Solid State

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Formulas

unitCellAtoms Apas Cloud

The atoms (lattice points) per unit cell: 1 for simple cubic, 2 for bcc, 4 for fcc (ccp), 6 for hcp, 2 for end-centred.

lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred

packingEfficiency Apas Cloud

The packing efficiency as a percentage: 52.4 for simple cubic (π/6), 68.0 for bcc (√3π/8), 74.0 for fcc and hcp (π/(3√2)).

lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred

atomRadius Apas Cloud

The radius of the touching spheres from the edge a: a/2 (sc), √3a/4 (bcc), a/(2√2) (fcc); for hcp a/2 with a the side of the hexagon. NCERT Ex 1.5: silver fcc, a = 407 pm gives r = 144 pm.

lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred
a (number) -- the edge, any unit

cellEdge Apas Cloud

The edge a from the atomic radius: 2r (sc), 4r/√3 (bcc), 2√2 r (fcc).

lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred
r (number) -- the radius

nearestNeighbour Apas Cloud

The distance between nearest neighbours, 2r, from the edge: a (sc), √3a/2 (bcc), a/√2 (fcc).

lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred
a (number) -- the edge

crystalDensity Apas Cloud

The density d = zM/(a³N_A) in g cm⁻³ from the atoms per cell, the molar mass in g mol⁻¹ and the edge in pm (NCERT Ex 1.4, exercises 1.13 to 1.15).

z (number) -- atoms (or formula units) per cell
M (number) -- the molar mass, g/mol
a_pm (number) -- the edge in pm

cellEdgeFromDensity Apas Cloud

The edge in pm from the density: a = (zM/(dN_A))^(1/3). NCERT Ex 1.4: niobium bcc, d = 8.55 g/cm³, M = 93 gives a = 330.5 pm, r = 143.1 pm.

z (number) -- atoms per cell
M (number) -- the molar mass, g/mol
d (number) -- the density, g/cm³

atomsFromDensity Apas Cloud

The atoms per cell z = d a³ N_A / M from the density, the molar mass and the edge in pm: 1, 2 or 4 tells the lattice.

d (number) -- the density, g/cm³
M (number) -- the molar mass, g/mol
a_pm (number) -- the edge in pm

radiusRatioSite Apas Cloud

The radius-ratio rule: the record `site` (linear, triangular, tetrahedral, octahedral, cubic or close packed), `coordination` (2, 3, 4, 6, 8, 12) and the range `lower` to `upper` of r⁺/r⁻ that gives it: 0.155, 0.225, 0.414, 0.732, 1.

ratio (number) -- r⁺/r⁻

coordinationNumber Apas Cloud

The coordination numbers of a structure as the record `cation` and `anion`: NaCl 6:6, CsCl 8:8, ZnS 4:4, CaF₂ 8:4, Na₂O 4:8; for a lattice both are its own (6, 8, 12, 12).

structure (string) -- NaCl (rock salt), CsCl, ZnS (zinc blende), CaF2 (fluorite), Na2O (antifluorite), diamond, or a lattice

latticeFormula Apas Cloud

The formula of a compound from the sites each species occupies, with each count per cell: corners count 1/8, faces 1/2, edges 1/4, the body centre 1, tetrahedral voids 1 each (8), octahedral voids 4 per cell. Species are separated by semicolons, a species' sites by commas; a site may be preceded by a number ("4 edges", "7 corners"), "half", "alternate" or "all". The record `formula` and one field per species.

sites (string) -- "X: corners, face centres; M: body centre, 4 edges"

voidCount Apas Cloud

The voids in a close-packed arrangement of n spheres: n octahedral, 2n tetrahedral.

kind (string) -- tetrahedral or octahedral
n (number) -- the spheres

3D pictures

unitCell Apas Cloud

A cubic unit cell in 3D: the cube's edges, the spheres at the lattice points and a legend. A lattice (sc, bcc, fcc, end-centred; hcp as its hexagonal prism) or a structure (NaCl, CsCl, ZnS, CaF₂, Na₂O, diamond) with its two species coloured. "open" (the default) draws small balls; "fill" the touching spheres at their true radii, part see-through. The picture is a space3d one: label3d and view3d add to it.

