This document derives expressions for inter-planar spacing (d) in different crystal structures. It defines d as the perpendicular distance from the origin of the unit cell to the nearest (hkl) plane. By representing a plane using its intercept form equation and substituting the intercepts, the document arrives at the general expression for d as 1/√[(h/a)2+(k/b)2+(l/c)2]. It then provides the specific expressions for d in tetragonal and orthorhombic crystal structures by substituting their unit cell parameters a, b, c.

1. DERIVATION OF
INTER-PLANAR SPACING AND
EXPRESSIONS FOR
INTER-PLANAR SPACING FOR
VARIOUS CRYSTAL
STRUCTURES
GERUGANTI SUDHAKAR
PHD(MATERIAL’S ENGINEERING)
UNIVERSITY OF HYDERABAD

2. Useful concept for crystallography & diffraction
Lattice planes
Think of sets of planes in lattice - each plane in set
parallel to all others in set. All planes in set
equidistant from one another
Infinite number of sets of planes in lattice
d
d -
interplanar
spacing

3. The interplanar distance dhkl is defined
to be the distance from
the origin of the unit cell to the (hkl) plane
nearest the origin along the normal to the
plane, i.e. the perpendicular distance from
the origin to the plane.

4. Equation of plane in intercept form is:
(x/a)+(y/b)+(z/c) = 1 ………….(I)
Where a,b,c are x-,y-,z-intercepts
respectively.
In the present case, PLANE is touching
x-axis=a/h,
y-axis=b/k,
z-axis=c/l
Substituting THIS INTERCEPTS in equation
(I); we have
[x/(a/h)]+[y/(b/k)]+[z/(c/l)] = 1;

5. On simplying we have,
(hx/a)+(ky/b)+(lz/c) = 1 ………(II)
PERPENDICULAR DISTANCE FROM
ORIGIN TO ABOVE PLANE (II)
IS “INTER-PLANAR SPACING: d”
Therefore,1/√[(h/a)^2+(k/b)^2+(z/c)^2] = d
d^2= 1/[(h/a)^2+(k/b)^2+(z/c)^2]

7. FOR:
1. TETRAGONAL a = b ≠ c;
2. ORTHO-RHOMBIC a ≠b ≠c.
Substituting the above in “INTER-PLANAR
SPACING : d”
We get expression for THOSE crystal structures