---
title: TDHF-QRx in Heavy Nuclear Reactions
url: https://www.emergentmind.com/topics/tdhf-qrx-approach
type: topic
---

# TDHF-QRx in Heavy Nuclear Reactions

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 [1710.06813; 1412.1755; 1902.01616; 1606.00699]. 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 [1412.1755; 1106.3492].

## 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 [1902.01616]. 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 \(R(t)\) [1606.00699]. 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 [1612.08917].

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 [1412.1755; 1902.01616]. 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 \(^{238}\mathrm{U}+{}^{238}\mathrm{U}\) [1412.1755; 1606.00699; 1710.06813].

## 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,
\[
h(\{\phi_\mu\})\,\phi_\lambda=i\hbar\,\dot{\phi}_\lambda,
\qquad \lambda=1,\dots,N
\]
[1710.06813; 1412.1755]. The formal derivation begins from the time-dependent variational principle with action
\[
S=\int_{t_1}^{t_2}dt\,\langle \Phi(t)|H-i\hbar\partial_t|\Phi(t)\rangle
\]
and yields deterministic mean-field dynamics for the occupied orbitals [1412.1755].

The nuclear papers emphasize **unrestricted** TDHF, meaning that the time evolution is performed on a full **3D Cartesian grid without symmetry restrictions** [1710.06813; 1412.1755; 1606.00699]. 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 [1710.06813].

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 \(30\) fm, or \(34\)–\(40\) fm in the \(^{238}\mathrm{U}+{}^{238}\mathrm{U}\) study—and then boosted for collision. Time propagation uses a Taylor expansion of the unitary mean-field propagator up to roughly order \(10\)–\(12\), with time step \(\Delta t=0.4\) fm/\(c\) [1710.06813; 1412.1755; 1606.00699]. 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 [1710.06813; 1412.1755; 1508.04638].

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 [1902.01616]. 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 [1412.1755; 1902.01616].

## 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 [1106.3492; 1412.1755]. The conceptual starting point is the constrained static state consistent with the instantaneous TDHF density and, in principle, current,
\[
\langle \Phi_{\rho,\mathbf j}|\hat\rho(\mathbf r)|\Phi_{\rho,\mathbf j}\rangle=\rho(\mathbf r,t), \qquad
\langle \Phi_{\rho,\mathbf j}|\hat{\mathbf j}(\mathbf r)|\Phi_{\rho,\mathbf j}\rangle=\mathbf j(\mathbf r,t),
\]
although in practice the current constraint is replaced by a static density-constrained state with zero current [1412.1755; 1106.3492].

The density-constrained energy is
\[
E_{\mathrm{DC}}=\langle \Phi_\rho|\hat H|\Phi_\rho\rangle,
\]
and the collective energy is written as
\[
E_{\mathrm{coll}} = E_{\mathrm{kin}}(\rho,\mathbf j)+E_{\mathrm{DC}}(\rho),
\]
with an approximate kinetic term
\[
E_{\mathrm{kin}\approx \frac{m}{2}\sum_q \int d^3r\,\frac{\mathbf j_q^2(t)}{\rho_q(t)}
\]
or, in an earlier formulation,
\[
E_{kin} \approx \frac{\hbar^2}{2m}\int d^3r\; \frac{\mathbf{j}(t)^2}{\rho(t)}
\]
[1412.1755; 1106.3492]. Subtracting the isolated binding energies yields the microscopic ion-ion potential
\[
V(R)=E_{\mathrm{DC}(\rho(\mathbf r,t))}-E_{A_1}-E_{A_2},
\]
as a function of the internuclear separation \(R(t)\) extracted from the TDHF trajectory [1412.1755; 1106.3492; 1606.00699].

The same framework yields a coordinate-dependent mass,
\[
M(R)=\frac{2\,[E_{\mathrm{c.m.}-V(R)]}{\dot R^2},
\]
which enters barrier-penetration calculations and allows fusion or capture cross sections to be computed with the **incoming-wave boundary condition (IWBC)** method [1106.3492; 1412.1755]. 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 [1412.1755; 1106.3492].

Another recurring DC-TDHF observable is the intrinsic or precompound excitation energy. In the \(^{48}\mathrm{Ca},{}^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}\) work, it is written as
\[
E^*(R(t)) = E_{TDHF} - E_{DC}(R(t)) - E_{kin}(R(t))
\]
[1606.00699]. 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 [1412.1755].

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 [1412.1755; 1106.3492].

## 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 [1710.06813; 1508.04638]. 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
\[
\rho=\rho_0/2=0.08\ \mathrm{fm}^{-3},
\]
and
\[
\tau_{\rm contact}=t_2-t_1
\]
or equivalently
\[
t_{\rm contact}=t_2-t_1
\]
[1710.06813; 1606.00699].

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 [1508.04638; 1606.00699; 1612.08917]. In the \(^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}\) and related Bk-target studies, fusion is identified when the contact time exceeds roughly \(25\text{--}35\ \mathrm{zs}\) or about \(35\) zs and the system reaches a mononuclear shape without neck formation [1606.00699; 1612.08917].

Other standard observables include fragment masses and charges, total kinetic energy (TKE), excitation energies, deformation evolution, moments of inertia, and mass-angle distributions (MADs) [1412.1755; 1508.04638]. The quadrupole tensor
\[
Q_{ij}=\int d^3r\,\rho_{\mathrm{TDHF}}(\mathbf r,t)\,\big(3x_i x_j-r^2\delta_{ij}\big)
\]
is diagonalized to obtain the principal-axis quadrupole moment and the deformation parameter
\[
\beta_2=\frac{4\pi}{3}\frac{Q_{20}}{A R_0^2}, \qquad R_0=1.2\,A^{1/3},
\]
which track elongation and breakup during quasifission [1412.1755].

