Neutron-Gamma Competition for $\Beta $-Delayed Neutron Emission

LA-UR-16-25845 Neutron-gamma competition for β-delayed neutron emission

M. R. Mumpower,∗ T. Kawano, and P. M¨oller Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: August 8, 2016) We present a coupled Quasi-particle Random Phase Approximation and Hauser-Feshbach (QRPA+HF) model for calculating delayed particle emission. This approach uses microscopic nuclear structure information which starts with Gamow-Teller strength distributions in the daughter nucleus, and then follows the statistical decay until the initial available excitation energy is exhausted. Explicitly included at each particle emission stage is γ-ray competition. We explore this model in the context of neutron emission of neutron-rich nuclei and ﬁnd that neutron-gamma competition can lead to both increases and decreases in neutron emission probabilities, depending on the system considered. A second consequence of this formalism is a prediction of more neutrons on average being emitted after β-decay for nuclei near the neutron dripline compared to models that do not consider the statistical decay.

PACS numbers: 23.40.-s, 24.60.Dr

I. INTRODUCTION ∼ 250 measurements exist on neutron emitters in the literature [4]. Delayed neutron emission is a common decay channel Very few measurements exist for the branching ra- found in neutron-rich nuclei. It was first discovered in tios or Pn values of multi-neutron emission. The largest 1939 during an investigation of the splitting of uranium known two-neutron emitter (86Ga) was observed at the and thorium when these elements were bombarded with Holifield Radioactive Ion Beam Facility with a P2n = neutrons [1]. This phenomenon is energetically possi- 20(10)% [7]. Future studies at radioactive beam facilities ble when a precursor nucleus with β−-decay energy win- will extend the reach of measureable nuclei well into the dow (Qβ) populates excited states in the daughter nu- regime of multi-neutron emission predicted by theory [8– cleus above the one neutron separation energy (Sn). The 10]. However, it will still be many years before there is excited states may decay by particle or γ-ray emission. sufficient data to fully benchmark theoretical models. Accurately accounting for β-delayed neutron emission is Delayed neutron emission is challenging to describe necessary in, for example, nuclear reactor operation, to theoretically, relying on the complex description of the the rapid neutron capture or r-process of nucleosynthesis. nucleus, its excited states, and the likelihood of particle For r-process nucleosynthesis, the predicted produc- emission. In particular it is difficult in any model to cal- tion pathway ventures far from the currently known nu- culate the energies of the high-lying states near the neu- clides towards the neutron dripline where Sn is small tron separation energy very accurately. Two approaches and Qβ is large. In this case, multiple-neutron emis- to the description of delayed neutron emission can be sion is possible and may even be the dominant decay found in the literature. The first approach involves mod- pathway for many nuclei near the neutron dripline. This els using systematics, such as the Kratz-Herman formula provides a secondary source of free neutrons during late- [11] or more recent approaches, see e.g. [12]. Models us- times when the nuclei decay back to stability [2]. See ing systematics produce a good global fit to known data Ref. [3] for a recent review and further discussions of the and improvements have recently been made by Miernik impact of β-delayed neutron emission on r-matter flow. et al. where they provide an approximation to neutron- arXiv:1608.01956v1 [nucl-th] 5 Aug 2016 Experimental studies of β-delayed neutron emission gamma competition [13]. The applicability of system- are difficult due to the challenge of producing short-lived atics to multi-neutron emission towards the dripline is neutron-rich nuclei. Studies of these nuclei are further unknown, as they are based on simple relations which compounded by the requirement of high efficiency neu- factor in Qβ and Sn. tron detectors [4]. Alternative techniques seek to sidestep A second approach is to use a microscopic descrip- the problem of developing high efficiency neutron detec- tion, usually starting with some form of Quasi-particle tors by using recoil-ion spectroscopy coupled with a β- Random Phase Approximation (QRPA) that provides Paul trap [5]. Only recently has it been possible to mea- the initial population of the daughter nucleus [8–10, 14– sure delayed neutron branching ratios for heavy nuclei 16]. Rapid progress has been made for the description near the N = 126 shell closure [6]. In total, roughly of half-lives using the Finite Amplitude Method which is equivalent to QRPA, but this method has not yet been applied to Pn predictions [17]. To calculate Pn values from the QRPA approaches, a schematic formula is used ∗Electronic address: [email protected] based off the available energy window [8–10]. This ap- 2 proach to calculating Pn values is sometimes referred to calculation we smooth the β-strength distribution by a the ‘cutoff’ method in the literature. For