# Expanding Nuclear Physics Horizons with the Gamma Factory

Dmitry Budker  Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany
Helmholtz-Institut, GSI Helmholtzzentrum für Schwerionenforschung, 55128
Mainz, Germany  Department of Physics, University of California, Berkeley,
California 94720, USA  Julian C. Berengut  School of Physics, University of
New South Wales, Sydney 2052, Australia  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Victor V. Flambaum  School of
Physics, University of New South Wales, Sydney 2052, Australia  Johannes
Gutenberg-Universität Mainz, 55128 Mainz, Germany  Helmholtz-Institut, GSI
Helmholtzzentrum für Schwerionenforschung, 55128 Mainz, Germany  The New
Zealand Institute for Advanced Study, Massey University Auckland, 0632
Auckland, New Zealand  Mikhail Gorchtein  Johannes Gutenberg-Universität
Mainz, 55128 Mainz, Germany  Junlan Jin  Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China  Felix
Karbstein  Helmholtz-Institut Jena, GSI Helmholtzzentrum für
Schwerionenforschung, 07743 Jena, Germany  Theoretisch-Physikalisches
Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-
Wien-Platz 1, 07743 Jena, Germany  Mieczyslaw Witold Krasny  LPNHE, Sorbonne
Université, CNRS/IN2P3, Paris; France  CERN, Geneva, Switzerland  Yuri A.
Litvinov  GSI Helmholtzzentrum für Schwerionenforschung, Planckstrasse 1,
64291 Darmstadt, Germany  Adriana Pálffy  Department of Physics, Friedrich-
Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany  Vladimir
Pascalutsa  Institut für Kernphysik, Johannes Gutenberg-Universität Mainz,
55128 Mainz, Germany  Alexey Petrenko  Budker Institute of Nuclear Physics,
Novosibirsk, Russia  Novosibirsk State University  Andrey Surzhykov
Physikalisch–Technische Bundesanstalt, D–38116 Braunschweig, Germany  Institut
für Mathematische Physik, Technische Universität Braunschweig, D–38106
Braunschweig, Germany  Laboratory for Emerging Nanometrology Braunschweig,
D-38106 Braunschweig, Germany  Peter G. Thirolf  Fakultät für Physik, Ludwig-
Maximilians-Universität München, 85748 Garching, Germany  Marc Vanderhaeghen
Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, 55128 Mainz,
Germany  Hans A. Weidenmüller  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Vladimir Zelevinsky  Department
of Physics and Astronomy and National Superconducting Cyclotron
Laboratiory/Facility for Rare Isotope Beams, Michigan State University, 640 S.
Shaw Lane, East Lansing, MI 48824, USA

###### Abstract

The Gamma Factory (GF) is an ambitious proposal, currently explored within the
CERN Physics Beyond Colliders program, for a source of photons with energies
up to $\approx 400\,$MeV and photon fluxes (up to $\approx 10^{17}$ photons
per second) exceeding those of the currently available gamma sources by orders
of magnitude. The high-energy (secondary) photons are produced via resonant
scattering of the primary laser photons by highly relativistic partially-
stripped ions circulating in the accelerator. The secondary photons are
emitted in a narrow cone and the energy of the beam can be monochromatized,
eventually down to the $\approx 1$ ppm level, via collimation, at the expense
of the photon flux. This paper surveys the new opportunities that may be
afforded by the GF in nuclear physics and related fields.

###### Contents

  1. 1 Introduction
    1. 1.1 The Gamma Factory
    2. 1.2 The goals and main results of this paper
  2. 2 Nuclear spectroscopy in the ion beam
    1. 2.1 Low-energy nuclear transitions
    2. 2.2 The lowest-lying nuclear isomer 229mTh
    3. 2.3 Interaction of nuclear and atomic degrees of freedom
    4. 2.4 Further experimental prospects involving nuclear transitions in the ion beam
      1. 2.4.1 Spectroscopy of isomers
      2. 2.4.2 Laser cooling with nuclear transitions
      3. 2.4.3 Production of higher-energy gamma rays
  3. 3 $P$\- and $CP$-violating Compton scattering of primary photons from stored ions
  4. 4 Nuclear physics with GF secondary-photon beam on fixed targets
    1. 4.1 Narrow resonances
    2. 4.2 Pygmy dipole resonances
    3. 4.3 GDR and multiphoton excitation
    4. 4.4 Photonuclear response above particle threshold
      1. 4.4.1 Neutron-capture cross sections of s-process branching nuclei
      2. 4.4.2 ($\gamma$,p), ($\gamma,\alpha$) cross sections for p-process nucleosynthesis
      3. 4.4.3 Direct measurement of astrophysical $S$-factors
      4. 4.4.4 Alpha clustering in heavy nuclei
      5. 4.4.5 Fano effect in nuclear gamma spectroscopy
    5. 4.5 Photofission with monochromatic gamma beams
    6. 4.6 Odd harmonics in angular distribution of fission fragments
    7. 4.7 Photoinduced processes on the proton and light nuclei
    8. 4.8 Pion photoproduction
      1. 4.8.1 Photoproduction of bound $\pi^{-}$
      2. 4.8.2 Neutron-skin measurements in coherent ($\gamma,\pi^{0}$) reactions
      3. 4.8.3 Pion decay constant from the Primakoff effect
    9. 4.9 Precision measurement of nuclear $E1$ polarizabilities
    10. 4.10 Delta-resonance region and continuum effects
    11. 4.11 Parity-violation in photo-nuclear processes
  5. 5 Experimental considerations with fixed targets
    1. 5.1 Thermal-load issues
    2. 5.2 Parallel spectroscopy with spatially resolved detection
    3. 5.3 Gamma-ray/X-ray and gamma-ray/gamma-ray pump-probe spectroscopy
    4. 5.4 Highly monochromatic gamma beams
  6. 6 Photophysics with a storage ring for radioisotopes
  7. 7 Colliding-beam opportunities
    1. 7.1 Photoabsorption structure functions
    2. 7.2 Production of ultrahigh-energy gamma rays
      1. 7.2.1 Scattering of secondary photons off stored relativistic ions
      2. 7.2.2 Scattering of secondary photons off a proton beam
  8. 8 Production of isotopes and isomers for medicine, dark matter search and gamma lasers
    1. 8.1 Production of medical isotopes via ($\gamma,n$) reactions
    2. 8.2 Production of nuclear isomers via ($\gamma,\gamma^{\prime}$) reactions
  9. 9 Induced gamma emission and gamma lasers
  10. 10 Gamma polarimetry
    1. 10.1 Polarimetry with narrow resonances
    2. 10.2 Other polarimetric techniques
  11. 11 Quantum vacuum effects
    1. 11.1 Vacuum birefringence
      1. 11.1.1 Quasi-constant magnetic field
      2. 11.1.2 High-intensity laser pulse
    2. 11.2 Photon Splitting
    3. 11.3 Photon Scattering
  12. 12 Nuclear physics with tertiary beams
    1. 12.1 Tertiary beams at the GF
    2. 12.2 Polarized electron, positron and muon sources
    3. 12.3 High-purity neutrino beams
    4. 12.4 Neutron and radioactive ion sources
    5. 12.5 Production of monoenergetic fast neutrons
    6. 12.6 Metrology with keV neutrons
  13. 13 Nuclear physics opportunities at the SPS
  14. 14 Speculative ideas and open questions
    1. 14.1 Applying the Gamma Factory ideas at other facilities
    2. 14.2 Nuclear waste transmutation
    3. 14.3 Laser polarization of PSI
    4. 14.4 Quark-gluon plasma with polarized PSI
    5. 14.5 Ground-state hyperfine-structure transitions in PSI
    6. 14.6 Detection of gravitational waves
  15. 15 Conclusions and optimistic outlook
  16. A Survey of existing and forthcoming gamma facilities
  17. B Gamma resonances in 13C
    1. B.1 The 8.86 MeV $M1$ resonance
    2. B.2 The 7.55 MeV $E2$ resonance
    3. B.3 The 3.09 MeV $E1$ resonance
  18. C Other reviews and nuclear databases

Figure 1: The Gamma Factory concept. Laser photons with momentum $\hbar k$
(the primary photon beam) impinge onto ultrarelativistic ions (relativistic
factor $\gamma$, mass $m$, velocity $v$) circulating in a storage ring.
Resonantly scattered photons, as seen in the laboratory frame, are emitted in
a narrow cone with an opening angle $\approx 1/\gamma$ in the direction of the
motion of the ions. The energy of these secondary photons is boosted by a
factor of up to $4\gamma^{2}$ with respect to the energy of the initial
photons.

##  1 Introduction

The Gamma Factory (GF) is a novel research tool proposed in Ref. [1] and
subsequently developed at CERN as part of its Physics Beyond Colliders studies
[2]. The main aim of the GF is the generation of high-intensity gamma-ray
beams of tunable energy and relatively small energy spread. At the same time,
the GF facility offers unique scientific opportunities by far not limited to
the various uses of the produced gamma rays (the secondary beam), but also
includes spectroscopy of stored relativistic ions, including radioactive ions,
with the (primary, energy boosted) laser photons that are used to produce the
gamma rays, as well as the production and use of various tertiary beams.

