What is the significance of cos²α+cos²β+cos²γ = 1 in crystallography?

[RK7]

On pg 6 of http://www.scribd.com/doc/3914281/Crystal-Structure, it quotes this result without proof. My notes from uni also quote this result but I can't see where it comes from. Does anyone know? Thanks

 

[mathman]

It looks like the Pythagorean theorem applied in three dimensions.

 

[HallsofIvy]

Well, it isn't true for just any angle $\alpha$, $\beta$, $\gamma$, of course! If, however, you draw a line through the origin of a three dimensional coordinate system, define $\alpha$ to be the angle the line makes with the x-axis, $\beta$ to be the angle the line makes with the y-axis, and $\gamma$ to be the angle the line makes with the z-axis, then this is true.

To see that, think of a vector of unit length in the direction of that line. If we drop a perpendicular from the tip of the vector to the x-axis, we have a right triangle in which an angle is $\alpha$ and the hypotenuse is 1, the length of the vector. Thus, the projection of the vector on the x-axis, and so the x-component of the vector is $cos(\alpha)$. Similarly, the y-component of the vector is $cos(\beta)$ and the z-component of the vector is $cos(\gamma)$. That is, $cos^2(\alpha)+ cos^2(\beta)+ cos^2(\gamma)$ is the square of the length of the vector which is, of course, 1.

By the way, look what happens if you do this in two dimensions. If you have a line in the plane through the origin making angles $\alpha$ with the x-axis and $\beta$ with the y- axis then $\beta= \pi/2- \alpha$ so $cos^2(\alpha)+ cos^2(\beta)= cos^2(\alpha)+ cos^2(\pi/2- \alpha)= cos^2(\alpha)+ sin^2(\alpha)= 1$.

 

[RK7]

That was embarrassing.. thanks

 

[CellCoree]

This equation is known as the cosine rule and is a fundamental relationship in crystallography. It is derived from the law of cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those two sides multiplied by the cosine of the angle between them. In the case of a crystal structure, the three sides of the triangle represent the lattice vectors, and the angles between them are the angles of the unit cell. When this equation is applied to all three angles in a unit cell, it reduces to the equation cos²α+cos²β+cos²γ = 1. This result is essential in understanding the symmetry and properties of crystal structures. If you would like to see a detailed proof of this equation, I recommend consulting a textbook on crystallography or speaking with a professor or expert in the field.