ease of cita- Gaussian, tion later in the paper, we define three reference mod- (k) 2 els: QRPA-1 (Gamow-Teller (GT) only) [8] and QRPA-2 X (k) 1 [E − Ex] ω(Ex) = C b √ exp − , (2) (additional inclusion of First-Forbidden (FF) transitions) 2πΓ 2Γ2 [9], and D3C∗ [10]. The latter is a microscopic approach k based off covariant density functional theory and proton- where b(k) are the branching ratios from the parent state neutron relativistic QRPA. (k) to the daughter states E , Ex is the excitation energy In this paper we present a further upgrade to the of the daughter nucleus, and the Gaussian width is taken microscopic framework of computing delayed neutron- to Γ = 100 keV, as in [18]. The normalization constant, emission probabilities. Our approach combines both the C, is the given by the condition QRPA and statistical methods into a single theoretical framework. This is achieved by first calculating the β- Z Qβ decay intensities to accessible states in the daughter nu- ω(Ex)dEx = 1, (3) cleus using QRPA. The subsequent decay of these states 0 by neutron or γ-ray emission is then treated in the sta- where Q is the β−-decay energy window. tistical Hauser-Feshbach (HF) theory. This work extends β Our QRPA solution may be fully replaced or supple- that of Ref. [18], which was limited to near stable iso- mented with experimental data when available. In the topes, to the entire chart of nuclides. case where the level structure and β-decay scheme are known completely, data are taken from the RIPL-3 / ENSDF and no QRPA strength is used. If the database is II. MODEL incomplete, we combine the QRPA solutions with renor- malized ENSDF data. In some cases, energy levels are The calculation of β-delayed neutron emission in known but the spin and parity assignments are uncertain. the Quasi-particle Random Phase Approximation plus In this case, or in the case of that no level information is Hauser-Feshbach (QRPA+HF) approach is a two step found in ENSDF, the QRPA solution is relied upon. process. It involves both the β-decay of the precursor When applicable, ground state masses are taken from nucleus (Z,A) where Z represents the atomic number, the latest compilation of the Atomic Mass Evaluation and A is the total number of nucleons, and the statisti- (AME2012) [22]. Ground state masses and deformations cal decay of the subsequent generations, (Z + 1,A − j) from the 1995 version of the Finite-Range Droplet Model where j is the number of neutrons emitted, ranging from (FRDM1995) are used unless otherwise noted [23]. 0 to many. The statistical decay continues until all the In the second stage of the calculation we assume the available excitation energy is exhausted. This is shown daughter nucleus is a compound nucleus initially pop- schematically in Fig 1. ulated by the β-decay strength from the QRPA calcu- Beginning with the precursor nucleus (Z,A), the de- lations. This assumption is based off the independence cay to the daughter (Z + 1, A) is calculated using the hypothesis from N. Bohr, from which it follows that the QRPA framework of Mölleret al. [8, 9, 19, 20]. In this compound nucleus depends only on its overall properties framework, the Schrödingerequation is solved for the nu- rather than the details of formation [24]. We follow the clear wave functions and single-particle energies using a statistical decay of this nucleus and subsequent genera- folded-Yukawa Hamiltonian. Residual interactions (pair- tions until all the available excitation energy is exhausted ing and Gamow-Teller) are added to obtain the decay using the Los Alamos CoH statistical Hauser-Feshbach matrix element from initial to final state, hf|βGT |ii.A code [18, 25, 26]. β-decay strength function is used to describe the behav- For calculation of the neutron transmission coefficients ior of the squares of overlap integrals of nuclear wave we employ the Koning-Delaroche global optical potential functions. The β-decay strength function is given by [21] that has an iso-spin dependent term [27]. This model has been shown to provide a very good description of I(E) neutron-rich nuclei, see e.g. Ref. [28, 29]. We have also Sβ(Ex) = (1) f(Z,Qβ − Ex)T1/2 used the Kunieda deformed optical potential [30], however we find only minor changes to our results and so do where I(Ex) β-intensity per MeV of levels at energy Ex not report on this model here. in the daughter nucleus, f is the usual Fermi function As shown in Fig. 1, the excited states in the subsequent and T1/2 is the half-life of the parent nucleus in seconds. generations can be discrete or in the continuum. Above −1 −1 The units of Sβ are s MeV . Here we only consider the last known level, or in the case of no known lev- the Gamow-Teller (GT) strength from QRPA, similar to els, the Gilbert-Cameron level density is used [31]. This Ref. [8]. hybrid formulation uses a constant temperature model The transition energies and rates for β-decay strongly at low energies and matches to a Fermi gas model in depend on the level structure in the participating nu- the high energy regime. Parameters for this formula are clei. Due to the natural uncertainty associated with this taken from systematics at stability and extrapolated as 3

