1 (edited by T. Rauscher 2013-10-11 17:47:26)

Topic: FAQ: Photodisintegration experiments for astrophysics

Can measurements of nuclear photodisintegration help to constrain astrophysical photodisintegration rates?

I get this or similar questions so often recently that I decided to write a short summary on this topic here.
The short answer is: No, only in a very limited manner.

The long answer, of course, is more differentiated. I cannot easily show equations or graphs here, therefore I try a verbal summary without those. For further details, please take a look at the literature at the end of this post. Here I just combine and summarize what is detailed in those articles. A review of all relevant effects is also found in [1] and photodisintegration in the gamma-process is discussed in [12]. Tables of sensitivities of cross sections and stellar rates to various nuclear properties and all ground-state (g.s.) contributions to the stellar rate (see below) are given in [10].

1. Experiments vs rates

The fact that photodisintegrations play a role in certain astrophysical processes often gives rise to the misconception that a photodisintegration measurement will directly help with determining the astrophysical reaction rate [7,8]. This is not the case. The reason is that, even when exciting the nucleus to an astrophysically relevant energy (and this is important because at higher energies so many additional effects are coming into play that it becomes difficult to impossible to extract the ones relevant in the astrophysical energy range), the photon-induced transition from the ground state with subsequent particle emissions is only a tiny fraction of all transitions appearing in the astrophysical case [1-4,10-13].

The majority of the relevant gamma-transitions have lower gamma-energy (about 2-4 MeV) and start at an already excited state of the nucleus (and therefore will probably also select different spin and parity quantum numbers) [1,2]. The ground state contribution to the stellar photodisintegration rate is always very small, often less than a percent [1,3,8,10-12]. Therefore, it is always recommended to measure the capture because there the ground state contribution is larger and thus the rate can be better constrained by experiment (although some theory may still be necessary) [1,3,10,13].

(The situation may be different for nuclei with very low particle separation energy (or reaction Q-value) but these nuclei are far from stability and not yet accessible by experiments [2,10].)

Why is this so? The reasons are somewhat hard to visualize without checking the equations. Three components act together [1]:
a) The principle of detailed balance in nuclear reactions states that there is a simple relation between forward and reverse reaction between a discrete initial and a final state. The cross sections are related by a simple phase space factor.
b) In astrophysics, most nuclei are in thermal equilibrium with the surrounding plasma and therefore are present also as excited nuclei, not just nuclei in their ground states. The fraction of nuclei in a specific excited state is given by its spin and a Boltzmann population factor depending on the energy of the excited state Ex and the plasma temperature: P=(2J+1)*exp(-Ex/(kT))
c) The projectile (or gamma) energies are also thermally distributed in a plasma. Obviously, the same energy distribution must act on a nucleus in the ground state and one in an excited state. This implies that the respective energy distributions (for example, Maxwell-Boltzmann or Planck distributions) are shifted relative to the ground state energy E0 and extending to higher excitation energy E0+Ex when acting on an excited state.

It is important to realize that a), b), and c) apply to all initial and final nuclei simultaneously in a reaction in an astrophysical environment.

Already from the notion that a capture reaction (with positive Q-value) releases gammas to a wider range of final states than particles (e.g. neutrons) are released in its reverse (photodisintegration) reaction we can see that a larger number of gamma transitions are relevant than particle transitions [2] (see also chapter 3 below). Astrophysical rates obey reciprocity between forward and backward reactions (in a rate all reactions on ground and excited target states are included and integrated over the energy distribution of the projectiles). This can be found by combining the principle of detailed balance a) and the thermal population of target states c). It also is easy to show that also in this case the number of participating transitions is larger in the exit channel of a reaction with positive Q-value [1,10,11].

