Nuclear resonance fluorescence of 208 Pb in heavy-ion colliders
aa r X i v : . [ nu c l - t h ] S e p EPJ manuscript No. (will be inserted by the editor)
Nuclear resonance fluorescence of
Pb in heavy-ion colliders
Uliana Dmitrieva and Igor Pshenichnov Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, 141700 Moscow Region, Russia Institute for Nuclear Research, Russian Academy of Sciences, 60-letia Okatyabrya, 7a, 117312 Moscow, RussiaReceived: date / Revised version: date
Abstract.
In ultraperipheral collisions (UPC) of nuclei the impact of Lorentz-contracted electromagneticfields of collision partners leads to their excitations. In case of heavy nuclei the emission of neutrons is amain deexcitation channel and forward neutrons emitted in UPC were detected at the Relativistic Heavy-Ion Collider (RHIC) and at the Large Hadron Collider (LHC) by means of Zero Degree Calorimeters.However, the excitation of low-lying discrete nuclear states is also possible in UPC below the neutronseparation energy. In this work by means of the Weizsacker-Williams method the data on nuclear resonancefluorescence (NRF) induced by real photons in
Pb are used to model the excitations of discrete levelsin colliding nuclei. Due to Lorentz boosts one can expect that de-excitation photons with energies up to40 GeV and 300 GeV are emitted in very forward direction, respectively, at the LHC and at the FutureCircular Collider (FCC-hh). Energy, rapidity and angular distributions of such photons are calculatedin the laboratory system, which can be used for monitoring of collider luminosity or triggering particleproduction in UPC.
PACS.
Since the 1950’s electromagnetic (Coulomb) excitation ofnuclei by charged particles have been used to probe nu-clear structure [1,2]. Such studies were mostly focused onthe properties of nuclear states excited by electromagneticfields because the properties of electromagnetic interac-tion were already well understood. A variety of nuclearand particle physics phenomena can be be studied in ul-traperipheral collisions (UPC) of nuclei, where the colli-sion impact parameter b exceeds the sum of the radii, R and R , of colliding nuclei: b > R + R [3]. In particular,properties of exotic nuclei far away from the valley of sta-bility have been investigated in UPC with stable nuclei [4,5]. Electromagnetic excitation of collective states like giantdipole resonances (GDR) in nuclei typically results in theirelectromagnetic dissociation (EMD) with emission of neu-trons and protons [6]. The excitation and decay of GDR isthe most important UPC channel and multiphonon GDRexcitations were successfully studied in UPC [7,8]. Morerecent studies include the investigations of EMD of nucleiwith a neutron skin [9,10]. Excited nuclear states belowthe GDR region are known as pygmy resonances [11,12],and their coupling to giant resonances have been also in-vestigated [13].A lot of data on Au–
Au, U– U and
Pb–
Pb collisions have been collected, respectively, at the
Correspondence to : [email protected] (I. Pshenichnov)
Relativistic Heavy-Ion Collider (RHIC) at Brookhaven Na-tional Laboratory (BNL) and at the Large Hadron Col-lider (LHC) at the European Organization for NuclearResearch (CERN). In addition to hadronic interactions ofnuclei with overlap of their nuclear densities research pro-grams at RHIC and the LHC include studies of UPC [14].As demonstrated by measurements and calculations [15],the uranium beam lifetime at RHIC is strongly influencedby the bound-free e + e − − pair production (BFPP) andEMD of U circulating in RHIC. At the LHC both pro-cesses are responsible for the production of secondary ionsfrom Pb which can potentially impact supercon-ducting magnets and cause their quenching [16,17]. Thecross sections of the most important channels of EMDof Pb calculated by RELDIS Monte Carlo model [18]agree well with the cross sections of forward neutron emis-sion measured by the ALICE experiment [19] at the LHC.As shown in these publications, the EMD of ultrarela-tivistic nuclei with emission of nucleons is presently wellunderstood. The detection of forward neutrons by ZeroDegree Calorimeters (ZDC) [20,21,22] helps to monitorthe collider luminosity [18] and can be also used as a trig-ger of UPC events with particle production [14].However, much less attention has been payed so far tolow-energy excitations of ultrarelativistic nuclei in UPCbelow the nucleon separation energy and thus resultingexclusively to emission of few MeV photons in the nu-cleus rest frame. It is expected that due to the Lorentzboost energetic forward photons will be emitted in the