Precursor Z+1, A Z+1, A-1 Z+1, A-2 Z+1, A-3

β-decay Continuum Continuum Continuum β delayed neutron delayed neutron delayed neutron Q γ

d n e S y a l e d n S

QRPA Hauser-Feshbach

FIG. 1: Schematic of the combined QRPA+HF approach. Initial population of the the daughter nucleus (Z+1,A) is determined by QRPA. Subsequent delayed-neutron and γ-ray emission are handled in the HF framework and shown by solid and dotted lines respectively. The statistical decay is followed until all available excitation energy is exhausted, denoted by trailing dots.

(j) one crosses the nuclear landscape [32]. Shell corrections mission coefficient for c , ρj(Ek) is the level density in (j) are included in the level density via the method proposed c at energy Ek and Nj(Ei) is a normalization factor de- by Ignatyuk et al. [33]. fined below. The neutron-emission transition probability The γ-ray transmission coefficients are constructed us- from c(j) to c(j+1) is defined as ing the generalized Lorentzian γ-strength function (γSF) 1 (j+1) (j) for E1 transitions. Higher order electric and magnetic qj(Ei,Ek0 ) = Tn (Ei − Sn − Ek0 )ρj+1(Ek0 ) transitions are also considered, but due to the strong Nj(Ei) (5) dependence of the transitions on multipolarity, lower L (j) transitions usually take precedence. where the transition is from an energy level Ei in c (j+1) (j) A key component of the HF calculations is the com- to an energy level Ek in c , Sn is the neutron sepa- (j) (j+1) petition of γ-ray emission at each stage of particle emis- ration energy of c , Tn is the neutron transmission (j+1) (j) sion. We now introduce several definitions to make this coefficient from c to c and ρj+1(Ek0 ) is the level (j+1) discussion more precise. We denote the compound state density in c evaluated at Ek0 . The prime on the k (j) (j) index indicates that the energy level is in a different com- as ck where c represents the j-th compound nucleus and k the index of the excited state in this nucleus which pound nucleus. The normalization factor, Nj is given by takes values from the highest excitation to the ground the sum over all possible exit channels state. Using this definition, c(0) would be shorthand for (j) Z Ei describing the daughter nucleus (Z +1,A) and c would (j) Nj(Ei) = Tγ (Ei − Ek)ρj(Ek)dEk describe (Z + 1,A − j). 0 The γ-emission transition probability in the j-th com- (j) Z Ei−Sn pound nucleus is defined as (j+1) (j) + Tn (Ei − Sn − Ek0 )ρj+1(Ek0 )dEk0 (6) 0 1 (j) pj(Ei,Ek) = Tγ (Ei − Ek)ρj(Ek) (4) where the integration in each case runs over the appro- Nj(Ei) priate energy window. In the above equations we have where the transition is from a level with high excitation explicitly omitted the indices of quantum numbers and (j) energy Ei to a level of energy Ek, Tγ is the γ-ray trans- assumed implicitly that all transitions obey spin-parity 4

selection rules. The units for q and p are in [1/MeV]. 1.0 Also note that in both Eq. (4) and (5) the transition neutrons probabilities do not depend on how the initial state was 0.8 γ-rays populated in accordance with the compound nucleus assumption. (j+1) 0.6 The level population in c for particular energy Ek is defined as 0.4 k−1 Probability X Pj+1(Ek) = Pj+1(Ei)pj+1(Ei,Ek) 0.2 i=0 k−1 X 0 0 0.0 + Pj(Ek )qj(Ek ,Ek) (7) 4.80 4.85 4.90 4.95 5.00 5.05 5.10 5.15 5.20 k0=0 Excitation Energy [MeV] where the summations run over all levels which may feed (j+1) 86 the compound state ck . The first term takes into ac- FIG. 2: (Color online) Neutron-gamma competition in Ge (j+1) after β-decay of 86Ga. The dotted line represents the neutron count γ emission in ck while the second term takes 86 (j) separation energy of Ge. into account neutron emission from ck0 . Both terms depend on all the possible pathways taken to populate Ek. Thus, the population of a particular level is a recursive function which can be handled naturally on a computer. In our case, the initial population comes from β-decay, 0.7 s-wave such that P0(Ek) = Sβ(Ek). 0.6 From these definitions, we can compute the j-th neu- p-wave tron emission probability by considering all the channels 0.5 d-wave that end in the j-th compound nucleus relative to all 0.4 possible pathways: n T 0.3 Pjn = Pj(Egs) (8) 0.2 where we evaluate the level population at the ground (j) state energy, Egs, in c . This equation is a recursive 0.1 convolution and therefore in general a P value may in- jn 0.0 crease or decrease relative to calculations based on the 0.0 0.2 0.4 0.6 0.8 1.0 older energy window argument. Beta-decay with no neu- Neutron Energy [MeV] tron emission is denoted as P0n and because of the normalization in the calculation of the level population we FIG. 3: (Color online) Neutron transmission coefficients for have a similar constraint on the neutron emission proba- s, p and d-wave neutron emitted from 86Ge. For p and d P bilities, j=0 Pjn = 1. waves the transmission for different spins are averaged. Delayed neutron spectra can also be obtained from the coupled QRPA+HF approach. This has already been explored in the context of the QRPA+HF model in a previous study [18] so we do not cover this here. A. Neutron-gamma competition

