arX

iv:1

403.

2671

v1 [

nucl

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11

Mar

201

4LA-UR-14-21210, NT@UW-14-06

Relativistic Coulomb excitation within Time Dependent Superfluid Local Density

Approximation

I. Stetcu,1 C. Bertulani,2 A. Bulgac,3 P. Magierski,3, 4 and K.J. Roche3, 5

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA2Department of Physics and Astronomy, Texas A & M University - Commerce, Commerce, TX 75429, USA

3Department of Physics, University of Washington, Seattle, WA 98195–1560, USA4Faculty of Physics, Warsaw University of Technology, ulica Koszykowa 75, 00-662 Warsaw, POLAND

5Pacific Northwest National Laboratory, Richland, WA 99352, USA

(Dated: March 12, 2014)

Within the framework of the unrestricted time-dependent density functional theory, we presentfor the first time an analysis of the relativistic Coulomb excitation of the heavy deformed open shellnucleus 238U. The approach is based on Superfluid Local Density Approximation (SLDA) formulatedon a spatial lattice that can take into account coupling to the continuum, enabling self-consistentstudies of superfluid dynamics of any nuclear shape. We have computed the energy deposited in thetarget nucleus as a function of the impact parameter, finding it to be significantly larger than theestimate using the Goldhaber-Teller model. The isovector giant dipole resonance, the dipole pygmyresonance and giant quadrupole modes were excited during the process. The one body dissipationof collective dipole modes is shown to lead a damping width Γ↓ ≈ 0.4 MeV and the number ofpre-equilibrium neutrons emitted has been quantified.

PACS numbers: 25.70.De, 21.60.Jz, 25.70.-z, 24.30.Cz

Coulomb excitation represents an ideal method toprobe the properties of large amplitude nuclear motion,because the excitation process is not obscured by uncer-tainties related to nuclear forces. The excitation prob-abilities are governed by the strength of the Coulombfield only and they can be fully expressed in terms of theelectromagnetic multipole matrix elements [1–6]. Fromthe theoretical point of view, Coulomb excitation can betreated as a textbook example of a nuclear system beingsubjected to an external, time-dependent perturbation.However, in order to be able to probe nuclear collectivemodes involving multi-phonon states for example [7, 8],a large amount of energy has to be transferred to thenuclear system. Thus the interaction time should be rel-atively short and the velocity of the projectile has tobe sufficiently large for an efficient excitation of nuclearmodes of frequency ω, the collision time τcoll = b/γvhas to fulfill the condition that ωτcoll ≃ 1. Here b isthe impact parameter, v is the projectile velocity, andγ = (1 − v2/c2)−1/2 is the Lorentz factor. One of thebest known examples of collective nuclear motion is theisovector giant dipole resonance (IVGDR). A reasonablygood estimate of the IVGDR vibrational frequency is~ω ≈ 80MeV/A1/3 for spherical nuclei. It implies thatthe excitation of a heavy nucleus to such energies requiresa relativistic projectile.

We report on a new and powerful method to studyrelativistic Coulomb excitation and nuclear large ampli-tude collective motion in the framework of Time De-pendent Superfluid Local Density Approximation (TD-SLDA). This is a fully microscopic approach to the prob-lem based on an extension of the Density FunctionalTheory (DFT) to superfluid nuclei and time-dependent

external probes, where all the nuclear degrees of free-dom are taken into account on the same footing, withoutany restrictions and where all symmetries (translation,rotation, parity, local Galilean covariance, local gaugesymmetry, isospin symmetry, minimal gauge coupling toelectromagnetic (EM) fields) are correctly implemented[9, 10]. The interaction between the impinging 238U pro-jectile and the 238U target is very strong (∝ ZpZtα ≈ 62,where α is the fine structure constant), which thus requirea non-perturbative treatment, and the excitation processis highly non-adiabatic. We assume a completely clas-sical projectile straight-line motion since its de Brogliewavelength is of the order of 0.01 fm for γ ∼ 1.5− 2. Inevaluating the EM-field created by the uranium projec-tile with a constant velocity v = 0.7c along the z-axis,we neglect its deformation. The projectile produces anEM-field described by scalar A0 and vector A Lienard-Wiechert potentials (note an additional e),

A(r, t) =v

cA0(r, t) =

v

c

γZe2√

(x− b)2 + y2 + γ2(z − vt)2,

with the reaction plane in the xz-plane. These fieldscouple to a deformed 238U target nucleus residing ona spatial lattice, see [11]. The interaction leads to aCM motion of the target as well as to its internal ex-citation and full 3D dynamical deformation of the tar-get. In order to follow the internal motion for a longenough trajectory that allows the extraction of useful in-formation, we perform a transformation to a system inwhich the lattice moves with the CM. The required trans-formation for each single particle wave function readsφn(r, t) = exp(iR(t)·p)ψn(r, t), with R(t) describing theCM motion and p the momentum operator. The equa-

2

tion of motion (simplified form here) for φn becomes

i~φn(r, t) =[

H (r+R(t), t) + R(t) · p]

φn(r, t), (1)

where R(t) =∫

d3r j(r, t)/M is the CM velocity andj(r, t) the total current density.The target nucleus is described within the SLDA and

its time evolution is governed by the TD meanfield-likeequations (spin degrees of freedom are not shown):

i~∂

∂t

(

U(r, t)V (r, t)

)

=

(

h(r, t) ∆(r, t)∆∗(r, t) −h∗(r, t)

)(

U(r, t)V (r, t)

)

.(2)

The single-particle Hamiltonian h(r, t) and the pairingfield ∆(r, t) are obtained self-consistently from an energyfunctional that is in general a function of various nor-mal, anomalous, and current densities, and which alsocontains the time-dependent external fields. The exter-nal electromagnetic (EM) field has the minimal gaugecoupling ∇A = ∇ − iA/~c (and similarly for the time-component) in all terms with currents, as well as in thedefinition of the momentum operator p in Eq. (1), de-tails in [11]. In the current calculation, the Skyrme SLy4energy functional [12] was adopted, with nuclear pairingas introduced in Ref. [13], which provides a very decentdescription of the IVGDR in 238U [10]. The couplingbetween the spin and the magnetic field was neglected.The Coulomb self-interaction between protons of the tar-get nucleus is taken into account using the modificationof the method described in Ref. [14], so as not to includecontributions from images in neighboring cells. For thedescription of the numerical methods see Refs. [15, 16]and many other technical details can be found in [11].The DFT approach to quantum dynamics has some pe-

culiar characteristics, which might appear strange at thebeginning. Unlike a regular quantum mechanics (QM)treatment one does not have access to wave functions,but instead to various one-body densities and currents.Within a DFT approach quantities, which are trivial toevaluate in QM, become basically impossible to calculate.For example, by solving the Schrodinger equation onecan evaluate at any time the probability that a systemremained in its initial state from P (t) = |〈Ψ(0)|Ψ(t)〉|2,where Ψ(t) is the solution of the Schrodinger equation (orsome of its approximations). Within a DFT frameworkone has access to the one-body (spin-)density ρ(r, t) andone-body (spin-)current j(r, t) and there is no formulayet for the probability P (t) that the system remained inits initial state. One can calculate for example a quantitysuch as

∫

d3rρ(r, 0)ρ(r, t), but there is no obvious wayto relate it to the probability P (t). One might try todefine P (t) instead through the overlap of the initial andcurrent “Slatter determinants” constructed through thefictitious single-particle wave functions entering the DFTformalism, which is a rather arbitrary postulate. One canfind quite often in literature various formulas used withinDFT treatment of nuclei, which are simply “copied” from

0.1

0.5

0.9

1.3(a)

0.0

0.4

0.8

1.2

1500 2000 2500 3000 3500 4000 4500

P(t

) [1

0-3 M

eV c

/fm

]

t [fm/c]

(b)

FIG. 1. (color online) The emitted energy rate via EM ra-diation. The results correspond to a collision with impactparameter b = 16.2 fm, for two orientations: nuclear symme-try axis perpendicular (a) and parallel (b) with respect to theincoming projectile. The dashed line corresponds to the timeaveraged quantity. The rate at which this quantity changesis directly related to the characteristic damping time, whichwe estimate at 500 fm/c, leading to a width Γ↓ ≈ 0.4 MeV.

various mean field approaches, without any solid justifi-cation provided. As we will show here, restrictions in-herent to a DFT approach, prevent us from being ableto calculate various quantities, which within a QM ap-proach are easy to evaluate. Even though we evaluateaccurate densities and currents well beyond the linearregime, within a DFT approach we cannot separate forexample the emission of one and two photons from anexcited nucleus, which however could be estimated rela-tive easily within a perturbative linear response approachsuch a (Q)RPA. On the other hand a DFT approach hasunquestionable advantages, allowing us to go far into thenon-linear regime and describe large amplitude collectivemotion.

