TDHF-QRx in Heavy Nuclear Reactions

• The TDHF-QRx approach is a TDHF-based reaction-analysis strategy that integrates density-constrained methods and auxiliary extensions to extract key observables in heavy-ion collisions.
• It employs unrestricted 3D time-dependent Hartree-Fock to simulate mean-field dynamics, capturing nucleon transfer, deformation, and quasifission processes.
• The framework facilitates the extraction of ion-ion potentials, contact times, and excitation energies crucial for understanding fusion, quasifission, and multinucleon transfer in superheavy systems.

Searching arXiv for “TDHF QRx” and closely related TDHF/quasifission references. The term “TDHF-QRx approach” does not denote a distinct formalism in the cited arXiv literature. In the relevant nuclear-reaction papers, the documented framework is standard time-dependent Hartree-Fock (TDHF), often augmented by density-constrained TDHF (DC-TDHF) and, in broader multinucleon-transfer contexts, by projection, fluctuation, statistical-decay, or pairing extensions rather than by any separately defined “QRx” equation or operator [(Umar et al., 2017); (Umar et al., 2014); (Sekizawa, 2019); (Umar et al., 2016)]. Within that literature, the expression can therefore be understood only as a loose label for a TDHF-centered reaction-analysis strategy in which unrestricted three-dimensional mean-field dynamics provide the dominant reaction trajectory, while auxiliary constructions are used to extract ion-ion potentials, contact times, excitation energies, fragment properties, and inputs relevant to quasifission and compound-nucleus formation [(Umar et al., 2014); (Umar et al., 2011)].

1. Terminological status and scope

The central terminological point is negative but important: the supplied literature repeatedly states that there is no explicit QRx formalism named as such in the relevant texts on multinucleon transfer, superheavy-element dynamics, or actinide collisions (Sekizawa, 2019). Likewise, the study of $$^{48}\mathrm{Ca},{}^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$$ explicitly notes that it does not introduce a distinct “TDHF-QRx” formalism, nor a separate “reaction-coordinate” or “quasi-reaction” framework beyond standard TDHF/DC-TDHF collective observables such as the internuclear distance $R(t)$ (Umar et al., 2016). The review on quasifission dynamics presents TDHF as one component of a broader strategy in which TDHF supplies microscopic trajectories and mean observables, while additional ingredients beyond mean field would be needed for widths, fluctuations, and branching probabilities (Umar et al., 2016).

This terminological ambiguity has practical consequences. In the nuclear-heavy-ion literature, the approach associated with the supposed label is not a new theory parallel to TDHF, but rather a family of TDHF-based analyses of capture, quasifission, multinucleon transfer, and superheavy-element formation [(Umar et al., 2014); (Sekizawa, 2019)]. A plausible implication is that “QRx” functions, at most, as an informal umbrella label for TDHF plus selected beyond-mean-field or post-processing extensions, rather than as a formally specified approximation scheme.

The scope of this TDHF-centered framework is broad. It is used to study capture cross sections, quasifission trajectories, mass-angle distributions, fragment excitation energies, moments of inertia relevant to $P_{\mathrm{CN}}$, and multinucleon-transfer pathways in systems ranging from $^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$ to $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$, $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$, $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$, and $^{238}\mathrm{U}+{}^{238}\mathrm{U}$ [(Umar et al., 2014); (Umar et al., 2016); (Umar et al., 2017)].

2. Mean-field foundation: unrestricted TDHF

At the base of the approach is the time-dependent Hartree-Fock approximation, in which the many-body state is constrained to remain a single time-dependent Slater determinant. The equations of motion are written as

$$h(\{\phi_{\mu}\})\,\phi_{\lambda}(r,t)=i\hbar\,\frac{\partial}{\partial t}\phi_{\lambda}(r,t), \qquad (\lambda=1,\dots,A),$$

or equivalently in the notation of the superheavy-element review,

$$h(\{\phi_\mu\})\,\phi_\lambda=i\hbar\,\dot{\phi}_\lambda, \qquad \lambda=1,\dots,N$$

[(Umar et al., 2017); (Umar et al., 2014)]. The formal derivation begins from the time-dependent variational principle with action

$R(t)$0

and yields deterministic mean-field dynamics for the occupied orbitals (Umar et al., 2014).

