Compound-nuclear reaction cross sections from surrogate measurements

Nuclear reaction cross sections are important for a variety of applications in the areas of astrophysics, nuclear energy, and national security. When these cross sections cannot be measured directly or predicted reliably, it becomes necessary to develop indirect methods for determining the relevant reaction rates. The surrogate nuclear reactions approach is such an indirect method. First used in the 1970s for estimating $(n,f)$ cross sections, the method has recently been recognized as a potentially powerful tool for a wide range of applications that involve compound-nuclear reactions. The method is expected to become an important focus of inverse-kinematics experiments at rare-isotope facilities. The present paper reviews the current status of the surrogate approach. Experimental techniques employed and theoretical descriptions of the reaction mechanisms involved are presented and representative cross section measurements are discussed.

Figure 1

Schematic representation of the desired (top) and surrogate (bottom) reaction mechanisms. The basic idea of the surrogate approach is to replace the first step of the desired reaction, $a+A$, by an alternative (surrogate) reaction, $d+D\to b+B^{*}$, that is experimentally easier to access yet populates the same compound nucleus. The subsequent decay of the compound nucleus into the relevant channel, $c+C$, can then be measured and used to extract the desired cross section. Three typical decay channels are shown here: neutron evaporation, fission, and $\gamma$ emission.

Figure 2

Yttrium reaction network showing isomeric states and partial level schemes of ${}^{86}\mathrm{Y}$, ${}^{87}\mathrm{Y}$, ${}^{88}\mathrm{Y}$, ${}^{89}\mathrm{Y}$, and ${}^{90}\mathrm{Y}$ and associated reaction channels. Most cross sections are unknown or poorly constrained.

Figure 3

Determination of fission coincidence probabilities for the ${}^{240}\mathrm{U}$ compound nucleus in an early surrogate experiment. The proton-singles spectra $N_{\delta }$ for ${}^{238}\mathrm{U}(t,p)$ is shown at the top with interpolated dashed lines to determine the spectrum under the clearly visible peaks from $(t,p)$ reactions on ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$. The ratio of this spectrum to the $N_{\delta f}$ spectrum shown below it results in $P_f$, shown at the bottom. From [49].

Figure 4

The $(n,f)$ cross sections determined using the Weisskopf-Ewing approximation to analyze $(t,pf)$ data. The solid lines show the results of the direct measurements that existed at the time. The agreement above 1 MeV is quite good. From [48].

Figure 5

The $(n,f)$ cross sections for ${}^{241,242,243}\mathrm{Am}$, ${}^{237}\mathrm{Np}$, and ${}^{231}\mathrm{Pa}$ determined using the Weisskopf-Ewing approximation to analyze $({}^3\mathrm{He},tf)$ data. The results were compared to direct $(n,f)$ cross sections and ENDF/B-IV evaluations. From [34].

Figure 6

Experimental setup at IPN Orsay. A ${}^3\mathrm{He}$ beam strikes a target at the center of the chamber. Silicon telescopes located at 90° and 130° with respect to the beam detect light ions and an array of photovoltaic cells detects fission fragments. From [119].

Figure 7

The energy deposited in a $\Delta E\mathrm{\text{−}}E$ silicon-detector telescope enables clear identification of the various light-ion exit channels. From [88].

Figure 8

Particle-singles spectra for various exit channels from ${}^3\mathrm{He}$ reactions on a ${}^{232}\mathrm{Th}$ target. The data on the left (right) correspond to the particles detected with the telescope at 90° (130°). The solid lines show data collected with the target and the dashed lines show data from the $50\mathrm{\text{−}}\mu \mathrm{g}/{\mathrm{cm}}^2$ carbon backing alone. From [119].

Figure 9

The ${}^{230}\mathrm{Th}(n,f)$, ${}^{231}\mathrm{Pa}(n,f)$, and ${}^{233}\mathrm{Pa}(n,f)$ cross sections determined using ${}^{232}\mathrm{Th}({}^3\mathrm{He},x)$ surrogate reactions ([119]) are compared to direct measurements ([122]; [105]; [153]; [151]), and the ENDF/B-VI and JENDL-3 evaluations. From [119].