As a unique and unprecedented research tool, the GF opens new possibilities
across several fields of physics, from atomic physics (with the opportunities
surveyed in [3]), to nuclear physics and related fields (the subject of the
present paper), to elementary-particle physics and searches for physics beyond
the standard model (SM). While the GF today is still a proposal, some of its
key components such as the production and storage of partially stripped ions
(PSI), for instance hydrogen- and helium-like Pb and phosphorus-like Xe [4, 5,
6], have been already demonstrated experimentally in the Super Proton
Synchrotron (SPS) and the Large Hadron Collider (LHC) at CERN. Moreover, the
proof-of-principle experiment [7], further discussed in Sec. 13, will
demonstrate the full GF concept at SPS. The design of the experiments is
complete, awaiting approval of the SPS Committee.

###  1.1 The Gamma Factory

The principle of the GF is illustrated in Fig. 1. Highly charged partially-
stripped ions circulate in a storage ring at ultrarelativistic speeds with a
Lorentz factor $\gamma=(1-\beta^{2})^{-1/2}$, with $\beta=v/c\approx 1$ being
the ion speed normalized by the speed of light. The electrons bound to these
ions interact with a (primary) laser beam and undergo transitions between the
atomic shells. The resonance fluorescence photons, as seen in the laboratory
frame, are emitted in a narrow cone with opening angle of $\approx 1/\gamma$,
and form the secondary beam. The photon energy is boosted by a factor of
$\approx 4\gamma^{2}$ with respect to the original laser photons used for the
electronic excitation. The anticipated secondary beam parameters are listed in
Table 1. In many respects, the GF will be a qualitative leap compared to the
existing gamma sources, as well as sources currently under construction. These
are briefly discussed in Appendix A.

Successful excitation of an atomic shell relies on a resonance condition
involving both the relativistic factor $\gamma$ of the PSI as well as the
primary (optical) laser frequency $\omega$. Thus, the tunability of the
secondary beam energy is achieved by simultaneously tuning $\gamma$ and
adjusting the wavelength or the incident angle $\theta_{l}$ of the primary
laser beam to maintain the resonance with an atomic transition in the PSI
($\hbar\omega^{\prime}$):

$\hbar\omega^{\prime}=\hbar\omega\gamma(1+\beta\cos\theta_{l}).$ (1)

The ion beam energy variation is a routine procedure in a storage ring.
Indeed, at the LHC, the ions are injected with an initial relativistic factor
of around $\gamma=220$, and are subsequently accelerated up to $\gamma=2900$.
Tunable-laser technologies are also well developed.

The secondary gamma-ray beam will have a strong correlation between the
emission angle $\theta$ (see Fig. 1) and the gamma-ray energy $E$ given by the
expression

$E(\theta)=\frac{E_{max}}{2\gamma^{2}(1-\beta\cos\theta)},$ (2)

where $E_{max}=(1+\beta)\gamma\hbar\omega^{\prime}$. For small angles this can
be approximated as

$E(\theta)\approx\frac{E_{max}}{1+\gamma^{2}\theta^{2}}\approx\frac{4\gamma^{2}\hbar\omega}{1+\gamma^{2}\theta^{2}}.$
(3)

A small fraction of photons are emitted with large angles, $\theta\sim 1$, and
in this case the emitted photon energy is comparable to the incident laser
photon energy. For instance,

$E\left(\frac{\pi}{2}\right)=\frac{E_{max}}{2\gamma^{2}}\approx 2\hbar\omega.$
(4)

For a given emission angle, the residual photon energy spread is mainly
determined by the angular spread of the PSI beam and its angular size as seen
from the observation point (the location of the target or detector). A typical
normalized transverse emittance of the ion beam in the LHC is
$\epsilon_{n}=\gamma\sigma_{x}\sigma_{x^{\prime}}\approx 1$ mm$\cdot$mrad. The
transverse beam size in the LHC can be varied from $\sim$1 mm down to $\sim$10
$\mu$m (in the collider interaction points). With a mm-wide ion beam,
$\sigma_{x}\approx 1$ mm and for $\gamma\approx 10^{3}$, this corresponds to
angular spread on the order of a microradian
$\sigma_{x^{\prime}}=\epsilon_{n}/\gamma\sigma_{x}\approx 10^{-6}$. Therefore,
the uncorrelated energy spread of the secondary gamma-photon beam will be on
the order of $\delta\theta/\theta\approx\sigma_{x^{\prime}}\gamma\approx
10^{-3}$ (see Fig. 2). However, for the visible angular size of the mm-wide
source of photons to drop below 1 $\mu$rad, the distance to the observation
point should be more than 1 km.

Laser cooling can potentially dramatically reduce the ion-beam emittance and
energy spread. The transverse-only laser cooling [8] can be applied to reduce
the transverse beam size and angular spread by one order of magnitude at which
point the energy spread in the ion beam will become important, since the
typical relative energy spread in the LHC ion beam is on the order of
$10^{-4}$. To improve the energy-angle correlation in the resulting gamma-
photon beam to better than $10^{-4}$, both transverse and longitudinal laser
cooling should be applied [9, 10, 8]. Deep laser cooling will be limited by
the intrabeam scattering and other collective effects.

In summary, an optimal trade-off between the gamma beam intensity and its
spectral resolution will have to be made for each of the usage cases of the GF
beams, including case-by-case choice of the number of ions per bunch,
interaction-point collision optics, beam emittance and the distance of the
gamma beam target to its production point.

Figure 2: Correlation between the energy and angle of emitted photon in one
particular case of the Gamma Factory based on hydrogen-like calcium ion beam
in the LHC. The top histogram shows the corresponding photon flux density
(photons per unit area per unit time) into a small solid angle. The right
histogram shows the flat energy spectrum. The ion beam parameters in this
example are: $\gamma=2048$, $\sigma_{x}=0.44$ mm, $\sigma_{x^{\prime}}=2.2$
$\mu$rad (normalized emittance
$\epsilon_{n}=\gamma\sigma_{x}\sigma_{x^{\prime}}=2$ mm$\cdot$mrad) [11].

Beside their energy and emission pattern, the secondary photons are
characterized by their polarization. The polarization depends not only on the
emission angle $\theta$, but also on the polarization state (e.g., magnetic
sublevel population) of the incident PSI and the polarization of incident
laser photons. A proper choice of these parameters results in emission of
secondary gamma rays with high degrees of circular or linear, and generally,
arbitrary polarization. A detailed analysis of the polarization of scattered
photons, as performed using the density-matrix approach, is presented in Ref.
[12].

###  1.2 The goals and main results of this paper

Realization of the Gamma Factory project will result in a new facility based
on the significant progress achieved in accelerator physics, laser technology,
detectors, control devices, computer control, and data-analysis techniques.
The state-of-the art modern instrumentation and measurement techniques will
enable the next step in our understanding of deep problems in nuclear, atomic,
and hadron physics.

Historically, new tools have led to new discoveries. Therefore, as we survey
the science opportunities offered by the GF, we attempt to identify novel,
potentially breakthrough directions. Here are some of these:

  * •

Physics opportunities with primary (Sec. 2) and secondary (Sec. 4) beams with
previously unattainable parameters;

  * •

Spectroscopy of nuclear gamma transitions on par with laser spectroscopy of
atoms (Sec. 2);

  * •

Investigation of the physics of exotic nuclei and of the mechanism of their
stability along the road to the drip lines (Sec. 6);

  * •

Direct measurements of astrophysical S-factors at relevant energies (Sec.
4.4.3);

  * •

Resolving narrow resonances in the photofission cross section of actinides via
state-selective high-resolution spectroscopy (Sec. 4.5);

  * •

Pionic (Sec. 4.8) and Delta-resonance (Sec. 4.10) physics on a qualitatively
and quantitatively new level of precision;

  * •

Precision measurement of parity violation (PV) in hadronic and nuclear systems
at previously inaccessible levels (Sec. 4.11);

  * •

Investigation of exotic radioactive nuclei in conjunction with the CERN ISOLDE
facility (Sec. 4) and/or a dedicated storage ring (Sec. 6);

  * •

Production of copious amounts of isotopes and nuclear isomers for medicine,
dark matter search and gamma lasers via photonuclear reactions (Sec. 8);

  * •

Gamma polarimetry at the $<10^{-6}$ rad level (Sec. 10.1);

  * •

Production of and physics opportunities with high-intensity, monoenergetic and
small-emittance tertiary beams (neutrons, muons, neutrinos, etc.; Sec. 12).

Parameter | Value
---|---
Ion $\gamma$ factor | 10111The lowest $\gamma$ factors are for the SPS. At the LHC, they are $\approx 200$. – 2900
Ion species | Pbq+ as an example
Transverse beam radius | 16 $\mu$m
Number of ions in a bunch | 108222A larger number can be expected for lighter PSI [13], for example, $3\cdot 10^{9}$ for He-like Ca.
Number of bunches in the ring | 592 – 1232
Effective repetition rate | 10 MHz33320 MHz planned after the LHC injector upgrade
Ion energy spread | 10-4
RMS bunch length | 7.9 cm
Normalized emittance | 1.6 $\mu$m
Circumference of the LHC | 26.7 km
Table 1: Representative parameters of the Gamma Factory at CERN with currently
available ion beams. $q$ denotes the charge state of the ions. The numbers are
presented for Pb ions. Note that the current optimization for the LHC is for
the collider mode. A dedicated optimization for the GF may lead to
improvements.

In this paper, we outline a possible nuclear physics program for the GF,
successively addressing the opportunities afforded by using the primary photon
beams (starting with spectroscopy of stored ions with primary photons in Sec.
2), the secondary photon beams (starting with a discussion of fixed-target
experiments in Sec. 4), followed with a discussion of production and use of
tertiary beams of various particles in Sec. 12.