Uliana Dmitrieva, Igor Pshenichnov: Nuclear resonance fluorescence of
Pb in heavy-ion colliders laboratory system. This process is similar to nuclear res-onance fluorescence (NRF), which has been studied in-tensively in experiments with real photons [23,24,25] andproposed, in particular, for non-destructive assays of spentnuclear fuel [26]. The properties of numerous discrete ex-cited nuclear states in , , , Pb below the neutronemission threshold were studied by detecting NRF pho-tons after the impact of bremmstrahlung radiation withthe endpoint energy of 6.75 MeV [27]. While the inter-est in detecting forward photons emitted by nuclear spec-tators in nucleus-nucleus collisions has been clearly ex-pressed [28], very forward photons at the LHC were de-tected exclusively by the LHCf experiment in pp and p–Pb collisions [29,30,31,32]. To the best of our knowledgethere are only two theory papers [33,34] devoted to theexcitation of discrete nuclear levels in UPC at the LHCand the respective photon emission. In the first work [33]a process of e + e − − pair production in fusion of virtualphotons in UPC of Ca nuclei followed by the excitationof discrete levels in Ca by the electron or positron with asubsequent photon emission has been investigated. In thesecond work [34] a process of production of a meson in γγ -fusion with simultaneous excitation of discrete nuclearlevels O ⋆ (2 + , 6.92 MeV) and Pb ⋆ (3 − , 2.62 MeV), re-spectively, in O– O and
Pb–
Pb UPC at the LHChas been studied.In the present work direct excitation by Weizsacker-Williams photons of discrete levels of
Pb in their UPCat the LHC and at the Future Circular Collider (FCC-hh) [35,36] is considered. All known excited states belowthe nucleon separation energy of
Pb are taken into ac-count. Their decays are only possible by emission of pho-tons, and the properties of this radiation are investigated.
Pb at the LHC and FCC-hh
As known, the cross section of photon absorption leadingto the transition from the nuclear ground state with thespin J to an isolated resonant nuclear state with the en-ergy E r , the total width Γ r and the spin J r is describedby the Breit-Wigner formula: σ ( E γ ) = π (cid:18) ~ cE r (cid:19) g Γ ( E γ − E r ) + Γ r / . (1)Here E γ is the photon energy, g = (2 J r + 1) / (2 J + 1)is the statistical factor, Γ is the partial width of the de-cay back to the ground state. The width Γ is typically offew eV. It is assumed that the direct nuclear resonancefluorescence dominates, Γ /Γ r ∼
1. This means that theexcited state mostly decays via a photon emission with adirect transition to the ground state, i.e., γ = γ ′ , and anytransitions to intermediate states J i can be neglected, seeFig.1. Fig. 1.
Nuclear resonance fluorescence with the excitation ofan isolated nuclear state with the energy E r and spin J r and decay to an intermedate ( J i ) or directly to theground state ( J ). The integral cross section for the considered excitedstate with the energy E r is calculated as: I ( E r ) = Z d E γ σ ( E γ ) = π (cid:18) ~ cE r (cid:19) g Γ Γ r = π Γ r σ max ( E r ) , (2)where the maximum cross section value at E r is: σ max ( E r ) = 2 π (cid:18) ~ cE r (cid:19) g Γ Γ r . (3)The calculation of the NRF cross section σ NRF ( E r ) inultraperipheral collisions for a given resonant state E r isstraightforward. The variations of the Weizsacker-Williamsspectrum n WW [3] within a very small resonance width Γ r can be safely neglected. Therefore, n WW can be repre-sented by its value at E r to calculate the respective inte-gral: σ NRF ( E r ) = Z d E γ σ ( E γ ) n WW ( E γ ) = n WW ( E r ) I ( E r )= π n WW ( E r ) Γ r σ max ( E r ) . (4)In Table 1 the values of energy E r , width Γ r and σ max for excitations of discrete levels in Pb below the neu-tron emission threshold in the energy range from 4 to8 MeV [37] are given. In total, 14 levels in
Pb aretaken into account. Their parameters were used to calcu-late NRF cross sections for each of these exited state in ul-traperipheral
Pb–
Pb collisions according to Eq. (4).The resulting cross sections for collisions at the LHC andFCC-hh at √ s NN = 5 .
