Find Approx Wavelength of Titanium Kbeta X-Rays

[negatifzeo]

Homework Statement

The X-ray spectrum for a typical metal is shown in Figure 31-22. Find the approximate wavelength of Kbeta X-rays emitted by titanium. (Hint: An electron in the M shell is shielded from the nucleus by the single electron in the K shell, plus all the electrons in the L shell.)
(The figure seems irrelevant to me, no useful data is privded by it, it's just a graph with no labels.)

Homework Equations

The Attempt at a Solution

I don't really know what to do here, there's several things I don't understand. I believe that K-beta X-rays are emitted when the electron drops from the 3rd excited state to the ground level. But how do you determine the energy level difference here? And when Titanium is in an "excited" state, does that mean that just one electron moved up 3 levels, or more than one? Any help in explaining this would be greatly appreciated.

 

[alphysicist]

Hi negatifzeo,

That's right; in this case, there is only one electron in the n=1 shell, and one of the electrons that is in the n=3 shell is dropping to fill the n=1 shell.

If this was a hydrogenic titanium atom with only one electron, and the electron was making a transition from n=3 to n=1, the energy levels would be found from:

$$E_n = -Z^2 \frac{13.6\mbox{ eV}}{n^2}$$

If the atom is not hydrogenic, then the equation above has to modified to account for the other electrons. The inner electrons will partly shield the nuclear charge. We account for this in the above equation by changing $Z\to Z_{\rm eff}$, where $Z_{\rm eff}$ is called the effective nuclear charge.

So mostly what this problem is about is that you need to find the effective Z values for an electron in the n=3 shell and for an electron in the n=1 shell of titanium.

Does this help?

 

[Moetasim]

Dear student,

Thank you for your question. I understand that this may seem confusing at first, but let me try to break it down for you.

Firstly, the figure provided is a graph of the X-ray spectrum for a typical metal. This means that it shows the different energies of X-rays emitted by a metal. The x-axis represents the wavelength (in nanometers) and the y-axis represents the intensity of the X-rays.

Now, let's focus on the Kbeta X-rays emitted by titanium. As you correctly stated, these X-rays are emitted when an electron in the M shell drops down to the K shell. However, in order for this to happen, the M shell electron must first be excited to a higher energy level. This can happen through various processes, such as collisions with other particles or absorption of energy.

When an electron is excited, it moves to a higher energy level. In the case of titanium, the Kbeta X-rays are emitted when an electron in the M shell drops down to the K shell. This means that the electron has moved from the third excited state to the ground state. The energy level difference between these two states can be calculated using the equation:

ΔE = E3 - E1 = (13.6 eV) * (Z^2 / n^2) * (1/n1^2 - 1/n2^2)

where Z is the atomic number (22 for titanium), n is the principal quantum number (3 for the third excited state and 1 for the ground state), and n1 and n2 are the quantum numbers of the initial and final states, respectively.

Once you have calculated the energy level difference, you can use the equation:

E = hc/λ

where E is the energy of the X-ray, h is Planck's constant, c is the speed of light, and λ is the wavelength of the X-ray.

I hope this helps to clarify things for you. If you have any further questions, please don't hesitate to ask. Good luck with your homework!