaced points* 
* *Translation from one point to another is a multiple of a basic unit vector* 

*2D lattice*

* *Add a unit vector in a new direction and stack copies of 1D lattice in this direction*

* *Translation vector* 

* *A vector whose magnitude and direction describes how far in what direction a translation is performed*

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*Fundamental region*

* *Smallest part of the pattern that can produce the entire pattern through all the symmetries in a group*

* *2D Lattices* 
* *Oblique*

* *A side does not equal B side* 
* *Y does not equal 60, 90 or 120*
* *Max point group symmetry is 2*

* *Rectangular (2D lattices)*

* *A side does not equal B side*
* *Y = 90*
* *Maximum point group symmetry is 2mm* 

*Diamond (2D Lattices)*

* *A side = B side* 
* *Y does not equal 60, 90 or 120*
* *Max point group symmetry is 2mm* 
* *Lattice usually represented by different unit cell* 
* *A centered rectangular cell* 
* *Non-primitive unit cell ©*
* *IF UNIT CELL LOOKS LIKE DIAMOND - ITS PROBABALY C_SOMETHING*

* *Hexagonal (2d lattices)*

* *A side = B side* 
* *Y = 60 or 120* 
* *If y is 120, max point group is 3m*
* *Primitive unit cell is used* 
* *Max point group symmetry is 6mm*

* *Square  (2D lattices)*

* *A side = B side* 
* *Y = 90* 
* *Max point group symmetry is 4mm*

* *Wallpaper symmetry elements*

* *2, 3, 4 or 6 fold rotations around a point*
* *Other folds cant cover plane without gaps*
* *Reflections in mirror lines* 
* *Translations along any lattice vector* 
* *Glide reflections* 
* *Combination reflection and translation* 
* *Now in multiple directions*

* *Centered Unit cells* 
* *Primitive (P)*

* *Lattice points all on corners*

* *Base-Centered (A, B or C)*

* *Lattice points on corners and the ends of lines in the middle of shape, parallel to A/B/C*

*Face-centered (F)*

* *Lattice points in corners and on every face*

* *Body-centered (I)*

* *Lattice points in every corner and one in the center of shape*

* *Rhombohedral ( R )*

* *One in every corner and two in the middle (diagonal line from two opposite corners)*

*Triclinic (unit cell requirements)*

* *A does not equal B, which does not equal C* 
* *Angles do not equal each other either*
* *No angle is 90* 

* *Monoclinic (cell reqs)*

* *A does not equal B, which does not equal C*
* *Angles a and y equal 90, b does not* 
* *Two fixed angles at 90*

* *Orthorhombic (cell reqs)*

* *A does not equal B, which does not equal C*
* *All angles (a, b, y) equal 90*

* *Tetragonal*

* *Edge A = edge B != edge C*
* *All angles (a, b, y) equal 90*

* *Rhombohedral / Trigonal (R)*

* \
* *Edge A = edge B = edge C*
* *All angles (a=b=y) and are NOT 90*

*Hexagonal*

* *Edge A = edge B != edge C*
* *Angles a and b equal 90, y equals 120*
* *Fixed 3rd angle is 120 and 2 equal sides (a=b)*

* *Cubic / Isometric*

* *Edge A = edge B = edge C* 
* *All angles (a, b, y) equal 90* 

* *Reading space group symbols*

* *First symbol says what type of unit cell the group has* 
* *Ex. C2/c, I4bar2d*
* *Maximum point group in the space group can be found by removing the capital letter, any subscripts, any 1’s (other than P1 or P1Bar) and changing any lower case letter to m*
* *Ex. C2/c -> 2/m, P6(3)/m*
* *The point group indicates the crystal system, together w the unit cell indicates the bravais lattice* 
* *C2/c -> Monoclinic C (since 2/m is monoclinic, and first symbol says unit cell)*
* *Each symbol after the capital indicates some elements of the group* 
* *Rotation, rotoinversion or screw axis in direction* 
* *11 of the 230 that have chiral screw axes and no mirrors or glides are actually identical*

* *Sphere packing*

* *Arrangement of non-overlapping (hard) spheres in a confined space*
* *Usually inorganic*
* *Packing fraction is the fraction of the space filled by spheres out of total space available*

*Helix / Screw Axis (Notation)*

* *Notation: n_m (n subscript m)* 
* *N is the rotation (must be either 2, 3, 4, 6)*
* *Rotate 360/n COUNTER CLOCKWISE around axis*
* *Translate m/n of a unit cell around axis*
* *M is greater than or equal to 1, and less than or equal to n-1*

Odd number / m is

2\*odd number bar

1 bar

180 clockwise + reflection

6 bar

120 clockwise + reflection OR 60 counter + inversion

4 bar

90 clockwise + reflection OR 90 counter clockwise + inversion

3 bar

60 clockwise + reflection

general symbol for inversion

rotoreflection : theta clockwise + reflection

rotoinversion: 180 - theta counter clockwise + inversion

order of inversions (nbar)

even = cyclic order n

odd = cyclic order 2n

regular polyhedron point group

MUST be one of the cubics