##  2 Nuclear spectroscopy in the ion beam

Laser spectroscopy of atomic transitions is well established in ion storage
rings, see, for example, Refs. [14, 15]. Precision spectroscopy of nuclear
transitions may also be possible with the GF.

With the initial laser-photon energy of up to 10 eV and the relativistic
factor at the LHC of up to $\gamma\simeq 3000$, in their reference frame, the
circulating ions “see” the photon energies up to 60 keV, high enough to excite
low-lying nuclear states. This opens up possibilities of nuclear spectroscopy
or even nuclear quantum optics already with the primary GF beam. For this
purpose, it is not necessary to use PSI and the experiments can be performed
with bare ions. The prospects of combining relativistically accelerated nuclei
as a target with presently available coherent light sources for first nuclear
quantum optics experiments were originally discussed in Ref. [16]. At that
time, a combination of X-ray free electron lasers and more moderate target
acceleration was envisaged. This in turn posed severe challenges due to the
geographical distance between the required large-scale facilities, with table-
top laser or accelerator alternatives still lacking in performance [17, 18].
The GF envisaged at CERN is in a unique position of achieving high photon
energies even with (table-top) lasers operating in the visible and ultraviolet
(UV) ranges, thus opening new possibilities for nuclear quantum optics
experiments.

In this Section, we give examples of low-lying nuclear transitions that can be
driven by the primary laser light of the GF, discuss the opportunities related
to nuclear isomers and possible couplings between the nuclear and atomic
degrees of freedom, and put forward further prospects for using nuclear
transitions for laser cooling or obtaining even higher gamma-ray energies at
the GF.

One needs to point out that direct spectroscopy of nuclear transitions at the
GF would require overcoming a number of specific challenges, including
ensuring the availability of ion sources with stable or radioactive isotopes
capable of producing beams of sufficient intensity, operation of the full
accelerator chain with radioactive species, as well as low excitation rates
and long decay times of the narrow excited states. None of these appear as
a-priori “show stoppers”, but would need to be carefully addressed on a case-
by-case basis.

###  2.1 Low-energy nuclear transitions

Examples of low-energy gamma transitions starting from a stable or long-lived
nuclear ground state are listed in Table 2. Typically, the low-lying nuclear
spectrum can be described by collective rotations and vibrations of the
nuclear surface [19], and such excited states frequently connect to the ground
state via electric-dipole-forbidden transitions [20]. Direct spectroscopy on
these transitions can provide detailed information, for example, on the
relative role of different transition multipoles (e.g., $M1$ and $E2$ mixing),
or provide accurate transition probabilities. In the context of nuclear
quantum optics, many of these transitions were theoretically investigated for
coherent driving with X-ray free electron laser sources [16, 20, 21].

We note that some of the transitions to the ground state listed in Table 2
have not been directly observed, indicating their strong degree of
suppression. For instance, due to its low energy, the radiative decay channel
of the 8 eV first excited state in 229Th has not been observed so far, despite
many experimental attempts of direct photoexcitation [22]. This peculiar
“isomeric” transition is discussed in more detail in the context of the GF in
the next section. In addition, the 13.034 and 51.697 keV excited states in
235U have only been observed to connect to the 76 eV long-lived first excited
state [23]. This is attributed to selection rules for the $K$-quantum number,
i.e., the nuclear spin projection on the symmetry axis. In Table 2 we
therefore list instead the available experimental data on the two transitions
to the isomeric state. Detailed investigations of strongly suppressed
transitions at the GF, for instance the ones connecting the 13.034 and 51.697
keV levels of 235U to the ground state, would contribute to a better
understanding of nuclear structure and the interplay between nuclear
collective and single-particle degrees of freedom.

In addition to transitions connecting the nuclear ground state with a low-
lying excited state, one can envisage the use of nuclei in metastable states
as PSI in the ion beam. Such nuclear metastable states are also known as
isomers and can store large amounts of energy over long period of time. Their
existence is typically attributed to large differences in spin, shape or $K$
quantum number, between the isomer and the lower-lying levels [24, 25]. In an
advantageous configuration of the nuclear excited levels, once the excitation
occurs from the isomeric state to an upper gateway level, the subsequent
nuclear decay may proceed directly to a state below the isomer, thus reaching
the ground state in a fast cascade. Such a process is called isomer depletion,
since it allows for the depopulation of the isomeric state and thus a
controlled release of the energy stored in the metastable state. A typical
example is the case of the 2.4 MeV 93mMo isomer, for which we present the
relevant partial level scheme in Fig. 3. An excitation scheme involving such a
$\Lambda$-like three-level system (of the states $I$, $G$ and $F$ in the Fig.
3 example), known as Raman-type excitation in atomic and molecular physics, is
advantageous since the excitation and decay photons have different energies
and can therefore be more easily distinguished in experiments. In the context
of isomers, an additional fascinating potential application is related to the
controlled release of nuclear energy on demand that would allow the design of
a clean nuclear energy storage solution without involving fission or fusion
[24]. A list of nuclear isomer parameters useful for potential isomer
depletion as well as other applications such as medical applications or gamma-
ray lasers are presented in Tables 7, 8, 9 and 10. An idea on how to use the
GF for spectroscopy of isomers and possible depleting levels above them is
sketched in Sec. 2.4.1.

Figure 3:  Partial level scheme of ${}^{93}_{42}\mathrm{Mo}$. The isomeric
state (I) can be excited to the gateway level (G) which subsequently decays
back to I or to a level F, initiating a cascade via different intermediate
states (dashed line) to the ground state. The direct I$\to$F decay is a
strongly hindered $E4$ transition, while I$\to$G and G$\to$F are prompt $E2$
transitions.