The nuclear papers emphasize unrestricted TDHF, meaning that the time evolution is performed on a full 3D Cartesian grid without symmetry restrictions [(Umar et al., 2017); (Umar et al., 2014); (Umar et al., 2016)]. This is not a minor numerical detail. For heavy deformed systems, especially actinide-actinide or actinide-based reactions, long contact times, neck formation, orientation effects, and symmetry breaking are dynamical and may be suppressed by artificial constraints (Umar et al., 2017).

The practical implementation is highly standardized across the cited work. Static Hartree-Fock or HF-BCS ground states are first generated for projectile and target, the nuclei are placed at large separation—typically about $R(t)$1 fm, or $R(t)$2–$R(t)$3 fm in the $R(t)$4 study—and then boosted for collision. Time propagation uses a Taylor expansion of the unitary mean-field propagator up to roughly order $R(t)$5–$R(t)$6, with time step $R(t)$7 fm/$R(t)$8 [(Umar et al., 2017); (Umar et al., 2014); (Umar et al., 2016)]. The energy density functional is Skyrme-based, with SLy4 or SLy4d depending on the calculation, and time-odd terms are retained for Galilean invariance and dissipation [(Umar et al., 2017); (Umar et al., 2014); (Umar et al., 2015)].

The physical content of baseline TDHF is likewise consistent across the literature. It is a microscopic, parameter-free, self-consistent theory of one-body dissipation and average reaction dynamics, especially appropriate for near-barrier heavy-ion collisions where two-body collisional dissipation is expected to be less important (Sekizawa, 2019). It naturally describes transfer, deformation, neck formation, deep-inelastic motion, quasifission, and fusion on the mean trajectory, but it does not generate a distribution of channels from a single initial condition [(Umar et al., 2014); (Sekizawa, 2019)].

3. Density-constrained extension and collective observables

A major extension repeatedly associated with the approach is density-constrained TDHF (DC-TDHF), which converts the time-dependent trajectory into collective energies, ion-ion potentials, and excitation measures [(Umar et al., 2011); (Umar et al., 2014)]. The conceptual starting point is the constrained static state consistent with the instantaneous TDHF density and, in principle, current,

$R(t)$9

although in practice the current constraint is replaced by a static density-constrained state with zero current [(Umar et al., 2014); (Umar et al., 2011)].

The density-constrained energy is

$P_{\mathrm{CN}}$0

and the collective energy is written as

$P_{\mathrm{CN}}$1

with an approximate kinetic term

$P_{\mathrm{CN}}$2

or, in an earlier formulation,

$P_{\mathrm{CN}}$3

[(Umar et al., 2014); (Umar et al., 2011)]. Subtracting the isolated binding energies yields the microscopic ion-ion potential

$P_{\mathrm{CN}}$4

as a function of the internuclear separation $P_{\mathrm{CN}}$5 extracted from the TDHF trajectory [(Umar et al., 2014); (Umar et al., 2011); (Umar et al., 2016)].

The same framework yields a coordinate-dependent mass,

$P_{\mathrm{CN}}$6

which enters barrier-penetration calculations and allows fusion or capture cross sections to be computed with the incoming-wave boundary condition (IWBC) method [(Umar et al., 2011); (Umar et al., 2014)]. For deformed systems, the orientation dependence of the barrier is treated explicitly, and the total cross section can be obtained by angle averaging with orientation weights from Coulomb alignment calculations [(Umar et al., 2014); (Umar et al., 2011)].