structure (string) -- the lattice or structure
mode (string) -- open or fill
yaw (number) -- the camera's turn in degrees (25 unless given)
pitch (number) -- its tilt (18)

unitCellSites Apas Cloud

A cubic cell with any occupation, for a JEE question: the species and their sites as latticeFormula reads them (the first n positions of a site when a number is given), the formula under it.

sites (string) -- "X: corners, faces; M: body, 4 edges"
yaw (number) -- the camera's turn (25)
pitch (number) -- its tilt (18)

atomShare Apas Cloud

The cells that share an atom at a site: a corner atom among 8 cells (1/8 to each), a face-centre atom between 2 (1/2), an edge-centre atom among 4 (1/4), a body-centre atom in its own cell (1).

site (string) -- corner, face, edge or body

closePacking3d Apas Cloud

Layers of close-packed spheres stacked in 3D and lettered: ABAB (hexagonal close packing, the third layer over the first) or ABCABC (cubic close packing, the third over the first layer's other voids).

order (string) -- ABAB (hcp) or ABCABC (ccp)
layers (number) -- 1 to 4 (3 unless given)

voidShape Apas Cloud

The void between touching spheres: tetrahedral (four spheres, the fourth over the hollow of three, r/R = 0.225) or octahedral (two triangles of three turned 60°, r/R = 0.414), the small sphere in it and the polyhedron of the centres drawn.

kind (string) -- tetrahedral or octahedral

cellVoids Apas Cloud

The fcc cell with its voids marked: the 8 tetrahedral voids a quarter along each body diagonal (the cell cut into eight small cubes, one void's four neighbours joined), or the 13 octahedral positions (the body centre and the 12 edge centres, 4 per cell) with the body centre's octahedron.

lattice (string) -- fcc
kind (string) -- tetrahedral or octahedral

2D pictures

packingSectionOn Apas Cloud

The section that gives a from r: the square face with the spheres touching along the edge (sc, a = 2r) or along the face diagonal (fcc, √2a = 4r), or the body-diagonal rectangle a × √2a of bcc (√3a = 4r).

d (diagram) -- the diagram
lattice (string) -- sc (simple cubic, primitive), bcc, fcc (ccp), hcp or end-centred

closePacking2dOn Apas Cloud

A layer of spheres packed in two dimensions: square close packing (AAA rows, coordination 4, square voids) or hexagonal (ABAB rows, coordination 6, triangular voids), the voids dotted.

d (diagram) -- the diagram
kind (string) -- square or hexagonal
rows (number) -- 1 to 12 (3)
columns (number) -- 2 to 16 (5)

radiusRatioOn Apas Cloud

The derivation of a radius-ratio limit: the touching anions, the cation in the void and the right triangle that gives it, with the working in the caption: triangular 2/√3 − 1 = 0.155, tetrahedral √(3/2) − 1 = 0.225, octahedral √2 − 1 = 0.414, cubic √3 − 1 = 0.732.

d (diagram) -- the diagram
void (string) -- triangular, tetrahedral, octahedral or cubic

crystalDefectOn Apas Cloud

A plane of an ionic crystal with a point defect marked and named: vacancy, interstitial, Schottky (a cation and an anion missing), Frenkel (a cation moved to an interstitial site), F-centre (an electron in an anion vacancy) or impurity (Sr²⁺ for two Na⁺ with a vacancy).

d (diagram) -- the diagram
defect (string) -- vacancy, interstitial, Schottky, Frenkel, F-centre or impurity
cation (string) -- the cation's symbol (Na⁺)
anion (string) -- the anion's symbol (Cl⁻)

bandDiagramOn Apas Cloud

Band theory: the valence and conduction bands of a conductor (overlapping), an insulator (a large gap) or a semiconductor (a small gap), or all three side by side.

d (diagram) -- the diagram
kind (string) -- conductor, insulator, semiconductor or all

dopedSiliconOn Apas Cloud

A silicon lattice with a dopant at its centre: n-type (As with its fifth electron free) or p-type (B with an electron hole in one bond).

d (diagram) -- the diagram
kind (string) -- n or p

magneticOrderOn Apas Cloud

The magnetic moments of a solid as a row of arrows: ferromagnetic (all aligned), antiferromagnetic (equal and opposed), ferrimagnetic (unequal and opposed) or paramagnetic (random).

d (diagram) -- the diagram
kind (string) -- ferromagnetic, antiferromagnetic, ferrimagnetic or paramagnetic