Question

could someone please hlep me to solve this Ex??

Exercise 5- Space group Imm2 Below, there is the representation of the Imm2 space group. The projections are in the (001) plane with the symmetry operations on the one hand and the general positions of the space group on the other hand. a 1-What is the crystal system, the Bravais lattice and the point group of this space group (SG)? 2- Name an give the direction of the point symmetry operations and the non-symmorphic operation (translationnal symmetry) of this space group. Is this space group centroymmetric or non-centrosymmetric? 3- The number of general positions in this SG is equal to 8. The coordinates are (1) x, y,z (4) i,y,z (2)E,y,z (3) x, y,z and the fourth other ones considering the Bravais lattice translations. Write these four latter coordinates and specify by which symmetries they are obtained from the starting coordinate (х, у, 2). 4- Give the multiplicity of special positions due to the mirrors (reflection plane) perpendicular to the direction b. Write these special positions 5-What is the site's symmetry of an atom whose coordinate is (0 1/2 z)? what is the multiplicity of the site? Write the equivalent special positions for this atom? 6- What are the selection rules for the (hkl) reflections? 8- Using the selection rules given below for the non-symmorphic symmetry, establish the list of the selection rules for the reflections with I single or 2 zero indices for the Imm2 space group. Don't forget the extra-conditions due to the effect of one Miller indice equal to zero in the selection rules previously found. 2

Answer 1

Crystal system is a method of classifying crystalline substances on the basis of their unit cell. There are seven unique crystal systems.

The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table: Required symmetries of point group Crystal family Crystal system (6) (7) Triclinic Point groups Space groups Bravais lattices Lattice system Triclinic None monoclinic 1 twofold axis of rotation or 1 mirror plane 3 2 monoclinic Orthorhombic 3 twofold axes of rotation or 1 twofold axis of rotation and 2 mirror planes Orthorhombic Tetragonal 1 fourfold axis of rotation 2 Tetragonal Rhombohedral 1 threefold axis of rotation Hexagonal Trigonal Hexagonal Hexagonal Cubic 7 1 sixfold axis of rotation 3 fourfold axes of rotation Total Cubic 230 14

Crystal family Lattice system Schönflies 14 Bravais Lattices Base-centered Body-centered Face-centered Primitive triclinic monoclinic orthorhombic tetragonal 07 rhombohedral D3d hexagonal r= 120 N hexagonal cubic