Another recurring DC-TDHF observable is the intrinsic or precompound excitation energy. In the $P_{\mathrm{CN}}$7 work, it is written as

$P_{\mathrm{CN}}$8

(Umar et al., 2016). In the broader superheavy-element literature, fragment excitation energies are extracted directly from the TDHF trajectory and used to assess how much entrance-channel kinetic energy is converted into internal excitation during quasifission (Umar et al., 2014).

This density-constrained machinery is not merely auxiliary. It is the main route by which the TDHF trajectory becomes quantitatively useful for barrier systematics, excitation partitioning, and inputs to models of compound-nucleus formation and decay [(Umar et al., 2014); (Umar et al., 2011)].

4. Quasifission diagnostics and dynamical observables

In the heavy-system applications, the approach is used primarily to characterize quasifission, defined as reseparation after contact and nucleon exchange but before the formation of an equilibrated compound nucleus (Umar et al., 2017, Umar et al., 2015). The central dynamical observable is the contact time, defined as the interval between the first merging of the nuclear surfaces and their subsequent reseparation. The surface is taken from the half-density isosurface

$P_{\mathrm{CN}}$9

and

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$0

or equivalently

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$1

(Umar et al., 2017, Umar et al., 2016).

The literature uses contact time as a practical discriminator among reaction classes. Very short contact implies quasielastic or weak-transfer dynamics; intermediate times with reseparation indicate quasifission; very long contact with evolution to a mononuclear shape without a neck is taken operationally as fusion (Umar et al., 2015, Umar et al., 2016, Umar et al., 2016). In the $^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$2 and related Bk-target studies, fusion is identified when the contact time exceeds roughly $^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$3 or about $^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$4 zs and the system reaches a mononuclear shape without neck formation (Umar et al., 2016, Umar et al., 2016).

Other standard observables include fragment masses and charges, total kinetic energy (TKE), excitation energies, deformation evolution, moments of inertia, and mass-angle distributions (MADs) [(Umar et al., 2014); (Umar et al., 2015)]. The quadrupole tensor

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$5

is diagonalized to obtain the principal-axis quadrupole moment and the deformation parameter

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$6

which track elongation and breakup during quasifission (Umar et al., 2014).

Mass-angle analysis uses the mass ratio

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$7

and the cited studies interpret the MAD as a clock for dinuclear rotation and contact duration [(Umar et al., 2014); (Umar et al., 2016)]. TKE values are compared with Viola systematics, with good agreement in the superheavy-element applications; this is taken as evidence that the modeled exit channels are indeed quasifission-like, with final kinetic energy determined mainly by Coulomb repulsion at scission [(Umar et al., 2014); (Umar et al., 2016)].

The approach also extracts rotational observables relevant to $^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$8 analyses. The moment-of-inertia tensor is

$^{40,48}\mathrm{Ca}+{}^{238}\mathrm{U}$9

whose principal moments give $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$0 and $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$1, from which an effective moment of inertia is constructed [(Umar et al., 2014); (Umar et al., 2016)]. These quantities connect TDHF trajectories to statistical or phenomenological analyses of fragment angular distributions, even though the papers caution that quasifission is not a fully equilibrated process (Umar et al., 2016).

5. Representative applications in superheavy and actinide systems

The approach has been applied most extensively to reactions relevant to superheavy-element formation and to actinide collisions. Across these studies, the recurring message is that the decisive quantities are not only the nominal barrier energy or capture threshold, but also deformation, orientation, shell effects, energy dissipation, and the competition between fusion and quasifission [(Umar et al., 2014); (Umar et al., 2017); (Umar et al., 2016)].

For $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$2, DC-TDHF reproduces capture cross sections with explicit orientation dependence of the deformed $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$3 target, and the results agree well with measured data (Umar et al., 2014). In the comparison between $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$4 and $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$5, the neutron-rich projectile exhibits less quasifission, less dissipation, and lower fragment excitation, which is presented as a microscopic explanation of why neutron-rich fusion reactions are more favorable for superheavy-element synthesis (Umar et al., 2014).

