## Interplanar Distance in Rhombohedral Crystal Lattice Solution

STEP 0: Pre-Calculation Summary
Formula Used
Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3))))))
d = sqrt(1/(((((h^2)+(k^2)+(l^2))*(sin(α)^2))+(((h*k)+(k*l)+(h*l))*2*(cos(α)^2))-cos(α))/(alattice^2*(1-(3*(cos(α)^2))+(2*(cos(α)^3))))))
This formula uses 3 Functions, 6 Variables
Functions Used
sin - Trigonometric sine function, sin(Angle)
cos - Trigonometric cosine function, cos(Angle)
sqrt - Square root function, sqrt(Number)
Variables Used
Interplanar Spacing - (Measured in Meter) - Interplanar Spacing is the distance between adjacent and parallel planes of the crystal.
Miller Index along x-axis - The Miller Index along x-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction.
Miller Index along y-axis - The Miller Index along y-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction.
Miller Index along z-axis - The Miller Index along z-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction.
Lattice parameter alpha - (Measured in Radian) - The Lattice parameter alpha is the angle between lattice constants b and c.
Lattice Constant a - (Measured in Meter) - The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis.
STEP 1: Convert Input(s) to Base Unit
Miller Index along x-axis: 9 --> No Conversion Required
Miller Index along y-axis: 4 --> No Conversion Required
Miller Index along z-axis: 11 --> No Conversion Required
Lattice parameter alpha: 30 Degree --> 0.5235987755982 Radian (Check conversion here)
Lattice Constant a: 14 Angstrom --> 1.4E-09 Meter (Check conversion here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
d = sqrt(1/(((((h^2)+(k^2)+(l^2))*(sin(α)^2))+(((h*k)+(k*l)+(h*l))*2*(cos(α)^2))-cos(α))/(alattice^2*(1-(3*(cos(α)^2))+(2*(cos(α)^3)))))) --> sqrt(1/(((((9^2)+(4^2)+(11^2))*(sin(0.5235987755982)^2))+(((9*4)+(4*11)+(9*11))*2*(cos(0.5235987755982)^2))-cos(0.5235987755982))/(1.4E-09^2*(1-(3*(cos(0.5235987755982)^2))+(2*(cos(0.5235987755982)^3))))))
Evaluating ... ...
d = 1.72733515814283E-11
STEP 3: Convert Result to Output's Unit
1.72733515814283E-11 Meter -->0.0172733515814283 Nanometer (Check conversion here)
0.0172733515814283 0.017273 Nanometer <-- Interplanar Spacing
(Calculation completed in 00.005 seconds)
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National Institute of Technology (NIT), Meghalaya
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## < 10+ Inter-planar distance and inter-planar angle Calculators