The incoming projectile excites various modes in thetarget nucleus and the axial symmetry of the initialground state is lost. Because 238U is highly deformed theenergy of the first 2+ is 45 keV, corresponding to a verylong rotational period, and thus during simulation timeconsidered here (≈ 10−20 sec.) it can be considered fixed.The identification of these modes requires certain care,since during the collision the system is beyond the lin-ear regime and the analysis using the response functionis not applicable in general. However, the informationabout the excited nuclear modes is carried in the sub-sequent EM radiation leading to nuclear de-excitation.De-excitation to the ground state via photon emission re-quires times of about 10−16 sec., which is four orders of

3

magnitude longer than in the current calculations. How-ever, it is possible to compute the spectrum of the pre-equilibrium neutrons and gamma radiation, which allowsthe identification of the excited nuclear modes. We canaccurately treat the system as a classical source of elec-tromagnetic radiation and the time dependence of protoncurrent governs the rate of emission, see Refs. [11, 17, 18]:

P (t+ r/c) =e2

πc

∑

l,m

∣

∣

∣

∣

∫ ∞

−∞

blm(k, ω)e−iωtdω

∣

∣

∣

∣

2

, (3)

with

blm(k, ω) =

∫

dtd3re−iωt∇× j(r, t)jl(kr)Y

∗lm(r), (4)

where ω = kc, jl(kr) is the spherical Bessel function of or-der l, and j(r, t) is the proton current. The emission rateP is plotted in Fig. 1. The magnitude of this quantityindicates that the total amount of radiated energy duringthe evolution time (about 2500 fm/c) is rather small com-pared to the total absorbed energy and does not exceed1 MeV, which is about 2 − 3% of the deposited energyreported in Table I below. This implies that the effectof damping of nuclear motion due to the emitted radi-ation can been neglected for such short time intervals.Consequently, the decreasing intensity of the radiation,see Fig. 1, is merely related to the rearrangements of theintrinsic structure of the excited nucleus caused by damp-ing of collective modes due to the one-body dissipationmechanism.TDSLDA provides the EM power spectrum [11, 17, 18],

dE

dω=

4e2

c

∑

l,m

|blm(k, ω)|2, (5)

arising from various multipoles in Eqs. (3, 4). This quan-tity is different from what one would compute withina linear response approach or first order perturbationtheory, see e.g. Refs. [1–6], which provides the exci-tation probability only ∝ |

∫

d3rρtr(r)Vext(r)|2, whereρtr(r) is the transition density and Vext(r) - the externalfield. dE/dω is proportional to the excitation probability,here in the non-linear regime, and the subsequent pho-ton emission probability as well. A typical example ofthe emitted EM radiation for a given impact parameteris shown in Fig. 2, here due to the internal excitation ofthe system alone. The EM radiation due the CM motionhas been calculated separately (see Table I and [11]).In Fig. 2(a) the emitted radiation shows a well defined

maximum at energy 10 − 12 MeV which corresponds tothe excitation of IVGDR. We have applied a smoothingof the original calculations using the half-width of 1 MeV.Therefore, the original separate peaks split due to the de-formation of 238U merge into a single wider peak. How-ever, at larger frequencies another local maximum existswhich we associate with the isovector giant quadrupole

10-4

10-3

10-2

10-1(a)

10-4

10-3

10-2

10-1

0 10 20 30 40

(b)

0246

4 6 8

x10-4

dE/d~ω

~ω [MeV]

dipole (⊥)dipole (‖)

totalquadrupole

FIG. 2. (color online) The energy spectrum of emitted EMat the impact parameter b = 12.2 fm. We show the dipolecontributions for both orientations, and the total quadrupolecontribution. The lower panel shows the radiation emittedfrom the target nucleus excited when only the dipole compo-nent of the projectile electromagnetic field is used.

resonance (IVGQR). In order to rule out other possibil-ities we have repeated the calculation by retaining onlythe dipole component of the electromagnetic field pro-duced by the projectile [11]. The results are shown inFig. 2(b). In this case, the high-energy structure above20 MeV disappears, evidence that the high energy peakis related to the IVGQR. Moreover, noticeable contribu-tion to the total radiation is coming from the quadrupolecomponent of radiated field.

At low energies a change of slope occurs at about~ω ≈ 7 MeV and it is also present at the same energyfor other impact parameters and orientations, see [11].It clearly indicates that there is a considerable amountof strength at low energies, giving rise to an additionalcontribution to the EM radiation. We attribute this ad-ditional structure to the excitation of the pygmy dipoleresonance (PDR). The inset of the figure 2 shows thespectrum of emitted radiation due to this mode. Thecontribution to the total radiated energy coming from thePDR is rather small and reads: 1.7, 1.5 and 0.8 keV forimpact parameters 12.2, 16.2 and 20.2 fm, respectively.It corresponds to about 0.07%, 0.14%, 0.15% of the emit-ted radiation (due to internal motion) respectively, whichis increasing with the impact parameter. The relativelysmall amount of E1 strength obtained in our calculations,in the region where the PDR is expected, agrees with theexperimental findings in the recent measurements per-formed on 238U [19].

The comparison between the average energy trans-ferred to the internal motion of the target nucleus for

4

TABLE I. Internal excitation energy in TD-SLDA (Eint) andin the Goldhaber-Teller model (EGT ) for three values of im-pact parameters b and two orientations of the nucleus withrespect to the beam. In the third column we list the ra-tios of the internal excitation energies to the total transferredenergy, while the energy radiated from the excited nucleusduring time interval δt = 2500 fm/c after collision is shownin the fourth column. The ratio of the energy emitted due tothe internal motion to the total energy radiated in the sameinterval is shown in column five.

b(fm) Eint (MeV) Eint/E Eintγ (MeV ) Eint

γ /Eγ EGT

12.2 || 25.11 0.588 0.5 0.941 17.05

16.2 || 8.966 0.470 0.217 0.939 7.33

20.2 || 3.798 0.367 0.106 0.934 3.47

12.2 ⊥ 39.29 0.668 0.911 0.960 19.33

16.2 ⊥ 12.87 0.547 0.411 0.963 8.6

20.2 ⊥ 5.413 0.444 0.199 0.961 4.21

three values of the impact parameter obtained withinTDSLDA and also within a simplified Goldhaber-Tellermodel [20], see Table I, shows that significantly moreenergy is deposited by the projectile within the TD-SLDA. The Goldhaber-Teller model is equivalent to alinear response result, assuming that all isovector tran-sition strength is concentrated in two sharp lines, cor-responding to an axially deformed target. An exactQRPA approach would therefore severely underestimatethe amount of internal energy deposited, one reason beingthe non-linearity of the response, naturally incorporatedin TDSLDA. The other reason being the fact that thepresent microscopic framework describing the target al-lows for many degrees of freedom, apart from pure dipoleoscillations, to be excited. At the same time, the CM tar-get energy alone is approximately the same as obtainedin a simplified point particles Coulomb recoil model ofboth the target and projectile.