Isotope | $T^{g}_{1/2}$ |  $E_{e}$ (keV) | $I_{g}$ | $I_{e}$ | $\lambda L$ |  $T^{rad}_{1/2}$ (s) | $\alpha(K)$ | $\alpha(L)$
---|---|---|---|---|---|---|---|---
229Th | 7880 y | 0.008444transition energy from Refs. [26, 27], and radiative transition rate deduced using theoretical predictions in Ref. [28] | 5/2+ | 3/2+ | $M1$ | $5.19\times 10^{3}$ | - | -
235U |  $7\times 10^{8}$ y | 0.076 | 7/2- | 1/2+ | $E3$ |  $7.03\times 10^{23}$555using theoretical predictions from Ref. [29] | - | -
201Hg | stable | 1.565 | 3/2- | 1/2- | $M1$ | $3.76\times 10^{-3}$ | - | -
205Pb |  $1.7\times 10^{7}$ y | 2.329 | 5/2- | 1/2- | $E2$ | $9.07\times 10^{2}$ | - | -
181Ta | stable | 6.238 | 7/2+ | 9/2- | $E1$ | $4.34\times 10^{-4}$ | - | -
239Pu |  $2.4\times 10^{4}$ y | 7.861 | $1/2^{+}$ | $3/2^{+}$ | $M1$ | $2.04\times 10^{-7}$ | - | -
169Tm | stable | 8.410 | 1/2+ | 3/2+ | $M1$ | $1.07\times 10^{-6}$ | - | -
83Kr | stable | 9.406 | 9/2+ | 7/2+ | $M1$ | $2.80\times 10^{-6}$ | - | 14
187Os | stable | 9.756 | 1/2- | 3/2- | $M1$ | $9.01\times 10^{-7}$ | - | -
137La |  $6\times 10^{4}$ y | 10.560 | 7/2+ | 5/2+ | $M1$ | $1.04\times 10^{-5}$ | - | 93.2
45Sc | stable | 12.400 | 7/2- | 3/2+ | ($M2$) | $1.96\times 10^{2}$ | 362 | 54
235U | -666The isomeric state at 0.076 keV is considered as the ground state for this transition. In PSI the IC channel of the isomeric decay is closed and the radiative half-life would be much longer than the ground-state $\alpha$ decay half-life. | 13.034 | 1/2+777isomeric state spin | 3/2+ | M1 |  $2.43\times 10^{-7}$888Radiative decay rate is given for the transition to the isomeric state at 0.076 keV. | - | -
73Ge | stable | 13.284 | 9/2+ | 5/2+ | $E2$ | $3.1\times 10^{-3}$ | 299 | 666
57Fe | stable | 14.413 | 1/2- | 3/2- | $M1$ | $9.32\times 10^{-7}$ | 7.35 | 0.78
151Eu |  $\geq 1.7\times 10^{18}$ y | 21.541 | 5/2+ | 7/2+ | $M1$ | $2.62\times 10^{-7}$ | - | 21.7
149Sm | stable | 22.507 | 7/2- | 5/2- | $M1$ | $2.24\times 10^{-7}$ | - | 22.2999theoretical values derived using the BrIcc database [30] based on Ref. [31] with corresponding mixing-ratio data from the ENSDF database [32]
119Sn | stable | 23.871 | 1/2+ | 3/2+ | $M1$ | $1.07\times 10^{-7}$ | - | 4.1
161Dy | stable | 25.651 | 5/2+ | 5/2- | $E1$ | $9.59\times 10^{-8}$ | - | 1.799
201Hg | stable | 26.272 | $3/2^{-}$ | $5/2^{-}$ | $M1$ | $4.61\times 10^{-8}$ | - | 55.99
129I |  $1.6\times 10^{7}$ y | 27.793 | $7/2^{+}$ | $5/2^{+}$ | $M1$ | $1.02\times 10^{-7}$ | - | 4.06
229Th | 7880 y | 29.190 | 5/2+ | 5/2+ | $M1$ |  $3.26\times 10^{-8}$101010using theoretical predictions from Ref. [33] | - | 1689
40K |  $1.2\times 10^{9}$ y | 29.830 | 4- | 3- | $M1$ | $5.47\times 10^{-9}$ | 0.269 | 0.0239
201Hg | stable | 32.145 | $3/2^{-}$ | $3/2^{-}$ | $M1$ |  $5.04\times 10^{-9}$111111calculated for the direct decay to the ground state only | - | 30.89
237Np |  $2.1\times 10^{6}$ y | 33.196 | $5/2^{+}$ | $7/2^{+}$ | $M1$ | $9.92\times 10^{-9}$ | - | 1319
125Te | stable | 35.492 | $1/2^{+}$ | $3/2^{+}$ | $M1$ | $2.15\times 10^{-8}$ | 11.69 | 1.602
189Os | stable | 36.200 | 3/2- | 1/2- | $M1$ | $1.09\times 10^{-8}$ | - | 15.6
121Sb | stable | 37.129 | $5/2^{+}$ | $7/2^{+}$ | $M1$ | $4.06\times 10^{-8}$ | 9.36 | 1.227
129Xe | stable | 39.578 | 1/2+ | 3/2+ | $M1$ | $1.25\times 10^{-8}$ | 10.27 | 1.41
233U |  $1.6\times 10^{5}$ y | 40.351 | $5/2^{+}$ | $7/2^{+}$ |  $M1$121212This transition has $M1+E2$ multipolarity with a large mixing ratio $\delta\approx 0.93$. $M1$ is the dominant radiative channel by a small marge. | $1.03\times 10^{-7}$ | - | 3749
243Am | 7364 y | 42.20 | $5/2^{-}$ | $7/2^{-}$ | $M1$ | $6.43\times 10^{-9}$ | - | 110
229Th | 7880 y | 42.435 | 5/2+ | 7/2+ | $M1$ | $2.59\times 10^{-8}$ | - | 99.39
240Pu | 6561 y | 42.824 | $0^{+}$ | $2^{+}$ | $E2$ | $1.55\times 10^{-7}$ | - | 658
246Cm | 4706 y | 42.852 | $0^{+}$ | $2^{+}$ | $E2$ | $1.31\times 10^{-7}$ | - | 7709
248Cm |  $3.5\times 10^{5}$ y | 43.400 | $0^{+}$ | 2+ | $E2$ | $1.22\times 10^{-7}$ | - | 7249
234U |  $2.5\times 10^{5}$ y | 43.498 | $0^{+}$ | $2^{+}$ | $E2$ | $1.80\times 10^{-7}$ | - | 520
244Pu |  $8.1\times 10^{7}$ y | 44.2 | $0^{+}$ | $2^{+}$ | $(E2)$ | $1.23\times 10^{-7}$ | - | 560
242Pu |  $3.7\times 10^{5}$ y | 44.54 | $0^{+}$ | $2^{+}$ | $E2$ | $1.20\times 10^{-7}$ | - | 5439
238U |  $4.5\times 10^{9}$ y | 44.916 | $0^{+}$ | $2^{+}$ | $E2$ | $1.26\times 10^{-7}$ | - | 444
236U |  $2.3\times 10^{7}$ y | 45.242 | 0+ | $2^{+}$ | $E2$ | $1.38\times 10^{-7}$ | - | 429
235U |  $7\times 10^{8}$ y | 46.103 | 7/2- | 9/2- | $M1$ | $7.15\times 10^{-10}$ | - | 40
183W |  $\geq 6.7\times 10^{20}$ y | 46.484 | 1/2- | 3/2- | $M1$ | $1.73\times 10^{-9}$ | - | 6.469
232Th |  $1.4\times 10^{10}$ y | 49.369 | 0+ | $2^{+}$ | $E2$ | $1.15\times 10^{-7}$ | - | 244
81Kr |  $2.3\times 10^{5}$ y | 49.57 | $7/2^{+}$ | $9/2^{+}$ | $M1$ | $9.41\times 10^{-9}$ | 1.1179 | 0.1699
235U | -6 | 51.697 | 1/2+7 | 5/2+ | $E2$ |  $8.34\times 10^{-8}$8 | - | 226
230Th |  $7.5\times 10^{4}$ y | 53.227 | 0+ | $2^{+}$ | $E2$ | $8.08\times 10^{-8}$ | - | 166.8
157Gd | stable | 54.536 | 3/2- | 5/2- | $M1$ | $1.74\times 10^{-9}$ | 9.50131313This value corresponds to neutral atoms. For He- or H-like ions, the ionization potential increases closing the IC channel. | 2
239Pu |  $2.4\times 10^{4}$ y | 57.275 | $1/2^{+}$ | $5/2^{+}$ | $E2$ | $3.58\times 10^{-8}$ | - | 161.1
237Np |  $2.1\times 10^{6}$ y | 59.540 | $5/2^{+}$ | $5/2^{-}$ | $E1$ |  $1.86\times 10^{-7}$11 | - | 0.3769
155Gd | stable | 60.010 | 3/2- | 5/2- | $M1$ | $2.04\times 10^{-9}$ | 7.25 | 1.48

Table 2: Examples of low-energy nuclear transitions, sorted by energy, with
the lower state being a stable or long-lived ground (or isomeric) state with
half-life $T^{g}_{1/2}$ and nuclear spin $I_{g}$. The excited state energy
$E_{e}$ and the dominant multipolarity $\lambda L$ are given together with the
excited state spin $I_{e}$. $T^{rad}_{1/2}$ corresponds to the calculated
radiative half-life of the excited state, and $\alpha(K/L)$ is the internal
conversion coefficient (of the transition) corresponding to the $K/L$-shell
electrons. Nuclear parameters were taken from the ENSDF database [32] unless
otherwise specified.

###  2.2 The lowest-lying nuclear isomer 229mTh

Considerable interest in both theory and experiment has been paid in the past
years to the lowest-known nuclear excited state, lying at only $\approx$8 eV
above the ground state of 229Th [26, 27]. 229Th belongs to the light actinide
nuclear mass region known for the presence of enhanced collectivity and shape-
dynamic properties [34]. The single-particle states of the odd neutron
determine the 229Th ground state with $K^{\pi}=5/2^{+}$ and the isomeric state
with $K^{\pi}=3/2^{+}$ based on the $5/2^{+}[633]$ and $3/2^{+}[631]$ single-
particle orbitals. Here $K$ refers again to the projection of the total
nuclear angular momentum on the body-fixed principal symmetry axis of the
system, $\pi$ is parity, and we use the usual Nilsson notation
$K^{\pi}[Nn_{z}\Lambda]$ with $N$, $n_{z}$ and $\Lambda$ being the asymptotic
Nilsson quantum numbers [35].

Because the isomeric transition energy is accessible with vacuum-ultraviolet
lasers, 229mTh is the best candidate for a nuclear clock, i.e., a clock based
on a nuclear rather than an electronic transition [36, 37, 38]. At the GF, the
low-energy isomeric transition can be in principle accessed with a primary
optical laser beam directed at an obtuse angle to the ion beam. In addition,
one can also make use of the higher levels of 229Th to populate the isomer.
Indeed, in a recent nuclear resonant scattering (NRS) experiment performed on
a solid-state thorium-oxide target at the synchrotron-radiation source at
SPring-8 in Japan, the 229mTh isomer was excited indirectly via population of
a higher-lying excited state [39]. Resonant X-rays were driving the transition
between the ground state $5/2^{+}[633]$ and the second excited state
$5/2^{+}[631]$ at 29.19 keV (see also Table 2). This excited state decays with
a branching ratio of $\approx$0.9 to the isomeric state $3/2^{+}[631]$. This
is another example of Raman-type excitation in a nuclear $\Lambda$ three-level
system similar to the isomer-depletion process discussed in Sec. 2.1, however,
now with the aim of populating instead of depleting the long-lived state.

The NRS experiment provided valuable information about the properties of the
second excited state of 229Th, whose excitation energy of about 29.2 keV has
been determined with sub–eV accuracy. In addition, it also opened a new route
for the determination of the 229mTh energy [39, 40], though so far less
precisely than the best prior measurements [26, 27]. A different approach at
the GF would be to employ a 229Th beam in a dedicated storage ring, see
Section 6. The intense GF photon flux would drive the 29 keV transition in the
relativistic Th ions, allowing for precision studies of excited state
properties. Another interesting advantage at the GF is the possibility to
study the properties of 229mTh in different atomic charge states. Here, due to
strong hyperfine splitting one may expect sizeable differences between odd-
and even charge states [41].

One of the drawbacks for NRS experiments is the use of solid-state targets, in
which the predominant nuclear decay is via internal conversion, quenching the
isomeric state in microseconds [38]. This is where the unique capabilities of
the GF can come into play. Using 229Thq+ ions with charge states $q=84-90$ in
the LHC, the internal conversion channel would be closed, since only
$2p_{3/2}$ orbitals and higher have ionization potentials smaller than 29 keV.
We may then envisage coherent nuclear driving with two narrowband UV primary
beams that can excite simultaneously the two $M1$ transitions
$5/2^{+}[633]\rightarrow 5/2^{+}[631]$ and $3/2^{+}[631]\rightarrow
5/2^{+}[631]$, both at $\approx 29$ keV. Varying the frequency of the second
laser beam driving the $3/2^{+}[631]\rightarrow 5/2^{+}[631]$ transition,
would provide an independent route to determine the energy of the isomeric
nuclear state 229mTh with $10^{-2}$ eV or perhaps even better accuracy. In
addition, the two secondary X-ray beams would open, for the first time, the
possibility to exploit quantum-optics schemes as known from atomic or
molecular $\Lambda$-systems directly with nuclear transitions.

