-
Research ArticleBremsstrahlung from Relativistic Heavy Ions in a
Fixed TargetExperiment at the LHC
Rune E. Mikkelsen, Allan H. Sørensen, and Ulrik I. Uggerhøj
Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, 8000 Aarhus C, Denmark
Correspondence should be addressed to Rune E. Mikkelsen;
[email protected]
Received 20 March 2015; Accepted 13 May 2015
Academic Editor: Gianluca Cavoto
Copyright © 2015 Rune E. Mikkelsen et al. This is an open access
article distributed under the Creative Commons AttributionLicense,
which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properlycited. The
publication of this article was funded by SCOAP3.
We calculate the emission of bremsstrahlung from lead and argon
ions in ultraperipheral collisions in a fixed target
experiment(AFTER) that uses the LHC beams. With nuclear charges of
Ze equal to 82e and 18e, respectively, these ions are acceleratedto
energies of 7 Tev × Z. The bremsstrahlung peaks around ≈100GeV and
the spectrum exposes the nuclear structure of theincoming ion. The
peak structure is significantly different from the flat power
spectrum pertaining to a point charge. Photons arepredominantly
emitted within an angle of 1/𝛾 to the direction of ion propagation.
Our calculations are based on the Weizsäcker-Williams method of
virtual quanta with application of existing experimental data on
photonuclear interactions.
1. Introduction
The structure of stable nuclei, in particular the
chargedistribution, may be investigated by impact of photons
andelectrons as, for example, shown in pioneering works
byMcAllister andHofstadter; see, for example,
[1].Thismethod,however, is not possible for unstable nuclei with
shortlifetimes as, for example, hypernuclei. Instead,
essentiallywith a change of reference frame, one may let the
nucleusunder investigation impinge on a suitable target, for
example,an amorphous foil, and measure the delta electrons
and/orphotons emitted in the process.The interaction thus
proceedsbetween the nucleus and a target electron or a virtual
photonsimilarly originating from the target. With this method
thecharge distribution may be measured, in this case of
theprojectile, which might be a nucleus of very short lifetime,𝛾𝑐𝜏
≃ 1mm, where 𝛾 is the Lorentz factor, 𝑐 the speed oflight, and 𝜏
the lifetime. With the proposal to extract protonsand heavy ions
from the LHC for fixed target physics, the so-called AFTER@LHC,
such measurements would in principleenable charge distributions, or
at least sphericity, for nucleiwith lifetimes down to femtoseconds
to be extracted. Wereport calculations of bremsstrahlung emission
from Pb andAr nuclei, with energies corresponding to the maximum
ofthe LHC.We are restricted to cases where the projectile is
left
intact, that is, to ultraperipheral collisions in which
projectileand target nuclei do not overlap. The interaction
betweenthe collision partners is electromagnetic but the structure
ofthe composite projectile nucleus, namely, the strong
nuclearforce, plays a significant role in the photon emission
throughthe giant dipole resonance.
2. Bremsstrahlung
We study bremsstrahlung emission by relativistic heavy ions.When
traversing an amorphous target, the projectile ionsinteract with
the target electrons and nuclei. This causesradiation emission and
energy loss to the projectiles. Wefocus on the radiation and assume
the ion beam to bemonoenergetic; that is, we consider targets
sufficiently thinthat the projectile energy loss is minimal. We
further assumeimpact parameters in excess of the sumof the radii of
collisionpartners.
To establish a reference value for the cross section, wefirst
consider the incoming ion as a point particle of electriccharge 𝑍𝑒
colliding with target atoms of nuclear charge 𝑍
𝑡𝑒.