Interplanar Distance in Triclinic Crystal Lattice
Interplanar Spacing = sqrt(1/((((Lattice Constant b^2)*(Lattice Constant c^2)*((sin(Lattice parameter alpha))^2)*(Miller Index along x-axis^2))+((Lattice Constant a^2)*(Lattice Constant c^2)*((sin(Lattice Parameter Beta))^2)*(Miller Index along y-axis^2))+((Lattice Constant a^2)*(Lattice Constant b^2)*((sin(Lattice Parameter gamma))^2)*(Miller Index along z-axis^2))+(2*Lattice Constant a*Lattice Constant b*(Lattice Constant c^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter Beta))-cos(Lattice Parameter gamma))*Miller Index along x-axis*Miller Index along y-axis)+(2*Lattice Constant b*Lattice Constant c*(Lattice Constant a^2)*((cos(Lattice Parameter gamma)*cos(Lattice Parameter Beta))-cos(Lattice parameter alpha))*Miller Index along z-axis*Miller Index along y-axis)+(2*Lattice Constant a*Lattice Constant c*(Lattice Constant b^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter gamma))-cos(Lattice Parameter Beta))*Miller Index along x-axis*Miller Index along z-axis))/(Volume of Unit Cell^2)))
Interplanar Angle for Hexagonal System
Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(0.5*((Miller Index along plane 1*Miller Index k along Plane 2)+(Miller Index h along plane 2*Miller Index k along Plane 1)))+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt(((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index along plane 1*Miller Index k along Plane 1)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 1^2)))*((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index h along plane 2*Miller Index k along Plane 2)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 2^2))))))
Interplanar Angle for Orthorhombic System
Interplanar Angle = acos((((Miller Index along plane 1*Miller Index h along plane 2)/(Lattice Constant a^2))+ ((Miller Index l along plane 1*Miller Index l along plane 2)/(Lattice Constant c^2))+ ((Miller Index k along Plane 1*Miller Index k along Plane 2)/(Lattice Constant b^2)))/ sqrt((((Miller Index along plane 1^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))*((Miller Index l along plane 1^2)/(Lattice Constant c^2)))* (((Miller Index h along plane 2^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))+((Miller Index l along plane 1^2)/(Lattice Constant c^2)))))
Interplanar Distance in Rhombohedral Crystal Lattice
Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3))))))
Interplanar Distance in Monoclinic Crystal Lattice
Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)/(Lattice Constant a^2))+(((Miller Index along y-axis^2)*(sin(Lattice Parameter Beta)^2))/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))-(2*Miller Index along x-axis*Miller Index along z-axis*cos(Lattice Parameter Beta)/(Lattice Constant a*Lattice Constant c)))/((sin(Lattice Parameter Beta))^2)))
Interplanar Angle for Simple Cubic System
Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2))))
Interplanar Distance in Hexagonal Crystal Lattice
Interplanar Spacing = sqrt(1/((((4/3)*((Miller Index along x-axis^2)+(Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis^2)))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Orthorhombic Crystal Lattice
Interplanar Spacing = sqrt(1/(((Miller Index along x-axis^2)/(Lattice Constant a^2))+((Miller Index along y-axis^2)/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Tetragonal Crystal Lattice
Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)+(Miller Index along y-axis^2))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Cubic Crystal Lattice
Interplanar Spacing = Edge Length/sqrt((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))

## Interplanar Distance in Rhombohedral Crystal Lattice Formula

Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3))))))
d = sqrt(1/(((((h^2)+(k^2)+(l^2))*(sin(α)^2))+(((h*k)+(k*l)+(h*l))*2*(cos(α)^2))-cos(α))/(alattice^2*(1-(3*(cos(α)^2))+(2*(cos(α)^3))))))

## What are Bravais Lattices?

Bravais Lattice refers to the 14 different 3-dimensional configurations into which atoms can be arranged in crystals. The smallest group of symmetrically aligned atoms which can be repeated in an array to make up the entire crystal is called a unit cell.
There are several ways to describe a lattice. The most fundamental description is known as the Bravais lattice. In words, a Bravais lattice is an array of discrete points with an arrangement and orientation that look exactly the same from any of the discrete points, that is the lattice points are indistinguishable from one another.
Out of 14 types of Bravais lattices some 7 types of Bravais lattices in three-dimensional space are listed in this subsection. Note that the letters a, b, and c have been used to denote the dimensions of the unit cells whereas the letters 𝛂, 𝞫, and 𝝲 denote the corresponding angles in the unit cells.

## How to Calculate Interplanar Distance in Rhombohedral Crystal Lattice?

Interplanar Distance in Rhombohedral Crystal Lattice calculator uses Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3)))))) to calculate the Interplanar Spacing, The Interplanar Distance in Rhombohedral Crystal Lattice, also called Interplanar Spacing is the perpendicular distance between two successive planes on a family (hkl). Interplanar Spacing is denoted by d symbol.

How to calculate Interplanar Distance in Rhombohedral Crystal Lattice using this online calculator? To use this online calculator for Interplanar Distance in Rhombohedral Crystal Lattice, enter Miller Index along x-axis (h), Miller Index along y-axis (k), Miller Index along z-axis (l), Lattice parameter alpha (α) & Lattice Constant a (alattice) and hit the calculate button. Here is how the Interplanar Distance in Rhombohedral Crystal Lattice calculation can be explained with given input values -> 0.172734 = sqrt(1/(((((9^2)+(4^2)+(11^2))*(sin(0.5235987755982)^2))+(((9*4)+(4*11)+(9*11))*2*(cos(0.5235987755982)^2))-cos(0.5235987755982))/(1.4E-09^2*(1-(3*(cos(0.5235987755982)^2))+(2*(cos(0.5235987755982)^3)))))).