In particular we can, in principle, envisage so-called $\pi$-pulses which swap
the population between the lower and the upper state for each of the
transitions, or stimulated Raman adiabatic passage (STIRAP) [42, 43] between
the ground state and the isomeric state. The latter would have the advantage
that the population is coherently and efficiently transferred from the ground
state $5/2^{+}[633]$ to the isomeric state $3/2^{+}[631]$ independently of the
branching ratio of the excited state. While already proposed for nuclear
coherent population transfer [17], the lack of suitable X-ray and gamma-ray
facilities has prevented STIRAP from being observed for nuclear transitions so
far. This could change with the advent of the GF. Laser phase fluctuations are
known to reduce the performance of STIRAP [44] as adiabaticity is disturbed
during phase jumps. Nearly transform-limited pulses are best suited for
experimental implementation of STIRAP [44, 42]. Such coherence properties of
the driving beams should be available from the primary UV lasers.

###  2.3 Interaction of nuclear and atomic degrees of freedom

The GF can be used to drive nuclear or atomic transitions in the ion beam with
energies of up to $\approx$60 keV. In this region, the coupling of the nuclear
and atomic degrees of freedom can be rather strong. Thus, the GF could become
a powerful tool to explore the interplay between nuclear excitation or decay
and electronic transitions. In the following, we outline the relevant
processes and experimental prospects.

We address separately weak-interaction and electromagnetic-interaction
processes involving atomic electrons. In the first category fall the electron-
capture (EC) and bound $\beta$ decay channels. EC cannot occur in bare ions
because of the absence of $K$-shell electrons. Furthermore, it was shown that
for the cases of ${}^{140}_{~{}59}$Pr and ${}^{142}_{~{}61}$Pm, the nuclear
lifetime of the one-electron Pr58+ (Pm60+) ion is shorter than the ones of the
corresponding two- or many-electron cases [45, 46]. This could be explained by
conservation of the total angular momentum, since only particular spin
orientations of the nucleus and of the captured electron can contribute to the
allowed EC decay [47]. This selection rule leads to the acceleration of the
allowed Gamow-Teller $1^{+}\rightarrow 0^{+}$ decay in one-electron Pr58+ and
Pm60+ ions, while it stalls $1^{+}\rightarrow 2^{+}$ decays in one-electron
${}^{122}_{~{}53}$I ions [48]. Bound $\beta$ decay, on the other hand, can
only occur if vacancies in the atomic shell are available for the $\beta^{-}$
particle to occupy. The process and corresponding nuclear lifetime are
sensitive to the electronic environment. For example, for bare 163Dy and 187Re
ions, the empty $K$ shells enable $\beta^{-}$ decay into these atomic orbits
with half-lives of 48$\pm 3$ d and 32.9$\pm 2.0$ y, respectively, while in the
corresponding neutral atoms the nuclei are stable (163Dy) or live $10^{9}$
times longer (187Re) [49, 50, 51]. The study of such processes, however, does
not require the primary GF photon beam to drive any nuclear transition, but
rather $\beta$-unstable ions in the storage ring and control of the electronic
states.

Studies of weak decays of highly-charged ions are routinely conducted at GSI
in Darmstadt [52, 53]. However, there are numerous cases of forbidden decays
of interest to astrophysics, for which such measurements are not presently
feasible at GSI due to insufficient production rates [54]. To give just one
example, ${}^{56}_{28}$Ni is among the most abundant products in supernova
explosions. The half-life of a neutral atom is 6.075(10) d [55] and the main
decay branch is the Gamow-Teller $0^{+}\rightarrow 1^{+}$ transition to the
1720.19 keV state in ${}^{56}_{27}$Co. In fully-ionized 56Ni28+ nuclei, the EC
decay channel is disabled. Experiments indicate a weak $\beta^{+}$ decay
branch [56] rendering 56Ni a possible cosmo-chronometer, in contradiction to
shell-model predictions [57]. The ISOL production method employed at ISOLDE
can be superior in providing high-intensity beams of some elements [58].
Furthermore, the production yields will be significantly improved within the
EPIC project (Exploiting the Potential of ISOLDE at CERN) [59], which will
facilitate conducting such experiments at the GF.

We now turn to the electromagnetic coupling between atomic and nuclear
transitions for decay channels such as internal conversion, bound internal
conversion or electronic bridge [60], which are suitable case studies that can
take full advantage of the GF. Internal conversion is typically the stronger
decay channel of low-lying excited nuclear states, in particular for
excitation energies below 60 keV. This in turn means that the lifetimes of
such excited states strongly depend on the electronic configuration of the
ion, both in terms of available electrons and spin couplings [61]. In a
pioneering study by Phillips and co-workers, the internal conversion rates for
the 14-keV Mössbauer level in 57Fe were studied in F-like to H-like ions
produced as secondary beams in Coulomb excitation. For H-like ions, effects
due to conservation of the total spin $F$ were revealed [62]. The
investigation of nuclear decays in highly ionized ${}^{125}_{~{}52}$Te44+ up
to ${}^{125}_{~{}52}$Te48+ ions led to the discovery of a new decay mode,
bound internal conversion (BIC) [63].

Studies of the influence of the electron shell on isomer lifetimes are
particularly noteworthy since they might provide valuable information about
energies and symmetry properties of nuclear states [64, 65]. Experimental
information is presently limited to only a few cases [66, 67, 68, 69, 70, 71].
In addition, a strong incentive comes from the field of astrophysics, since
nucleosynthesis in stellar plasmas proceeds at high temperatures and,
therefore, at a high degree of ionization [51, 72, 73].

With the unique capabilities of the GF, we can design, for the first time,
scenarios in which both the nuclear excited states and electronic excited
states are populated by the primary laser beam Lorentz-boosted in the
reference frame of the accelerated ions. An additional degree of freedom for
the experiment is the charge state of the ions. In order to investigate the
effects of the atomic shell on nuclear decay lifetimes, PSI with few remaining
electrons would be of interest. Nuclear transitions below 60 keV such as those
listed in Table 2 could be driven by one laser beam, while a second laser beam
could be used to control the electronic shell. For convenience, we provide
also the $K$-shell and $L$-shell internal conversion coefficients, i.e., the
ratios between the internal conversion (for the respective electronic shell)
and the radiative decay rates, for the transitions listed in Table 2. Due to
low excitation energies and the rather high ionization potentials of the $K$
and $L$-shell electrons, for many of these transitions the respective internal
conversion channels are closed; these are indicated with “-” in the last two
columns of Table 2.

We also note that although the special cases of the low-lying 229mTh and 235mU
isomers are well-known for having large internal conversion coefficients for
the outer-shell electrons, the corresponding low-charge states are not
accessible at the GF which will operate with beams of partially stripped but
nevertheless highly charged ions. For example, the internal conversion
coefficient for the 8 eV isomeric transition in 229mTh is $\approx 10^{9}$ for
the neutral atom [64], but the channel is closed already for Th+ ions. From
Table 2 we identify interesting cases with large internal conversion
coefficients for the $K$\- and $L$-shell electrons, which are appealing
candidates for future studies at the GF. Focusing on internal conversion
coefficients larger than 100, we identify as promising candidates the
transitions in 45Sc, 73Ge, 237Np (the 33 keV transition), 243Am, together with
almost all listed transitions in Th and U isotopes, and with all listed
transitions in Pu and Cm isotopes. For these nuclei, studies at the GF of the
dependence of the nuclear excited-state lifetime on the charge state and spin
coupling of the electrons would be very interesting.

A different and promising prospect coupling the atomic and nuclear degrees of
freedom at the GF involves the so-called electronic bridge process. Thereby
the interaction of a nucleus with a photon is mediated by atomic electrons. In
the electronic bridge process, the nuclear decay is accompanied by virtual
excitation of an electron into a higher bound orbital. The electron then de-
excites by photoemission. The process is third-order in QED, but can
nevertheless be the dominant channel for the decay of a nuclear isomer.
Electronic bridge processes involving nuclear excitation at the expense of the
initially excited electronic shell are sometimes called “inverse electronic
bridge” [74], however, many works use the same term for both excitation and
decay channels. The electronic bridge may dominate, for example, in low-energy
transitions where the hyperfine interaction between the electron and the
nucleus is less suppressed by selection rules than the direct gamma-decay,
since the photon coupling will include a lower power of the photon energy. In
this case the electrons can act as an effective “bridge” for the interaction,
particularly if the resonance condition is met [75].

The electronic bridge has been studied for several nuclear transitions
including the 76 eV 235mU isomer [74, 76, 77, 29], the 8 eV 229mTh isomer [78,
79, 80, 81, 82, 83, 84, 85], and the 3.05 keV nuclear transition starting from
the $6^{-}$ 464 keV isomer of 84Rb [86]. Laser-induced electronic bridge has
been proposed to determine the excitation energy of the 229Th isomer in
various charge states [87, 88, 89, 90]. However, to date, the electronic
bridge process has not been experimentally observed [91].

