Best Formula for Fitting Nuclear Charge Radius Data

[kelly0303]

Hello! Can someone point me towards some papers/readings providing formulas (derived theoretically or based on experimental data) for the nuclear charge radius? Almost all the papers where I found a formula for that are of the form $aA^b+c$, where a, b and c are constants and A is the mass number (N+Z) of the given nucleus. I guess these work well in certain situations, but the formula is obviously wrong (switching the number of protons and neutrons gives the same value of A, but obviously the charge radius will be very different). So are there some formulas more general (and hence hopefully closer to the true formula)? Thank you!

 

[hutchphd]

switching the number of protons and neutrons gives the same value of A

This is a small brain perturbation. Don't worry I get them all the time (of course I'm old!) . Atomic number

 

[kelly0303]

This is a small brain perturbation. Don't worry I get them all the time (of course I'm old!) . Atomic number

Fixed it :)

 

[hutchphd]

Oh so the problem remains...too bad I am totally ignorant of nuclear physics. Seems odd to me too then.

 

[mfb]

(switching the number of protons and neutrons gives the same value of A, but obviously the charge radius will be very different)

Why is this so obvious? The size of the nucleus will be very similar, the relative distribution of the protons will be very similar, this leads to a very similar charge radius. The charge radius doesn't tell you how many charges are in the nucleus, it just gives you the average "distance" to the center.

 

[kelly0303]

Why is this so obvious? The size of the nucleus will be very similar, the relative distribution of the protons will be very similar, this leads to a very similar charge radius. The charge radius doesn't tell you how many charges are in the nucleus, it just gives you the average "distance" to the center.

One of the main properties of interest in nuclear physics (and astrophysics) is the symmetry energy. One of the best ways to calculate it is to correlate it to the neutron skin of certain elements, which is defined as the difference between the neutron radius and charge radius for a given nucleus. Measuring he proton radius is relatively easy, but measuring the neutron one is extremely complicated. The best way nowadays to extract the data experimentally is to measure the charge radius for a normal nucleus (N,Z) then create artificially (at a radioactive beam facility for example) the mirror nucleus (Z,N) i.e. the number of protons and neutrons is switched and measure the charge radius of this new nucleus. From there, taking into account correction to the fact that the nuclear force is not perfectly identical between protons and neutrons, you can infer the neutron radius of (N,Z) from the charge radius of (Z,N). According to your argument (and hence assuming that the formula I mentioned is good enough), the (N,Z) and (Z,N) nuclei will all have the exactly same charge radius (as they have the same A) hence the neutron skin of any nucleus will be basically zero, hence we can't extract any information about symmetry energy. There are a few more cases where assuming (N,Z) and (Z,N) have the same charge radius is fundamentally wrong. The point is the same: that formula is fundamentally flawed and my questions still remains: are there any other more realistic, theoretical or empirical formulas for the charge radius? Thank you!

 

[mfb]

If you think everyone uses a bad formula and you think you know better then publish that.
Otherwise I'm not sure what you hope to get.

 

[kelly0303]

If you think everyone uses a bad formula and you think you know better then publish that.
Otherwise I'm not sure what you hope to get.

If I knew a better formula, I wouldn't post here asking for a better formula... I am just saying that the formula I mentioned above is obviously flawed. There are actually many paper saying that and proposing improved formulas (which I would be glad to share if you want) i.e. I don't "think" it's a bad formula, it was proven it's a bad formula (simply by calculating the rms error it predicts along the nuclear chart). I was asking if someone, ideally working in the area, knows of a better formula. So what I hope to get is a better formula for the charge radius.

 

[Vanadium 50]

The charge distribution is a distribution. Converting this to a single number, the charge radius, loses information. I think you're stuck with this fact.

 

[kelly0303]

The charge distribution is a distribution. Converting this to a single number, the charge radius, loses information. I think you're stuck with this fact.

The charge distribution is a distribution indeed. But that is not what people are using in practice. What they are using is the RMS charge radius, which is a number for each element/isotope, for which people are trying to find a formula which is what I am asking about. Here are some papers that hopefully will make clear what I am asking for: https://journals.aps.org/prc/pdf/10.1103/PhysRevC.94.064315?casa_token=U_I7qSde7ekAAAAA%3AzdLxMoZIA6BLFomfDOhpaiXqcDhdd7t3yno5J3l7FS9by0n7AhMhTGum5pJqYtEampg8pFybu45AcmQ, paper2, https://journals.aps.org/prc/pdf/10.1103/PhysRevC.88.011301?casa_token=qxPWthdy5i4AAAAA%3A2_OTCQHwFKDO9rG23Un92Hv2kmItyehPgM-6_Mu0V1t-inRsYNrABM3xraZgyU43UmSL9WIxJrZqoU8, paper4 . Of course there are a lot more papers out there that i don't know about, trying to find better formulas for the nuclear charge radius, so again, my questions is if people can point me towards some of these papers. Is that such a bad question to ask on this forum?