III. RESULTS Figure 2 shows a representative example of neutron- gamma competition for 86Ge (Z = 32) which is popu- We have performed QRPA+HF calculations to ob- lated after the β-decay of 86Ga (Z = 31). From the tain new delayed-neutron emission probabilities for all transition probabilities near the one neutron separation neutron-rich nuclei from the valley of stability towards energy in 86Ge we see that the neutron-gamma competi- the neutron dripline. Our aim in this section is to present tion extends roughly 100 keV above Sn = 4.93. a general overview of the model results. We also discuss The neutron transmission coefficients for s, p and d- quantitative benchmarks of the model performance, but waves are shown in Fig. 3. For p and d wave neutrons we caution that most measured systems do not exhibit the transmissions for different spins are averaged. A large neutron-gamma competition and so we do not ex- quick rise in p-wave neutrons produces the sharp cut-off pect a large deviation of our results from previous mod- of neutron-gamma competition observed in Fig. 2. De- els [8, 9]. However, far from the stable isotopes we find spite this small energy range for neutron-gamma com- substantial deviations from older model predictions and petition, there is still an impact on the predicted Pn discuss the significance of this below. values. The QRPA-1 model predicts P1n ∼ 61% and 5

102 101 100 107Tc 10-1 1 86Ga 10 10-2 10-3 -4 100 10 10-5 10-6 10-1 10-7 10-8 GT intensity [%] Spectrum [1/MeV] -9 -2 10 10 -10 10 S1n S2n S3n 10-11 -12 10-3 10 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16 Excitation energy [MeV] Eγ [MeV]

FIG. 4: (Color online) Gaussian broadened Gamow-Teller FIG. 5: (Color online) The γ-ray spectrum after the β-decay β-decay intensity of 93As from the QRPA-1 model of Ref. [8]. of 107Tc and 86Ga.