The Gamma Factory provides a new opportunity to explore the electronic bridge
process, particularly as a means of driving nuclear excitation. Not only are
there many suitable transitions (see Table 2), but one can also tune the
charge state of the PSI to optimize the resonance condition for the electronic
excitation. As a concrete example we may consider the 2.329 keV nuclear
excited state in 205Pb, which in neutral atoms is an isomer with 24 $\mu$s
half-life [32]. Understanding the decay properties of this state is important
for nuclear astrophysics, where a much faster EC-decay is expected to 205Tl
than the 17 My decay of the 205Pb ground state [92, 93]. This process affects
the very end of s-process nucleosynthesis in the Tl-Bi region [94].
Furthermore, since 205Pb is the only short-lived cosmic radioactivity produced
exclusively in the s-process, the evidence for which was found in meteorites,
it turns to be important for constraining cosmochemical simulations [95].

Once the outer N-, O- and P-shell electrons are stripped off, the internal
conversion channel is closed and the decay can only proceed via an $E2$
radiative decay, leading to a half-life of approx. 3 h. In PSI, the $M1$
electronic fine-structure transition $2p_{1/2}$-$2p_{3/2}$ has approximately
the same energy as the nuclear transition. Therefore, in suitably chosen PSI,
electronic bridge decay paths will open where the $2p_{1/2}$-$2p_{3/2}$
transition mediates the nuclear transition. For example, in N-like Pb, the
electronic bridge rate will exceed the radiative rate by several orders of
magnitude, reducing the lifetime to minutes or seconds, depending on how well
the resonance condition is met. This electronic bridge process may be directly
observed in the GF by comparing excitation rates in different PSI where the
resonant electronic transition is available (such as N-like) against those for
which it is not (such as He-like).

###  2.4 Further experimental prospects involving nuclear transitions in the
ion beam

In addition to the aspects discussed so far, several proposals on how to use
low-energy nuclear transitions for isomer spectroscopy, laser cooling or
production of even higher gamma-ray energies in the secondary beams have been
put forward. We shortly address these ideas here.

####  2.4.1 Spectroscopy of isomers

The fact that the stored ions can be accelerated and decelerated in the
storage ring opens the possibility to perform spectroscopy of appropriately
long-lived isomeric states using two subsequent photoinduced transitions in a
scenario reminiscent of pump-probe spectroscopy. A primary laser with a fixed
photon energy is first used. The ion relativistic factor $\gamma$ is adjusted
to excite nuclear transitions which populate a long-lived isomeric state. This
could be either a direct transition or excitation of the long-lived isomeric
state via intermediate states.

After a sufficient amount of isomeric ions are stored, one can change the
relativistic factor of the ions to a different value $\gamma^{\prime}$. At
this energy, with the same primary laser, transitions depleting the isomeric
state would be driven, and secondary photons detected. This provides a method
for performing high-resolution spectroscopy of isomeric states and possible
gateway levels around it in a broad range of frequencies, effectively driving
($\gamma,\gamma^{\prime}$) reactions as the ones discussed in Sec. 8.2 in the
context of isomer production for medical applications. A limitation is that,
with primary-beam photon energies of up to $\approx$10 eV, the energy of the
laser-driven gamma transitions needs to be below 60 keV. One candidate with an
adequate level scheme below this energy is 229mTh, for which transitions from
the 29 keV level connect to both the isomeric and ground states. With the
development of short-wave laser and mirror technologies and/or the advent of
the LHC high-energy upgrade, the limitation of 60 keV as maximum photon energy
will be overcome, significantly enlarging the set of potential candidates.

####  2.4.2 Laser cooling with nuclear transitions

Cooling via scattering of laser photons near-resonant to gamma transitions in
bare nuclei was suggested in Ref. [96], where the authors show that reasonable
damping times (of less than an hour) can be achieved for several bare ions at
the LHC. While laser cooling of PSI using electron transitions is generally
much more efficient, laser cooling of bare nuclei may be advantageous for
colliding-beam applications to avoid beam losses due to mutual stripping of
the colliding PSI beams [96]. Candidate transitions for laser cooling can be
selected from those listed in Table 2 or from the original proposal in Ref.
[96].

####  2.4.3 Production of higher-energy gamma rays

In principle, the GF concept can be extended to produce photons with much
higher energies than 400 MeV that can be generated by scattering conventional
laser photons on relativistic PSI. This could be achieved by replacing the
optical laser source, which excites the electronic shells of the primary beam,
with an X-ray laser source driving nuclear transitions of the relativistic
ions. Suitable nuclear resonances include the so-called Giant Dipole
Resonances (GDR) which are discussed in Sec. 4.3. We can envisage an efficient
excitation up to nuclear energies of $\approx$15 MeV, where the dipole
response for stable medium to heavy nuclei exhausts almost 100$\%$ of the
Thomas-Reiche-Kuhn sum rule. Much higher energies would lead to particle loss
and reduced radiative decay of the nuclear excitation. In turn, 15 MeV
corresponds to $\approx$2.6 keV photon energy for the primary beam and
$\approx$87 GeV photon energy for the secondary beam considering
$\gamma=2900$.

X-ray free electron sources such as the LCLS [97], SACLA [98] or the European
XFEL [99] can easily cover the region of interest of a few keV. In addition,
since the GDR is broad, the limited temporal coherence of the X-ray source
should not play an important role. The difficulty when envisaging such a setup
is that the GF and the X-ray facilities would need to be co-located, which is
an obvious limiting factor. This has been so far a major impediment for other
proposed combinations of relativistic acceleration and coherent X-ray sources
in nuclear quantum optics [16, 17, 18]. However, this impediment might be
solved by using table-top plasma-driven sources [100].

Further opportunities for producing higher-energy gamma rays may be afforded
by scattering off relativistic beams of the secondary photons produced by the
GF itself. This is discussed in Sec. 7.2.

##  3 $P$\- and $CP$-violating Compton scattering of primary photons from
stored ions

Searching for exotic signatures of violation of symmetry under spatial
inversion $P$ and time reversal $T$ in Compton scattering,
$\gamma(\vec{q})+N(\vec{p})\to\gamma(\vec{q}^{\,\prime})+N(\vec{p}^{\,\prime})$,
is a natural task for the GF. A clear distinction of the GF (where photons are
scattered from stored ion beams) from existing gamma sources based on laser-
light backscattering from an electron beam is the possibility to probe
specifically hadronic $P$-violating, $T$-conserving (PVTC) and $P$\- and
$T$-violating (PVTV) interactions [101]. The experiment will consist in
counting the number of the generated secondary photons as a function of the
circular polarization of the primary light.

We start this discussion with the case of the proton beam at the LHC. For the
laser-photon energy of $\sim 10$ eV and the relativistic factor
$\gamma_{p}\approx 7000$ (approximately double that of the ion beam), the
photon energy in the photon-proton c.m. frame of $\sim 140$ keV can be
reached. The theory of Compton scattering on a nucleon at low energies with
PVTC was laid out in Refs. [102, 103], and the PVTV case was considered in
Ref. [104]. Long-wavelength photons only interact with the bulk properties of
the nucleon: its charge, mass, magnetic moment, and, in the case of a PVTV
process, the electric dipole moment (EDM). Additionally, polarizabilities
parametrize the response of the internal structure of the nucleon (which also
depends on PVTC and PVTV interactions) to a quasistatic electromagnetic field.
Peculiar for PVTC and PVTV Compton processes, the effect of polarizabilities
dominates over the respective ground-state contribution, the opposite to the
case of Compton scattering without symmetry violation [104]. This opens up a
possibility to search for a new class of PVTV interactions other than those
generating the EDM. Disregarding the effects of the nucleon spin for this
discussion, the PVTC and PVTV signatures in the Compton scattering process
with circularly polarized incoming laser photons are given by the
$\vec{S}_{\gamma}\cdot(\vec{q}+\vec{q}^{\,\prime})$ and
$\vec{S}_{\gamma}\cdot(\vec{q}-\vec{q}^{\,\prime})$ terms, respectively, where
$\vec{S}_{\gamma}=i\vec{\varepsilon}_{\lambda}(\vec{q})\times\vec{\varepsilon}^{*}_{\lambda\prime}(\vec{q}^{\,\prime})$
is the photon spin defined in terms of the initial and final photon
polarization vectors $\vec{\varepsilon}_{\lambda}(\vec{q})$ and
$\vec{\varepsilon}^{*}_{\lambda\prime}$, respectively. The presence of such
PVTC and PVTV terms leads to a single-spin asymmetry [104]

$A_{\gamma}\equiv\frac{\sigma^{+}-\sigma^{-}}{\sigma^{+}+\sigma^{-}}=A_{\gamma}^{\rm
PVTC}\cos^{4}\frac{\theta}{2}+A_{\gamma}^{\rm PVTV}\sin^{4}\frac{\theta}{2},$
(5)

where $\sigma^{\pm}$ stands for the differential Compton cross section with
left/right circular polarization of the incident photon. Asymmetries
$A_{\gamma}^{\rm PVTC}$ and $A_{\gamma}^{\rm PVTV}$ encode the respective
polarizabilities. Importantly, both PVTC and PVTV signals arise in the same
observable, and the only difference is in its angular dependence. This feature
follows from the fact that the $P$-even, $T$-odd vector $\vec{S}_{\gamma}$
projects onto either the $P$-odd, $T$-odd combination
$(\vec{q}+\vec{q}^{\,\prime})$, or the $P$-odd, $T$-even combination
$(\vec{q}-\vec{q}^{\,\prime})$, so that the single-spin asymmetry is purely
PVTC in the forward direction, purely PVTV in the backward direction, and a
mixture of the two in between.

The size of the asymmetry $A_{\gamma}$ in Eq. (5) depends on the model used to
generate $A_{\gamma}^{\rm PVTC}$ and $A_{\gamma}^{\rm PVTV}$. For the nucleon
case, the relevant mechanism is due to parity-violating couplings of lightest
mesons, most notably the pions.

