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(hkl) => Plane

{hkl} => Family of Planes

[hkl] => Direction

<hkl> => Family of direction

Steps to calculate miller indices :

Steps | X | Y | Z |
---|---|---|---|

Intercept | \[ 1 \] | \[ 1 \] | \[ \infty \] |

Reciprocal | \[ 1 \] | \[ 1 \] | \[ 0 \] |

Rationalise | \[ 1 \] | \[ 1 \] | \[ 0 \] |

Miler Indices | \[ (1 1 0) \] |

Steps | X | Y | Z |
---|---|---|---|

Intercept | \[ 1 \] | \[ 1 \] | \[ \frac{1}{2} \] |

Reciprocal | \[ 1 \] | \[ 1 \] | \[ 2 \] |

Rationalise | \[ 1 \] | \[ 1 \] | \[ 2 \] |

Miler Indices | \[ (1 1 2) \] |

Here the axis is not passing through the intercept. In such cases we have to shift the origin.

Steps | X | Y | Z |
---|---|---|---|

Intercept | \[ \frac{1}{2} \] | \[ 1 \] | \[ \infty \] |

Reciprocal | \[ 2 \] | \[ 1 \] | \[ 0 \] |

Rationalise | \[ 2 \] | \[ 1 \] | \[ 0 \] |

Miler Indices | \[ (2 1 0) \] |

Show Answer

Here all the axis is not passing through the intercept. In such cases we have to shift the origin.

Steps | X | Y | Z |
---|---|---|---|

Intercept | \[ -1 \] | \[ 1 \] | \[ 2 \] |

Reciprocal | \[ -1 \] | \[ 1 \] | \[ \frac{1}{2} \] |

Rationalise | \[ -2 \] | \[ 2 \] | \[ 1 \] |

Miler Indices | \[ [-2 2 1] \] |

- Plane which are close to origin have higher miller indices than plane which are far away from origin
- If a plane is parallel to any axis then its corresponding miller index on that axis wll be zero
- Two parallel planes will have same Miller Indices . Note :- Sign may change depending on there position.
- Angle between two planes having miller indices \( ( h_{1} k_{1} l_{1}) \) and \( ( h_{2} k_{2} l_{2}) \) will be - \[ cos (\theta) = \frac{ h_{1}h_{2} + k_{1}k_{2} + l_{1}l_{2}}{ \sqrt{h_{1}^2 + k_{1}^2 + l_{1}^2} \times \sqrt{h_{2}^2 + k_{2}^2 + l_{2}^2} } \]
- Two planes having miller indices \( ( h_{1} k_{1} l_{1}) \) and \( ( h_{2} k_{2} l_{2}) \) will be perpendicular to each other when - \[ h_{1}h_{2} + k_{1}k_{2} + l_{1}l_{2} = 0 \]

Interplanar spacing is the separation between sets of parallel planes formed by the individual atoms in the lattice structure considering one plane passing through origin. \[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] Where a = Lattice parameter

Show Answer

Answer : d = 300nm

As given a = 900nm

And Miller indices of plane is (2 2 1)

=> \( d = \frac{900}{\sqrt{2^2 + 2^2 + 1^2}} nm \)

=> \( d = \frac{900}{ 3 } \)

=> \( d = 300nm \)

For HCP, the miller indices are denoted as (uvtw) for planes and [uvtw] for direction.