Figure 10

Results for ${}^{241}\mathrm{Am}(n,f)$, ${}^{243}\mathrm{Cm}(n,f)$, and ${}^{242}\mathrm{Cm}(n,f)$ determined from detecting ${}^4\mathrm{He}$, $d$ and $t$ exit channels, respectively, from ${}^3\mathrm{He}$ bombardment of an ${}^{243}\mathrm{Am}$ target. The surrogate-reaction results by Kessedjian et al. are compared to direct measurements ([51]; [164]; [63]; [66]) and evaluations. From [88].

Figure 11

A cross section of a typical STARS/LiBerACE experimental configuration is shown. The scattering chamber houses the targets and particle detectors. The LiBerACE HPGe detectors surround the scattering chamber.

Figure 12

The proton-gated fission spectra from a ${}^{235}\mathrm{U}(d,p)$ surrogate experiment. The light and heavy mass fission-fragments peaks are indicated. From [3].

Figure 13

The ${}^{237}\mathrm{Np}(n,f)$ cross section determined from a surrogate experiment at STARS/LiBerACE. (a) Results from [11], [135], ENDF/B-VII.0 and JENDL3.3, for the ${}^{237}\mathrm{Np}(n,f)$ cross section from 10–20 MeV. (b) A comparison of the surrogate ratio result to measurements by [152].

Figure 14

The ${}^{237}\mathrm{U}(n,f)$ cross section determined from a ${}^{238}\mathrm{U}(\alpha ,{\alpha }^{\prime }f)$ surrogate experiment ([37]) (shown as triangles with statistical and systematic uncertainties) is compared to an earlier result ([180]) (shown as squares without uncertainties). From [37].

Figure 15

Cross section for ${}^{233}\mathrm{Pa}(n,f)$ determined using a hybrid surrogate approach (dots) and compared to other results from experiment and the EMPIRE-2.19 code. From [114].

Figure 16

${}^{236}\mathrm{U}(n,f)$ cross section obtained from a Weisskopf-Ewing analysis of surrogate ${}^{238}\mathrm{U}({}^3\mathrm{He},\alpha )$ measurements. Data represented by open squares correspond to events for which the outgoing $\alpha$ particle was observed at 36° to 45°, while filled circles correspond to an angular range of 57° to 62°. The solid line is the cross section that results from averaging over all experimentally accessible angles, 36° to 62°. The fact that the extracted cross section depends on the angle of the outgoing $\alpha$ particle is an indication that the Weisskopf-Ewing approximation is violated.

Figure 17

Experimental test of the surrogate ratio method for neutron-induced fission, for ${}^{233,235}\mathrm{U}$ targets, for which the compound-formation cross sections are expected to be almost identical. The ratio of measured fission probabilities (squares) is compared to the ratio $\sigma [{}^{233}\mathrm{U}(n,f)]/\sigma [{}^{235}\mathrm{U}(n,f)]$ of evaluated fission cross sections (solid line). From [91].

Figure 18

Calculated branching ratios $G_{\mathrm{fission}}^{\mathrm{CN}}(E,J,\pi )$ for fission of ${}^{236}\mathrm{U}^{*}$, as a function of the laboratory neutron energy in the ${}^{235}\mathrm{U}+n$ system. Results are shown for positive-parity states with total angular momenta $J=0$, 5, 10, 15, 20 (top panel) and $J=0$, 1, 2, 3, 4, 5 (bottom panel) in the compound nucleus ${}^{236}\mathrm{U}^{*}$. From [60].

Figure 19

Calculated branching ratios $G_{\mathrm{fission}}^{\mathrm{CN}}(E,J,\pi )$ for fission of ${}^{239}\mathrm{U}^{*}$, as a function of the laboratory neutron energy in the ${}^{238}\mathrm{U}+n$ system. Results are shown for positive-parity states with total angular momenta $J=1/2,\dots 21/2$, for neutron energies $E_n=0–5\mathrm{MeV}$. From [42].