The average energy radiated due to the internal exci-tation represents only part of the total radiated energy.(One should remember that a straightforward DFT ap-proach provides no measure for the variance.) Also, be-cause of the spreading of the strength due to one-bodydissipation only a fraction of the energy Γγ/Γ (whereΓγ is the EM-width alone and Γ the total width of theIVGDR) is emitted as a pulse, as shown in Fig. 1. A sub-sequent pulse of reduced amplitude is to be expected aftera delay ≈ Γ/(ΓγωIV GDR) ≈ 105 . . . 106 fm/c. Since oursimulations times are much shorter we are not able to seeemission of the second photon, as reported in experiment[7, 8], where two photons were measured in coincidence.In our calculations we have followed the nuclear evolutionduring approximately 2500 fm/c after collision, see [11].The other component of the EM radiation arises from theCM acceleration as a result of collision (Bremsstrahlung).This process takes part only during the relatively short

144.5

145.0

145.5

146.0

1500 2000 2500 3000 3500 4000

N(t

)

t [fm/c]

FIG. 3. (color online) The number of neutrons inside thesphere of radius 15 fm around the target nucleus as a func-tion of time. Both the perpendicular (solid lines) and parallel(dashed lines) are shown, for three values of the impact pa-rameters: 12.2, 16.2, and 20.2 fm.

time interval τcoll = b/vγ, see [11]. The radiation emittedfrom the internal motion has a much longer time scale.We can estimate the cross section for the emission of

radiation by means of

σγ ∝ 2π

∫ (

2

3

Eγ⊥(b)

E⊥(b)+

1

3

Eγ||(b)

E||(b)

)

bdb ≈ 127 mb, (6)

where E⊥(b) and E||(b) are the total energies transferredto the target nucleus during the collision at the impactparameter b and for two orientations of the target nu-cleus: perpendicular and parallel to the beam, respec-tively. A detailed comparison of intensities of radiationfor various impact parameters and orientations is shownin Table I. It is evident that although the intensity of ra-diation decreases with increasing impact parameter, theratio between the intensities due to the internal modeswith that of the CM motion remains fairly constant. It isalso slightly different for two orientations. In the case ofthe perpendicular orientation the target nucleus is moreefficiently excited which results in a larger ratio. Interest-ingly, the ratio of the energy radiated (fourth column) tothe energy deposited (second column) increases with in-creasing impact parameter. Eq. (6) can be decomposedinto contributions from various frequencies of emitted ra-diation and results are shown in [11].The average energy deposited in the target nucleus is

of the order of the neutron separation energy. In Fig. 3we plot the total number of neutrons inside a (smoothed)sphere of radius 15 fm which is slightly larger than the nu-clear diameter (see [11] for details). For all these impactparameters neutrons can leak from the excited system.Since more energy is deposited in the nucleus with per-pendicular orientation with respect to the beam, the rateof emitted neutrons is larger in that case.We thank G.F. Bertsch for a number of discussions and

reading the manuscript. We acknowledge support un-der U.S. DOE Grants DE-FC02-07ER41457, DE- FG02-08ER41533, and Polish National Science Centre (NCN)

5

Grant, decision no. DEC-2013/08/A/ST3/00708. Partof this work was performed under the auspices of theNational Nuclear Security Administration of the US De-partment of Energy at Los Alamos National Laboratoryunder contract No. DE-AC52-06NA25396. This researchused resources of the National Energy Research ScientificComputing Center, which is supported by the Office ofScience of the U.S. Department of Energy under Con-tract No. DE-AC02-05CH11231 and of the Oak RidgeLeadership Computing Facility at the Oak Ridge Na-tional Laboratory, which is supported by the Office ofScience of the U.S. Department of Energy under ContractNo. DE-AC05-00OR22725. Some of the calculations re-ported here have been performed at the University ofWashington Hyak cluster funded by the NSF MRI GrantNo. PHY-0922770. KJR was supported by the DOE Of-fice of Science, Advanced Scientific Computing Research,under award number 58202 ”Software Effectiveness Met-rics” (Lucille T. Nowell).

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[4] C. Bertulani and V. Ponomarev, Phys. Rep. 321, 139(1999).

[5] E. Lanza, M. Andres, F. Catara, P. Chomaz, andC. Volpe, Nucl. Phys. A 654, 792c (1999).

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[7] K. Boretzky, A. Grunschloß, S. Ilievski, P. Adrich, T. Au-

mann, C. A. Bertulani, J. Cub, W. Dostal, B. Eber-lein, T. W. Elze, H. Emling, M. Fallot, J. Holeczek,R. Holzmann, C. Kozhuharov, J. V. Kratz, R. Ku-lessa, Y. Leifels, A. Leistenschneider, E. Lubkiewicz,S. Mordechai, T. Ohtsuki, P. Reiter, H. Simon,K. Stelzer, J. Stroth, K. Summerer, A. Surowiec, E. Wa-jda, and W. Walus ((LAND Collaboration)), Phys. Rev.C 68, 024317 (2003).

[8] S. Ilievski, T. Aumann, K. Boretzky, T. W. Elze,H. Emling, A. Grunschloß, J. Holeczek, R. Holzmann,C. Kozhuharov, J. V. Kratz, R. Kulessa, A. Leisten-schneider, E. Lubkiewicz, T. Ohtsuki, P. Reiter, H. Si-mon, K. Stelzer, J. Stroth, K. Summerer, E. Wajda, andW. Walus (LAND Collaboration), Phys. Rev. Lett. 92,112502 (2004).

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of Physics 81, 1151 (2003), arXiv:physics/0012024.[15] A. Bulgac and K. J. Roche, J. Phys.: Conf. Series 125,

012064 (2008).[16] A. Bulgac, Y.-L. Luo, P. Magierski, K. J. Roche, and

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6

Supplemental material to:

Relativistic Coulomb excitation within Time Dependent Superfluid Local DensityApproximation

I. Stetcu, C. Bertulani, A. Bulgac, P. Magierski and K.J. Roche

Density functional in TDSLDA and coupling to electromagnetic field

Here we present various definitions and conventions which we have used in the manuscript. The density functionalis constructed from the following local densities and currents:

• density: ρ(r) = ρ(r, r′)|r=r′

• spin density: ~s(r) = ~s(r, r′)|r=r′

• current: ~j(r) = 12i (~∇− ~∇′)ρ(r, r′)|r=r′

• spin current (2nd rank tensor): J(r) = 12i(~∇− ~∇′)⊗ ~s(r, r′)|r=r′

• kinetic energy density: τ(r) = ~∇ · ~∇′ρ(r, r′)|r=r′

• spin kinetic energy density: ~T (r) = ~∇ · ~∇′~s(r, r′)|r=r′

• anomalous density: χ(r) = χ(rσ, r′σ′)|r=r′,σ=1,σ′=−σ

where

ρ(r, r′) =∑

µ

(

V ∗µ (r,+)Vµ(r

′,+) + V ∗µ (r,−)Vµ(r

′,−))

sx(r, r′) =

∑

µ

(

V ∗µ (r,+)Vµ(r

′,−) + V ∗µ (r,−)Vµ(r

′,+))

sy(r, r′) = i

∑

µ

(

V ∗µ (r,+)Vµ(r

′,−)− V ∗µ (r,−)Vµ(r

′,+))

sz(r, r′) =

∑

µ

(

V ∗µ (r,+)Vµ(r

′,+)− V ∗µ (r,−)Vµ(r

′,−))

τ(r, r′) =∑

µ

(

~∇V ∗µ (r,+) · ~∇Vµ(r′,+) + ~∇V ∗

µ (r,−) · ~∇Vµ(r′,−))

~j(r) = −Im(

∑

µ

V ∗µ (r,+) · ~∇Vµ(r,+) + V ∗

µ (r,−) · ~∇Vµ(r,−)

)

=

= Im

(

∑

µ

Vµ(r,+) · ~∇V ∗µ (r,+) + Vµ(r,−) · ~∇V ∗

µ (r,−)

)

Jx(r) = Im

(

∂

∂ysz(r, r

′)− ∂

∂zsy(r, r

′)

)

|r=r′

Jy(r) = Im

(

∂

∂zsx(r, r

′)− ∂

∂xsz(r, r

′)

)

|r=r′

Jz(r) = Im

(

∂

∂xsy(r, r

′)− ∂

∂ysx(r, r

′)

)

|r=r′χ(rσ, r′σ′) =

∑

µ

V ∗µ (r, σ)Uµ(r

′, σ′) (7)

7

The coupling of the nuclear system to the electromagnetic field:

