If the planes of a crystal are 3.50 Å (1 Å = 10-10 m = 1 Ångstrom unit) apart, (a) what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs when the waves strike the planes at an angle of 22.0°, and in what part of the electromagnetic spectrum do these waves lie?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. An emerging class of two dimensional material termed mxene has ac lattice parameter. A planar spacing of 9.23 years. If the electromagnetic ray is made incident on the planes at an angle of 30 degrees, determine its wavelength for the first interference maximum, also determine the region in which the electromagnetic ray falls in terms of its wavelength. Awesome. OK. So we're given some multiple choice answers. Let's read them off to see what our final answer might be. A is LAMBDA equals 450 nanometers lies in the this region. B is lambda equals 1.1 0.51 nanometers lies in the IR infrared region. C is lambda equals 0.108 nanometers lies in the X ray region and D is lambda equals 0.923 nanometers lies in the X ray region. OK. So first off, let us recall Bragg's Law equation. So Bragg's Law equation states that two multiplied by the interp planar spacing multiplied by sine theta is equal to the nth order multiplied by a wavelength. So N is some integer hole number for the order. OK. So let's write down all of our known and unknown variables. So D equals 9.23 years, but we need to convert to meters. So when we convert that to meters, we should get 9.23 multiplied by 10 to the power of negative 10 m. We know that the in this case equals 30 this is given to us in the problem, we do not know what the wavelength is. We're trying to find the wavelength for the interference maximum. And we know that the end value is equal to one because we're dealing with a first order condition. OK. So let's plug in all of our known variables to solve for the wavelength. So since we know that N equals one, so it'll just be one multiplied by lambda the wavelength. So we know that the wavelength is equal to two multiplied by D which was 9.23 multiplied by 10 to the power of negative 10 meters multiplied by sign of 30 degrees. OK. So we plug into a calculator, we should get that Lambda is equal to 9.23 multiplied by 10 to the power of negative 10 m. But when we convert this two nanometers, which all we have to do is just multiply by 10 to the power of nine, we should get 0.923 nanometers. So that is our lambda. So lambda equals the wavelength equals 0.923 nanometers. OK. So when we refer to electromagnetic spectrum chart or looking up our nanometer value, it determines that it lies within the X ray region. So OK. So it lies in the X ray region hooray, we did it, we finished solving this problem and this one was a quick one. OK. So that means that our multiple choice answer, the correct one is D wavelength equals. So lambda equals 0.923 nanometers and it lies in the X ray region. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.