Figure 20

(a) Schematic distributions of total angular momentum for the compound nucleus ${}^{236}\mathrm{U}^{*}$. The mean angular momentum is $⟨J⟩=7.03$, 10.0, 12.97, and 3.30 for distributions $a$, $b$, $c$, and $d$, respectively; positive and negative parities are taken to be equally probable. The distributions were chosen solely to perform a sensitivity study. (b) Distributions of total angular momentum for ${}^{236}\mathrm{U}^{*}$ produced in the neutron-induced reaction $n+{}^{235}\mathrm{U}(n,f)$, for selected neutron energies.

Figure 21

(a) Weisskopf-Ewing and (b) ratio estimates of the ${}^{235}\mathrm{U}(n,f)$ cross section, using the distribution of angular momenta shown in Fig. 20. The crosses represent the “reference” ${}^{235}\mathrm{U}(n,f)$ cross section from the fit. From [60].

Figure 22

Surrogate fission probabilities and $(n,f)$ cross sections for ${}^{235}\mathrm{U}$ (top) and ${}^{239}\mathrm{U}$ (bottom). (a), (c) Measured and predicted $(t,pf)$ coincidence probabilities and (b), (d) the extracted cross sections. (b) Results by [178], whose analysis accounted for spin-parity differences between the desired and surrogate reactions, to the earlier analysis by Cramer et al., which employed the Weisskopf-Ewing approximation, and to the ENDF/B-VI evaluation ([167]). (d) The ${}^{239}\mathrm{U}(n,f)$ cross section, compared to the Weisskopf-Ewing result from Cramer et al. From [178].

Figure 23

Estimate of the isomer-to-ground state $(n,f)$ cross section ratio for the ${}^{235}\mathrm{U}(n,f)$ reaction, deduced by [177], from ${}^{234}\mathrm{U}(t,pf)$ surrogate fission probabilities and by [99] from ${}^{235}\mathrm{U}(d,pf)$ surrogate data. From [179].

Figure 24

Estimate of the ${}^{241}\mathrm{Am}(n,f)$ cross section deduced from ${}^{242}\mathrm{Pu}({}^3\mathrm{He},tf)$ surrogate data ([34]; [181]). The surrogate results are compared to the ENDF/B-VI and ENDL99 evaluations. From [181].

Figure 25

Summary of surrogate-method $(n,f)$ cross section results. For each target nucleus, the cross section was averaged over an energy range where it is relatively flat [$E_n=1–3\mathrm{MeV}$ for $(t,pf)$, 1–6 MeV for $({}^3\mathrm{He},xf)$ results], and that single value is plotted as an open circle for the surrogate result, and a filled circle for directly measured data. From [179].

Figure 26

Comparison of the calculated ${}^{235,237,239}\mathrm{U}(n,f)$ cross sections with the ENDF/B-VI evaluation. From [180].

Figure 27

${}^{233}\mathrm{Pa}(n,\gamma )$ cross section from two separate surrogate experiments, compared to evaluations. The triangles show the result of a Hauser-Feshbach calculation with parameters adjusted to fit the ${}^{233}\mathrm{Pa}(n,f)$ cross section that was obtained from a ${}^{232}\mathrm{Th}({}^3\mathrm{He},pf)$ surrogate experiment. The large, filled circles show the results of a Weisskopf-Ewing analysis of a ${}^{232}\mathrm{Th}({}^3\mathrm{He},p\gamma )$ surrogate measurement. The solid and dotted curves show prior JENDL3.3 and ENDF/B-VI.6 results. Adapted from [28].

Figure 28

[3] used the Internal Surrogate Ratio (ISR) approach to determine the ${}^{235}\mathrm{U}(n,\gamma )$ cross section. Their measured coincidence ratio $P_{(d,p),\gamma }(E)/P_{(d,p),f}(E)$ is compared to the ratio $\sigma [{}^{235}\mathrm{U}(n,\gamma )]/\sigma [{}^{235}\mathrm{U}(n,f)]$ of cross sections from ENDF/B-VII evaluations. The latter are based on direct measurements. The comparison provides a test of the ISR method which is based on the assumption that the Weisskopf-Ewing is approximately valid. Under this circumstance, the cross section expressions factorize as shown in Eq. (11), and the measured coincidence ratio $P_{(d,p),\gamma }(E)/P_{(d,p),f}(E)$ should equal the cross section ratio for the neutron-induced reactions. Adapted from [3].

