## Interplanar Distance in Triclinic Crystal Lattice Solution

STEP 0: Pre-Calculation Summary
Formula Used
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)))
d = sqrt(1/((((b^2)*(c^2)*((sin(α))^2)*(h^2))+((alattice^2)*(c^2)*((sin(β))^2)*(k^2))+((alattice^2)*(b^2)*((sin(γ))^2)*(l^2))+(2*alattice*b*(c^2)*((cos(α)*cos(β))-cos(γ))*h*k)+(2*b*c*(alattice^2)*((cos(γ)*cos(β))-cos(α))*l*k)+(2*alattice*c*(b^2)*((cos(α)*cos(γ))-cos(β))*h*l))/(Vunit cell^2)))
This formula uses 3 Functions, 11 Variables
Functions Used
sin - Sine is a trigonometric function that describes the ratio of the length of the opposite side of a right triangle to the length of the hypotenuse., sin(Angle)
cos - Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle., cos(Angle)
sqrt - A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number., sqrt(Number)
Variables Used
Interplanar Spacing - (Measured in Meter) - Interplanar Spacing is the distance between adjacent and parallel planes of the crystal.
Lattice Constant b - (Measured in Meter) - The Lattice Constant b refers to the physical dimension of unit cells in a crystal lattice along y-axis.
Lattice Constant c - (Measured in Meter) - The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis.
Lattice parameter alpha - (Measured in Radian) - The Lattice parameter alpha is the angle between lattice constants b and c.
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.
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.
Lattice Parameter Beta - (Measured in Radian) - The Lattice Parameter Beta is the angle between lattice constants a and c.
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.
Lattice Parameter gamma - (Measured in Radian) - The Lattice Parameter gamma is the angle between lattice constants a and b.
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.
Volume of Unit Cell - (Measured in Cubic Meter) - Volume of Unit Cell is is defined as the space occupied within the boundaries of unit cell.
STEP 1: Convert Input(s) to Base Unit
Lattice Constant b: 12 Angstrom --> 1.2E-09 Meter (Check conversion ​here)
Lattice Constant c: 15 Angstrom --> 1.5E-09 Meter (Check conversion ​here)
Lattice parameter alpha: 30 Degree --> 0.5235987755982 Radian (Check conversion ​here)
Miller Index along x-axis: 9 --> No Conversion Required
Lattice Constant a: 14 Angstrom --> 1.4E-09 Meter (Check conversion ​here)
Lattice Parameter Beta: 35 Degree --> 0.610865238197901 Radian (Check conversion ​here)
Miller Index along y-axis: 4 --> No Conversion Required
Lattice Parameter gamma: 38 Degree --> 0.66322511575772 Radian (Check conversion ​here)
Miller Index along z-axis: 11 --> No Conversion Required
Volume of Unit Cell: 105 Cubic Angstrom --> 1.05E-28 Cubic Meter (Check conversion ​here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
d = sqrt(1/((((b^2)*(c^2)*((sin(α))^2)*(h^2))+((alattice^2)*(c^2)*((sin(β))^2)*(k^2))+((alattice^2)*(b^2)*((sin(γ))^2)*(l^2))+(2*alattice*b*(c^2)*((cos(α)*cos(β))-cos(γ))*h*k)+(2*b*c*(alattice^2)*((cos(γ)*cos(β))-cos(α))*l*k)+(2*alattice*c*(b^2)*((cos(α)*cos(γ))-cos(β))*h*l))/(Vunit cell^2))) --> sqrt(1/((((1.2E-09^2)*(1.5E-09^2)*((sin(0.5235987755982))^2)*(9^2))+((1.4E-09^2)*(1.5E-09^2)*((sin(0.610865238197901))^2)*(4^2))+((1.4E-09^2)*(1.2E-09^2)*((sin(0.66322511575772))^2)*(11^2))+(2*1.4E-09*1.2E-09*(1.5E-09^2)*((cos(0.5235987755982)*cos(0.610865238197901))-cos(0.66322511575772))*9*4)+(2*1.2E-09*1.5E-09*(1.4E-09^2)*((cos(0.66322511575772)*cos(0.610865238197901))-cos(0.5235987755982))*11*4)+(2*1.4E-09*1.5E-09*(1.2E-09^2)*((cos(0.5235987755982)*cos(0.66322511575772))-cos(0.610865238197901))*9*11))/(1.05E-28^2)))
Evaluating ... ...
d = 1.53891539382534E-11
STEP 3: Convert Result to Output's Unit
1.53891539382534E-11 Meter -->0.0153891539382534 Nanometer (Check conversion ​here)
0.0153891539382534 0.015389 Nanometer <-- Interplanar Spacing
(Calculation completed in 00.020 seconds)
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## <Inter Planar Distance and Inter Planar Angle Calculators

