Problem statement: The Bragg angles of a certain reﬂection from copper is 47.75◦ at 20◦C but is 46.60◦ at 1000◦C. What is the coeﬃcient of linear expansion of copper? (Note: the Bragg angle θ is half of the measured diﬀraction (deﬂection) angle 2θ). Attempt at solution: Using \begin{equation} 2d sin( \theta )= n \lambda \end{equation} to find the ratio of d(T=1000) and d(T=20) and saying that this is equal to the lattice constant ratio for those temperatures I found that: \begin{equation} \frac{a(T=1000)}{a(T=20)}=\frac{sin( \theta (T=20) )}{sin( \theta ( T= 1000))} \end{equation} Which when used in the equation for the linear expansion coefficient, kappa: \begin{equation} \kappa = \frac{a(T=1000)}{a \Delta T} \end{equation} gives a value of 10^{-3} per kelvin, which is about 100 times too large when I compared it to the actual data. I know this is a basic question, but I can't see what I'm wondering what I'm doing wrong. Thanks in advance! \end{equation}
The coefficient of thermal expansion is not given by that last formula. You should have a (Δa) in the formula, the difference between the lattice constants at the two temperatures.