Sometimes it leads to confusion when only looking at the Boltzmann population factors of the excited states as given in b). They seem to give similar weights to excited states in the initial and final channel. However, it is incorrect to take only these factors as a measure of the importance of initial states. Rather, folding these factors with the shifted energy distributions of the projectiles (or gammas) of c) and expressing everything in an energy scale relative to the ground state leads to a transformed weight: W=(2J+1)(1-Ex/Eeff), where Eeff is in the effectively relevant energy range for an astrophysical rate (aka the Gamow window) [1]. This has two implications. Firstly, it is a linear weight, more slowly declining than the exponentially declining Boltzmann factor. Secondly, the weight is relative to the Gamow energy which is higher in a reaction with negative Q-value. (When this energy is EG in the capture, then it is EG+|Q| in the photodisintegration.) Therefore, the weights decline more slowly and reach higher in the case of photodisintegration than in the case of capture.

Finally, folding the actual astrophysical weights W with the relevant gamma transition strengths (which are - close to stability - roughly about halfway between the ground state and the particle separation energy [2]) shows that the ground state contribution to astrophysical photodisintegration rates is small.

2. Implementation in reaction networks

For this and other (numerical) reasons, in astrophysical reaction network calculations only reactions with positive Q-value are included (also only measurements of such, with few exceptions) and the reverse direction is calculated using the reciprocity of stellar rates (which can be derived from a)+b)+c) above) [1]. Therefore, even if a measurement of a rate with negative Q-value would be able to constrain the astrophysical rate, it would have to be converted to its reverse first before being used in an astrophysical calculation [1,3,10,13].

3. Relevant energies

There are two important types of energies appearing in capture/photodisintegration studies. They are also connected.

The first type of relevant energy range are the projectile energies which give most of the contributions to the reaction rate integral. These are always given relative to the ground state of the target nucleus. They also specify the most important formation energies of the compound nucleus. The simple formula for the Gamow window of a reaction is only applicable to light nuclei, for intermediate and heavy targets, the reaction rate integrand (for the stellar rate including thermally excited target states) has to be examined [9]. The maxima and widths of the integrands are tabulated for a large number of nuclei (see this topic). The tables are for reactions with positive Q-value. As pointed out above, the relevant energy window (relative to the ground state of the final nucleus) for the reverse reaction can be computed easily by adding the Q-value to the tabulated energy window.

The other interesting energy range is that of the gamma-energies determining the gamma-width appearing in the reaction cross section. This is not necessarily the one given by the Gamow window. Rather, the Gamow window determines the excitation energy range in which the compound nucleus is formed and from which it decays (by gamma and/or particle emission). Although transitions from EG+Q to an excited state with Ex above the ground state are weaker because the relative gamma energy EG+Q-Ex is smaller, there are more levels available due to the exponentially increasing level density with increasing Ex. Thus, there is a competition between the decreasing individual gamma strengths and the increasing level density, leading to a maximum in the summed gamma strengths at about 3-4 MeV below EG+Q [2]. (This may only be different in nuclei with low level density up to EG+Q (perhaps because of a small Q) where single gamma transitions are important.) This is only slightly modified by the linear weights W acting on the excited state transitions in the astrophysical rate.

4. How can experiments help?

From the above, it is found that photodisintegration (with real photons or by Coulomb excitation) cannot directly constrain the astrophysical reaction rate. Capture reactions are much better suited for this as their ground state contribution to the stellar rate is much larger than for photodisintegrations. See [10] for complete tables of g.s. contributions across the nuclear chart; it can be seen that STELLAR photodisintegration rates almost always have extremely tiny g.s. contributions (with very few exceptions [13]). Therefore what is measured is closer to the value required in astrophysics. Nevertheless, further corrections from theoretical calculations may still be needed.