~E = −~∇φ− 1

c

∂ ~A

∂t(8)

~B = ~∇× ~A (9)

(10)

requires the following transformation of proton densities and currents (subscript A denotes the quantities in thepresence of electromagnetic field): The coupling of the nuclear system to the electromagnetic field:

e ~E = −~∇A0 −1

c

∂ ~A

∂t(11)

e ~B = ~∇× ~A (12)

(13)

(we included the electron charge e into the definition of EM potentials) requires the following transformation of protondensities and currents:

• density: ρA(r) = ρA(r)

• spin density: ~sA(r) = ~s(r)

• current: ~jA(r) = ~j(r) − 1~c~Aρ(r)

• spin current (2nd rank tensor): JA(r) = J(r)− 1~c~A⊗ ~s(r)

• spin current (vector): ~JA(r) = ~J(r)− 1~c~A× ~s(r)

• kinetic energy density: τA(r) =(

~∇− i 1~c~A)

·(

~∇′ + i 1~c~A)

ρ(r, r′)|r=r′

= τ(r) − 2 1~c~A ·~j(r) + e2

~2c2 | ~A|2ρ(r) = τ(r) − 2 1~c~A ·~jA(r) − e2

~2c2 | ~A|2ρ(r)

• spin kinetic energy density: ~TA(r) =(

~∇− i 1~c~A)

·(

~∇′ + i 1~c~A)

~s(r, r′)|r=r′

= ~T (r) − 2 1~c~AT · J(r) + e2

~2c2 | ~A|2~s(r) = ~T (r)− 2 1~c~AT · JA(r) − e2

~2c2 | ~A|2~s(r)

As a result the proton single particle hamiltonian has the form:

hA(r) = −~∇A ·(

B(r) + ~σ · ~C(r))

~∇A + UA(r) +1

2i

(

~W (r) · (~∇A × ~σ) + ~∇A · (~σ × ~W (r)))

+~UAσ (r) · ~σ +

1

i

(

~∇A · ~UA∆(r) + ~UA

∆(r) · ~∇A

)

(14)

and

UA(r) = U(r)− C∇J 1

~c~∇ · [ ~A× ~s(r)]− Cτ

(

21

~c~A ·~j(r) + e2

~2c2| ~A|2ρ(r)

)

(15)

~UAσ (r) = ~Uσ(r)− C∇J 1

~c~∇× [ ~Aρ(r)]− CsT

(

21

~c~AT · J(r) + e2

~2c2| ~A|2~s(r)

)

(16)

~UA∆(r) = ~U∆(r)− Cj 1

~c~Aρ(r) (17)

~∇A ·(

B(r) + ~σ · ~C(r))

~∇A =[

~∇A

(

B(r) + ~σ · ~C(r))]

· ~∇A +

(

B(r) + ~σ · ~C(r))

[

∆− i1

~c

(

~A · ~∇A + ~∇A · ~A)

+e2

~2c2| ~A|2

]

(18)

We neglect in our approach the coupling of the magnetic field to nucleon magnetic moments as it represents thehigher order correction and does not influence the results presented in the manuscript. Consequently the neutronsingle particle hamiltonian is not affected by the electromagnetic field.

8

Numerical Implementation

We build a spatial three-dimensional Cartesian grid in coordinate space with periodic boundary conditions, andderivatives evaluated in momentum (Fourier-transformed) space. This method represents a flexible tool to describelarge amplitude nuclear motion as it contains the coupling to the continuum and between single-particle and collectivedegrees of freedom. For the present problem, we have considered a box size of 403 with the lattice constant 1 fm.The time step has been set to 0.076772 fm/c with a total time interval of about 4000 fm/c. The projectile is initiallyplaced at such a distance from the target nucleus that the collision occurs after 1600−1700 fm/c. Even though iniallythe projectle is far enough from the target and hence the EM fields are weak, spurious excitations produced by asudden switch of the EM interaction at t = 0 are possible. They were avoided by multiplying the EM potentials inEqs. (29, 30) by the smoothing function f(t) = 1/ [1 + exp((r(t) −R0)/a0)], where R0 = 800 fm, a0 = 120 fm. Thisensures that the EM field varies smoothly within the distance a0, but stay approximately equal to its physical valuewithin the distance 2R0.

Coulomb potential on the lattice

The Coulomb self-interaction of protons on the lattice has to be treated with care in order to avoid spuriousinteraction with the proton density images in the neighboring cells. Such effect arises due to the long tail of theCoulomb term. Here we present the method used to describe the Coulomb self-interaction of the target nucleus.Consider the charge distribution eρ(r):

∇2A0(r) = 4πe2ρ(r) (19)

A0(r) =

∫

d3r′e2ρ(r)

|r,−r′| (20)

Note that above we have defined A0 as eA0 (note e2 in the formula). After the Fourier transform one gets:

A0(r) =

∫

d3k

(2π)3e2ρ(~k)

k2exp(i~k · r) (21)

The above prescription generates however the spurious interaction between neighboring cells.Therefore we define the modified potential (Nx, Ny, Nz denote number of equidistant lattice points in each direction,

Li = Ni∆x, i = x, y, z, ∆x is the lattice constant):

f(r) = 1/r for r <√

L2x + L2

y + L2z

f(r) = 0 otherwise (22)

Clearly the Fourier transform is:

f(k) = 4π1− cos(k

√

L2x + L2

y + L2z)

k2(23)

and moreover

A0(r) =

∫

d3k

(2π)3e2ρ(~k)

k2exp(i~k · r) = 1

27NxNyNz

∑

~k∈LxLyLz

e2ρ(~k)f(k) exp(i~k · r) (24)

where in the last term ρ(~k) is the Fourier transformed density on the lattice 27LxLyLz. In practice it means that onehas to perform forward and backward Fourier transforms on the lattice three times bigger in each direction.This may however be avoided if one realizes that the Fourier transform of the density in the larger lattice can be

expressed through the Fourier transforms in the smaller lattices:

ρklm(~k) =∑

r∈L3

ρ(x, y, z) exp

(

−i(

k2π

3Lxx+ l

2π

3Lyy +m

2π

3Lzz

))

exp(−i~k · r) (25)

and we need to perform 27 FFTs on the smaller lattice L for k, l,m = 0, 1, 2 of the following quantities:

ρ(x, y, z) exp(

−i(

k 2πLx

x+ l 2πLy

y +m 2πLz

z))

.

9

FIG. 1. Coordinate system used to describe the reaction.

Subsequently we obtain the potential through the relation:

A0(r) =

=1

27NxNyNz

2∑

k,l,m=0

∑

~k∈L3

e2ρklm(~k)f

(

~k +

(

k2π

3Lx, l

2π

3Ly,m

2π

3Lz

))

exp(i~k · r)

× exp

(

i

(

k2π

3Lxx+ l

2π

3Lyy +m

2π

3Lzz

))

. (26)

Dipole component of the electromagnetic field produced by the projectile

Coordinates (see fig. 1):

R = (b, 0, vt), r = (x, y, z). (27)

The interaction energy:

VE(t) =

∫

d3rA0(r, t)ρc(r, t), (28)

where

A0(r, t) =γZe2

√

(x− b)2 + y2 + γ2(z − vt)2, (29)

and γ = (1 − v2/c2)−1/2. ρc(r) = Ψ∗(r)Ψ(r) is the charge density of the nucleus at location r and Ψ(r) are protonwavefunctions. The vector potential is given by

A(r, t) =v

cA0(r, t). (30)

In order to extract the dipole component we used the interaction Hamiltonian:

Hint(r, t) =γZe2

√

(x− b)2 + y2 + γ2(z − vt)2− γZe2√

b2 + γ2v2t2, (31)

where one subtracts the second term which is responsible for the c.m. scattering (i.e. monopole field).

10

Consequently the dipole term reads:

HE1m(r, t) =

√

2π

3rY1m (r)

γZe2

(b2 + γ2v2t2)3/2

∓b, (if m = ±1)√2vt (if m = 0)

, (32)

where r is the coordinate of one of the protons in the target. A sum over m has to be performed, i.e.

