Solving Math Problem: Why is m=2, not 1?

[hidemi]

Homework Statement

An x-ray beam of wavelength 3 x~ 10^−11 m is incident on a calcite crystal of lattice spacing 0.3 nm. The smallest angle between crystal planes and the x-ray beam which will result in constructive interference is:

A. 2.9◦
B. 5.7◦
C. 12◦
D. 23◦
E. none of these

The answer is A



2 d sinθ = m λ

2x0.3x10^-9 sinθ = 2.3 x 10^-11

However, why is the m=2 ?

Relevant Equations

2 d sinθ ＝ m λ

I calculate by :
2 *3*10^−9* sinθ = 1 * 3 *10^−11
but can not get a correct answer.

If I substitute the m by 2:
2 *3 *10^−9 * sinθ = 2* 3 * 10^−11
then the answer : θ=29 is correct

I wonder why the m is 2 not 1. Could someone help me?

 

[Steve4Physics]

I calculate by :
2 *3*10^−9* sinθ = 1 * 3 *10^−11
but can not get a correct answer.

If I substitute the m by 2:
2 *3 *10^−9 * sinθ = 2* 3 * 10^−11
then the answer : θ=29 is correct

I wonder why the m is 2 not 1. Could someone help me?

You have used d = 3*10^-9m instead of d = 0.3*10^-9 m.

And (when using m=2) your equation "2 *3 *10^−9 * sinθ = 2* 3 * 10^−11" gives θ = 0.57º, not "θ=29". So you are making some arithmetic error as well (incorrect use of brackets on calculator?).

With m=1:
mλ/(2d) = 1 * 3*10⁻¹¹ / (2 * 0.3*10⁻⁹) = 0.05
θ = sin⁻¹(mλ/(2d)) = sin⁻¹(0.05) = 2.9º (Answer A).

 

[hidemi]

You have used d = 3*10^-9m instead of d = 0.3*10^-9 m.

And (when using m=2) your equation "2 *3 *10^−9 * sinθ = 2* 3 * 10^−11" gives θ = 0.57º, not "θ=29". So you are making some arithmetic error as well (incorrect use of brackets on calculator?).

With m=1:
mλ/(2d) = 1 * 3*10⁻¹¹ / (2 * 0.3*10⁻⁹) = 0.05
θ = sin⁻¹(mλ/(2d)) = sin⁻¹(0.05) = 2.9º (Answer A).

Thanks for pointing out my mistake.