Mass-angle analysis uses the mass ratio
\[
M_R=\frac{m_1}{m_1+m_2},
\]
and the cited studies interpret the MAD as a clock for dinuclear rotation and contact duration [1412.1755; 1606.00699]. 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 [1412.1755; 1606.00699].

The approach also extracts rotational observables relevant to \(P_{\mathrm{CN}}\) analyses. The moment-of-inertia tensor is
\[
\Im_{ij}/m = \int d^3r\,\rho_{\mathrm{TDHF}}(\mathbf r,t)\,(r^2\delta_{ij}-x_i x_j),
\]
whose principal moments give \(\Im_\parallel\) and \(\Im_\perp\), from which an effective moment of inertia is constructed [1412.1755; 1612.08917]. 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 [1612.08917].

## 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 [1412.1755; 1710.06813; 1606.00699].

For **\(^{48}\mathrm{Ca}+{}^{238}\mathrm{U}\)**, DC-TDHF reproduces capture cross sections with explicit orientation dependence of the deformed \(^{238}\mathrm{U}\) target, and the results agree well with measured data [1412.1755]. In the comparison between **\(^{40}\mathrm{Ca}+{}^{238}\mathrm{U}\)** and **\(^{48}\mathrm{Ca}+{}^{238}\mathrm{U}\)**, 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 [1412.1755].

For **\(^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}\)** and **\(^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}\)**, 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 [1606.00699; 1508.04638]. Quantitatively, the DC-TDHF barrier heights for \(^{48}\mathrm{Ca}+{}^{249}\mathrm{Bk}\) are
\[
E_B(\text{tip}) = 191.22\ \mathrm{MeV}, \qquad R_B(\text{tip})=15.04\ \mathrm{fm},
\]
\[
E_B(\text{side}) = 204.36\ \mathrm{MeV}, \qquad R_B(\text{side})=12.47\ \mathrm{fm},
\]
while for \(^{50}\mathrm{Ti}+{}^{249}\mathrm{Bk}\) they are
\[
E_B(\text{tip}) = 211.2\ \mathrm{MeV}, \qquad R_B(\text{tip})=14.48\ \mathrm{fm},
\]
\[
E_B(\text{side}) = 224.6\ \mathrm{MeV}, \qquad R_B(\text{side})=12.96\ \mathrm{fm}
\]
[1606.00699]. These calculations conclude that both projectiles can fuse with \(^{249}\mathrm{Bk}\) under suitable side-contact conditions and that the experimentally poorer prospects for \(Z=119\) with \(^{50}\mathrm{Ti}\) are not explained by a dramatic entrance-channel TDHF disadvantage alone [1606.00699].

The most extreme transfer scenario studied in the supplied literature is **\(^{238}\mathrm{U}+{}^{238}\mathrm{U}\)** [1710.06813]. Here the goal is not only superheavy-element formation but also the possible production of neutron-rich high-\(Z\) 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
\[
Z \approx 98\text{--}101,\qquad A \approx 252\text{--}260
\]
up to about \(E_{\rm c.m.}=1100\) MeV, while at the highest energy studied,
\[
E_{\rm c.m.}=1350\ \mathrm{MeV},
\]
the heavy fragment reaches about
\[
Z \simeq 124,\qquad A \simeq 325
\]
[1710.06813]. For tip-tip collisions, several qualitatively distinct exit topologies occur: ternary quasifission with a small neck fragment at \(E_{\rm c.m.}=875\) MeV, reseparation into two excited \(^{238}\mathrm{U}\)-like fragments at \(1100\) MeV, a contact-time peak near \(1237\) MeV with a heavy fragment around \(Z\simeq 134,\ A\simeq 353\), and renewed ternary breakup at \(1300\) MeV with a heaviest central fragment around \(Z\simeq 103,\ A\simeq 274\) [1710.06813].

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 [1412.1755; 1710.06813; 1606.00699].

## 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 [1902.01616].

**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 [1902.01616]. 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 [1902.01616].

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** [1902.01616]. 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 [1902.01616].

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 [1902.01616]. 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 [1902.01616]. 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 [1710.06813]. In the Bk-target studies, BCS pairing with fixed occupations is included for \(^{50}\mathrm{Ti}\) to restore spherical density, while pairing is omitted for \(^{48}\mathrm{Ca}\) for computational speed [1606.00699].

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 [1902.01616; 1412.1755].

## 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 [1412.1755; 1902.01616]. As a result, it does not naturally yield full fragment mass distributions, event-by-event fluctuations, or complete compound-nucleus formation probabilities \(P_{\mathrm{CN}}\) [1412.1755; 1612.08917]. 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 \(^{238}\mathrm{U}+{}^{238}\mathrm{U}\) 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 \(E_{\rm c.m.}\), making their survival unlikely [1710.06813]. 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 [1606.00699].

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 \(K\)-distribution widths and rotational effects [1412.1755; 1612.08917].

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 [1412.1755; 1106.3492; 1710.06813]. 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 \(P_{\mathrm{CN}}\) and to multinucleon-transfer production pathways [1412.1755; 1902.01616].

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 [1412.1755; 1902.01616; 1710.06813].

Source: https://www.emergentmind.com/topics/tdhf-qrx-approach