Interplanar Angle for Hexagonal System Calculator

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Inter-planar distance and inter-planar angle

Atomic Packing Factor

Density of different Cubic Cell

Lattice

Lattice Direction

Volume of Different Cubic Cell

✖The Miller Index along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 1.ⓘ Miller Index along plane 1 [h₁]			+10% -10%
✖The Miller Index h along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 2.ⓘ Miller Index h along plane 2 [h₂]			+10% -10%
✖The Miller Index k along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 1.ⓘ Miller Index k along Plane 1 [k₁]			+10% -10%
✖The Miller Index k along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 2.ⓘ Miller Index k along Plane 2 [k₂]			+10% -10%
✖The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis.ⓘ Lattice Constant a [a_lattice]			+10% -10%
✖The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis.ⓘ Lattice Constant c [c]			+10% -10%
✖The Miller Index l along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 1.ⓘ Miller Index l along plane 1 [l₁]			+10% -10%
✖The Miller Index l along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 2.ⓘ Miller Index l along plane 2 [l₂]			+10% -10%

✖The Interplanar Angle is the angle, f between two planes, (h1, k1, l1) and (h2, k2, l2).ⓘ Interplanar Angle for Hexagonal System [θ]

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Interplanar Angle for Hexagonal System

Formula

`"θ" = acos((("h"_{"1"}*"h"_{"2"})+("k"_{"1"}*"k"_{"2"})+(0.5*(("h"_{"1"}*"k"_{"2"})+("h"_{"2"}*"k"_{"1"})))+((3/4)*(("a"_{"lattice"}^2)/("c"^2))*"l"_{"1"}*"l"_{"2"}))/(sqrt((("h"_{"1"}^2)+("k"_{"1"}^2)+("h"_{"1"}*"k"_{"1"})+((3/4)*(("a"_{"lattice"}^2)/("c"^2))*("l"_{"1"}^2)))*(("h"_{"2"}^2)+("k"_{"2"}^2)+("h"_{"2"}*"k"_{"2"})+((3/4)*(("a"_{"lattice"}^2)/("c"^2))*("l"_{"2"}^2))))))`

Example

`"3.145155°"=acos((("5"*"8")+("3"*"6")+(0.5*(("5"*"6")+("8"*"3")))+((3/4)*((("14A")^2)/(("15A")^2))*"16"*"25"))/(sqrt(((("5")^2)+(("3")^2)+("5"*"3")+((3/4)*((("14A")^2)/(("15A")^2))*(("16")^2)))*((("8")^2)+(("6")^2)+("8"*"6")+((3/4)*((("14A")^2)/(("15A")^2))*(("25")^2))))))`

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Interplanar Angle for Hexagonal System Solution

STEP 0: Pre-Calculation Summary

Formula Used

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))))))
θ = acos(((h₁*h₂)+(k₁*k₂)+(0.5*((h₁*k₂)+(h₂*k₁)))+((3/4)*((a_lattice^2)/(c^2))*l₁*l₂))/(sqrt(((h₁^2)+(k₁^2)+(h₁*k₁)+((3/4)*((a_lattice^2)/(c^2))*(l₁^2)))*((h₂^2)+(k₂^2)+(h₂*k₂)+((3/4)*((a_lattice^2)/(c^2))*(l₂^2))))))
This formula uses 3 Functions, 9 Variables

Functions Used

cos - Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle., cos(Angle)
acos - The inverse cosine function, is the inverse function of the cosine function. It is the function that takes a ratio as an input and returns the angle whose cosine is equal to that ratio., acos(Number)
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 Angle - (Measured in Radian) - The Interplanar Angle is the angle, f between two planes, (h1, k1, l1) and (h2, k2, l2).
Miller Index along plane 1 - The Miller Index along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 1.
Miller Index h along plane 2 - The Miller Index h along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 2.
Miller Index k along Plane 1 - The Miller Index k along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 1.
Miller Index k along Plane 2 - The Miller Index k along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 2.
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 Constant c - (Measured in Meter) - The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis.
Miller Index l along plane 1 - The Miller Index l along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 1.
Miller Index l along plane 2 - The Miller Index l along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 2.

STEP 1: Convert Input(s) to Base Unit

Miller Index along plane 1: 5 --> No Conversion Required
Miller Index h along plane 2: 8 --> No Conversion Required
Miller Index k along Plane 1: 3 --> No Conversion Required
Miller Index k along Plane 2: 6 --> No Conversion Required
Lattice Constant a: 14 Angstrom --> 1.4E-09 Meter (Check conversion here)
Lattice Constant c: 15 Angstrom --> 1.5E-09 Meter (Check conversion here)
Miller Index l along plane 1: 16 --> No Conversion Required
Miller Index l along plane 2: 25 --> No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

