Bragg Reflection w/ X-Ray tube voltage

[howin]

Homework Statement

When a beam of monochromatic x-rays is incident on a particular NaCl crystal, Bragg reflection in the first order (ie w/ m=1) occurs at theta=20 degrees. The value of d = .28nm. What is the minimum voltage at which the x-ray tube can be operating?

Homework Equations

2dsin theta = m lambda (Bragg condition)
lamba = hc/eV
lambda (min) = 1240 nm / V

The Attempt at a Solution

m = 1 so the Bragg condition gives lambda=.19
then I put that into V=hc/e lambda = 1240eV nm / e(.19nm) = 6526 V
This gives me an answer with the right units, but I didn't take into account the difference between just lambda and lambda minimum. When I tried to put lambda into the equation for lambda min it just spit out the same thing I had (obviously).

 

[Jhero]

it is important to accurately interpret and solve problems. In this case, the Bragg condition equation is correct, but the value of lambda in the equation should be the minimum wavelength (lambda min) that can produce Bragg reflection in the first order. This minimum wavelength can be calculated using the formula lambda min = 2dsin theta, where d is the lattice spacing of the crystal and theta is the Bragg angle. So, in this case, the correct value of lambda min would be 2(.28 nm)sin(20 degrees) = .19 nm. This value can then be plugged into the equation for lambda min = 1240 nm / V to solve for the minimum voltage.

Additionally, it is important to note that the value of lambda in the equation V=hc/e lambda is the wavelength of the x-rays being emitted from the x-ray tube, not the wavelength required for Bragg reflection. So, using the minimum wavelength in this equation will not give the correct answer.

Correct Solution:

To find the minimum voltage at which the x-ray tube can operate, we can use the formula V = 1240 nm / lambda min, where lambda min is the minimum wavelength required for Bragg reflection in the first order.

Using the given values, we can calculate lambda min = 2dsin theta = 2(.28 nm)sin(20 degrees) = .19 nm.

Substituting this value into the equation for V, we get V = 1240 nm / .19 nm = 6526 V.

Therefore, the minimum voltage at which the x-ray tube can operate is 6526 V.