02 and 39.4 TeV, respectively, aregiven in Table 1.The calculated NRF cross sections are also shown inFig. 2. As can be seen from this figure, the contributionof low-lying levels is enhanced because the Weizsacker-Williams spectrum n W W ( E γ ) changes with equivalent pho-ton energy E γ approximately as 1 /E γ . As a result, thecontribution of first four levels ( ∼ liana Dmitrieva, Igor Pshenichnov: Nuclear resonance fluorescence of Pb in heavy-ion colliders 3
Table 1.
Characteristics of excited states in
Pb below neutron emission threshold and corresponding NRF cross sections forultraperipheral
Pb–
Pb collisions at the LHC and FCC-hh at √ s NN = 5 .
02 and 39.4 TeV, respectively.Level Energy Width Spin, Photoabsorption cross NRF cross sectionnumber E r Γ r parity section σ max ( E r ) σ NRF ( E r )(MeV) (eV) J πr (b) (b) √ s NN = 5 .
02 TeV √ s NN = 39 . collision energies. The sum of NRF cross sections for allconsidered levels amounts to 4.15 b and to 5.16 b, respec-tively, at √ s NN = 5 .
02 and 39.4 TeV, see Table 1. level number ( b ) NR F σ .
085 4 .
842 5 .
293 5 .
512 5 .
846 5 .
948 6 .
264 6 .
312 6 .
363 6 .
720 7 .
063 7 .
083 7 .
279 7 . Level energy in MeV = 5.02 TeV (LHC) NN s = 39.4 TeV (FCC) NN s Fig. 2.
NRF cross sections for ultraperipheral
Pb–
Pbcollisions at the LHC and FCC-hh, respectively, at √ s NN =5 .
02 TeV (solid histogram) and at √ s NN = 39 . There are two features of ultraperipehral collisions of ul-trarelativistic nuclei which simplify the calculations of theangular distribution of photons emitted by these nuclei.Firstly, because of various orientations of the reaction plane, spin states of excited nuclei are not aligned and one can as-sume that photons from nuclear de-excitation are emittedisotropically in the rest frame of emitting nucleus. Sec-ondly, because of negligible changes of the total energy ofa nucleus E A = γM A due to the absorption of an equiva-lent photon, one can safely assume that this nucleus prop-agates further along the beam direction with initial veloc-ity. This is because of the energy of equivalent photons isrestricted by E max ∼ γ/R and this value represents onlya negligible fraction of the total energy of the nucleus.For example, for heavy nuclei r = E max /E A ≈ /RM A ∼ − . In this estimation, as well as in the following textnatural units are used: ~ = c = 1 .One can consider an excited nucleus with the Lorentz-factor γ which emits a photon with the energy E r andmomentum p r = E r in its rest frame. In this referenceframe the angle between the photon momentum and thebeam direction can be denoted as θ r . In the laboratoryframe the photon energy E , its longitudinal p L and trans-verse p T momentum components with respect to the beamdirection are given by the corresponding Lorentz boost: E = γ ( E r + βp r cos θ r ) p L = γ ( βE r + p r cos θ r ) p T = p r sin θ r (5)Thus, the angle θ of photon emission in the laboratoryframe can be calculated from the relation:tan θ = p T p L = sin θ r γ ( β + cos θ r ) (6)because of γ ≫ β ≈ θ = 1 γ tan θ r . (7)This indicates that the angular distribution of photonsemitted in the nucleus rest frame is extremely contracted Uliana Dmitrieva, Igor Pshenichnov: Nuclear resonance fluorescence of
Pb in heavy-ion colliders by factor of 1 /γ in the laboratory system in the case ofultrarelativistic nuclei. This is a well-known projector ef-fect which makes insignificant the differences in angulardistributions of emitted photons for individual levels withvarious J πr . In other words, all these photons can be de-tected by a quite compact forward detector.The photon energy E in the laboratory system dependson the angle of emission θ r in the nucleus rest frame. Withthe condition γ ≫ β ≈ E on emis-sion angle θ in the laboratory system can be simplified: E = γ ( E r + βp r cos θ r ) ≈ γE r (1 + cos θ r ) = 2 γE r cos θ r
2= 2 γE r θ r = 2 γE r γ tan θ , (8)where E r is the photon energy in the nucleus rest frame.It is convenient to express the ratio E/E r as a functionof θ , as shown in Fig. 3 for Pb beams at the LHC,because this dependence is valid for NRF from any levelin
Pb at a given γ . In the case of very forward photon (rad) θ r E / E = 5.02 TeV NN s Fig. 3.