HE1(r, t) =∑

i=protons

∑

m

HE1m(ri, t) =∑

i=protons

∑

m

rliY1m(ri)f1m(t), (33)

where f1m(t) is the part of the interaction which does not involve the intrinsic structure of the nucleus:

f1m(t) =

√

2π

3

γZe2

(b2 + γ2v2t2)3/2

∓b, (if m = ±1)√2vt (if m = 0)

. (34)

Goldhaber-Teller model of Coulomb excitation

We present here the simple model for the description of the Coulomb excitation based on the Goldhaber-Tellermodel. The results obtained within this model have been listed in Table 1 in the manuscript and served as a referencefor a fully microscopic treatment of the Coulomb excitation. Within the model it is assumed that both protons andneutrons are represented by rigid density distributions which can oscillate against each other. Thus the parametersdescribing the nucleus are represented by masses of neutrons and protons Mn,Mp, proton charge Ze and frequenciesof IVGDR: ωx, ωz. The Coulomb excitation can therefore by treated in terms of classical motion of two bodiesrepresenting protons and neutrons bound by the harmonic potential and perturbed by an external field produced bythe incoming projectile. In turn the classical equation of motion reads:

Mpd2xpdt2

= Fx − dzpdtBy − µω2

x(xp − xn)

Mpd2zpdt2

= Fz +dxpdt

By − µω2z(zp − zn)

Mnd2xndt2

= µω2x(xp − xn)

Mnd2zndt2

= µω2z(zp − zn)

(35)

where µ =MpMn

Mn+Mp

is the reduced mass and

Fx = − d

dx(f(t)A0(r, t))

Fz = − d

dz(f(t)A0(r, t))

By = (∇× v

cf(t)A0(r, t))y .

(36)

Smoothing function f(t) has been described in the section: Numerical implementation.In this model the target nucleus possess only two types of degrees of freedom: those related to the CM motion

and those describing the internal harmonic excitation of GDR. In fig. 2 the results of the ultrarelativistic Coulombexcitation within this model have been shown. It corresponds to the collision of 238U on 238U (the same as describedin the manuscript) at the impact parameter b = 12.2 fm. The frequencies of the IVGDR have been assumed to be~ωx = 10 MeV and ~ωx = 12 MeV. The figure 3 shows the energy deposited in the target nucleus and correspondingto the internal motion (i.e. vibrations of protons against neutrons) as a function of the impact parameter.

11

−2000 −1000 0 1000 2000 3000−40

−30

−20

−10

0

10

20

30

40

disp

lace

men

t (fm

)

a)

x

z

(x2+z2)1/2

−200 −100 0 100 200 300

−100

−80

−60

−40

−20

0

20

40

forc

e (M

eV/fm

)

b)

transversal

longitudinal

−500 0 500 10000

0.005

0.01

0.015

0.02

0.025

velo

city

(in

uni

ts o

f c)

c)

|vp|

|vn|

0 100 2000

5

10

15

20

25

30

35

t (fm/c)

Ene

rgy

depo

site

d in

the

targ

et (

MeV

)

d)

Etot

Ecm

Eint

0 500 1000

−2

−1.5

−1

−0.5

0

0.5

1

1.5

t (fm/c)

disp

lace

men

t (fm

)

e)

xp

zp

xn

zn

−1000 −500 0 500 10000

0.05

0.1

0.15

0.2

0.25

0.3

t (fm/c)

disp

lace

men

t (fm

)

f)

|Rp−R

cm|

|Rn−R

cm|

FIG. 2. (color online) Coulomb excitation within the Goldhaber-Teller model at the impact parameter b = 12.2fm corre-sponding to the perpendicular configuration (see manuscript). Subfigure a) shows the time evolution of the CM of the target.Subfigure b) shows forces acting on the target nucleus during the collision. In the subfigure c) the velocities of protons andneutrons are shown. In the subfigure d) the energies of the target nucleus have been plotted as a function of time. In thesubfigure e) the position of the CM of protons and neutrons have been shown as a function of time. Finally the subfigure f)shows the relative position of CM of protons and neutrons with respect to the CM of the target nucleus.

Electromagnetic radiation from a nucleus described within TDSLDA

In this section we will use the notation (φ, ~A) instead of (A0 = eφ, A) to denote the EM potentials.Let us consider the proton density and current (we use Gauss units):

ρ(r, t) =

∫ ∞

−∞

dω

2πρ(r, ω) exp(−iωt) (37)

~j(r, t) =1

2π

∫ ∞

−∞

dω~j(r, ω) exp(−iωt) (38)

where

ρ(r, ω) =

∫

d3k

(2π)3ρ(~k, ω) exp(i~k · r) (39)

~j(r, ω) =

∫

d3k

(2π)3~j(~k, ω) exp(i~k · r) (40)

Maxwell equations:

∇ · ~E(r, t) = 4πeρ(r, t) (41)

∇ · ~B(r, t) = 0 (42)

∇× ~E(r, t) = −1

c

∂

∂t~B(r, t) (43)

∇× ~B(r, t) =1

c

(

∂

∂t~E(r, t) + 4πe~j(r, t)

)

(44)

12

10 12 14 16 18 20 22 2410

0

101

102

impact parameter (fm)

Dep

osite

d E

nerg

y (M

eV)

FIG. 3. (color online) Energy deposited in the target nucleus 238U for three values of the impact parameter: 12.2, 16.2, 20.2 fmand for two nuclear orientations: nuclear symmetry axis being parallel (squares) and perpendicular (circles) to the trajectoryof incoming projectile. The same quantity is shown for the Goldhaber-Teller model, assuming that the frequencies of the dipoleoscillations are ~ω = 10 MeV and ~ω = 12 MeV parallel (blue-dashed line) and perpendicular (red-solid line) to the nuclearsymmetry axis, respectively.

and spatial Fourier transforms:

i~k · ~E(~k, t) = 4πeρ(~k, t) (45)

i~k · ~B(~k, t) = 0 (46)

i~k × ~E(~k, t) = −1

c

∂

∂t~B(~k, t) (47)

i~k × ~B(~k, t) =1

c

(

∂

∂t~E(~k, t) + 4πe~j(~k, t)

)

(48)

Hence clearly:

~E = ~E|| + ~E⊥ (49)

~B = ~B⊥ (50)

where

∇× ~E||(r, t) = 0 (51)

∇ · ~E⊥(r, t) = 0 (52)

∇ · ~B⊥(r, t) = 0 (53)

and

~E = −∇φ(r, t)− 1

c

∂

∂t~A(r, t) (54)

~B = ∇× ~A(r, t) (55)

13

Clearly

~E||(r, t) = −∇φ(r, t) − 1

c

∂

∂t~A||(r, t) (56)

~E⊥(r, t) = −1

c

∂

∂t~A⊥(r, t) (57)

~B = ∇× ~A⊥(r, t) (58)

Therefore one has a freedom to choose ~A||(r, t) (gauge transformation) whereas ~A⊥(r, t) is the gauge invariant partof the vector potential.We choose the Coulomb gauge:

~A||(r, t) = 0 ⇔ ~A(r, t) = ~A⊥(r, t) (59)

Hence

e ~E||(r, t) = −∇φ(r, t) (60)

e ~E⊥(r, t) = −1

c

∂

∂t~A⊥(r, t) (61)

e ~B = ∇× ~A⊥(r, t) (62)

and only perpendicular components of electric and magnetic fields are responsible for emission of radiation. Theimportant equation in this case is the fourth Maxwell equation:

∇× ~B⊥(r, t) =1

c

(

∂

∂t~E⊥(r, t) + 4πe~j⊥(r, t)

)

+1

c

(

∂

∂t~E||(r, t) + 4πe~j||(r, t)

)

(63)

Since the lhs represents the vector of type ⊥ therefore:

∂

∂t~E||(r, t) + 4πe~j||(r, t) = 0 (64)

and

∇× ~B⊥(r, t) =1

c

(

∂

∂t~E⊥(r, t) + 4πe~j⊥(r, t)

)

(65)

Substituting the potential ~A:

∇×∇× ~A⊥(r, t) = −∇2 ~A⊥(r, t) =1

c2

(

− ∂

∂t2~A⊥(r, t) + 4πce2~j⊥(r, t)