P ∼ 13% while QRPA+HF yields P ∼ 69% and 2n 1n petition in the daughter nucleus 70Ni (Z = 28) in the P ∼ 6%. While there is a larger discrepancy with ex- 2n QRPA+HF model, however, our results are dependent on periment, P ∼ 20(10)%, this result is still within the 2n the uncertain spin-parity assignments in ENSDF, which mean model error, which we will discuss below. We fur- makes it difficult to draw any concrete conclusions. If ther note that it is likely that FF transitions play a role in the Gilbert-Cameron level density is used for this cal- this decay, which have not been considered in either calculation, neutron emission will dominant as there is al- culation. When FF transitions are considered, using the ways an open pathway for neutron emission to proceed QRPA-2 model, there is a shift to larger neutron emis- at high excitation energies. This nucleus highlights the sion for 86Ga: P ∼ 20% and P ∼ 44% This example 1n 2n importance of using realistic energy levels as well as spin- highlights the delicate dependency of P predictions on n parity assignments in the calculation of delayed neutron the β-strength function. emission probabilities. It is difficult to predict whether or not Pn values will increase or decrease relative to the QRPA-1 model due The TAGS technique coupled with the production po- to the convolution involved in Eq. (8). Consider the case tential of radioactive beam facilities will allow for the of 93As. The Gaussian broadened GT β-decay intensity exploration of nuclear structure through β-decay mea- is shown in Fig. 4. From the energy window argument, surements far from stability. To this end the QRPA+HF one obtains P1n ∼ 84%, P2n ∼ 14% and P3n ∼ 2% with framework can help to isolate interesting nuclei by con- the QRPA-1 model. The new QRPA+HF model predicts structing the expected γ-spectrum associated with the an overall shift to smaller neutron emission: P1n ∼ 94%, decay. We show the capacity of the QRPA+HF in pre- 107 P2n ∼ 6% and P3n ∼ 0%, which is similar to the previous dicting γ-spectrum in Fig. 5 for select nuclei: Tc and 86 case of the β-decay of 86Ga. Ga. A clear distinction can be seen between these two 107 The reason for this reduction typically comes from the cases. The β-decay of Tc produces no neutron emis- suppression of neutron emission just above threshold due sion resulting in the distinct drop off of the γ-spectrum 86 to spin-parity conservation. For the vast majority of nu- around 4.65 MeV whereas Ga is our familiar two neutron emitter discussed above. The long tail at high ener- clei near stability, roughly 75%, we find a reduction of P1n relative to the QRPA-1 model. In contrast, we find that gies in the γ-spectrum of this decay comes from the com- some nuclei, roughly 25%, have their neutron emission in- petition of neutron and γ emission while the sharp drops creased. An example is 145Cs where the QRPA-1 model are an indication that neutron emission has occurred. predicts P0n ∼ 98% and P1n ∼ 2%. The QRPA+HF model predicts P0n ∼ 78% and P1n ∼ 22%, which is exp closer to the measured value of P1n ∼ 15%. Most recently it has been shown that there is sig- B. Global model results nificant neutron-gamma competition above the neutron threshold associated with the β-decay of 70Co (Z = 27) using the Total Absorption Gamma Spectroscopy To show our results globally across the chart of nuclides (TAGS) technique [34]. We do find neutron-gamma com- in a concise manner we compute the average number of 6 delayed neutrons per β-decay TABLE I: Number of P1n theory predictions within ±x% of X hni = jP (9) measurements for various models. The label QRPA+HF rep- jn resents the results of this work. j=0 5% 10% 20% 30% 40% >40% where j denotes the number of neutrons emitted, Pjn is the computed probability to emit j-neutrons as defined in QRPA+HF 115 21 29 21 10 29 Eq. (8) and the sum of Pjn is unity. The delayed-neutron QRPA-1 113 23 34 16 11 28 emission predictions using the QRPA+HF framework QRPA-2 117 24 35 20 9 20 throughout the chart of nuclides are shown in Fig. 6. We find that neutron-emission channel begins on average D3C∗ 98 28 34 26 12 27 roughly five or so units in neutron number from the last stable isotope, in agreement with measurements. Figure 6 shows a fairly smooth trend in the number of delayed other means of analysis must be considered when dealing neutrons as neutron excess increases, which mirrors the with such quantities [9]. results of older model calculations [8, 9]. To gauge the average residuals of a given model we For the majority of nuclei near stability, we find neu- compute tron emission to dominate over γ emission above the neu- N tron threshold. For these nuclei, it is okay to assume 1 X e¯ = |P (i) − P exp(i)| (10) that decays to energies above the separation energy al- N 1n 1n ways lead to delayed neutron emission. To date, this has i=1 been the basis for model calculations of delayed-neutron where the sum runs over N = 225 measured P1n values. emission probabilities, see e.g. Eq. (2) in Ref. [9]. To- The QRPA-1 model of Ref. [8] yields an average devia- wards the dripline however, j-neutron separation ener- tione ¯QRP A−1 = 15.4% and for our new QRPA+HF we gies, Sjn, may overlap causing the assumption to become obtaine ¯QRP A+HF = 15.3%. As expected, the inclusion difficult to implement as it is not clear how to partition of neutron-gamma competition does not greatly impact the β-decay strength without double counting. The older the results near stability. The latest fully microscopic models, QRPA-1 and QRPA-2, handled this by mapping approach from Ref. [10] gives comparable predictabil- any additional neutron emission above j-neutrons into ity,e ¯D3C∗ = 15.4%. This microscopic model also in- the j-neutron emission probability, where j = 3. cludes FF transitions. If FF transitions are considered in We illustrate this scenario by the transition between the macroscopic-microscopic approach, as in the QRPA- (Z + 1,A − 2) and (Z + 1,A − 3) in Fig. 1. Here, the 2 model of Ref. [9], a substantial improvement is seen energy window argument becomes invalid and only the in the mean errore ¯QRP A−2 = 12.2%. This suggests that QRPA+HF model provides the correct treatment via the FF transitions play an important factor in improving the statistical decay. Thus, proper treatment of delayed neu- description of Pjn values. An update to the QRPA-2 tron emission probabilities requires neutron-gamma com- model which includes the neutron-gamma competition petition as one approaches the neutron dripline. This can discussed here will be the subject of an upcoming publi- be seen most notably after magic neutron numbers, indication. cated by solid vertical lines in Fig. 6. We find an enhance- It is also instructive to consider how many P1n the- ment in the number of neutrons emitted in these regions ory predictions are within ±x% of measurements. This compared to the aforementioned method. This is where metric is summarized for several models in Table I. An the largest deviation from older models occurs, with neu- inspection of the table reveals that roughly 50% of P1n tron emission exceeding hni ∼ 5 or more in QRPA+HF. predictions are within ±5% of measurements. The re- Whether this has an impact on r-matter flow will depend mainder of P1n predictions fall in the tail of the distribu- on the shape of the neutron dripline among other things. tion, greater than ±10%, which contribute significantly Complete reaction network calculations must be used in to the mean model error. The cause of strong outliers order to determine the full impact on the r-process and greater than 40% is most likely due to a poor reproduc- we reserve this effort for a future study. tion of the β-strength. There are many contributing factors that may impact the calculation of Pn values. We now consider the sensi- C. Model comparisons tivity of our results to the γSF. An increase to the γSF of roughly one order of magnitude has been recently sug- We now discuss our results in a quantitative fashion. gested from emission of γ-ray in 87,88Br and 94Rb. by Since Pn values range from 0% to 100%, it is easiest to J. Tain et al. [35]. We have performed the suggested consider model residuals or differences between theoret- increase of one order of magnitude to the γSF used in ical calculations and experimental data. Other physical QRPA+HF and find our results are mostly unchanged. quantities, for example half-lives, vary orders of magni- The largest change is roughly ∆P1n = 3%, with nearly tude. Further, the differences between calculation and all nuclei showing little to no change in Pn values. This is experiment can also vary orders of magnitude. Ergo, quite predictable outcome, as γ-ray transmission is gen- 7

