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Interplanar Angle for Hexagonal System Formula

GoInterplanar Angle for Simple Cubic System Formula

GoInterplanar Angle for Orthorhombic System Formula

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The Interplanar Angle is the angle, f between two planes, (h1, k1, l1) and (h2, k2, l2). Check FAQs

θ = a \cos (\frac{(h 1 \cdot h 2) + (k 1 \cdot k 2) + (0.5 \cdot ((h 1 \cdot k 2) + (h 2 \cdot k 1))) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot l 1 \cdot l 2)}{\sqrt{(({h 1}^{2}) + ({k 1}^{2}) + (h 1 \cdot k 1) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot ({l 1}^{2}))) \cdot (({h 2}^{2}) + ({k 2}^{2}) + (h 2 \cdot k 2) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot ({l 2}^{2})))}})

θ - Interplanar Angle?h₁ - Miller Index along plane 1?h₂ - Miller Index h along plane 2?k₁ - Miller Index k along Plane 1?k₂ - Miller Index k along Plane 2?a_lattice - Lattice Constant a?c - Lattice Constant c?l₁ - Miller Index l along plane 1?l₂ - Miller Index l along plane 2?

Interplanar Angle for Hexagonal System Example

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Here is how the Interplanar Angle for Hexagonal System equation looks like with Values.

Here is how the Interplanar Angle for Hexagonal System equation looks like with Units.

Here is how the Interplanar Angle for Hexagonal System equation looks like.

3.1452 = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot (25^{2})))}})

3.1452° = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot (25^{2})))}})

3.1452 = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{14^{2}}{15^{2}}) \cdot (25^{2})))}})

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Inter Planar Distance and Inter Planar Angle » fx Interplanar Angle for Hexagonal System

Interplanar Angle for Hexagonal System Solution

Follow our step by step solution on how to calculate Interplanar Angle for Hexagonal System?

FIRST Step Consider the formula

θ = a \cos (\frac{(h 1 \cdot h 2) + (k 1 \cdot k 2) + (0.5 \cdot ((h 1 \cdot k 2) + (h 2 \cdot k 1))) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot l 1 \cdot l 2)}{\sqrt{(({h 1}^{2}) + ({k 1}^{2}) + (h 1 \cdot k 1) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot ({l 1}^{2}))) \cdot (({h 2}^{2}) + ({k 2}^{2}) + (h 2 \cdot k 2) + ((\frac{3}{4}) \cdot (\frac{{a lattice}^{2}}{c^{2}}) \cdot ({l 2}^{2})))}})

Next Step Substitute values of Variables

∴

θ = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{{14A}^{2}}{{15A}^{2}}) \cdot (25^{2})))}})

Next Step Convert Units

∴

θ = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9m}^{2}}{{1.5E-9m}^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9m}^{2}}{{1.5E-9m}^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9m}^{2}}{{1.5E-9m}^{2}}) \cdot (25^{2})))}})

Next Step Prepare to Evaluate

∴

θ = a \cos (\frac{(5 \cdot 8) + (3 \cdot 6) + (0.5 \cdot ((5 \cdot 6) + (8 \cdot 3))) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9}^{2}}{{1.5E-9}^{2}}) \cdot 16 \cdot 25)}{\sqrt{((5^{2}) + (3^{2}) + (5 \cdot 3) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9}^{2}}{{1.5E-9}^{2}}) \cdot (16^{2}))) \cdot ((8^{2}) + (6^{2}) + (8 \cdot 6) + ((\frac{3}{4}) \cdot (\frac{{1.4E-9}^{2}}{{1.5E-9}^{2}}) \cdot (25^{2})))}})

Next Step Evaluate

∴

θ = 0.0548933107110509rad

Next Step Convert to Output's Unit

∴

θ = 3.14515502724408°

LAST Step Rounding Answer

∴

θ = 3.1452°

Interplanar Angle for Hexagonal System Formula Elements

Variables

Functions

Interplanar Angle

The Interplanar Angle is the angle, f between two planes, (h1, k1, l1) and (h2, k2, l2).

Symbol: θ

Measurement: AngleUnit: °

Note: Value can be positive or negative.

Formulas to find Interplanar Angle

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.

Symbol: h₁

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index along 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.

Symbol: h₂

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index h along 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.

Symbol: k₁

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index k along 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.