The major part of the radiation is due to the interaction ofthe
projectile with target nuclei which, in turn, are screenedby target
electrons at distances beyond the Thomas-Fermi
Hindawi Publishing CorporationAdvances in High Energy
PhysicsVolume 2015, Article ID 625473, 4
pageshttp://dx.doi.org/10.1155/2015/625473
-
2 Advances in High Energy Physics
distance, 𝑎TF. The cross section differential in energy for
theemission of bremsstrahlung photons from an ion with atomicnumber
𝐴 then reads [2]
𝑑𝜒
𝑑ℏ𝜔
=
163𝑍2𝑡𝑍4
𝐴2 𝛼𝑟
2𝑢𝐿, (1)
where 𝛼 ≡ 𝑒2/ℏ𝑐 is the fine structure, 𝑒 the unit
electriccharge, and ℏ the reduced Planck constant. The
classicalnucleon radius is defined as 𝑟
𝑢≡ 𝑒
2/𝑀𝑢𝑐2, where 𝑀
𝑢
is the atomic mass unit. Expression (1) gives the radiationcross
section or power spectrum; it is the number spectrumweighted by the
photon energy, ℏ𝜔. The factor 𝐿 is given by
𝐿 ≈ ln( 233𝑀𝑍1/3𝑡𝑚
)−
12[ln (1+ 𝑟2) − 1
1 + 𝑟−2] ,
𝑟 =
96ℏ𝜔𝛾𝛾𝑍1/3𝑡𝑚𝑐
2,
(2)
where 𝛾 ≡ 𝐸/𝑀𝑐2, 𝛾 ≡ (𝐸 − ℏ𝜔)/𝑀𝑐2, 𝑚 is the electronmass, and 𝑀
is the mass of the projectile. The material-dependent factor 𝐿
accounts for the electronic screeningof the target nuclei. It is
essentially the logarithm of theratio between the effective maximum
(∼2𝑀𝑐) and minimum(∼ℏ/𝑎TF) momentum transfers to the scattering
center. Thereference power spectrum extends all the way up to
theenergy of the primary ion and varies only slightly with
energy.However, as we will study in this paper, photons with
energyℏ𝜔 ≳ 𝛾ℏ𝑐/𝑅 have wavelengths smaller than the radius of
theions which cause the emission, making them sensitive to
thenuclear structure and collective dynamics of the
constituentprotons. Taking this into account causes significant
change inthe shape of the bremsstrahlung spectrum.
3. The Weizsäcker-Williams Method
When the emitted bremsstrahlung photons have a wave-length that
is small compared to the nucleus, the sizeand structure of the
nucleus affect the emission. We caninvestigate this by using the
Weizsäcker-Williams methodof virtual quanta [2–5]. In this
approach, we represent amoving charged particle by a spectrum of
virtual photonswhich scatter on a stationary charged particle.
There arecontributions to bremsstrahlung from scattering both
ontarget constituents and on the projectile ion. The latter isthe
dominant contribution. Hence we change to a referenceframe in which
the incoming ion is at rest and where werepresent the screened
target nuclei by a bunch of virtual pho-tons. These photons scatter
off the ion and are subsequentlyLorentz boosted back to the
laboratory frame—resulting inan energy increase by a factor of up
to 2𝛾. This means thatthe bremsstrahlung can be calculated using
the Weizsäcker-Williams method of virtual quanta, photonuclear
interactiontheory, and a Lorentz transformation. For the cross
sectionwetake the elastic photonuclear interaction cross section as
thisensures that the scatterer remains intact; that is, the
incomingion does not disintegrate in the process of radiation
emission.The spectrumof virtual photons is given in [2,
6].Multiplying
this with the photonuclear scattering cross section
differentialin angle results in the scattering cross section
differentialin energy and angle. The transformation to the
laboratoryframe is performed by utilizing an invariance relation
[2] andproduces the bremsstrahlung power spectrum differential
inenergy and angle; for details, see [6–8].
In [6], one of us used this procedure to calculate
thebremsstrahlung spectrum of relativistic bare lead ions. Thiswas
possible by using the photonuclear interaction dataprovided in [9].
However, data for other nuclei is notabundantly available. We
therefore developed a procedure toderive the necessary elastic
scattering cross sections takingtotal photonuclear absorption cross
sections as input; theseare available in the ENDF database for
about 100 differentnuclei [10]. We obtain the elastic scattering
cross sections atlow to moderate energies, that is, at energies
covering thegiant dipole resonance by applying the optical theorem
and adispersion relation to the total photonuclear absortion
data;see [7]. At higher energies additional constraints are
invokedto ensure coherence.With this construct, the
bremsstrahlungspectrum can be calculated for any ion for which the
totalphoton absorption cross section is known. Due to theavailable
data, our approach is most exact for lead ions whichhave already
been successfully accelerated to 4 TeV×𝑍 in theLHCmachine. Also,
this allowed us to cross-check the earliercalculations for the
bremsstrahlung from lead. It has not beenfinally decided if other
ions will ever be used in the LHC.But argon ions are frequently
discussed as a possibility if thephysics case requires lower mass
ions to be accelerated [11].Supporting this idea, in 2015, the CERN
accelerator complexis successfully accelerating argon ions in the
Super ProtonSynchrotron at energies up to 150GeV/n. We therefore
alsoprovide bremsstrahlung calculations for argon ions at LHCenergy
in the next section.
