The stipulated conventional cell is a square of side

2 a

. Although introduced as a centered rectangular lattice, this equal-side case is a square Bravais lattice viewed with a rotated primitive basis. A corner-to-center separation is

d_{nn} = a^{2} + a^{2} = 2 a,

(1)

which is shorter than the

2 a

separation between adjacent conventional corners. Consequently the first contact of equal circles occurs at

2 r = d_{nn}, r = \frac{a}{2} .

(2)

The conventional square represents two lattice points: one from its four shared corners and one from its center. Its packing fraction is therefore

η = \frac{2 π r ^{2}}{( 2 a ) ^{2}} = \frac{π}{4} ≃ 0.7854 .

(3)

Equivalently, one circle occupies each primitive area

2 a^{2}

. This is square circle packing; the drawing does not define a triangular close-packed lattice.

Solution

 0  0

A unit cell is a repeat region whose lattice translations tile the plane. A primitive unit cell represents exactly one lattice point and has the minimum repeat area for a lattice basis.

Cell A is nonprimitive; cell B is primitive. A represents two points and has area

4 a^{2}

. B has four shared corner points and no additional interior point, represents one point and has area

2 a^{2}

. Fractional corner counting refers to the tiling, not to fractional physical atoms.

Solution

 0  0

Using the printed

x, y

axes, take a lower vertex of cell B as origin. One allowed pair of primitive lattice translation vectors is

a_{1} = a (\hat{x} + \hat{y}), a_{2} = a (- \hat{x} + \hat{y}) .

(1)

Both vectors connect nearest lattice points, have length

2 a

, and are perpendicular. Their determinant has magnitude

2 a^{2}

, consistent with a primitive area. Exchanging them or reversing their directions gives equivalent basis conventions; the corresponding reciprocal lattice vectors must then use the same convention.

Solution

 0  0

A visibly different primitive parallelogram is generated by

a_{1}^{'} = a_{1} = a (1, 1), a_{2}^{'} = a_{1} + a_{2} = a (0, 2) .

(1)

Its vertices, relative to one lattice point, are

(0, 0)

(a, a)

(a, 3 a)

and

(0, 2 a)

. Its area is

A_{c}^{'} = det (a a 0 2 a) = 2 a^{2} .

(2)

The basis change has integer matrix

(1011)

with determinant one. It is invertible over the integers, so the new vectors generate every original lattice point, not merely a sublattice. The cell is therefore primitive despite being sheared relative to cell B.