Answer
The second-order reflection occurs from the
same family of reflecting planes at the angle of $$\theta=6.77^{\circ}$$
Work Step by Step
Bragg's law is given by the equation
$$n\lambda=2dsin\theta$$
which governs the diffraction of waves from a crystal. $n$ is the order of diffraction, $\lambda$ is the wavelength of the wave, $d$ is the spacing between planes of a single family off which the waves are being reflected, and $\theta$ is the angle at which the reflected waves constructively interfere and form a maxima on a screen.
We are given that when $n=1$, $\theta=3.4^{\circ}$, and we need to find out the value of $\theta$ for $n=2$. We are also told that the second order reflection also takes place from the same family of planes, and since the family of planes is the same, the distance between them remains the same. Therefore the value of $d$ does not change. Our value for $\lambda$ will also not change since the experimental setup remains the same.
Using information about the first order reflection, we can calculate the value of $d$ and $\lambda$.
$$1*\lambda=2dsin(3.4)$$
So,
$$\frac{\lambda}{2d}=0.059$$
Hence,
$$\frac{\lambda}{d}=0.118$$
This ratio can now be used to find the angle for the second order diffraction (or reflection). We now have $n=2$
$$2\lambda=2dsin\theta'$$
where $\theta'$ is the angle at which the second order reflection occurs.
From the above equation we get
$$\frac{\lambda}{d}=sin\theta'$$
But we have established before that
$$\frac{\lambda}{d}=0.118$$
Thus,
$$sin\theta'=0.118$$
Thus, $$\theta'=6.77^{\circ}$$
Therefore the second order reflection from the same family of planes for this systems occurs at $6.77$ degrees.