100 10

8 80

6 60 >

5 n < 4 40 Proton Number (Z) 3

20 1

0 20 40 60 80 100 120 140 Neutron Number (N)

FIG. 6: (Color online) Average number of neutrons emitted after a single β-decay using the QRPA+HF approach with FRDM1995 masses and GT strength data from Ref. [8]. Closed shells shown by parallel straight lines with stable nuclei shown in black.

erally orders of magnitude weaker than neutron trans- ities. mission. We conclude from the perspective of delayed We explored this new framework in the context of de- neutron emission that an enhancement of an order of layed neutron emission and find only a slight improve- magnitude is not warranted for the γSF. We recognize ment in neutron emission probabilities near stability. To- that such an enhancement would have a larger effect on wards the neutron dripline we find substantial differences the neutron capture rates of neutron-rich nuclei. This from older model predictions due to the competition of is because cross sections are proportional to TnTγ and Tn+Tγ neutrons and γ-rays above the neutron threshold, which Tn ∼ Tn + Tγ . Thus, a relatively small change to Tγ will will likely have an influence on r-matter flow. We found have a considerable impact on the cross section. the prediction of Pn values depends strongly on the β- strength function and thus on the associated model used to construct the initial population in the daughter nu- IV. SUMMARY cleus. The γSF plays a lesser role, only impacting Pn values up to ∼ 3% and we observed a negligible impact We have presented a more microscopic method to cal- from the choice of the HF neutron optical model. culate delayed particle emission. This approach couples A proper theoretical description of Pn values depends the QRPA and HF formalisms into one theoretical frame- on many facets of nuclear physics from nuclear struc- work (QRPA+HF) which permits the calculation of γ- ture to the reaction mechanism. Future Pn measure- spectra as well as delayed particle spectra and probabil- ments will provide great insight into inner workings of the 8 atomic nucleus and with sufficient data lead to the ability recent Pn measurements. We thank Scott Marley for to properly benchmark theoretical models. Prospective helpful discussions regarding current experimental tech- studies are planned to explore the impact the QRPA+HF niques. This work was carried out under the auspices of model has on nucleosynthesis calculations. the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Labora- tory under Contract No. DE-AC52-06NA25396. V. ACKNOWLEDGEMENTS

We thank Iris Dillmann for helpful discussions and her student, Stephine Ciccone, for providing a database of

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