)

(66)

∇2 ~A⊥(r, t)−1

c2∂

∂t2~A⊥(r, t) = −1

c4πe2~j⊥(r, t) (67)

where

~j⊥(r, t) = ~j(r, t)−~j||(r, t) = ~j(r, t) +1

4πe

∂

∂t~E||(r, t) = ~j(r, t)− 1

4πe2∂

∂t∇A0(r, t) (68)

Therefore in the Coulomb gauge

~A(r, t) = ~A⊥(r, t) =1

c

∫

d3r′e2~j⊥(r

′, t− |r− r′|/c)|r− r′| =

1

c

∫

d3r′e2~j⊥(r

′, ω) exp(ik|r− r′|)|r− r′| (69)

and in the far zone r >> r′:

~A(r, ω) =exp(ikr)

r

1

c

∫

d3r′e2~j⊥(r′, ω) exp(−i~k · r′) = exp(ikr)

r

1

ce2~j⊥(~k, ck) (70)

where ~k = k r

r and ω = ck and consequently:

~A(r, t) =1

2π

∫ ∞

−∞

dω ~A(r, ω) exp(−iωt) = e2

2π

∫ ∞

−∞

dk~j⊥(~k, ck)exp(ik(r − ct)

r(71)

14

Consequently since ~B = ∇× ~A we get:

e ~B(r, ω) = ∇× ~A(r, ω) = ∇× 1

c

∫

d3r′e2~j⊥(r

′, ω) exp(ik|r− r′|)|r− r′| =

=1

c

∫

d3r′ik(r− r′)× e2~j⊥(r

′, ω) exp(ik|r− r′|)|r− r′|2 − 1

c

∫

d3r′(r− r′)× e2~j⊥(r

′, ω) exp(ik|r− r′|)|r− r′|3

=1

c

∫

d3r′ik(r− r′)× e2~j(r′, ω) exp(ik|r− r′|)

|r− r′|2 − 1

c

∫

d3r′(r− r′)× e2~j(r′, ω) exp(ik|r− r′|)

|r− r′|3 , (72)

where in the last line we have used the fact that rotation of the vector of type || is zero. For the electric field:

e ~E⊥(r, ω) =iω

c~A(r, ω) =

iω

c2

∫

d3r′e2~j⊥(r

′, ω) exp(ik|r− r′|)|r− r′| (73)

e ~E||(r, ω) = −∇φ(r, ω) = −∇∫

d3r′e2ρ(r′, ω)

|r− r′| (74)

Hence in the far zone r >> r′ one gets:

~B(r, ω) =ie

c

exp(ikr)

r

∫

d3r′~k ×~j⊥(r′, ω) exp(−i~k · r′) =ie

c

exp(ikr)

r~k ×~j(~k, ω) (75)

~E(r, ω) = ~E⊥(r, ω) + ~E||(r, ω) =exp(ikr)

r

iω

c2e~j⊥(~k, ck) =

ie

c

exp(ikr)

rk~j⊥(~k, ck) (76)

=ie

c

exp(ikr)

rk(~j(~k, ck)−~j||(~k, ck)) =

ie

c

exp(ikr)

r

(

(~k · rr)~j(~k, ck)− (~k ·~j(~k, ck))r

r

)

(77)

=ie

c

exp(ikr)

r

r

r×(

~j(~k, ω)× ~k)

(78)

and consequently:

~B(r, t) =1

2π

∫ ∞

−∞

dω ~B(r, ω) exp(−iωt) = ie

2π

∫ ∞

−∞

dk~k ×~j(~k, ck)exp(ik(r − ct))

r(79)

=ie

c

∫ ∞

−∞

dω

2π~k ×~j(~k, ω)exp(−iω(t− r/c))

r(80)

~E(r, t) =1

2π

∫ ∞

−∞

dω ~E(r, ω) exp(−iωt) = ie

2π

r

r×∫ ∞

−∞

dk(

~j(~k, ck)× ~k) exp(ik(r − ct))

r(81)

=ie

c

r

r×∫ ∞

−∞

dω

2π

(

~j(~k, ω)× ~k) exp(−iω(t− r/c))

r(82)

Note that in the above expressions ~k and ω are related: ω = c|~k|.Poynting vector reads ~S = c

4π~E × ~B and thus:

~S(t) =c

4π(2π)2

∫ ∞

−∞

dω

∫ ∞

−∞

dω′ ~E(r, ω)× ~B(r, ω) exp(−i(ω + ω′)t)

=c

4π(2π)2

∫ ∞

−∞

dω

∫ ∞

−∞

dω′ ~E(r, ω)× ~B∗(r, ω′) exp(−i(ω − ω′)t)

=e2

4π(2π)2cr2r

r

∫ ∞

−∞

dω

∫ ∞

−∞

dω′(~k ×~j(~k, ω)) · (~k′ ×~j∗(~k′, ω′)) exp(−i(ω − ω′)t+ i(k − k′)r) (83)

=e2

4πcr2r

r

∣

∣

∣

∣

∫ ∞

−∞

dω

2π(~k ×~j(~k, ω)) exp(−iωt+ ikr)

∣

∣

∣

∣

2

(84)

=c

4π

r

r

∣

∣

∣

∣

∫ ∞

−∞

dω

2π~B(r, ω) exp(−iωt)

∣

∣

∣

∣

2

(85)

Energy per unit time emitted to the angle dΩ reads:

dP (t) = ~S(t) · rrr2dΩ =

e2

4πc

∣

∣

∣

∣

∫

dω

2π(~k ×~j(~k, ω)) exp(−iωt+ ikr)

∣

∣

∣

∣

2

dΩ (86)

15

Hence

dP

dΩ(t) =

e2

4πc

∣

∣

∣

∣

∫ ∞

−∞

dω

2π(~k ×~j(~k, ω)) exp(−iω(t− r/c))

∣

∣

∣

∣

2

=c

4πr2∣

∣

∣

~B(r, t)∣

∣

∣

2

(87)

Note that the radiation at time t is given by the current at time t− r/c, thus a simple time shift. Therefore the totalamount of radiated energy at the angle dΩ reads:

∫ ∞

−∞

dP

dΩ(t)dt =

c

4πr2∫ ∞

−∞

∣

∣

∣

~B(r, t)∣

∣

∣

2

dt =c

4πr2∫ ∞

−∞

~B(r, t) ·∫ ∞

−∞

dω

2π~B(r, ω) exp(−iωt)dt

=c

4πr2∫ ∞

−∞

dω

2π~B(r,−ω) · ~B(r, ω) =

c

4πr2∫ ∞

−∞

dω

2π~B∗(r, ω) · ~B(r, ω) (88)

which gives the spectral decomposition of emitted radiation:∫ ∞

−∞

dP

dΩ(t)dt =

c

8π2r2∫ ∞

−∞

dω∣

∣

∣

~B(r, ω)∣

∣

∣

2

=c

4π2r2∫ ∞

0

dω∣

∣

∣

~B(r, ω)∣

∣

∣

2

(89)

Hence the energy emitted at the angle Ω at frequency ω reads:

dE

dΩdω(ω) =

e2

4π2c

∣

∣

∣

~k ×~j(~k, ω)∣

∣

∣

2

=e2

4π2c

∣

∣

∣

∣

∫

d3r(

∇×~j(r, ω))

exp(−i~k · r)∣

∣

∣

∣

2

(90)

In order to calculate the quantities given by the expressions: (87) (90) we use the multipole expansion. Namely, letus consider eq. (90):

dE

dω=

∫

dE

dΩdω(ω)dΩ =

∫

e2

4π2c

∣

∣

∣

~k ×~j(~k, ω)∣

∣

∣

2

dΩ (91)

=e2

4π2c

∫∣

∣

∣

∣

∫

d3r

∫ ∞

−∞

dt(

~∇×~j(r, t))

exp(−i~k · r,+iωt)∣

∣

∣

∣

2

dΩ (92)

Let us denote:

~∇×~j(r, t) = ~b(r, t) (93)

~∇×~j(r, ω) = ~b(r, ω) (94)

