High-Precision Sigma_Coulex / Sigma_Rutherford

[JMAllmond]

I've a question about GOSIA regarding ratios of Sigma_Coulex(2+) / Sigma_Rutherford = Sigma_Rutherford * P(2+) / Sigma_Rutherford = P(2+); this is for a simple two-state system like that of 124Sn (0+ ground and 2+ first excited state). i.e., I'm studying 124Sn on a 12C target!

When I calculate Sigma_Coulex(2+) and Sigma_Rutherford with NCM=2 (lab ---(inelastic)--->cm) versus Sigma_Coulex(2+) and Sigma_Rutherford with NCM=1 (lab ---(elastic)--->cm), the cross sections change as one would expect but the ratios, Sigma_Coulex(2+)/Sigma_Rutherford, remain unchanged. According to the GOSIA manual (and according to what one would logically think), it seems that one should calculate Sigma_Coulex(2+) with NCM=2 and Sigma_Rutherford with NCM=1; this gives a Sigma_Coulex(2+)/Sigma_Rutherford ratio that is nearly 3% to 5% different than if the NCM flags were the same for both calculations.

After some thought, it seems to me that when you calculate Sigma_Coulex(2+) with NCM=2, the P(2+) component is calculated with respect to NCM=2 (as it should be), but the Sigma_Rutherford component of Sigma_Coulex(2+) is also calculated with respect to NCM=2 (at least this seems to be how GOSIA is handling it), which then wouldn't cancel with the Sigma_Rutherford in the denominator calculated with NCM=1. E.g., Gosia seems to be doing Sigma_Coulex(2+)(NCM2)/Sigma_Rutherford(NCM1) = Sigma_Rutherford(NCM2)*P(2+)(NCM2)/Sigma_Rutherford(NCM1) and hence the Sigma_Rutherford's will not cancel when calculating Sigma_Coulex and Sigma_Rutherford with different NCM values. Because my data is particle-gamma/particle = P(2+), shouldn't I be comparing my data to GOSIA calculations using the same NCM flag so that the Rutherfords cancel, Sigma_Coulex(2+)(NCM2)/Sigma_Rutherford(NCM2)
=Sigma_Rutherford(NCM2)*P(2+)(NCM2)/Sigma_Rutherford(NCM2)
=P(2+)(NCM2)?

Maybe I'm just misunderstanding the problem but it seems to me that Gosia isn't calculating Sigma_Coulex properly b/c it's using NCM=2 for the Sigma_Rutherford component of Sigma_Coulex...... Granted this is a small error but the error in my data is small enough to where this is a concern for me.

Cheers,

Mitch

[hayes]

Hi Mitch,

I'm still trying to completely understand the output you're getting. Here is something to think about that might get us closer to an answer. I've also asked for some help in understanding the problem; maybe someone else will give you some better ideas...

1. Usually, when someone says the Rutherford cross section is measured, it is really the inelastic cross section that is measured. I think that you got the actual Rutherford cross section by comparing the particle counts with the p-gamma counts--is that right?
2. In the expression Sigma_Coulex(2+) / Sigma_Rutherford = Sigma_Rutherford * P(2+) / Sigma_Rutherford = P(2+), the numerator on the right really is Integral[differential_Sigma_Rutherford * P(2+)], since the probability and differential Rutherford c.s. change with scattering angle and beam energy through the target.
3. Choosing an effective Q-value using NCM>1 is a necessary approximation in the semiclassical code, since the probabilities would have to be known before the calculation to get the exact scattering kinematics.
4. In the "typical" Gosia problems, the NCM state used for the inelastic cross section is not a concern, because Gosia usually is used to analyze relative gamma-ray yields.

I think that much is clear, but let me know if you disagree.

Since Gosia seems to be using the same NCM value for both the integrated gamma-ray cross section and the Rutherford/inelastic cross section, can you run two separate calculations, one for each? I think that you are fitting by hand to normalize to the Rutherford c.s., so this might not make for much more work. You could write a script to call Gosia twice, once with NCM=1 and once with NCM=2.

Let me know if I'm way off.

Adam

[JMAllmond]

Indeed, the particle singles is really the Rutherford + Coulex. However, for 124Sn on 12C, the Rutherford cross section is on the order of 384 mb while the Coulex cross section is on the order of 2 mb. Therefore, particle singles = Rutherford + Coulex = Rutherford; not including the Coulex component only introduces a 0.5% error (0.25% in the extracted M.E.).

You are correct, I am doing particle-gamma yield / particle yield and comparing this to GOSIA calculations/simulations "by hand". I've done this with gosia for
Coulex(NCM=2) / Rutherford (NCM=1),
Coulex(NCM=2) / Rutherford (NCM=2),
Coulex(NCM=1) / Rutherford (NCM=1) .

The last two gosia calculations (Coulex and Rutherford done with the same NCM flag) give the exact same ratio value. The first of the gosia calculations gives a value 3% to 5% different than the last two (i.e., 1.5% to 2.5% in the extracted M.E.). Yes, these are all very small effects but I care nonetheless!!!