Crystal family	Crystal system	Point group / Crystal class	Schönflies	Hermann–Mauguin	Orbifold	Coxeter	Point symmetry	Order	Abstract group
triclinic		pedial	C1	1	11	[ ]+	enantiomorphic polar	1	trivial {\displaystyle \mathbb {Z} _{1}}${\mathbb {Z}}_{1}$
		pinacoidal	Ci (S2)	1	1x	[2,1+]	centrosymmetric	2	cyclic {\displaystyle \mathbb {Z} _{2}}$\mathbb {Z} _{2}$
monoclinic		sphenoidal	C2	2	22	[2,2]+	enantiomorphic polar	2	cyclic {\displaystyle \mathbb {Z} _{2}}$\mathbb {Z} _{2}$
		domatic	Cs (C1h)	m	*11	[ ]	polar	2	cyclic {\displaystyle \mathbb {Z} _{2}}$\mathbb {Z} _{2}$
		prismatic	C2h	2/m	2*	[2,2+]	centrosymmetric	4	Klein four {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}${\mathbb {V}}={\mathbb {Z}}_{2}\times {\mathbb {Z}}_{2}$
orthorhombic		rhombic-disphenoidal	D2 (V)	222	222	[2,2]+	enantiomorphic	4	Klein four {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}${\mathbb {V}}={\mathbb {Z}}_{2}\times {\mathbb {Z}}_{2}$
		rhombic-pyramidal	C2v	mm2	*22	[2]	polar	4	Klein four {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}${\mathbb {V}}={\mathbb {Z}}_{2}\times {\mathbb {Z}}_{2}$
		rhombic-dipyramidal	D2h (Vh)	mmm	*222	[2,2]	centrosymmetric	8	{\displaystyle \mathbb {V} \times \mathbb {Z} _{2}}${\mathbb {V}}\times {\mathbb {Z}}_{2}$
tetragonal		tetragonal-pyramidal	C4	4	44	[4]+	enantiomorphic polar	4	cyclic {\displaystyle \mathbb {Z} _{4}}$\mathbb {Z} _{4}$
		tetragonal-disphenoidal	S4	4	2x	[2+,2]	non-centrosymmetric	4	cyclic {\displaystyle \mathbb {Z} _{4}}$\mathbb {Z} _{4}$
		tetragonal-dipyramidal	C4h	4/m	4*	[2,4+]	centrosymmetric	8	{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}${\mathbb {Z}}_{4}\times {\mathbb {Z}}_{2}$
		tetragonal-trapezohedral	D4	422	422	[2,4]+	enantiomorphic	8	dihedral {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{8}={\mathbb {Z}}_{4}\rtimes {\mathbb {Z}}_{2}$
		ditetragonal-pyramidal	C4v	4mm	*44	[4]	polar	8	dihedral {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{8}={\mathbb {Z}}_{4}\rtimes {\mathbb {Z}}_{2}$
		tetragonal-scalenohedral	D2d (Vd)	42m or 4m2	2*2	[2+,4]	non-centrosymmetric	8	dihedral {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{8}={\mathbb {Z}}_{4}\rtimes {\mathbb {Z}}_{2}$
		ditetragonal-dipyramidal	D4h	4/mmm	*422	[2,4]	centrosymmetric	16	{\displaystyle \mathbb {D} _{8}\times \mathbb {Z} _{2}}${\mathbb {D}}_{8}\times {\mathbb {Z}}_{2}$
hexagonal	trigonal	trigonal-pyramidal	C3	3	33	[3]+	enantiomorphic polar	3	cyclic {\displaystyle \mathbb {Z} _{3}}$\mathbb {Z} _{3}$
		rhombohedral	C3i (S6)	3	3x	[2+,3+]	centrosymmetric	6	cyclic {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}${\mathbb {Z}}_{6}={\mathbb {Z}}_{3}\times {\mathbb {Z}}_{2}$
		trigonal-trapezohedral	D3	32 or 321 or 312	322	[3,2]+	enantiomorphic	6	dihedral {\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{6}={\mathbb {Z}}_{3}\rtimes {\mathbb {Z}}_{2}$
		ditrigonal-pyramidal	C3v	3m or 3m1 or 31m	*33	[3]	polar	6	dihedral {\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{6}={\mathbb {Z}}_{3}\rtimes {\mathbb {Z}}_{2}$
		ditrigonal-scalenohedral	D3d	3m or 3m1 or 31m	2*3	[2+,6]	centrosymmetric	12	dihedral {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{{12}}={\mathbb {Z}}_{6}\rtimes {\mathbb {Z}}_{2}$
	hexagonal	hexagonal-pyramidal	C6	6	66	[6]+	enantiomorphic polar	6	cyclic {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}${\mathbb {Z}}_{6}={\mathbb {Z}}_{3}\times {\mathbb {Z}}_{2}$
		trigonal-dipyramidal	C3h	6	3*	[2,3+]	non-centrosymmetric	6	cyclic {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}${\mathbb {Z}}_{6}={\mathbb {Z}}_{3}\times {\mathbb {Z}}_{2}$
		hexagonal-dipyramidal	C6h	6/m	6*	[2,6+]	centrosymmetric	12	{\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2}}${\mathbb {Z}}_{6}\times {\mathbb {Z}}_{2}$
		hexagonal-trapezohedral	D6	622	622	[2,6]+	enantiomorphic	12	dihedral {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{{12}}={\mathbb {Z}}_{6}\rtimes {\mathbb {Z}}_{2}$
		dihexagonal-pyramidal	C6v	6mm	*66	[6]	polar	12	dihedral {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{{12}}={\mathbb {Z}}_{6}\rtimes {\mathbb {Z}}_{2}$
		ditrigonal-dipyramidal	D3h	6m2 or 62m	*322	[2,3]	non-centrosymmetric	12	dihedral {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}${\mathbb {D}}_{{12}}={\mathbb {Z}}_{6}\rtimes {\mathbb {Z}}_{2}$
		dihexagonal-dipyramidal	D6h	6/mmm	*622	[2,6]	centrosymmetric	24	{\displaystyle \mathbb {D} _{12}\times \mathbb {Z} _{2}}${\mathbb {D}}_{{12}}\times {\mathbb {Z}}_{2}$
cubic		tetartoidal	T	23	332	[3,3]+	enantiomorphic	12	alternating {\displaystyle \mathbb {A} _{4}}${\mathbb {A}}_{4}$
		diploidal	Th	m3	3*2	[3+,4]	centrosymmetric	24	{\displaystyle \mathbb {A} _{4}\times \mathbb {Z} _{2}}${\mathbb {A}}_{4}\times {\mathbb {Z}}_{2}$
		gyroidal	O	432	432	[4,3]+	enantiomorphic	24	symmetric {\displaystyle \mathbb {S} _{4}}${\mathbb {S}}_{4}$
		hextetrahedral	Td	43m	*332	[3,3]	non-centrosymmetric	24	symmetric {\displaystyle \mathbb {S} _{4}}${\mathbb {S}}_{4}$
		hexoctahedral	Oh	m3m	*432	[4,3]	centrosymmetric	48	{\displaystyle \mathbb {S} _{4}\times \mathbb {Z} _{2}}${\mathbb {S}}_{4}\times {\mathbb {Z}}_{2}$