4. Results: Bremsstrahlung
In this section, we present calculations of
bremsstrahlungspectra in ultraperipheral collision for bare argon
and leadions at 7 TeV × 𝑍 incident on a fixed target. Figure 1
showsthe power spectrum of bremsstrahlung for lead ions aimedat a
lead target. The spectrum obtained by integrating overall emission
angles has a pronounced peak around a photonenergy of about 80GeV.
This corresponds to the collectiveinteraction of the projectile
nucleons with the virtual photonsof the target—the giant dipole
resonance. This well-knownresonance in the photonuclear cross
sections is also apparentin the bremsstrahlung spectrum, albeit
here multiplied bya factor of 2𝛾 from the Lorentz boost. At
energies abovethe peak, significantly fewer photons are predicted
by thecurrent model. This is because coherence is restricted toa
decreasing range of photon scattering angles such thatmost
scattering events correspond to incoherent interactionwith the
projectile protons. Incoherent interaction of anenergetic virtual
photon with a target proton generally leadsto breakup of the
nucleus and hence does not contribute tothe spectrum. This
decreasing contribution from coherentscattering leads to a
depletion of the elastic scattering crosssection at higher
energies, which is apparent also in the
-
Advances in High Energy Physics 3
0 100 200 300 4000
200
400
600
800
Nuclear structurePointlike particles
ℏ𝜔 (GeV)
d𝜒/dℏ𝜔
(mba
rn)
Figure 1: Bremsstrahlung calculations for a Pb-208 projectile
withenergy 7 TeV × 𝑍 incident on a lead target. The dashed line
showsreference cross section (1) and the full drawn curve shows
thepresent results.
0 100 200 300 400 500 6000
5
10
15
20
Nuclear structurePointlike particles
ℏ𝜔 (GeV)
d𝜒/dℏ𝜔
(mba
rn)
Figure 2:The same as Figure 1 except for the projectile which is
hereAr-40.
bremsstrahlung spectrum. Since the dashed line in Figure
1extends all the way up to the energy of the ion, the
integrateddifference between the two curves is very large. Note
thatFigure 1 pertains to cases where the projectile is left
intact.The photoproduction in collisions with nuclear overlap
isdrastically different [6].
The spectrum for argon projectiles is shown in Figure 2. Itis
largely similar to that for lead except that the overall valuesare
much lower. The peak height is approximately 50 timeslower, and it
must be noted that there is no simple scalingbetween the heights
and shapes of the two spectra (scaling theheights with𝑍2 off-shoots
by a factor of 2-3 here).The lack ofsuch scaling is traced back to
differences in the photonuclearscattering cross sections. The argon
spectrum is somewhatbroader than that for lead and the high energy
tail extends tolarger energies than for lead. This difference
reflects a similardifference at the elastic scattering cross
sections and is dueto the different shapes of the argon and lead
nuclei. Lead isalmost spherically symmetric and this leads to a
very narrowgiant dipole resonance peak. For argon on the other
hand, thenucleus is much less symmetric, and the photonuclear as
wellas the bremsstrahlung cross sections actually consist of
twoindividual but close-lying peaks (for bremsstrahlung this
willshow through collimation).
0.5 1 1.5 20
0.5
1
1.5
2
𝜃 (mrad)
×108
d𝜒/d𝜃
(mba
rn×
GeV
)
Figure 3:The angular distribution of bremsstrahlung from a
Pb-208projectile with energy 7 TeV ×𝑍 incident on a lead target.
Note thatthe cross section is shown to be differential in the polar
emissionangle 𝜃 rather than in solid angle and hence includes a
factor of2𝜋sin(𝜃). The majority of the bremsstrahlung photons are
indeedemitted in the very near forward direction.
Figure 3 shows the bremsstrahlung spectrumobtained forlead ions
by integration over emission energy, that is, differ-ential in
angle instead of energy. As expected, there is a peakin the
radiation intensity at an angle corresponding to 1/𝛾.For LHC beam
energies this means that the bremsstrahlungphotons are emitted at
angles of order less than 1mrad.
5. Summary and Conclusions
We have presented calculations of the bremsstrahlung emis-sion
from relativistic heavy ions with energy correspondingto that of
the LHC beam. The calculations, which arerestricted to
ultraperipheral collisions between projectileand target nuclei, are
novel in the way that knowledge onnuclear structure is taken into
account using existing data onphotonuclear cross sections. In
addition, our approach hasnot before been applied to energies above
that available atthe SPS. We demonstrate that substantial cross
sections forthe emission of high energy photons are expected
aroundenergies of ℏ𝜔 ≈ 100GeV. Here our model produces aradiation
peak which overshoots the result for a pointlikeparticle of the
same charge and mass by a factor of roughly2 and 6 for argon and
lead, respectively. In the energy regionsbelow and above the peak,
however, our model producessignificantly fewer bremsstrahlung
photons than what isotherwise expected. If this holds true against
experiments,it implies that the energy loss of the ions through
thebremsstrahlung channel is much less severe than
previouslyexpected by some authors (cf. [6]).