Figure 4: Representative Feynman diagrams responsible for generating PVTC
polarizabilities. The square denotes the PVTC pion-nucleon coupling
$h^{1}_{\pi}$, while the dotted vertex denotes the two-photon coupling to the
pion.

The diagrams of Fig. 4 yield [102, 103]:

$A_{\gamma}^{\rm PVTC}\sim 3\times 10^{-8}\left(\frac{h^{1}_{\pi}}{5\times
10^{-7}}\right)\left(\frac{\omega}{100\,\mathrm{MeV}}\right)^{3},$ (6)

where $\omega$ stands for the c.m. photon energy. A recent measurement of the
PV asymmetry in $\vec{n}+p\to d+\gamma$ by the NPDGamma collaboration [105]
obtained

$h^{1}_{\pi}=(2.6\pm 1.2\pm 0.2)\times 10^{-7},$ (7)

with the first and second uncertainties being statistical and systematic,
respectively. This, together with the maximum energy of $140$ keV in this GF
setting, makes the PVTC asymmetry on a proton beam too small to be observed.

Figure 5: The $\pi^{0}$-pole contribution to the PVTV polarizabilities. The
square denotes the PVTV pion-nucleon coupling $\bar{g}_{0}$.

Similarly, the PVTV pion-nucleon coupling $\bar{g}_{0}$ generates PVTV
polarizabilities, for example, via $\pi^{0}$ exchange as shown in Fig. 5. This
mechanism leads to an estimate [104]

$A_{\gamma}^{PVTV}\sim\bar{g}_{0}\left(\frac{\omega}{m_{\pi}}\right)^{2}.$ (8)

The natural size of $\bar{g}_{0}$ is constrained by the neutron EDM [106],
$\bar{g}_{0}\lesssim 10^{-11}$.

To access the PVTC and PVTV signatures via laser backscattering at the GF, we
therefore turn our attention to systems in which i) the characteristic energy
scale is (ideally) comparable to the c.m. photon energy available, i.e. $60$
keV for the ion beam, and ii) the natural size of symmetry violation is
enhanced by the presence of parity doublets, nearly degenerate pairs of states
of opposite parity [107]. We denote the average excitation energy of the
parity doublet by $E_{\rm pd}\approx E_{1}\approx E_{2}$ and the energy
splitting within the parity doublet by $\Delta E=E_{2}-E_{1}$. If the typical
nuclear energy scale $E_{N}\approx 10$ MeV is much larger than this splitting,
an enhancement factor $R\sim E_{N}/\Delta E\gg 1$ arises, and the expectation
for the size of the symmetry-violating asymmetries in the “polarizability
regime” $\omega\leq E_{\rm pd}$ is

$\displaystyle A_{\gamma}^{\rm PVTC}$ $\displaystyle\sim
10^{-8}R\left(\frac{\omega}{E_{\rm pd}}\right)^{3},$ (9) $\displaystyle
A_{\gamma}^{\rm PVTV}$ $\displaystyle\lesssim
10^{-11}R\left(\frac{\omega}{E_{\rm pd}}\right)^{2}.$ (10)

We list a few known examples of parity doublets in light nuclei in Table 3.
Given that the photon energy in the center-of-momentum of the laser photon and
the stored beam is constrained to be below $60$ keV, while the typical nuclear
excitations reside at a few MeV, we can still operate in the polarizability
regime. The enhancement due to the small energy splitting within a doublet
comes in linearly, whereas energy suppression appears quadratically or
cubically. Because of this, one should aim at the smallest energy denominator
in Eqs. (9), (10) making 18,19F and 21Ne promising candidates.

Until now, these nuclei were only used for looking for PVTC signals. If a
small PVTV component is present additionally to the parity mixing (PV is a
prerequisite for PVTV effects), a backward-peaked component will arise. Note
that due to a quadratic energy suppression rather than cubic for PVTC, the
PVTV signal will have an additional enhancement with respect to the PVTC case,
adding to the motivation for looking for such tiny asymmetries.

Isotope | $T_{1/2}$ |  $E_{1}$ (keV) | $I_{1}^{P}$ |  $E_{2}$ (keV) | $I_{2}^{P}$ |  $\Delta E$ (keV) | $R$ | 108 $A^{\rm PVTC}_{\gamma}$ | FOM (s-1)
---|---|---|---|---|---|---|---|---|---
18F ($1^{+}$) | 109.77(5) min | 1042 | 0+ | 1081 | 0- | 39 | 256 | 0.05 | $2.5\times 10^{-8}$
19F ($\frac{1}{2}^{+}$) | stable | 0 | $\frac{1}{2}^{+}$ | 110 | $\frac{1}{2}^{-}$ | 110 | 91 | 15 | 2$\times 10^{-3}$
20Ne ($0^{+}$) | stable | 11255 | $1^{-}$ | 11258 | $1^{+}$ | 3.2 | 3125 | $5\times 10^{-4}$ | $2.5\times 10^{-22}$
21Ne ($\frac{3}{2}^{+}$) | stable | 2789 | $\frac{1}{2}^{+}$ | 2795 | $\frac{1}{2}^{-}$ | 5.7 | 1754 | 0.02 | $4\times 10^{-9}$
Table 3: Parameters of the low-lying parity doublets in isotopes of fluorine
and neon, from the ENSDF database [32].

The longitudinal polarization of the relativistic ions (those that have non-
zero spin) will allow to address the spin-dependent PVTC and PVTV
polarizabilities. Following Ref. [102] for the PVTC case worked out for the
polarized proton, we expect the PV longitudinal single-spin asymmetry

$\displaystyle A_{\rm Ion\;spin}^{\rm PVTC}$ $\displaystyle\sim
10^{-8}R\left(\frac{\omega}{E_{\rm pd}}\right)^{2}.$ (11)

The lower power of energy in this estimate suggests that if using a polarized
ion beam is a viable option the PVTC effects Compton scattering might be
easier to access than the photon asymmetry. In contrast, the contribution of
PVTV spin polarizabilities to this observable is suppressed by an extra power
of the photon energy, and involving the ion spin does not offer better
sensitivity to PVTV.

The figure of merit (FOM) is defined as [108]

$\textrm{FOM}=\mathrm{Rate}\times A^{2},$ (12)

and corresponds to the inverse time necessary for an observation of the
asymmetry. Above, the rate is proportional to the cross section and fluxes of
the colliding particles, and $A$ denotes the asymmetry. The cross section for
low-energy Compton scattering (Thomson scattering) is

$\sigma=\frac{8\pi}{3}\left(\frac{Z^{2}\alpha\hbar}{Mc}\right)^{2}.$ (13)

Here $Z$ and $M$ are the charge and mass of the ion. The cross-section is
$\sigma\approx 3$ $\mu$b for the case of 19F. The event rate is obtained as a
product of the laser photon rate of $N_{\gamma}\approx 10^{25}$ per second,
number of ions per bunch $N_{I/b}\approx 10^{10}$ (see Table 1), and the ratio
of the process cross section $\sigma$ to the cross section of the laser beam
$S\approx(20\,\mu\mathrm{m})^{2}$:

$\mathrm{Rate}=N_{I/b}\times N_{\gamma}\times\frac{\sigma}{S}\approx
10^{11}\,\mathrm{s}^{-1}.$ (14)

Here we assume that the time structure of the laser pulses is matched to that
of the ion bunches. Numerical estimates of the FOM for the four ions under
consideration are summarized in Table 3. These results indicate that a
promising candidate nucleus is 19F, for which a 10% measurement of the
asymmetry can be achieved with a day of statistics accumulation.

A final note in this section concerns the background not associated with
symmetry-violating effects. Final-state interaction (FSI) due to
electromagnetic rescattering will generate a non-zero phase of the Compton
amplitude if even a tiny linear polarization component $P$ is present, leading
to a false asymmetry due to a correlation
$\vec{S}_{\gamma}\cdot[\vec{q}\times{\vec{q}\,}^{\prime}]$. In Ref. [104] this
background was estimated as $A_{\gamma}^{\rm
FSI}\sim\alpha\frac{\omega^{2}}{M_{N}^{2}}\,P\,\sin^{2}\theta\cos 2\phi$. The
maximal degree of misalignment of circular polarization of laser photons can
reasonably be assumed to be $P\lesssim 10^{-6}$, and for $\omega\leq 60$ keV
the false asymmetry should be small. The modulation with the azimuthal angle
$\phi$ can be used to further suppress this undesired background: when
integrated over the full $2\pi$ range, this non-symmetry-violating background
drops out of the asymmetry. The above estimate relies on the assumption that
the photon energy lies below the inelastic threshold. For $\omega\leq 60\,$keV
this requirement suggests that a fully stripped ion beam has to be used to
avoid that atomic excitations and interferences thereof mimic the PVTC or PVTV
signatures. An additional PVTV-type correlation may be generated by FSI on top
of a PVTC signature. We expect this effect at the level $10^{-5}$ of PVTC or
smaller.

The experiment at the GF will entail measuring the flux of secondary photons
depending on the circular polarization of the laser photons. The detector will
have to be $2\pi$-symmetric in order to eliminate the $\cos 2\phi$-modulated
electromagnetic background.