We expand exp(−i~k · r):

exp(−i~k · r) = 4π∑

l,m

(−i)ljl(kr)Ylm(k)Y ∗lm(r) (95)

and consequently we get

dE

dω=

e2

4π2c

∫

∣

∣

∣

∣

∣

∣

∫

d3r

∫ ∞

−∞

dt

~b(r, t)4π

∑

l,m

(−i)ljl(kr)Ylm(k)Y ∗lm(r)

exp(iωt)

∣

∣

∣

∣

∣

∣

2

dΩ (96)

=e2

4π2c

∫

∣

∣

∣

∣

∣

∣

∫ ∞

−∞

dt4π∑

l,m

(−i)l~blm(~k, t)Ylm(k) exp(iωt)

∣

∣

∣

∣

∣

∣

2

dΩ (97)

=e2

4π2c

∫

∣

∣

∣

∣

∣

∣

4π∑

l,m

(−i)l~blm(~k, ω)Ylm(k)

∣

∣

∣

∣

∣

∣

2

dΩ (98)

where

~blm(k, t) =

∫

d3r~b(r, t)jl(kr)Y∗lm(r) (99)

~blm(k, ω) =

∫ ∞

−∞

~blm(k, t) exp(iωt)dt (100)

16

Note that ~blm is a function of k (not ~k) and

dE

dω=

e2

4π2c(4π)2

∫

∑

l,m,l′,m′

(−i)lil′(

~blm(k, ω) ·~b∗l′m′(k, ω))

Ylm(k)Y ∗l′m′(k)

dΩ (101)

=e2

4π2c(4π)2

∑

l,m

|~blm(k, ω)|2 =4e2

c

∑

l,m

|~blm(k, ω)|2 (102)

The above equation is used to calculate the spectrum of emitted radiation. In practice one needs only few multipoles.The contribution coming from l = 4 term is already negligibly small.In order to determine the rate of emitted radiation let us consider eq. (87):

P (t+ r/c) =

∫

dP

dΩ(t+ r/c)dΩ =

e2

4πc

∫∣

∣

∣

∣

∫ ∞

−∞

dω

2π(~k ×~j(~k, ω)) exp(−iωt)

∣

∣

∣

∣

2

dΩ (103)

=e2

4πc

∫∣

∣

∣

∣

∫ ∞

−∞

dω

2π

(∫

d3r(~∇×~j(r, ω)) exp(−i~k · r))

exp(−iωt)∣

∣

∣

∣

2

dΩ (104)

=e2

4πc

∫

∣

∣

∣

∣

∣

∣

∫ ∞

−∞

dω

2π

∫

d3r4π∑

l,m

(−i)l~b(r, ω)jl(kr)Ylm(k)Y ∗lm(r)

exp(−iωt)

∣

∣

∣

∣

∣

∣

2

dΩ (105)

=e2

4πc

∫

∣

∣

∣

∣

∣

∣

∫ ∞

−∞

dω

2π

4π∑

l,m

(−i)l~blm(k, ω)Ylm(k)

exp(−iωt)

∣

∣

∣

∣

∣

∣

2

dΩ (106)

=e2

4πc

∫ ∞

−∞

dω

2π

∫ ∞

−∞

dω′

2π

(4π)2∫

dΩ∑

l,m,l′,m′

(−i)lil′(

~blm(k, ω) ·~b∗l′m′(k′, ω′))

Ylm(k)Y ∗l′m′(k)

×

× exp(−i(ω − ω′)t) (107)

Note that in the last two lines of the above expression k = k′ because the vectors ~k,~k′ differ only by length (ω =ck, ω′ = ck′) but have the same direction specified by the angle Ω. Therefore:

P (t+ r/c) =e2

4πc

∫ ∞

−∞

dω

2π

∫ ∞

−∞

dω′

2π

(4π)2∑

l,m

(

~blm(k, ω) ·~b∗lm(k′, ω′))

exp(−i(ω − ω′)t) (108)

=e2

πc

∑

l,m

∣

∣

∣

∣

∫ ∞

−∞

~blm(k, ω) exp(−iωt)dω∣

∣

∣

∣

2

(109)

The last equation is used in practice to calculate the rate of emitted radiation.The above prescriptions work efficiently if one considers the radiation emitted due to internal nuclear excitation.

However in order to determine the contribution coming from the CM motion of the nucleus the simpler formula canbe derived. In this case the proton current reads:

~jp(r, t) = ~V (t)δ(r,−r0(t)) (110)

Then

~A(r, ω) =exp(ikr)

r

Ze2

c

∫

d3r′~jp(r′, ω) exp(−i~k · r′) (111)

=exp(ikr)

r

1

cZe2

∫ ∞

−∞

dt exp(iωt)~V (t) exp(−i~k · r0(t)) (112)

where

~jp(~k, ω) =

∫ ∞

−∞

dt exp

(

iω

(

t− 1

c~n · r0(t)

))

~V (t) (113)

≈∫ ∞

−∞

dt exp(iωt)~V

(

t+1

c~n · r0(t)

)

(114)

≈∫ ∞

−∞

dt exp(iωt)~V (t) = ~V (ω), (115)

17

where ω = ck. The approximation was made above that the recoil velocity is small and the total displacement ofthe nucleus is negligible. Therefore the perturbation of the radiation due to the change of nucleus position can beneglected. Consequently:

e ~B(r, ω) =iZe2

c

exp(ikr)

r~k × ~V (ω) (116)

and

~B(r, t) =

∫ ∞

−∞

dω

2π~B(r, ω) exp(−iωt) (117)

=iZe

c21

r

∫ ∞

−∞

dω

2πexp

(

−iω(

t− r

c

))

ω~n× ~V (ω) (118)

= −Zec2

1

r

d

dt

∫ ∞

−∞

dω

2πexp

(

−iω(

t− r

c

))

~n× ~V (ω) (119)

= −Zec2

1

r~n× d

dt~V(

t− r

c

)

(120)

Therefore

dP

dΩ(t) =

c

4πr2∣

∣

∣

~B(r, t)∣

∣

∣

2

=1

4πc3(Ze)2

∣

∣

∣

∣

∣

dV(

t− rc

)

dt

∣

∣

∣

∣

∣

2

sin2 θ (121)

and

P (t) =2

3

(Ze)2

c3

∣

∣

∣

∣

∣

dV(

t− rc

)

dt

∣

∣

∣

∣

∣

2

(122)

Spectral decomposition:

dE

dΩdω=

(Ze)2

4π2c|~k × ~V (ω)|2 (123)

and integrating over angles

dE

dω=

2

3π

(Ze)2

ck2|~V (ω)|2 =

2

3π

(Ze)2

c3ω2|~V (ω)|2 =

2

3π

(Ze)2

c3

∣

∣

∣

∣

∣

d~V

dt(ω)

∣

∣

∣

∣

∣

2

(124)

where d~Vdt (ω) is the Fourier transform of acceleration:

d~V

dt(ω) =

∫

dtd~V

dt(t) exp(iωt) (125)

The above derivation assumes that the moving nucleus can be treated as a point-like particle. This is a reasonableapproximation although it is not difficult to include suitable corrections. Let us consider the proton current in theform:

~jp(r, t) = ~V (t)ρ(r,−r0(t)) (126)

Using the same assumption as before, ie. that the motion is nonrelativistic and movement in space is negligible onegets:

~jp(~k, ω) = ~V (ω)ρ(~k) (127)

and (see (90)):

∫

dE

dΩdωdΩ =

(Ze)2

4π2c|~k × ~V (ω)ρ(~k)|2dΩ =

(Ze)2

4π2c

1

c2

∫

|~n× ω~V (ω)ρ(~k)|2dΩ (128)

=(Ze)2

4π2c3

∫

∣

∣

∣

∣

∣

~n× d~V

dt(ω)ρ(~k)

∣

∣

∣

∣

∣

2

dΩ (129)

18

where ~n = r

r . In the case of spherical density distribution it simplifies to:

∫

dE

dΩdωdΩ =

(Ze)2

4π2c3

∫

∣

∣

∣

∣

∣

~n× d~V

dt(ω)

∣

∣

∣

∣

∣

2

|ρ(k)|2dΩ =2

3π

(Ze)2

c3

∣

∣

∣

∣

∣

d~V

dt(ω)

∣

∣

∣

∣

∣

2

|ρ(k)|2 (130)

The above expressions can be used to determine the spectrum of emitted radiation. In the Figures 4 and 5 thecontributions to the energy spectrum coming from dipole and quadrupole terms are plotted for 3 values of impactparameter. The difference between the figures originates from two different smoothing widths that have been applied.Namely, the original curves have been convoluted with gaussians of widths 1 MeV (Fig. 4) and 0.5 MeV (Fig. 5 ).

