Exploring Moseley's Law & Electron Screening

[big man]

Ok I've had this problem with this last part of this one question.

Show that Moseley's Law for Ka radiation may be expressed as sqrt(f)=[sqrt((3/4)*(13.6/h))]*(Z-1) where f is the x-ray frequency and z is the atomic number. (b) Check the agreement of the original 1914 data shown in Figure 9.18 with Moseley's law. Do this by comparing the least-squares slope and intercept of the Ka line in Figure 9.18 to the theoretical slope and intercept predicted by Moseley's law. (c) Is the screened charge seen by the L shell electron equal to Z-1?

Now the first two parts were really easy, but I'm not so sure on the last part. The book does say that an electron in the L shell will be partially screened by the one remaining K shell electron and so it 'sees' a nuclear charge of only Z-1. I mean this fact was needed to do the first part of the question, and I kind of understand that since there is one electron in the K shell when a vacancy is left behind that the atomic number would appear to be Z-1. But does this mean that if the K shell is filled the the screened charge seen by the L shell electron is equal to Z-2?

Sorry if this is a stupid question, but I'd just like to understand this concept.

 

[Gokul43201]

What text did this question appear in ? I'd be very disappointed to see something like this in a physics textbook.

What is shown in fig. 9.18 - only K-lines or does it also include L-lines and so on ?

Moseley's law is NOT theoretical - it is empirical. Moseley found that the square root plots looked pretty darn straight and fit a straight line to them. Even within the Bohr formulation, if you do a careful calculation, you WILL NOT end up with Moseley's Law (at least, I did not), but a more general quadratic form for f.

For part (a), when you did the calculation of the energy difference that gives rise to the K-alpha line, you unconsciously assumed that the L-shell electron (n=2, initial state) and the K-shell electron (n=1, final state) see the same screened nuclear charge Z' = Z-1. Right ? (Ignoring for now, that trouble with understanding the screened nuclear charge seen by the K-electron) If this gave you the correct answer, then the answer to (c) must be "Yes", since this is the assumption you made in (a).

However, as you point out, the above assumption was made in the context that the L-shell had a missing electron. If the L-shell does have both electrons, will the screening change ?

The answer is "yes". But will it become Z-2 ? Most probably not. The screened charge is actually a quite complicated number to calculate, and is not given simply by the value of Z minus the number of inner electrons.

What appears, in my opinion, to be hidden behind every introductory discussion of Mosely's law is a LOT of stuff that has been conveniently swept under the rug. Moseley's Law is just an approximation to empirical data that seems to work reasonably well for the K-lines. Trying to use the Bohr model to explain an empirical relationship for large multielectron atoms is a bad idea. This is way beyond the scope of the Bohr model.

The Moseley screening constant for the L-lines is about 7.4. How would you explain that number from the Bohr model ?

As for the question (c) you have to answer, I'm not sure what the best approach is. The answer depends on the context. If it is in the context of a K-alpha line, then it might be wise to answer "yes". If the plot also shows L-lines, then the question might be in the context of one of these. In that case, the answer would be "no".

That the answer to an objective question much change based on some context is really an insult to the objectivity of the physical sciences. But that's what happens when the science is used incorrectly.

 

[big man]

The question appeared in Modern Physics (third edition) by Serway, Moses and Moyer.

The diagram does actually include the L lines as well, but I'd assume that part c is really referring to the Ka line...but now I'm not so sure. By the way the paragraph in the book was worded (below) I would have assumed that the question referred to the Ka line.

This is the assumption that had to be made to do part a (exactly from the text)

For the Ka line, the K shell vacancy is filled by an electron from the L shell (n=2). But an electron in the L shell is partially screened from the nucleus by the one remaining K shell electron and so sees a nuclear charge of only Z-1.

 

[Gokul43201]

Serway is usually pretty good about not misrepresenting stuff. I'll take a look at it later today (don't have a copy nearby right now).

 

[Astronuc]

The screened charge seen by the L shell electron is effectively equal to Z-1, however the relationship becomes slightly less than (Z-1) starting around Yttrium.

For the L lines, the M electrons would be 'shielded' by 2 K electrons and 7 L electrons, but in reality the effective charge is (Z-7.4).

One has to look at the atom in terms of quantum mechanics to understand why.

 

[big man]

Thanks for taking the time to help me with this Gokul and Astronuc.

I now understand that it is more complex than I had previously thought and that it is obviously not within the scope of my unit that I'm doing, but I'm still interested in knowing more about it. I don't mean for you guys to spend your time telling me obviously , but I'm wondering if you know of any good resources that discuss this in a little more detail.

Thanks again

 

[Gokul43201]

Here's a link to Slater's Rules. The rules are a short synthesis of very careful quantum mechanical calculations of the screening of nuclear charge by other electrons.

http://www.unb.ca/fredericton/science/chem/2201/Slater's_rule.html

The number $\sigma$ on that page refers to the difference between the total nuclear charge and the screened nuclear charge.

$$Z_{eff} = Z - \sigma$$

 

[big man]

Great! thanks for that.
Really appreciate your help guys.