X-radiation from a molybdenum target (0.626 A) is incident on a crystal with adjacent atomic planes spaced 4.00x10(-10) m apart. Find three smallest angles at which intensity maxima occur in the diffracted beam?
To find the three smallest angles at which intensity maxima occur in the diffracted beam, we can use Bragg's law. Bragg's law relates the angle of incidence, the angle of diffraction, and the spacing of atomic planes in a crystal.
Let's start by understanding Bragg's law:
Bragg's law is given by the equation: nλ = 2d sin(θ),
where:
- n is the order of the maximum (1 for the first maximum, 2 for the second maximum, and so on)
- λ is the wavelength of the incident radiation
- d is the spacing between two adjacent atomic planes in the crystal
- θ is the angle between the incident beam and the diffracted beam
In this case:
- The wavelength of the incident radiation, λ, is given as 0.626 Å (angstroms), which is equivalent to 0.626 x 10^(-10) m.
- The spacing between adjacent atomic planes, d, is given as 4.00 x 10^(-10) m.
Now, to find the three smallest angles at which intensity maxima occur, we need to substitute the values into Bragg's law and solve for θ for the first three orders (n = 1, 2, and 3). Since the question provides information only about molybdenum as the target material, we'll assume it's a single crystal without any specific orientation.
First Maximum (n = 1):
nλ = 2d sin(θ)
1 * (0.626 x 10^(-10)) = 2 * (4.00 x 10^(-10)) * sin(θ1)
sin(θ1) = (0.626 x 10^(-10)) / (2 * 4.00 x 10^(-10))
Now, calculate sin(θ1) and then take the inverse sine to find θ1:
θ1 = sin^(-1)([(0.626 x 10^(-10)) / (2 * 4.00 x 10^(-10))])
Repeat this process for the second (n = 2) and third (n = 3) maxima by substituting the respective values of n in the Bragg's law equation and solving for θ2 and θ3.
Calculating the angles at which intensity maxima occur using Bragg's law will yield the three smallest angles at which the diffracted beam will have intensity maxima.
To find the three smallest angles at which intensity maxima occur in the diffracted beam, we can use Bragg's Law. Bragg's Law states that for constructive interference to occur, the path difference between waves diffracted from adjacent atomic planes of a crystal should be an integer multiple of the wavelength.
The formula for Bragg's Law is given as: nλ = 2dsinθ
Where:
- n is the order of the diffraction (integer)
- λ is the wavelength of the X-ray radiation incident on the crystal (0.626 Å converted to meters)
- d is the spacing between adjacent atomic planes (4.00x10^(-10) m)
- θ is the angle of diffraction
Let's calculate the three smallest angles using Bragg's Law:
1. For the first maximum (n=1):
0.626 x 10^(-10) m = 2 x (4.00 x 10^(-10) m) x sin(θ1)
sin(θ1) = 0.626 / (2 x 4.00)
θ1 = sin^(-1) (0.626 / (2 x 4.00))
2. For the second maximum (n=2):
0.626 x 10^(-10) m = 2 x (4.00 x 10^(-10) m) x sin(θ2)
sin(θ2) = 0.626 / (2 x 4.00)
θ2 = sin^(-1) (0.626 / (2 x 4.00))
3. For the third maximum (n=3):
0.626 x 10^(-10) m = 2 x (4.00 x 10^(-10) m) x sin(θ3)
sin(θ3) = 0.626 / (2 x 4.00)
θ3 = sin^(-1) (0.626 / (2 x 4.00))
Now, you can use a calculator to find the values of θ1, θ2, and θ3 by taking the inverse sine (sin^(-1)) of the ratios.
These angles will give you the three smallest angles at which intensity maxima occur in the diffracted beam.