The ratio
E/E r as a function of photon emission angle θ in the laboratory frame for Pb beams colliding at the LHCat √ s NN = 5 .
02 TeV. emission at θ = 0, the photon energy reaches its maximumvalue of E max = 2 γE r . This means that energy of NRFphotons in the laboratory frame extends up to ∼
40 GeVand ∼
300 GeV for ultraperipheral
Pb–
Pb collisionsat √ s NN = 5 .
02 TeV and 39.4 TeV at the LHC and FCC-hh, respectively.As explained above, an isotropic distribution of photonemission in the rest frame of
Pb can be safely assumed:d N/ d Ω r = 1 / π , or, after the integration over azimuthalangle, d N/ d θ r = − sin θ r /
2. From the condition that thenumber of photons emitted within a given angle elementin both reference systems is an invariant, the angular dis-tribution in the laboratory system can be obtained:d N d θ = − sin θ r θ r d θ . (9) Then, by calculatingsin θ r = 2 tan θ r /
21 + tan θ r / γ tan θ γ tan θ = 2 γ sin θ (1 + γ tan θ ) cos θ (10)andd θ r d θ = 2 d(arctan( γ tan θ ))d θ = 2 γ (1 + γ tan θ ) cos θ , (11)One finally obtains for the interval of θ from 0 to π/ N d θ = − γ sin θ (1 + γ tan θ ) cos θ = − γ sin θ cos θ (1 + ( γ −
1) sin θ ) . (12)Alternatively, for the same interval of θ the angular dis-tribution can be expressed only via tan θ :d N d θ = − γ tan θ (1 + tan θ )(1 + γ tan θ ) . (13)The distribution of emitted photons can be expressedin terms of pseudorapidity η = − ln(tan θ/
2) instead ofangle θ . With d θ d η = − η e η + 1 = − η , (14)and tan θ = 2 tan θ/ − tan θ/ − η − e − η = 1sinh η , (15)using Eq.(13) the pseudorapidity distribution is obtained:d N d η = 2 γ − η (1 + γ sinh − η ) sinh η cosh η = 2 γ sinh η cosh η ( γ + sinh η ) . (16)This pseudorapidity distribution is shown in Fig.4 forultraperipheral Pb–
Pb collisions at the LHC at √ s NN =5 .
02 TeV. Since it is calculated for massless photons, d N/ d η ≡ d N/ d y . As expected, the distribution has a distinct max-imum at the very forward direction corresponding to thebeam rapidity y beam = 8 .
6. While this distribution is quitewide and extends from η = 6 to η = 12, for six units ofpseudorapidity, such photons can interact only with for-ward detectors. In particular, in ALICE experiment NRFphotons can hit only ZDC [21,22] ( | η | > .
8) or one side(ADC, − < η < − .