 

[Vanadium 50]

I understand that people don't use the distribution. My point is that anything that reduces the distribution to a single number, whether it be the RMS, the radius in which 90% of the charge is enclosed, or something else loses in formation. The loss comes from this reduction, not the particular number that the distribution is reduced to,

 

[kelly0303]

I understand that people don't use the distribution. My point is that anything that reduces the distribution to a single number, whether it be the RMS, the radius in which 90% of the charge is enclosed, or something else loses in formation. The loss comes from this reduction, not the particular number that the distribution is reduced to,

This has nothing to do with my questions... People DO use that one number (the charge radius) in almost all the nuclear calculations (whether it loses information or not). And if you looked over the papers I mentioned, some approximations are significantly better than others (especially than the one I mentioned in the original post). Of course at the end of the day they are all approximations (as we don't have a nuclear theory starting from first principles i.e. QCD), but some approximations are better than others. So my question is simply if people can point me towards better approximations of that one number (charge radius), similar to the stuff in the papers I mentioned. I am not asking if using that number is the best way to do calculations, or what is the best way to do them. I am just asking for the best approximations for that one particular parameter.

 

[Vanadium 50]

This has nothing to do with my questions...

It actually does.

 

[kelly0303]

It actually does.

I want to know the formula for a parameter that is used in calculations in the nuclear physics community. For example, most of the programs used for many-body calculations in nuclear physics require a formula as input for the nuclear charge radius (again I can provide plenty of papers to show that). Whether using that one number (the nuclear charge radius) is the best way to do it or not is a totally different problem. Actually what are the best parameters to use in these simulations is still an open problem and this is not what I am asking i.e. I don't want to know what are the best parameters to use in simulations/calculations, I just want to know the best approximations for a parameter that is already used. If you don't have an answer for that questions is totally fine, but, again, I don't see how being told that is not the best parameters to use or not helps me (for example) in using a program that requires that specific parameter as input. Again, your arguments are totally valid in the grand scheme of nuclear physics. But I am not aiming to come up with better theoretical models, I just need an approximation for a parameters that people are using already in the theoretical models.

 

[BillKet]

This has nothing to do with my questions... People DO use that one number (the charge radius) in almost all the nuclear calculations (whether it loses information or not). And if you looked over the papers I mentioned, some approximations are significantly better than others (especially than the one I mentioned in the original post). Of course at the end of the day they are all approximations (as we don't have a nuclear theory starting from first principles i.e. QCD), but some approximations are better than others. So my question is simply if people can point me towards better approximations of that one number (charge radius), similar to the stuff in the papers I mentioned. I am not asking if using that number is the best way to do calculations, or what is the best way to do them. I am just asking for the best approximations for that one particular parameter.

That's right. The formula you mentioned comes mainly from assuming that the nucleus is a liquid drop i.e. it has a constant density (initially the formula was simply $r_0A^{1/3}$, with $r_0$ being a constant). People still use this formula in many situation simply because it is easier to use and one can get some quick results. However, as you said, it has many flaws. Beside the obvious lack of dependence on the nuclear charge Z, it completely ignores other effects such as shell effects, pairing or short term correlations and if you need accurate results it becomes quite useless. Here is a paper giving a formula that I am currently using (it reduces the errors from 0.09 fm to 0.02 fm). It's not perfect (of course!), but oh well, better than nothing until we figure out how to derive it from QCD ;)

 

[mfb]

The best way to reduce the distribution to a single number will depend on the way this single number is used.

 

[BillKet]

The best way to reduce the distribution to a single number will depend on the way this single number is used.

I am not sure what you mean by that. If you mean that the charge radius might not always be the best parameter to use, then that is right. No one is claiming that, and there are of course other parameters that enters the calculations beside the nuclear charge radius. BUT, if you need that one parameter, which is the nuclear charge radius, how well you approximate it has nothing to do with how you use it i.e. what theoretical model you are testing, but it has to do with how well it fits the data. The way we do it is to first find a formula that approximates the measured charge radius as well as possible (here is a link with the latest measurements of charge radius), then we use that formula in the calculations, not the other way around. I think that what @kelly0303 is asking is simply for the best formula to fit the data from the link I just provided. It's simply a matter of finding the best fit to the some data points i.e. the one giving the best chi-square (although we use the rms error). It has nothing to do (in the context of this question) with the implications of actually using this measured data as input to theoretical calculations.

 

[kelly0303]

I am not sure what you mean by that. If you mean that the charge radius might not always be the best parameter to use, then that is right. No one is claiming that, and there are of course other parameters that enters the calculations beside the nuclear charge radius. BUT, if you need that one parameter, which is the nuclear charge radius, how well you approximate it has nothing to do with how you use it i.e. what theoretical model you are testing, but it has to do with how well it fits the data. The way we do it is to first find a formula that approximates the measured charge radius as well as possible (here is a link with the latest measurements of charge radius), then we use that formula in the calculations, not the other way around. I think that what @kelly0303 is asking is simply for the best formula to fit the data from the link I just provided. It's simply a matter of finding the best fit to the some data points i.e. the one giving the best chi-square (although we use the rms error). It has nothing to do (in the context of this question) with the implications of actually using this measured data as input to theoretical calculations.

Thank you so much for this! Yes! That's exactly what I need, a fit for that data. Also thank you for the link, I was using an older data set. I found that paper that you mentioned, too, but thank you for it. Please let me know if you find a better one.