For $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$6 and $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$7, orientation of the prolate Bk target dominates the dynamics. Side orientation yields higher barriers but longer contact times and fusion-favorable conditions, whereas tip orientation yields lower barriers but shorter contact times and quasifission-dominated outcomes (Umar et al., 2016, Umar et al., 2015). Quantitatively, the DC-TDHF barrier heights for $^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$8 are

$^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}$9

$^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$0

while for $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$1 they are

$^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$2

$^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$3

(Umar et al., 2016). These calculations conclude that both projectiles can fuse with $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$4 under suitable side-contact conditions and that the experimentally poorer prospects for $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$5 with $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$6 are not explained by a dramatic entrance-channel TDHF disadvantage alone (Umar et al., 2016).

The most extreme transfer scenario studied in the supplied literature is $^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$7 (Umar et al., 2017). Here the goal is not only superheavy-element formation but also the possible production of neutron-rich high-$^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$8 fragments through massive transfer in actinide-actinide collisions. Tip-side collisions generally have longer contact times than tip-tip collisions, and for central tip-side collisions the heavy fragment reaches approximately

$^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}$9

up to about $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$0 MeV, while at the highest energy studied,

$^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$1

the heavy fragment reaches about

$^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$2

(Umar et al., 2017). For tip-tip collisions, several qualitatively distinct exit topologies occur: ternary quasifission with a small neck fragment at $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$3 MeV, reseparation into two excited $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$4-like fragments at $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$5 MeV, a contact-time peak near $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$6 MeV with a heavy fragment around $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$7, and renewed ternary breakup at $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$8 MeV with a heaviest central fragment around $^{54}\mathrm{Cr}+{}^{186}\mathrm{W}$9 (Umar et al., 2017).

These applications collectively establish the empirical content of the approach: strong orientation dependence, large mass and charge rearrangements, shell-influenced fragment partitions, and quasifission as the dominant mechanism limiting compound-nucleus formation in very heavy systems [(Umar et al., 2014); (Umar et al., 2017); (Umar et al., 2016)].

6. Extensions beyond baseline TDHF

Because standard TDHF is deterministic and follows only the dominant mean trajectory, the literature repeatedly supplements it with controlled extensions. The multinucleon-transfer review is particularly explicit that the relevant beyond-TDHF ingredients are particle-number projection, TDHF+GEMINI, TDRPA, stochastic mean-field (SMF) methods, and pairing-enabled TDHFB/TDSLDA schemes, rather than any separately defined QRx equation (Sekizawa, 2019).

Particle-number projection is used to convert the final TDHF Slater determinant into probabilities for specific transfer channels after a collision. This addresses a structural limitation of TDHF: the theory gives a deterministic final mean field, but transfer observables require channel-resolved probabilities (Sekizawa, 2019). For comparison with measured yields, the review also highlights TDHF+GEMINI, where primary TDHF fragments are passed to the statistical evaporation code GEMINI++ for secondary deexcitation; this substantially improves agreement with measured cross sections because primary transfer products can be strongly modified by neutron evaporation (Sekizawa, 2019).

To remedy the severe underestimation of mass-distribution widths in deep-inelastic collisions, the literature turns to time-dependent random-phase approximation (TDRPA) derived from the Balian–Vénéroni variational principle (Sekizawa, 2019). In this formulation, TDHF provides the average trajectory while TDRPA includes one-body fluctuations around it. The review states that fragment-width predictions are substantially improved and may even agree quantitatively with experiment, although the version discussed there is restricted to symmetric reactions (Sekizawa, 2019).

A more general fluctuation framework is the stochastic mean-field (SMF) approach, in which initial fluctuations are sampled stochastically to produce an ensemble of trajectories. For multinucleon transfer, this can be formulated as a Fokker–Planck transport problem with drift and diffusion coefficients determined by TDHF single-particle orbitals (Sekizawa, 2019). This supplies the broader transfer distributions and rare-channel production probabilities absent in mean-field dynamics.