9) of ALICE Diffractive detector(AD) [38]. In the latter case photons which hit ADC arerather soft (
E < E = 2 γE r γ sinh − η , (17)obtained from Eqs. (8) and (15). E/E r as a function of η is shown in Fig. 5. liana Dmitrieva, Igor Pshenichnov: Nuclear resonance fluorescence of Pb in heavy-ion colliders 5 η η / d N d = 5.02 TeV NN s Fig. 4.
Pseudorapidity distribution of NRF photons calcu-lated for ultraperipheral
Pb–
Pb collisions at the LHC at √ s NN = 5 .
02 TeV. η r E / E = 5.02 TeV NN s Fig. 5.
The ratio
E/E r as a function of pseudorapidity η for Pb beams colliding at the LHC at √ s NN = 5 .
02 TeV.
Since the considered low-lying levels in
Pb are charac-terized by finite lifetimes, their de-excitations do not takeplace exactly at the interaction point, but rather awayfrom it. A typical distance between
Pb–
Pb inter-action and photon emission point can be estimated forthe level with the largest NRF cross section. Its energyis E r = 5 . Γ r = 32 .
91 eV.Therefore, the lifetime of the excited state in the nucleusrest frame is τ = ~ /Γ r ≈ · − s, while in the labora-tory frame it is t = γ · τ ≈ · − s. During this time t an ultrarelativistic Pb moves only for a small distanceof l = c · t ≈ µ m. One can note for comparison thatit is shorter than a characteristic diameter ( ∼ µ m) ofLHC beams delivered to the interaction point [39]. Forthe narrowest level with E r = 4 .
085 MeV, Γ r = 0 .
783 eV this distance is 42 times longer, l ≈ . l is smaller than the un-certainty of determination of the longitudinal position ofprimary interaction vertex [40]. Therefore, in the case oftheir detection NRF photons can be used to improve thedetermination of vertex position. We considered the process of nuclear resonance fluores-cence (NRF) induced by Weizsacker-Williams photons inultraperipheral collisions of ultrarelativistic
Pb at theLHC and FCC-hh. The interest to such a phenomenon istrully interdisciplinary, because the nuclear structure of
Pb, which is usually studied in low-energy physics, canbe probed in high-energy physics experiments. A more so-phisticated approach to probe the levels in
Pb by irra-diating beam nuclei at the LHC with a free electron laserhas been proposed in Ref. [37] to make an intense sourceof 1 MeV < E γ <
400 MeV photons. The status of thephoton-nucleus collider proposed in Ref. [37] is unclear,but it certainly presumes very high costs of the installa-tion and operation. In contrast, the registration of NRFphotons considered in the present work requires only thecreation of traditional electromagnetic calorimeters, pos-sibly with advanced transverse segmentation suitable formeasurements of a sharply-forward angular distributionsof these photons.As shown in the present work, photons with energy upto 40 GeV are frequently emitted in the forward directionbecause of the de-excitation of discrete levels in
Pb fol-lowing their ultraperipheral collisions at the LHC. Thiscan be also considered as a source of monochromatic pho-tons of higher energies compared to Ref. [37]. Such photonemission is also expected at the FCC-hh, with photon en-ergies up to 300 GeV. The total cross section of photonemission is estimated as large as 4.15 b and 5.16 b at theLHC and FCC-hh, respectively. Convenient energy, an-gular and pseudorapidity distributions of NRF photonshave been obtained analytically. Similarly to the stud-ies [33,34] of other processes of photon emission due to thede-excitation of discrete levels in ultraperipheral nucleus-nucleus collisions such distributions can be used for es-timating the impact on various detectors or monitoringcollision rate.Finally, one can note that the experiment LHCf [32]at the LHC makes possible to measure forward photonsstarting from 50 GeV. In the case of this low-energy cut isreduced to 10–40 GeV, NRF photons can be detected byLHCf. As photon energy is proportional to beam energy,further rise of beam energy in next LHC runs make feasi-ble the detection of NRF photons with the present LHCfequipment.This work has been carried out with financial supportof RFBR within the project 18-02-40035-mega.
Uliana Dmitrieva, Igor Pshenichnov: Nuclear resonance fluorescence of
Pb in heavy-ion colliders
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