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Why is the hexagonal closed packed structure not a Bravais lattice?

How can one readily say that a particular lattice is Bravais lattice or not?

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    $\begingroup$ A Bravais lattice is a well defined concept in solid state physics, covered in many textbooks. A HCP structure is a simple hexagonal Bravais lattice with a two-atom basis. $\endgroup$
    – Jon Custer
    Aug 28, 2018 at 18:33

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Definition of Bravais lattice is given by:

1.Bravais lattice is an infinite array of discrete points, appears exactly same from the point where the lattice is observed.
2. Position vector of any discrete point in the lattice can be written in the form $$\mathbf{R}=n_1 \mathbf{a_1}+ n_2 \mathbf{a_2}$$ where $\mathbf{a_1} ,\mathbf{a_2}$ are primitive, linearly independent vectors which spans the whole lattice.

Even though the hexagonal lattice follows the first definition, it doesn't follows the second one. Not all points in in the lattice can't be written as the linear combination of primitive vectors of hexagonal lattice. Thus hexagonal lattice is not Bravais.

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In a lattice the neighborhood of all spheres or points should be identical in all directions. Based on this idea and how we make an hcp and ccp lattice by stacking closed packed 2D layers, in this video you can see how hcp is not a lattice. https://www.youtube.com/watch?v=9zgdk-MVWY4

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