Angular momentum in fission fragments

We suggest that the angular momentum in fission fragments is generated by statistical excitation at scission. The magnitude of the angular momentum is determined by excitation energy and shell structure in the level density. Treating the prescission shape evolution as a diffusive process, implemented as a Metropolis walk on a five-dimensional potential-energy surface, the average magnitudes of the fission fragment angular momenta are calculated for ${}^{235}\mathrm{U}(n_{\mathrm{th}},\mathrm{f})$, assuming that they are perpendicular to the fission axis. The sawtooth behavior of the average angular momentum magnitude as function of mass number is discussed in connection with the similar observed behavior of the average neutron multiplicity, and a good understanding is achieved. The magnitudes of the angular momenta of light and heavy fragments are found to have a weak negative correlation, in accordance with recent experimental results. This correlation arises from the microcanonical sharing of excitation energy by the fragments at scission, where each energy provides a distribution of angular momenta.

Figure 1

Illustration of the angular momentum vectors at scission. Both the individual fragment angular momenta, ${\mathit{I}}_{\mathrm{L}}$ and ${\mathit{I}}_{\mathrm{H}}$, and the orbital angular momentum $\mathit{L}$ are perpendicular to the fission axis. The left drawing presents a side view, while the right drawing shows a view along the fission direction.

Figure 2

Estimates of the average angular-momentum magnitude from Eq. (8) (dashed curves in upper parts) calculated at excitation energies derived from estimated/measured average neutron multiplicities (lower parts) for ${}^{238}\mathrm{U}({\mathrm{n}}_{\mathrm{th}},\mathrm{f})/{}^{252}\mathrm{Cf}$(sf). The experimental data were taken from Ref. [7] for panels (a) and (b), and the estimates for ${}^{238}\mathrm{U}$ from thermal fission of ${}^{233}\mathrm{U}$ and ${}^{235}\mathrm{U}$ from Ref. [34] for (c) and from Ref. [35] for (d).

Figure 3

Intrinsic excitation energy at scission versus fragment mass number: (a) the average intrinsic energy, $\langle E_{\mathrm{intr}}^{*}\rangle$, (b) the associated dispersion, ${\sigma }_{E^{*}}$, and (c) the relative dispersion, ${\sigma }_{E*}/\langle E_{\mathrm{intr}}^{*}\rangle$. The values for each fragment are indicated by red dots connected by dashed lines, while the values for the combined system are shown by blue dots connected by solid lines. The black dot-dashed line in (c) shows the constant value 0.25.

Figure 4

The distributions of the angular momentum magnitude for four selected fragments with $E_{\mathrm{intr}}^{*}=10\mathrm{MeV}$. Solid curves: combinatorial level density for a spherical shape; dashed curves: Fermi-gas expression.

Figure 5

The mean angular-momentum magnitude for spherical fragments as a function of the mass number $A$. The left panel shows results obtained without pairing, while the right panel is for pairing included with standard strength. The excitation energies (from below) are $E_{\mathrm{intr}}^{*}=5$, 10, and 20 MeV. The dashed curves show the corresponding Fermi-gas approximation.

Figure 6

Angular momentum distributions at scission, $P\left(I_L\right)$ and $P\left(I_H\right)$ for fragment masses $A_L=98$ (deformed) and $A_H=138$ (spherical), and the corresponding distribution of the orbital angular momentum, $P\left(L\right)$.

Figure 7

Calculated average intrinsic excitation energy versus fragment mass number (blue squares connected by solid lines). Level densities are calculated assuming spherical (upper panel) or by deformed (lower panel) scission configurations. Red circles connected dy dashed lines show the excitation energy as calculated by the event-by-event method assuming Ericsson state densities [26].

Figure 8

Calculated average angular momentum versus fragment mass number. Level densities are calculated assuming spherical (upper panel) or deformed (lower panel) scission configurations. Filled squares are used when the deformed/spherical condition applies (see Table 1) and open squares are used when it does not apply. Dashed lines show the Fermi-gas result.

Figure 9

The average angular momentum magnitude for spherical fragments as a function of mass number for the three isotopic sequences having $Z=48$, 50, and 52, calculated for the same average intrinsic excitation energy, $\langle E_H^{*}\rangle =8\mathrm{MeV}$. The fragments included in the present study (see Table 1) are marked by circles. The data from [7] are included (dots with error bars).

Figure 10

Upper panel: The average angular momentum magnitude versus the fragment mass number. The data (red dots with error bars) are from Ref. [7]. The calculated values used level densities for either spherical (blue squares) or deformed (blue diamonds) fragments based on the degree of quadrupole deformation at scission (see Table 1). The Fermi-gas results are also shown (dashed curve). Lower panel: The corresponding intrinsic fragment excitation energy. Values obtained in Ref. [26] are also shown (red dots connected by the dashed curve).

Figure 11

Upper panel: The average angular momentum magnitudes (blue diamonds) calculated on the basis of the average excitation energies obtained from freya simulations and with spherical or deformed fragment shapes as listed in Table 1. The Fermi-gas result (dashed curve) and the data from Wilson et al. [7] (red dots with error bars) are also shown. Middle panel: The fragment excitation energy determined from the measured neutron multiplicity $\langle \nu \rangle$. Lower panel: The experimental neutron multiplicities (a smooth curve drawn throughout several experimental results as shown in [26]) used as input in the freya simulation.

Figure 12

Contour plot (linear scale) of the probability distributions of excitation energy (left-hand panel) and angular momentum (right-hand panel), for the mass division 98:138. The dispersion in total excitation energy is rather small, 0.1 times the average total energy, resulting in a strong negative correlation of the energies, $-0.80$. The correlation coefficient between the two angular momenta is $-0.059$.

Figure 13

Same as Fig. 12 but with the dispersion in total excitation energy taken in accordance with experiment, as 0.25 times the average total energy. This results in a negative correlation coefficient between the energies, $-0.22$, and a weaker correlation coefficient between the angular momenta, $-0.021$.

Figure 14

Correlation in magnitudes of the angular momenta of the two fission fragments, as measured [7] for the fission of ${}^{239}\mathrm{U}$ in the reaction ${}^{238}\mathrm{U}(n,\mathrm{f})$ (left-hand side), and as here calculated for the fission of ${}^{236}\mathrm{U}$ in ${}^{235}\mathrm{U}(n_{\mathrm{th}},\mathrm{f})$ (right-hand side).

Figure 15

Slopes (left-hand part) and intercepts at angular momentum $I=2$ for fragment 2 (right-hand part) obtained from Fig. 14. Experimental values are shown by filled red circles with error bars, and calculated values are shown by filled blue squares.