A (g,n) experiment thus tests the g-ray strength function (or gamma width) at much larger gamma-ray energy than the one of the g-rays actually contributing mostly to the STELLAR photodisintegration rate. In the past, a few (g,n) cross sections were measured (see, e.g., [6-8]) and an astrophysical rate was derived by renormalizing the predicted rate by the same factor as found when comparing the measured result with a prediction in the same model as used to calculate the rate. This implicitly assumes that any discrepancy found between experiment and theory for the much larger gamma energy of the g.s. photodisintegration equally applies also to the actually relevant transitions with smaller relative energy (or are due to the particle transitions in the exit channel). Usually, this is not a good assumption because the photon strength function behaves differently at low energy and the allowed partial waves in the particle transitions may have different relative angular momentum than the ones appearing in the astrophysical rate. Therefore the experiment does not really constrain the astrophysical rate.

Studying the behavior of gamma strength functions itself helps the predictions of astrophysical rates, both for captures and photodisintegrations. As mentioned above, the relevant gamma energies are of the order of 3-4 MeV [2]. Changes in the strength function within this energy range directly affect astrophysical capture and photodisintegration rates, changes outside that range are of smaller importance [2,12]. Unfortunately, this gamma-energy range can only be accessed indirectly, requiring a combination of experiment and theory. For example, (g,g') data can help to constrain the low-energy gamma-strength or -width. The (g,n) data cannot be used for this by themselves. Recent experiments (e.g., [14]) realized this and fit theoretical (g,g') data with theoretical models to use such  models for the calculation of the stellar photodisintegration rate. Note that in such experiments also any (g,n) cross section, if measured, does not contribute much to the actual constraint of the stellar rate.

Because of the relevance of reactions proceeding on thermally excited target states in an astrophysical plasma, it is important to study (particle) transitions from these excited states. This can be done by studying the inverse transitions to such excited states in a particle exit channel. Photodisintegrations can achieve this, as can inelastic particle scattering (e.g., (n,n') [5,11,12]). One has to be careful in the interpretation of such experiments, however, because starting from a specific ground state introduces a selection of possible quantum numbers (relative angular momenta) which may be different from the astrophysically relevant ones. Nevertheless, this may allow testing the predictions of particle transitions to/from excited states by theoretical models and also the predicted ratios to the ground state transitions (e.g., (g,n1)/(g,n0), (g,n2)/(g,n0), ...; (g,a1)/(g,a0), (g,a2)/(g,a0), ...). [12]

(Additional information required in predictions of captures and photodisintegrations are low-lying discrete states and nuclear level densities above those discrete states.)

What is essential in all kind of experiments is to really measure within the relevant energy region! The gamma strength function has to be known around 3-4 MeV [2]. The compound formation energy is given by the Gamow window [9]. For neutron captures, the upper end of the relevant energy window is at most at 0.2 MeV, even at the high plasma temperatures encountered in explosive burning environments. This translates into an excitation energy of the compound nucleus of S_n+0.2 MeV (with the neutron separation energy S_n). The energy window of charged particles is shifted to slightly higher energy due to the Coulomb barrier but does not exceed a few MeV [1,9]. Measuring at much larger energy than the astrophysical one does not yield much relevant information in most cases. This is because at higher energy the cross sections show a different sensitivity to photon- and particle-strengths than at astrophysical energies and higher partial waves also play a role [1,10]. Furthermore, additional reaction mechanisms may occur which are not appearing in the astrophysical energy range [1]. At high energy, so many additional effects are coming into play that it becomes difficult to impossible to extract and compare the ones relevant in the astrophysical energy range.

5. Conclusion
Much can be learned from photodisintegration experiments (including experiments with real photons and Coulomb excitation). However, it should be clear that there are severe limitations in this method regarding astrophysical rates. Without further information (such as (g,g') data), a (g,n) measurement cannot be used to constrain the astrophysical rate or to even only test the reliability of model predictions of such rates. Data showing the relative particle emission to g.s. and excited states, though, may test to a certain extent the prediction of thermal modification of the stellar capture reaction. The situation is similar in photodisintegrations emitting charged particles. If measurements could study, however, charged particle emission below the Coulomb barrier, this would constrain the rate because at low relative energy, the cross section is determined by the charged particle width. Unfortunately, such cross sections are very small and outside the reach of current methods, especially for unstable nuclei.