θ = acos(((h₁*h₂)+(k₁*k₂)+(0.5*((h₁*k₂)+(h₂*k₁)))+((3/4)*((a_lattice^2)/(c^2))*l₁*l₂))/(sqrt(((h₁^2)+(k₁^2)+(h₁*k₁)+((3/4)*((a_lattice^2)/(c^2))*(l₁^2)))*((h₂^2)+(k₂^2)+(h₂*k₂)+((3/4)*((a_lattice^2)/(c^2))*(l₂^2)))))) --> acos(((5*8)+(3*6)+(0.5*((5*6)+(8*3)))+((3/4)*((1.4E-09^2)/(1.5E-09^2))*16*25))/(sqrt(((5^2)+(3^2)+(5*3)+((3/4)*((1.4E-09^2)/(1.5E-09^2))*(16^2)))*((8^2)+(6^2)+(8*6)+((3/4)*((1.4E-09^2)/(1.5E-09^2))*(25^2))))))

Evaluating ... ...

θ = 0.0548933107110509

STEP 3: Convert Result to Output's Unit

0.0548933107110509 Radian -->3.14515502724408 Degree (Check conversion here)

FINAL ANSWER

3.14515502724408 ≈ 3.145155 Degree <-- Interplanar Angle

(Calculation completed in 00.004 seconds)

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< 10+ Inter-planar distance and inter-planar angle Calculators

Interplanar Distance in Triclinic Crystal Lattice

Go 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

Go 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

Go 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

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 Monoclinic Crystal Lattice

Go 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

Go 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

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 Orthorhombic Crystal Lattice

Go 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

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 Angle for Hexagonal System Formula

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 Angle for Hexagonal System?

Interplanar Angle for Hexagonal System calculator uses 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)))))) to calculate the Interplanar Angle, The Interplanar angle for Hexagonal system is the angle between two planes (h1, k1, l1) and (h2, k2, l2) in a Hexagonal system. Interplanar Angle is denoted by θ symbol.

How to calculate Interplanar Angle for Hexagonal System using this online calculator? To use this online calculator for Interplanar Angle for Hexagonal System, enter Miller Index along plane 1 (h₁), Miller Index h along plane 2 (h₂), Miller Index k along Plane 1 (k₁), Miller Index k along Plane 2 (k₂), Lattice Constant a (a_lattice), Lattice Constant c (c), Miller Index l along plane 1 (l₁) & Miller Index l along plane 2 (l₂) and hit the calculate button. Here is how the Interplanar Angle for Hexagonal System calculation can be explained with given input values -> 180.2041 = acos(((5*8)+(3*6)+(0.5*((5*6)+(8*3)))+((3/4)*((1.4E-09^2)/(1.5E-09^2))*16*25))/(sqrt(((5^2)+(3^2)+(5*3)+((3/4)*((1.4E-09^2)/(1.5E-09^2))*(16^2)))*((8^2)+(6^2)+(8*6)+((3/4)*((1.4E-09^2)/(1.5E-09^2))*(25^2)))))).

FAQ

What is Interplanar Angle for Hexagonal System?

The Interplanar angle for Hexagonal system is the angle between two planes (h1, k1, l1) and (h2, k2, l2) in a Hexagonal system and is represented as θ = acos(((h₁*h₂)+(k₁*k₂)+(0.5*((h₁*k₂)+(h₂*k₁)))+((3/4)*((a_lattice^2)/(c^2))*l₁*l₂))/(sqrt(((h₁^2)+(k₁^2)+(h₁*k₁)+((3/4)*((a_lattice^2)/(c^2))*(l₁^2)))*((h₂^2)+(k₂^2)+(h₂*k₂)+((3/4)*((a_lattice^2)/(c^2))*(l₂^2)))))) or 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)))))). The Miller Index along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 1, The Miller Index h along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 2, The Miller Index k along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 1, The Miller Index k along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 2, The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis, The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis, The Miller Index l along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 1 & The Miller Index l along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 2.

How to calculate Interplanar Angle for Hexagonal System?

The Interplanar angle for Hexagonal system is the angle between two planes (h1, k1, l1) and (h2, k2, l2) in a Hexagonal system is calculated using 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)))))). To calculate Interplanar Angle for Hexagonal System, you need Miller Index along plane 1 (h₁), Miller Index h along plane 2 (h₂), Miller Index k along Plane 1 (k₁), Miller Index k along Plane 2 (k₂), Lattice Constant a (a_lattice), Lattice Constant c (c), Miller Index l along plane 1 (l₁) & Miller Index l along plane 2 (l₂). With our tool, you need to enter the respective value for Miller Index along plane 1, Miller Index h along plane 2, Miller Index k along Plane 1, Miller Index k along Plane 2, Lattice Constant a, Lattice Constant c, Miller Index l along plane 1 & Miller Index l along plane 2 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 Angle?

In this formula, Interplanar Angle uses Miller Index along plane 1, Miller Index h along plane 2, Miller Index k along Plane 1, Miller Index k along Plane 2, Lattice Constant a, Lattice Constant c, Miller Index l along plane 1 & Miller Index l along plane 2. We can use 2 other way(s) to calculate the same, which is/are as follows -

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 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)))))

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