Finally, the review emphasizes the role of pairing extensions, including TDSLDA and TDHFB-type schemes, for pair transfer, gauge-angle effects, and modified fusion/quasifission dynamics (Sekizawa, 2019). In the actinide-collision work, pairing already enters at the static level through HF-BCS partial occupations used to reproduce the deformed uranium ground state more accurately (Umar et al., 2017). In the Bk-target studies, BCS pairing with fixed occupations is included for $^{238}\mathrm{U}+{}^{238}\mathrm{U}$0 to restore spherical density, while pairing is omitted for $^{238}\mathrm{U}+{}^{238}\mathrm{U}$1 for computational speed (Umar et al., 2016).

Taken together, these extensions clarify what a broadened “TDHF-QRx” label could plausibly encompass in practice: TDHF as the dynamical backbone, DC-TDHF for collective observables, projection for channel probabilities, TDRPA/SMF for fluctuations, statistical deexcitation for observable yields, and pairing dynamics for superfluid effects [(Sekizawa, 2019); (Umar et al., 2014)].

7. Limitations, interpretation, and significance

The main limitation is intrinsic to TDHF itself. The theory is deterministic, retains only a single time-dependent Slater determinant, and therefore provides the dominant reaction path rather than a statistical ensemble of possible outcomes [(Umar et al., 2014); (Sekizawa, 2019)]. As a result, it does not naturally yield full fragment mass distributions, event-by-event fluctuations, or complete compound-nucleus formation probabilities $^{238}\mathrm{U}+{}^{238}\mathrm{U}$2 [(Umar et al., 2014); (Umar et al., 2016)]. This is why the literature consistently frames TDHF as a source of microscopic inputs and average trajectories rather than as a complete theory of yields.

A second limitation concerns survival probabilities of the heaviest products. The $^{238}\mathrm{U}+{}^{238}\mathrm{U}$3 study shows that very large charge and mass transfers are possible, especially in tip-side geometry, but the resulting heavy fragments are predicted to carry excitation energies in the hundreds of MeV at high $^{238}\mathrm{U}+{}^{238}\mathrm{U}$4, making their survival unlikely (Umar et al., 2017). Similarly, the Bk-target studies emphasize that fusion or massive transfer at the TDHF level does not by itself guarantee successful evaporation-residue production, because post-contact statistical decay remains decisive (Umar et al., 2016).

A third limitation is methodological. Even when DC-TDHF provides microscopic potentials, excitation energies, and inertia parameters, further assumptions are needed to convert them into measurable cross sections or angular distributions, especially when coupling to statistical models or phenomenological formulas for $^{238}\mathrm{U}+{}^{238}\mathrm{U}$5-distribution widths and rotational effects [(Umar et al., 2014); (Umar et al., 2016)].

Despite these limitations, the significance of the approach in the cited literature is unambiguous. It provides a fully microscopic, parameter-free, self-consistent description of the entrance-channel and contact-stage dynamics of heavy-ion collisions, including deformation, orientation, neck formation, nucleon transfer, and one-body dissipation [(Umar et al., 2014); (Umar et al., 2011); (Umar et al., 2017)]. It reproduces capture barriers and cross sections, explains orientation effects in deformed actinide systems, identifies quasifission through contact-time and fragment observables, and provides microscopic inputs relevant to $^{238}\mathrm{U}+{}^{238}\mathrm{U}$6 and to multinucleon-transfer production pathways [(Umar et al., 2014); (Sekizawa, 2019)].

The strongest general conclusion supported by the supplied papers is therefore not the existence of a formally defined TDHF-QRx method, but rather the maturity of a TDHF-based microscopic program for heavy nuclear reaction dynamics. In that program, unrestricted 3D TDHF supplies the real-time reaction path; DC-TDHF extracts collective structure from that path; and projection, fluctuation, pairing, and deexcitation extensions are invoked when probabilities, widths, or observable post-collision yields are required [(Umar et al., 2014); (Sekizawa, 2019); (Umar et al., 2017)].