Figure 29

Calculated $\gamma$-decay probabilities $G_{\gamma }^{\mathrm{CN}}(E,J,\pi )$, for ${}^{92}\mathrm{Zr}$, ${}^{156}\mathrm{Gd}$, and ${}^{236}\mathrm{U}$. Shown is the probability that the compound nucleus, when produced with a specific $J\pi$ combination, decays via the $\gamma$ channel. The excitation energies shown correspond to incident-neutron energies of 0–4 MeV. The decay probabilities also depend on parity, only positive-parity results are shown here. More results can be found in [64]; [61].

Figure 30

Weisskopf-Ewing estimates for the (a) ${}^{155}\mathrm{Gd}(n,\gamma )$ and (b) ${}^{235}\mathrm{U}(n,\gamma )$ cross sections, extracted from analyses of simulated surrogate experiments, for the four different compound-nuclear $J\pi$ distributions shown in Fig. 31. For the gadolinium case, results from a Weisskopf-Ewing analysis of measured surrogate ${}^{156}\mathrm{Gd}(p,p^{\prime })$ data from [133] are also shown. The reference cross sections were obtained by adjusting the parameters for the Hauser-Feshbach calculation to reproduce direct $(n,\gamma )$ measurements.

Figure 31

Schematic spin-parity distributions, selected to simulate the compound nucleus created in the surrogate reaction. Positive and negative parity states are assumed to be populated with equal probability.

Figure 32

Ratios $G_{\gamma }^{{}^{198}\mathrm{Au}}(E,J,\pi )/G_{\gamma }^{{}^{194}\mathrm{Ir}}(E,J,\pi )$ of decay probabilities for the compound nuclei ${}^{198}\mathrm{Au}$ and ${}^{194}\mathrm{Ir}$, as a function of energy. Results for individual spins, $J\pi =0^{+},2^{+},\dots ,{10}^{+}$, are compared to the ratio relevant to the neutron-capture reaction. From [42].

Figure 33

Cross section ratios obtained from surrogate data are compared to ratios of evaluated cross sections. The four different compound-nuclear $J\pi$ distributions of Fig. 31 were used to simulate surrogate data. In addition, experimental results from the surrogate measurement by [133] are plotted for gadolinium. (a) External surrogate ratio approach for the ${}^{157}\mathrm{Gd}(n,\gamma )$ cross section, (b) external surrogate ratio approach for the ${}^{235}\mathrm{U}(n,\gamma )$ cross section, and (c) internal surrogate ratio approach for the ${}^{235}\mathrm{U}(n,\gamma )$ cross section.

Figure 34

Calculated probabilities from [146] of compound-nucleus formation as a function of energy across the neutron escape threshold, for the reaction $d+{}^{239}\mathrm{Pu}\to p+{}^{240}\mathrm{Pu}$ at $E_d=15\mathrm{MeV}$. This information is used to obtain the thick-dashed curve shown in Fig. 35.

Figure 35

Fission probabilities for ${}^{240}\mathrm{Pu}^{*}$ as extracted from surrogate $(d,pf)$ experiments, compared with that obtained from evaluated $(n,f)$ cross sections based on direct measurements, as a function of equivalent neutron energy (see text). The thick-dashed curve is the new result using the probabilities shown in Fig. 34.

Figure 36

Compound-formation probability for radiative capture of 19.6 MeV neutrons to unbound final states in the ${}^{89}\mathrm{Y}(n,\gamma ){}^{90}\mathrm{Y}^{*}$ reaction as a function of the orbital angular momentum of the final-state neutron ([56]). This process is similar to deposition of the neutron in the $(d,p)$ reaction. Results are shown for three values of the energy above the escape threshold for the final-state neutron.

Figure 37

Ratios of the yields of various $\gamma$-ray transitions in the ground-state band of ${}^{236}\mathrm{U}$ to the total production of ${}^{236}\mathrm{U}$, for the four schematic spin distributions shown in Fig. 31.

Figure 38

Hauser-Feshbach calculation illustrating the effect of preequilibrium neutron emission. The solid lines represent the fit to the known ${}^{235}\mathrm{U}(n,f)$ cross section, as discussed by [60]. The dashed lines are a calculation with identical parameters except that preequilibrium is turned off.