Our calculations can of course also be performed at
lowerenergies; see [7] for calculated cross sections for energies
ofabout 150GeV/n.The SPS can accelerate to about 450GeV×𝑍so that
experiments located in the SPS fixed target hall shouldbe able to
see this signal. The COMPASS experiment [12]may be a candidate. See
also [13] for calculations on deltaelectron emission from a similar
experimental condition asdiscussed here. Along with bremsstrahlung
measurements,such electrons would be highly sensitive to the
nuclear chargestructure.
-
4 Advances in High Energy Physics
If a fixed target facility using the LHC beams is con-structed,
one could hope for the option to produce secondarybeams.With such
beams, one could study the bremsstrahlungfrom short-lived nuclei.
If traversing a 1mm target, thelifetime would be 𝛾𝑐𝜏 = 1mm, which
leads to 𝜏 ≈ 4 ⋅ 10−16 s.
Exploiting the Lorentz time dilation, the nuclear structureof
these rare nuclei could be exposed. Potentially, along withthese
short-lived species, one may also study exotic beamswhere the
nucleus contains a strange quark. It is presentlyunknown whether
the presence of a strange quark increasesor decreases the radius of
the nuclear charge. Whereas sucheffects may be visible in the
bremsstrahlung spectrum, theywould certainly be impossible to study
using conventionalelectron scattering.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
References
[1] R. W. McAllister and R. Hofstadter, “Elastic scattering of
188-Mev electrons from the proton and the alpha particle,”
PhysicalReview, vol. 102, no. 3, pp. 851–856, 1956.
[2] J. D. Jackson, Classical Electrodynamics, John Wiley &
Sons,New York, NY, USA, 2nd edition, 1975.
[3] E. J. Williams, “Nature of the high energy particles of
penetrat-ing radiation and status of ionization and radiation
formulae,”Physical Review, vol. 45, p. 729, 1934.
[4] E. J. Williams, “Correlation of certain collision problems
withradiation theory,” Mathematisk-Fysiske Meddelelser, vol. 13,
no.4, pp. 1–4, 1935.
[5] C. F. Weizsäcker, “Ausstrahlung bei Stößen sehr
schnellerElektronen,” Zeitschrift für Physik, vol. 88, no. 9-10,
pp. 612–625,1934.
[6] A. H. Sørensen, “Bremsstrahlung from relativistic heavy ions
inmatter,” Physical Review A, vol. 81, Article ID 022901, 2010.
[7] R. E. Mikkelsen, A. H. Sørensen, and U. I. Uggerhøj,
Submittedto Physical Review C.
[8] T. V. Jensen and A. H. Sørensen, “remsstrahlung from
rela-tivistic bare heavy ions: nuclear and electronic contributions
inamorphous and crystallinematerials,”Physical ReviewA, vol. 87,no.
2, Article ID 022902, 15 pages, 2013.
[9] K. P. Schelhaass, J. M. Henneberg, M. Sanzone-Arenhövel
etal., “Nuclear photon scattering by 208Pb,”Nuclear Physics A,
vol.489, pp. 189–224, 1988.
[10] M. B. Chadwick, M. Herman, P. Obložinský et al.,
“ENDF/B-VII.1 nuclear data for science and technology: cross
sections,covariances, fission product yields and decay data,”
NuclearData Sheets, vol. 112, no. 12, pp. 2887–2996, 2011.
[11] M. Benedikt, P. Collier, V. Mertens, J. Poole, and K.
Schindl,LHC Design Report (CERN), vol. 3, chapter 33, 2004.
[12] D. vonHarrach, “TheCOMPASS experiment at
CERN,”NuclearPhysics A, vol. 629, no. 1-2, pp. 245–254, 1998.
[13] A. H. Sørensen, “Electron-ion momentum transfer at
ultra-relativistic energy,” in Proceedings of the 20th
InternationalConference on the Physics of Electronic and Atomic
Collisions, F.Aumayr andH.Winter, Eds., p. 475,World Scientific,
Singapore,1998.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Superconductivity
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
GravityJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Physics Research International
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Soft MatterJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
PhotonicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Journal of
Biophysics
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
ThermodynamicsJournal of