Analysis of X-rays Reflected Off Copper Powder

In summary, to determine the lattice parameter (a) for the cubic lattice of copper powder, we can use the Bragg equation and the given X-ray diffraction data to calculate the lattice spacing (d) for each diffraction peak. Taking the average of these values will give us an estimate for a, the length of the edge of the unit cell. The cubic lattice is a type of crystal structure where the lattice points form a cube, and in this problem, the lattice points are occupied by copper atoms.
  • #1
jbowers9
89
1

Homework Statement



X-rays 1.54 angstrom reflected off copper powder
@ 21.65º
25.21º
37.06º
44.96º
47.58º

Homework Equations



nλ = 2d sin(Θ) ; the Bragg equation

The Attempt at a Solution



The cubic latice is face centered at d110? 5 planes?
I tried plotting sin(Θ) vs. n to get the slope and calculate a, but it doesn't seem right?
How do I find the length of an edge of the unit cell and what is the cubic lattice?
 
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  • #2


Hello, thank you for your post. It seems like you are trying to determine the lattice parameter (a) of the cubic lattice based on the given X-ray diffraction data for copper powder. Here are some suggestions to help you solve this problem:

1. First, let's review the Bragg equation, nλ = 2d sin(Θ), which relates the diffraction angle (Θ) to the lattice spacing (d) and the wavelength of the X-rays (λ). In this case, n represents the diffraction order (1, 2, 3, etc.), λ is the wavelength of the X-rays (1.54 angstroms), and Θ is the diffraction angle (21.65º, 25.21º, etc.).

2. Next, we need to identify the diffraction peaks in the given data. From the angles provided, we can see that there are five diffraction peaks at 21.65º, 25.21º, 37.06º, 44.96º, and 47.58º. These correspond to the first five diffraction orders (n=1, 2, 3, 4, 5).

3. Now, we can use the Bragg equation to solve for the lattice spacing (d) for each diffraction peak. We have five equations (one for each peak) and five unknowns (d for each peak). We can rearrange the Bragg equation to solve for d: d = nλ / 2sin(Θ). Plugging in the values for n (1, 2, 3, 4, 5), λ (1.54 angstroms), and Θ (21.65º, 25.21º, etc.), we can solve for d for each peak.

4. Once we have the values for d, we can take the average to get an estimate for the lattice spacing (a) of the cubic lattice. Since we are dealing with a cubic lattice, we can assume that the lattice parameters (a, b, c) are all equal.

5. Finally, to answer your question about the cubic lattice, it is a type of crystal structure where the lattice points form a cube. In this case, the lattice points are occupied by copper atoms, and the X-rays are interacting with the copper atoms in the lattice to produce the diffraction pattern.

I hope this
 
  • #3


I would approach this problem by first familiarizing myself with the Bragg equation and its applications in X-ray diffraction. I would also research the properties of copper powder and its crystal structure in order to understand how X-rays interact with it.

Based on the given data, it appears that the X-rays have been diffracted at different angles, indicating the presence of different planes within the crystal structure of copper powder. The Bragg equation relates the diffraction angle (Θ) to the wavelength (λ) and the spacing between the crystal planes (d). By rearranging the equation, we can solve for the lattice spacing (d) using the known values of Θ and λ.

In order to determine the length of an edge of the unit cell, we need to know the crystal structure of copper powder. Copper has a face-centered cubic (FCC) crystal structure, meaning that the copper atoms are arranged in a cube with an atom at each corner and an atom in the center of each face. The lattice parameter (a) for FCC structures can be calculated using the formula a = 4d/sqrt(3), where d is the lattice spacing.

To find the lattice spacing, we can use the given data and the Bragg equation to calculate the value of d for each reflection angle. Then, using the formula for a, we can calculate the length of an edge of the unit cell. It is important to note that the given data only provides a limited number of reflection angles, so the calculated value for a may not be entirely accurate.

In addition, it is important to consider the limitations and sources of error in X-ray diffraction experiments. The quality of the X-rays, the properties of the sample, and the experimental setup can all affect the accuracy of the results. I would also consider these factors and try to minimize any potential sources of error in my analysis.

Overall, the given data on X-ray reflections off copper powder can provide valuable information about its crystal structure and properties. However, further experimentation and analysis may be needed to fully understand and characterize the sample.
 

1. What is the purpose of analyzing X-rays reflected off copper powder?

The purpose of this analysis is to study the structure and composition of the copper powder. By analyzing the X-ray diffraction patterns, we can determine the crystal structure of the powder and its chemical composition, as well as any impurities present.

2. How are X-rays reflected off copper powder?

X-rays are directed at the copper powder and are reflected off its surface. The X-rays interact with the atoms in the copper powder, causing them to scatter in different directions. This scattering produces a diffraction pattern that can be analyzed to determine the structure and composition of the powder.

3. What information can be obtained from the analysis of X-rays reflected off copper powder?

The analysis of X-rays reflected off copper powder can provide information such as the crystal structure, lattice parameters, and orientation of the powder. It can also reveal the presence of any impurities or defects in the powder.

4. How is the data from X-ray analysis of copper powder interpreted?

The data from X-ray analysis is interpreted by comparing the diffraction pattern to known patterns of different crystal structures and compositions. By matching the peaks in the pattern to those in a database, we can determine the structure and composition of the copper powder.

5. Are there any limitations to analyzing X-rays reflected off copper powder?

Yes, there are limitations to this method of analysis. X-rays can only penetrate a certain depth into the powder, so the analysis is limited to the surface layers. Additionally, the quality of the data can be affected by factors such as sample preparation, instrument calibration, and the presence of other materials in the sample.

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