Indeed, gosia must do int(Rutherford*P)/int(Rutherford), which in this specific case is equivalent to the true form of int(Rutherford*P)/[int(Rutherford) + int(Rutherford*P)] b/c int(Rutherford)>>int(Rutherford*P). However, b/c you say that it is a necessary approximation to treat Rutherford in the numerator with NCM=2 (obviously, P must be treated with NCM=2), wouldn't that approximation also make it necessary to approximate the Rutherford in the denominator with NCM=2 as well to get an "effective" cancelation....? Despite that the real Rutherford should be done with NCM=1.

E.g., If I were to integrate a distribution to get an average via <w>=int(w*dOmega)/int(dOmega), you'd expect dOmega to take the same form in both the numerator and denominator or you'll not get <w>. Therefore, if I had to approximate dOmega in the numerator to dOmega', I'd think that you'd need to do the same in the denominator. Am I wrong with this analogy?

Cheers,

Mitch

[Cline]

Hi Mitch:
What you mention is a well-known problem in reaction kinematics in that the Jacobian for the transformation from the lab to CM coordinates is Q-value dependent. We all make measurements at a fixed lab angle whereas some reaction codes, such as the semi-classical approximation, assume a fixed CM angle which defines a single fixed trajectory. To be rigorously correct you have to run the semi-classical codes for each value of NCM since you are measuring the cross section at slightly different CM angles for each state, that is, for slightly different trajectories. As a result one needs to run Gosia for each important value of NCM and select the appropriate cross sections.
The above set of calculations can be used to compare with the experimental result but this depends on what you are observing. For example in the 1960's we performed reorientation measurements by detecting the scattered ions using a high-resolution magnetic spectrometer for which both the ground state and excited states yields were completely resolved. In this case I computed the ratio of the 2+ cross section(calculated for NCM=2) divided by the ground state cross section calculated for NCM=1 which was the measured observable. For most modern experiments the gammas resolve the excited states but the particle singles detector cannot resolve the excited states from the ground state in the singles spectrum and thus the singles peak is a sum of the cross sections of the ground state for NCM=1, plus the first excited state for NCM=2, plus the cross section the state 3 for NCM=3, etc. In practice this complicated procedure is unnecessary since the excitation probability is large only when the first excited state excitation energy is low, i.e. rotational nuclei, for which the Q value is a very small fraction of the bombarding energy. For example, for strongly-deformed nuclei the negative Q value is a few hundred keV whereas the bombarding energy is around a GeV. When the excitation energy is high enough to be a significant fraction of the bombarding energy then the excitation probability is very small and thus the excited state is a negligible component in computing the singles yield.

In conclusion Gosia can be used to calculate the experimental observable that you measured by summing the values computed using the appropriate values of NCM corresponding to your experiment.

The high sensitivity to the Q-value that you are encountering is because you are using extreme inverse kinematics near the maximum scattering angle where the lab-to-CM angle transformation is very sensitive to the lab angle. The development of OP, INTI by Nigel Warr was induced in order to handle this angle sensitivity of the maximum scattering angle to the Q-value.

I hope that this helps.

Best regards

Doug

[warr]

To add to Doug's comment. It isn't just the Jacobian. The Jacobian is part of the Dsig term which is calculated in CMLAB:

Code: Select all

         Dsig = 250.*r3*SQRT(EP(lexp)/(EP(lexp)-ared*EN(NCM)))
     &          *dista*dista*(EPS(lexp))**4


where r3 is the Jacobian, EPS(lexp) is the eccentricity, Ep the energy of the meshpoint, ared is the reduced mass, En(NCM) is the energy of the state NCM. All three of these terms depend on NCM implicitly. On the other hand, dista varies with Ep, but not NCM.

The four plots attached below show the ratio between the NCM = 1 and NCM = 2 values for the Jacobian, the eccentricity raised to the fourth power and the square root term in Dsig as well as Dsig itself.

compare.png (107.24 KiB) Viewed 5629 times

So the switch between NCM = 1 and NCM = 2 really specifies a different trajectory, which has a different eccentricity and a different particle speed (which determines how long it has to interact). It also modifies the mapping of solid angle in the center of mass frame into the lab frame.

The change is of a few percent, so do not mix NCM = 1 and NCM = 2 values in the same calculation.

Nigel

[JMAllmond]

Thank you Nigel for such a detailed response.

So what you're saying is that even though the true Rutherford is obviously with NCM=1, when comparing Gosia calculations to experimental p-gamma / p data, one should use the same NCM value for the calculated Sigma_Coulex / Sigma_Rutherford ratio?

Thanks a lot for your help! I very much appreciate it!

Cheers,

Mitch

[JMAllmond]

I want to post a follow up that was discussed outside of the user forum in case someone takes this discussion to heart; the conclusion above is wrong!

Essentially, when comparing particle-gamma / particle data to Gosia calculations, you must do

Sigma_Coulex(NCM=2) / Sigma_Rutherford(NCM=1)

[hayes]

Thanks for the follow-up, Mitch. This would be a good topic for the Wiki. I did a quick search, and the closest topic I can find is this one:

http://www-user.pas.rochester.edu/~gosi ... le_singles

It would be great if you could add even the brief summary to the Wiki. We still don't have all Latex math extensions properly installed, but you can enter Latex, and we can upgrade the rendering later.

Adam