Interplanar Distance in Rhombohedral Crystal Lattice
​ Go 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 Hexagonal Crystal Lattice
​ Go 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 Tetragonal Crystal Lattice
​ Go 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
​ Go 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 Triclinic Crystal Lattice Formula

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)))
d = sqrt(1/((((b^2)*(c^2)*((sin(α))^2)*(h^2))+((alattice^2)*(c^2)*((sin(β))^2)*(k^2))+((alattice^2)*(b^2)*((sin(γ))^2)*(l^2))+(2*alattice*b*(c^2)*((cos(α)*cos(β))-cos(γ))*h*k)+(2*b*c*(alattice^2)*((cos(γ)*cos(β))-cos(α))*l*k)+(2*alattice*c*(b^2)*((cos(α)*cos(γ))-cos(β))*h*l))/(Vunit cell^2)))

## 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 Triclinic Crystal Lattice?

Interplanar Distance in Triclinic Crystal Lattice calculator uses 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))) to calculate the Interplanar Spacing, The Interplanar Distance in Triclinic 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 Triclinic Crystal Lattice using this online calculator? To use this online calculator for Interplanar Distance in Triclinic Crystal Lattice, enter Lattice Constant b (b), Lattice Constant c (c), Lattice parameter alpha (α), Miller Index along x-axis (h), Lattice Constant a (alattice), Lattice Parameter Beta (β), Miller Index along y-axis (k), Lattice Parameter gamma (γ), Miller Index along z-axis (l) & Volume of Unit Cell (Vunit cell) and hit the calculate button. Here is how the Interplanar Distance in Triclinic Crystal Lattice calculation can be explained with given input values -> 1.5E+7 = sqrt(1/((((1.2E-09^2)*(1.5E-09^2)*((sin(0.5235987755982))^2)*(9^2))+((1.4E-09^2)*(1.5E-09^2)*((sin(0.610865238197901))^2)*(4^2))+((1.4E-09^2)*(1.2E-09^2)*((sin(0.66322511575772))^2)*(11^2))+(2*1.4E-09*1.2E-09*(1.5E-09^2)*((cos(0.5235987755982)*cos(0.610865238197901))-cos(0.66322511575772))*9*4)+(2*1.2E-09*1.5E-09*(1.4E-09^2)*((cos(0.66322511575772)*cos(0.610865238197901))-cos(0.5235987755982))*11*4)+(2*1.4E-09*1.5E-09*(1.2E-09^2)*((cos(0.5235987755982)*cos(0.66322511575772))-cos(0.610865238197901))*9*11))/(1.05E-28^2))).

### FAQ

What is Interplanar Distance in Triclinic Crystal Lattice?
The Interplanar Distance in Triclinic 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/((((b^2)*(c^2)*((sin(α))^2)*(h^2))+((alattice^2)*(c^2)*((sin(β))^2)*(k^2))+((alattice^2)*(b^2)*((sin(γ))^2)*(l^2))+(2*alattice*b*(c^2)*((cos(α)*cos(β))-cos(γ))*h*k)+(2*b*c*(alattice^2)*((cos(γ)*cos(β))-cos(α))*l*k)+(2*alattice*c*(b^2)*((cos(α)*cos(γ))-cos(β))*h*l))/(Vunit cell^2))) or 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))). The Lattice Constant b refers to the physical dimension of unit cells in a crystal lattice along y-axis, The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis, The Lattice parameter alpha is the angle between lattice constants b and c, The Miller Index along x-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction, The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis, The Lattice Parameter Beta is the angle between lattice constants a and c, The Miller Index along y-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction, The Lattice Parameter gamma is the angle between lattice constants a and b, The Miller Index along z-axis form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction & Volume of Unit Cell is is defined as the space occupied within the boundaries of unit cell.
How to calculate Interplanar Distance in Triclinic Crystal Lattice?
The Interplanar Distance in Triclinic 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/((((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))). To calculate Interplanar Distance in Triclinic Crystal Lattice, you need Lattice Constant b (b), Lattice Constant c (c), Lattice parameter alpha (α), Miller Index along x-axis (h), Lattice Constant a (alattice), Lattice Parameter Beta (β), Miller Index along y-axis (k), Lattice Parameter gamma (γ), Miller Index along z-axis (l) & Volume of Unit Cell (Vunit cell). With our tool, you need to enter the respective value for Lattice Constant b, Lattice Constant c, Lattice parameter alpha, Miller Index along x-axis, Lattice Constant a, Lattice Parameter Beta, Miller Index along y-axis, Lattice Parameter gamma, Miller Index along z-axis & Volume of Unit Cell 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 Lattice Constant b, Lattice Constant c, Lattice parameter alpha, Miller Index along x-axis, Lattice Constant a, Lattice Parameter Beta, Miller Index along y-axis, Lattice Parameter gamma, Miller Index along z-axis & Volume of Unit Cell. We can use 3 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))))
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