0 10 20 30 4010

−4

10−2

0 10 20 30 4010

−4

10−2

dE

dhω

0 10 20 30 4010

−4

10−2

hω [MeV]

FIG. 4. (color online) The energy spectrum of emitted electromagnetic radiation due to internal excitation of the targetnucleus, caused by the collision at the impact parameters b = 12.2 fm (upper subfigure), b = 16.2 and b = 20.2 (lowestsubfigure) . The contributions from two orientations of the target nucleus are shown: perpendicular (squares) and parallel(circles) with respect to the incoming projectile. Dotted dashed line represents the dipole component of the radiation. Dashedline represents the quadrupole component of the radiation. In this case the smoothing width of the original curves was set to1 MeV.

For the radiation caused by the CM acceleration after collision the decomposition into multipoles is not useful andone can apply instead eqs. (122–124). The emission occurs within much shorter time scale governed by the collisiontime τcoll =

bvγ . The results are plotted in the Figs. 6, 7, 8, 9, 10.

Dipole dynamics and neutron emission

The framework of TDSLDA allows to calculate various one body observables. In this case the most important isthe nuclear dipole moment. Only two components of the dipole moment, lying in the reaction plane, can oscillate asa result of collision. In the Figs. 11, 12, 13 these two components of the dipole moment have been plotted.

19

0 10 20 30 4010

−4

10−2

0 10 20 30 4010

−4

10−2

dE

dhω

0 10 20 30 4010

−4

10−2

hω [MeV]

FIG. 5. (color online) The same as in the Fig. 4, but in this case the smoothing width of the original curves was set to 0.5MeV.

During the time evolution the nucleus can emit particles. In order to investigate this effect we have calculated thenumber of neutrons/protons within shells of various radii. As one can see from the Figs. 14, 15, 16, 17 the numberof protons in the shells outside the nucleus is negligible. Moreover this proton number is approximately constantwhich indicates that we rather probe the tail of the proton distribution than the emission process. On the contrarythe situation is different for neutrons. The number of neutrons in the smaller shell is much larger, although it is alsoapproximately constant. However in the larger shell the number of neutrons is constantly increasing in time with afairly constant average rate. It indicates that the neutron emission occurs as a result of Coulomb excitation process.

20

1500 1600 1700 18000

0.5

1

1.5

2

2.5x 10

−3

Time [fm/c]

P(t

) [

MeV

/ (fm

/c)

]

0 20 400

0.5

1

1.5

2

2.5x 10

−3

hω [MeV]

dE

/dhω

FIG. 6. (color online) The gamma emission rate (left panel) due to bremsstrahlung for the collision at the impact parameterb = 12.2fm. The right panel shows the energy spectrum emitted. Solid and dashed lines correspond to the perpendicular andparallel orientation of the target nucleus with respect to incoming projectile.

1500 1600 1700 18000

0.5

1

1.5

2

2.5x 10

−3

Time [fm/c]

P(t

) [

MeV

/ (fm

/c)

]

0 20 400

0.5

1

1.5

2

2.5x 10

−3

hω [MeV]

dE

/dhω

FIG. 7. (color online) The same as in the Fig. 6, but for the impact parameter b = 16.2fm.

21

1500 1600 1700 18000

0.5

1

1.5

2

2.5x 10

−3

Time [fm/c]

P(t

) [

MeV

/ (fm

/c)

]

0 20 400

0.5

1

1.5

2

2.5x 10

−3

hω [MeV]

dE

/dhω

FIG. 8. (color online) The same as in the Fig. 6, but for the impact parameter b = 20.2fm.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

dσ

dhω

[mb/M

eV]

hω [MeV]

FIG. 9. (color online) The contributions to the cross section with respect to gamma emission during 2500 fm/c after collision.The dashed line represents the Bremsstrahlung contribution. The solid line shows the contribution from intrinsic excitationmodes.

22

1500 2000 2500 3000 3500 400010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

time [fm/c]

|∆ R

CM

|2 [fm

2 ]

FIG. 10. (color online) The distance squared between the CM of protons and the total nuclear CM: |∆RCM |2 as a functionof time. The impact parameter is b = 12.2 fm, and the nuclear symmetry axis of the target is perpendicular to the projectile’strajectory. The slope does not depend on the orientation and the impact parameter. The numerical fit to the maxima (squaredamplitudes of dipole oscillations) with the function exp(−t/τ ) yields τ ≈ 500 fm/c.

0 2000 4000−1

−0.5

0

0.5

time [fm/c]

Dip

ole

mom

ent/e

[fm

]

Dy

Dz

0 2000 4000−0.4

−0.2

0

0.2

0.4

time [fm/c]

Dip

ole

mom

ent/e

[fm

]

−0.2 0 0.2−1

−0.5

0

0.5

Dy [fm]

Dz [f

m]

−0.5 0 0.5−0.4

−0.2

0

0.2

0.4

Dy [fm]

Dz [f

m]

FIG. 11. (color online) Two components of the dipole moment: Dz and Dy as a function of time. The left and right panelscorrespond to the collision with projectile moving along the y-axis and z-axis , respectively. Impact parameter: b = 12.2fm.

23

0 2000 4000−0.4

−0.2

0

0.2

0.4

time [fm/c]

Dip

ole

mom

ent/e

[fm

]

Dy

Dz

0 2000 4000−0.4

−0.2

0

0.2

0.4

time [fm/c]

Dip

ole

mom

ent/e

[fm

]−0.2 0 0.2

−0.4

−0.2

0

0.2

0.4

Dy [fm]

Dz [f

m]

−0.5 0 0.5−0.2

−0.1

0

0.1

0.2

Dy [fm]

Dz [f

m]

FIG. 12. (color online) The same as in the Fig. 11, but for the impact parameter b = 16.2fm.

0 2000 4000−0.3

−0.2

−0.1

0

0.1

0.2

time [fm/c]

Dip

ole

mom

ent/e

[fm

]

Dy

Dz

0 2000 4000−0.2

−0.1

0

0.1

0.2

time [fm/c]

Dip

ole

mom

ent/e

[fm

]

−0.1 0 0.1−0.3

−0.2

−0.1

0

0.1

0.2

Dy [fm]

Dz [f

m]

−0.2 0 0.2−0.15

−0.1

−0.05

0

0.05

0.1

Dy [fm]

Dz [f

m]

FIG. 13. (color online) The same as in the Fig. 11, but for the impact parameter b = 20.2fm.

24

0 1000 2000 3000 40001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

time [fm/c]

∆ N

(t)

FIG. 14. (color online) The number of neutrons present within the shell with inner radius 10fm and outer radius 15fm (redline). The number of neutrons present within the shell with inner radius 10fm and outer radius 20fm (blue line). Thin linecorresponds to the actual number of neutrons, whereas the thick line denotes the average value. The plot corresponds tothe collision with the target nucleus symmetry axis perpendicular to the trajectory of the incoming projectile. The impactparameter b = 12.2 fm.

25

0 1000 2000 3000 40001

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

time [fm/c]

∆ N

(t)

FIG. 15. (color online) The same as in the Fig. 14, but for the nuclear orientation parallel with respect to the incomingprojectile.

0 1000 2000 3000 40000.2

0.25

0.3

0.35

0.4

0.45

time [fm/c]

∆ N

(t)

FIG. 16. (color online) The same as in the Fig. 14, but for protons.

26

0 1000 2000 3000 40000.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

time [fm/c]

∆ N

(t)

FIG. 17. (color online) The same as in the Fig. 15, but for protons.