### FAQ

What is Interplanar Distance in Rhombohedral Crystal Lattice?
The Interplanar Distance in Rhombohedral Crystal Lattice, also called Interplanar Spacing is the perpendicular distance between two successive planes on a family (hkl) and is represented as d = sqrt(1/(((((h^2)+(k^2)+(l^2))*(sin(α)^2))+(((h*k)+(k*l)+(h*l))*2*(cos(α)^2))-cos(α))/(alattice^2*(1-(3*(cos(α)^2))+(2*(cos(α)^3)))))) or Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3)))))). The Miller Index along x-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction, The Miller Index along y-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction, The Miller Index along z-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction, The Lattice parameter alpha is the angle between lattice constants b and c & The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis.
How to calculate Interplanar Distance in Rhombohedral Crystal Lattice?
The Interplanar Distance in Rhombohedral Crystal Lattice, also called Interplanar Spacing is the perpendicular distance between two successive planes on a family (hkl) is calculated using Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3)))))). To calculate Interplanar Distance in Rhombohedral Crystal Lattice, you need Miller Index along x-axis (h), Miller Index along y-axis (k), Miller Index along z-axis (l), Lattice parameter alpha (α) & Lattice Constant a (alattice). With our tool, you need to enter the respective value for Miller Index along x-axis, Miller Index along y-axis, Miller Index along z-axis, Lattice parameter alpha & Lattice Constant a and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Interplanar Spacing?
In this formula, Interplanar Spacing uses Miller Index along x-axis, Miller Index along y-axis, Miller Index along z-axis, Lattice parameter alpha & Lattice Constant a. We can use 6 other way(s) to calculate the same, which is/are as follows -
• Interplanar Spacing = Edge Length/sqrt((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))
• Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)+(Miller Index along y-axis^2))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
• Interplanar Spacing = sqrt(1/((((4/3)*((Miller Index along x-axis^2)+(Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis^2)))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
• Interplanar Spacing = sqrt(1/(((Miller Index along x-axis^2)/(Lattice Constant a^2))+((Miller Index along y-axis^2)/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
• Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)/(Lattice Constant a^2))+(((Miller Index along y-axis^2)*(sin(Lattice Parameter Beta)^2))/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))-(2*Miller Index along x-axis*Miller Index along z-axis*cos(Lattice Parameter Beta)/(Lattice Constant a*Lattice Constant c)))/((sin(Lattice Parameter Beta))^2)))
• Interplanar Spacing = sqrt(1/((((Lattice Constant b^2)*(Lattice Constant c^2)*((sin(Lattice parameter alpha))^2)*(Miller Index along x-axis^2))+((Lattice Constant a^2)*(Lattice Constant c^2)*((sin(Lattice Parameter Beta))^2)*(Miller Index along y-axis^2))+((Lattice Constant a^2)*(Lattice Constant b^2)*((sin(Lattice Parameter gamma))^2)*(Miller Index along z-axis^2))+(2*Lattice Constant a*Lattice Constant b*(Lattice Constant c^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter Beta))-cos(Lattice Parameter gamma))*Miller Index along x-axis*Miller Index along y-axis)+(2*Lattice Constant b*Lattice Constant c*(Lattice Constant a^2)*((cos(Lattice Parameter gamma)*cos(Lattice Parameter Beta))-cos(Lattice parameter alpha))*Miller Index along z-axis*Miller Index along y-axis)+(2*Lattice Constant a*Lattice Constant c*(Lattice Constant b^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter gamma))-cos(Lattice Parameter Beta))*Miller Index along x-axis*Miller Index along z-axis))/(Volume of Unit Cell^2))) Let Others Know