Of course, new ideas for directly measuring relevant quantities are welcome.

For further details see:

[1] T. Rauscher, The Path to Improved Reaction Rates for Astrophysics, Int. J. Mod. Phys. E 20 (2011) 1071 (arXiv:1010.4283; the arXiv version includes a table of contents)

[2] T. Rauscher, Astrophysical relevance of gamma transition energies, Phys. Rev. C 78 (2008) 032801(R) (arXiv:0807.3556)

[3] T. Rauscher, G. G. Kiss, Gy. Gyürky, A. Simon, Zs. Fülöp, E. Somorjai, Suppression of the stellar enhancement factor and the reaction 85Rb(p,n)85Sr, Phys. Rev. C 80 (2009) 035801 (arXiv:0908.3195)

[4] T. Rauscher, P. Mohr, I. Dillmann, R. Plag, Opportunities to constrain astrophysical reaction rates for the s-process through determination of the ground state cross sections, ApJ 738 (2011) 143 (arXiv:1106.1728; note: although this focuses on s-process neutron capture, the same considerations with respect to the ground state contribution to the stellar rate can be applied to photodisintegration reactions)

[5] M. Mosconi, M. Heil, F. Käppeler, R. Plag, A. Mengoni, Neutron Physics of the Re/Os clock. II. The (n,n') cross section of 187Os at 30 keV neutron energy, Phys. Rev. C 82 (2010) 015803

[6] K. Sonnabend, P. Mohr, K. Vogt, A. Zilges, A. Mengoni, T. Rauscher, H. Beer, F. Käppeler, R. Gallino, The s-process branching at 185W, ApJ 583 (2003) 506 (arXiv:astro-ph/0209527)

[7] P. Mohr, K. Vogt, M. Babilon, J. Enders, T. Hartmann, C. Hutter, T. Rauscher, S. Volz, A. Zilges, Experimental simulation of a stellar photon bath by Bremsstrahlung: the astrophysical gamma-process, Phys. Lett. B 488 (2000) 127 (arXiv:nucl-ex/0007003)

[8] P. Mohr, T. Rauscher, K. Sonnabend, K. Vogt, A. Zilges, Photoreactions in Nuclear Astrophysics, Nucl. Phys. A718 (2003) 243

[9] T. Rauscher, Relevant energy ranges for astrophysical reaction rates, Phys. Rev. C 81 (2010) 045807 (arXiv:1003.2802)

[10] T. Rauscher, Sensitivity of astrophysical reaction rates to nuclear uncertainties, Ap. J. Suppl. 201 (2012) 26 (arXiv:1205.0685)

[11] T. Rauscher, Formalism for inclusion of measured reaction cross sections in stellar rates including uncertainties and its application to neutron capture in the s-process, Ap. J. Lett. 755 (2012) L10 (arXiv:1207.1664; note: although this applies the derived formalism to s-process neutron capture, the same considerations with respect to the ground state contribution to the stellar rate can be applied to any other reaction, including stellar photodisintegration reactions)

[12] T. Rauscher, N. Dauphas, I. Dillmann, C. Fröhlich, Zs. Fülöp, Gy. Gyürky, Constraining the astrophysical origin of the p-nuclei through nuclear physics and meteoritic data, Rep. Prog. Phys. 76 (2013) 066201 (arXiv:1303.2666)

[13] T. Rauscher, Suppression of Excited-State Contributions to Stellar Reaction Rates, Phys. Rev. C 88 (2013) 035803 (arXiv:1308.5816)

[14] H. Utsonomiya, et al., Photoneutron cross sections for Mo isotopes: A step toward a unified understanding of (g,n) and (n,g) reactions, Phys. Rev. C 88 (2013) 015805