Symbol: k₂

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index k along Plane 2

Lattice Constant a

The Lattice Constant a refers to the physical dimension of unit cells in a crystal lattice along x-axis.

Symbol: a_lattice

Measurement: LengthUnit: A

Note: Value can be positive or negative.

Formulas to find Lattice Constant a

Lattice Constant c

The Lattice Constant c refers to the physical dimension of unit cells in a crystal lattice along z-axis.

Symbol: c

Measurement: LengthUnit: A

Note: Value can be positive or negative.

Formulas to find Lattice Constant c

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.

Symbol: l₁

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index l along 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.

Symbol: l₂

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Miller Index l along plane 2

cos

Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle.

Syntax: 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.

Syntax: 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.

Syntax: sqrt(Number)

Other Formulas to find Interplanar Angle

Go Interplanar Angle for Simple Cubic System

θ = a \cos (\frac{(h 1 \cdot h 2) + (k 1 \cdot k 2) + (l 1 \cdot l 2)}{\sqrt{({h 1}^{2}) + ({k 1}^{2}) + ({l 1}^{2})} \cdot \sqrt{({h 2}^{2}) + ({k 2}^{2}) + ({l 2}^{2})}})

Go Interplanar Angle for Orthorhombic System

θ = a \cos (\frac{(\frac{h 1 \cdot h 2}{{a lattice}^{2}}) + (\frac{l 1 \cdot l 2}{c^{2}}) + (\frac{k 1 \cdot k 2}{b^{2}})}{\sqrt{((\frac{{h 1}^{2}}{{a lattice}^{2}}) + (\frac{{k 1}^{2}}{b^{2}}) \cdot (\frac{{l 1}^{2}}{c^{2}})) \cdot ((\frac{{h 2}^{2}}{{a lattice}^{2}}) + (\frac{{k 1}^{2}}{b^{2}}) + (\frac{{l 1}^{2}}{c^{2}}))}})

Other formulas in Inter Planar Distance and Inter Planar Angle category

Go Interplanar Distance in Cubic Crystal Lattice

d = \frac{a}{\sqrt{(h^{2}) + (k^{2}) + (l^{2})}}

Go Interplanar Distance in Tetragonal Crystal Lattice

d = \sqrt{\frac{1}{(\frac{(h^{2}) + (k^{2})}{{a lattice}^{2}}) + (\frac{l^{2}}{c^{2}})}}

How to Evaluate Interplanar Angle for Hexagonal System?

Interplanar Angle for Hexagonal System evaluator 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 evaluate 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 evaluate Interplanar Angle for Hexagonal System using this online evaluator? To use this online evaluator 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.

FAQs on Interplanar Angle for Hexagonal System

What is the formula to find Interplanar Angle for Hexagonal System?

The formula of Interplanar Angle for Hexagonal System is expressed as 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)))))). Here is an example- 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)))))).

How to calculate Interplanar Angle for Hexagonal System?

With 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) we can find Interplanar Angle for Hexagonal System using the formula - 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)))))). This formula also uses Cosine (cos)Inverse Cosine (acos), Square Root (sqrt) function(s).

What are the other ways to Calculate Interplanar Angle?

Here are the different ways to Calculate Interplanar Angle-

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)))))

Can the Interplanar Angle for Hexagonal System be negative?

Yes, the Interplanar Angle for Hexagonal System, measured in Angle can be negative.

Which unit is used to measure Interplanar Angle for Hexagonal System?

Interplanar Angle for Hexagonal System is usually measured using the Degree[°] for Angle. Radian[°], Minute[°], Second[°] are the few other units in which Interplanar Angle for Hexagonal System can be measured.

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Interplanar Angle for Hexagonal System Formula

​GoInterplanar Angle for Simple Cubic System Formula

​GoInterplanar Angle for Orthorhombic System Formula

Interplanar Angle for Hexagonal System Example

Interplanar Angle for Hexagonal System Solution

Interplanar Angle for Hexagonal System Formula Elements

Other Formulas to find Interplanar Angle

Other formulas in Inter Planar Distance and Inter Planar Angle category

How to Evaluate Interplanar Angle for Hexagonal System?

FAQs on Interplanar Angle for Hexagonal System

Variable Name

GoInterplanar Angle for Simple Cubic System Formula

GoInterplanar Angle for Orthorhombic System Formula