##  4 Nuclear physics with GF secondary-photon beam on fixed targets

This Section addresses the opportunities provided by the GF secondary beam for
nuclear spectroscopy and hadron physics in the energy range from a few to
hundreds of MeV. A wide variety of targets can be used in conjunction with the
GF secondary beam, benefiting from the vast experience and infrastructure at
CERN. Apart from stable or long-lived nuclear species, unique opportunities to
use rare radioactive elements may be opened by the close physical proximity to
the Isotope mass Separator On-Line facility (ISOLDE) [109], one of the world’s
leading sources of radioactive nuclides. Further opportunities may be rendered
possible by a dedicated storage ring for rare isotopes discussed in Sec. 6.

Perhaps the most obvious application of the GF is nuclear spectroscopy, which
will benefit from the narrow spectral bandwidth achievable with collimated GF
photons, their energy tunability and high intensities. Figure 6 displays a
schematic overview of possible photonuclear reaction pathways following the
absorption of an impinging photon. Nuclear spectroscopy experiments have been
performed since the advent of betatron particle accelerators and Schiff’s
seminal paper from 1946 [110], proposing to use electron bremsstrahlung
(converted from energetic electron beams) to detect nuclear resonance
fluorescence (NRF) off bound excited nuclear states. NRF is discussed in Sec.
4.1 on gamma spectroscopy of narrow resonances at the GF.

Figure 6: Possible photonuclear reactions following the absorption of a photon
provided by the GF: excitation below the particle separation threshold will
allow for nuclear spectroscopy via nuclear resonance fluorescence (NRF),
potentially followed by $\beta$ decay (photoactivation). Excitation into the
continuum above the particle-separation threshold will be followed by
photodissociation (i.e. emission of neutrons, protons or $\alpha$ particles)
or by photofission.

As indicated in Fig. 6, in parallel to NRF, deexcitation of an intermediate
level may proceed not directly towards the ground state of the excited
nucleus, but via $\beta$ decay to a daughter nucleus (photoactivation).
Alternatively, an excitation into the continuum above the particle-separation
threshold will lead to subsequent deexcitation via particle emission
(neutrons, protons, $\alpha$ particles), hence to photodissociation, addressed
in more detail in Sec. 4.4 or to photofission, discussed in Sec. 4.5.

In Fig. 7, possible strong (electric and magnetic) dipole excitations in
heavy, deformed nuclei are schematically shown as a function of their
excitation energy. We concentrate on dipole excitations since at the few MeV
energies of interest, the gamma-ray wavelength is much larger than the nuclear
size. On the magnetic dipole ($M1$) side, at lower energies, the orbital
magnetic dipole excitation corresponds in a simplistic geometric, macroscopic
picture to a scissors-like vibration of the deformed proton and neutron fluids
against each other [111]. Therefore this $M1$ mode is referred to as the
‘scissors mode’, the magnetic analogue of the GDR. We further discuss the
importance of these transitions in the context of the GF in Sec. 4.1. At
higher energies of $\approx$10 MeV, Gamow–Teller (GT) transitions [112]
appearing as $M1$ excitation strength are related to the common weak-
interaction processes of spin–isospin-type ($\sigma\tau$, i.e., a product of
the spin and isospin operators) in atomic nuclei. These are of interest not
only in nuclear physics, but also in astrophysics; they play an important role
in supernovae explosions and nucleosynthesis [113, 114, 115].

Figure 7: Illustration of the photoresponse of deformed atomic nuclei as a
function of the photon energy, divided into phenomena involving electric
($E1$) and magnetic ($M1$) excitations. Figure by A. Zilges.

On the electric dipole ($E1$) side of Fig. 7, the GDR represents a collective
excitation mode where protons and neutrons vibrate against each other along or
perpendicular to the symmetry axis of the nucleus. The double-humped structure
of the GDR in deformed nuclei appears typically at energies of around 77$\cdot
A^{-1/3}$ MeV ($A$ is the mass number), which corresponds to about 14 MeV in
rare earth nuclei. This isovector GDR was initially found as a spectral
feature corresponding to the excitation of the nuclear ground state, but soon
Brink and Axel concluded that giant resonances could be associated with any
nuclear state, independent of the microscopic structure of this state [116,
117]. This led to intense studies of GDR properties of excited states, the so-
called GDR in hot nuclei [118]. Further prospects of GF studies of the GDR and
potential multiphoton excitation are presented in Sec. 4.3.

In addition, enhanced electric dipole excitations are expected at lower
excitation energies. Several experimental studies using electromagnetic probes
have found an enhancement of electric dipole strength between about 5 and 10
MeV. This phenomenon is denoted as pygmy dipole resonance (PDR) and has been
intensively studied in recent years [119, 120, 121, 122, 123, 124, 125, 126].
In a simplified geometrical picture, the PDR is described as neutron skin
oscillating against a $N=Z$ core. The PDR phenomenon is discussed in more
detail in Sec. 4.2.

Moving towards even higher secondary-beam energies, and towards hadron
physics, the second part of the Section goes beyond traditional gamma
spectroscopy to address Compton and photo-induced processes on light nuclei in
Sec. 4.7, pion photoproduction in Sec. 4.8, and the Delta-resonance region and
continuum effects in Sec. 4.10. The Section concludes with prospects of more
exotic processes at the GF involving parity-violation 4.11.

###  4.1 Narrow resonances

Narrow resonances in a wide energy range up to the particle-separation
threshold are the subject of NRF. Resonance fluorescence in general refers to
the resonant excitation of an upper state by absorption of electromagnetic
radiation and decay of the excited level by emission of radiation. Via NRF,
nuclear states are excited in photon-scattering experiments and their
deexcitation through electromagnetic transitions is studied via gamma-ray
spectroscopy [127, 111]. From these experiments, photoabsorption cross
sections and the related photon strength functions [128, 129] are deduced,
which are important for the description of photonuclear reactions and the
inverse radiative-capture reactions.

In NRF experiments, the nucleus is excited by the absorption of a real photon.
As these photons only carry a small angular momentum, excitation favors
transitions where the change of the nuclear angular momentum is small. The
strongest transitions are typically of dipole character ($E1$ or $M1$). The
excited states may subsequently decay either directly or via intermediate
lower-lying states back to the ground state. The latter transitions are
typically weak and difficult to resolve from the background when using
energetically continuous (broadband) bremsstrahlung. Therefore, for example,
the determination of the total transition width $\Gamma$ is not always
possible with bremsstrahlung beams. An alternative way to perform NRF
experiments uses quasi-monochromatic photon beams as provided by existing or
upcoming Compton-backscattering facilities [130, 131] and in the future by the
GF.

In general, for all nuclear shapes where the centers of mass and charge do not
coincide, one expects an electric dipole moment leading to enhanced electric
dipole transitions. Examples are deformed shapes due to octupole deformations
or any kind of cluster configurations. Rotations on top of the octupole
vibrations coupled to the quadrupole deformed core lead to the so-called
octupole vibrational bands. In spherical nuclei close to magic proton or
neutron shells the lowest-lying excitations are quadrupole and octupole
vibrations of the nuclear shape. A coupling of these two single-‘phonon’
excitations leads to a two-phonon quintuplet with spins
$J^{\prime}=1^{-}-5^{-}$. The 1- member of this multiplet can, in principle,
be excited via an electric dipole transition from the ground state in NRF
experiments. The energies of all these dipole excitations ($E1$ and $M1$) lie
in the same range of roughly 2–4 MeV, see also Fig. 7. This emphasizes the
crucial necessity for parity assignments for the interpretation of observed
excitations. NRF is by far the most sensitive tool to detect such dipole
excitations and to (model independently) determine their characteristics:
excitation energies, spins, parities, decay energies, level widths, lifetimes,
decay branchings, multipole mixing ratios and absolute transition strengths.

There are many specific examples of transitions that can be studied at the GF
taking advantage of its monochromaticity, tunability and unsurpassed photon
flux. An interesting case is the one of 13C, which exhibits many accessible
resonances displayed in the partial level scheme in Fig. 8. The case of 13C
will be discussed in the context of polarimetry with narrow resonances in Sec.
10.1. The interesting neutron emission channels at 7.55 MeV and 8.86 MeV can
be selectively excited by the GF and used in turn to generate tertiary
monoenergetic fast neutron beams. This topic is the subject of Sec. 12.5.

Figure 8: Nuclear energy levels below 8.86 MeV in 13C. Energies, $I^{P}$ and
decay modes [55] are given for the levels discussed in this work. The arrows
show excitation to be induced by the secondary GF photons. For the lowest-
energy gamma resonance, the double-sided arrow indicates that the only decay
channel is via photon emission.

Another interesting case is the longest lived ($\tau_{1/2}\gtrsim 1.2\cdot
10^{15}$ y) isomer 180mTa lying 77 keV above a short-lived ground state (see
also Table 9). This isomer has been extensively studied in order to understand
its decay mechanisms via photoexcitation to a gateway level above the isomer
[132], following the scheme presented in Sec. 2.1. Previous experiments [133]
using broadband bremsstrahlung radiation could neither identify the gateway
levels nor could they provide information about the involved transition
probabilities. The high flux and excellent monochromaticity of the GF
secondary beam could finally shed light on this matter.

Finally, we note that scattering of linearly polarized photons on unpolarized
or spin-zero nuclei is an effective method to determine the parity of excited
nuclear states, see, for example, Ref. [134]. The advantage of the GF for such
studies is the ability to resolve relatively narrow resonances combined with
high statistical sensitivity and tunability of the photon energy.

###  4.2 Pygmy dipole resonances

The dipole response for stable (medium to heavy) nuclei with a moderate
neutron excess is almost entirely concentrated in the giant dipole resonance
(GDR) that exhausts almost 100$\%$ of the Thomas-Reiche-Kuhn (TRK) sum rule
(which is proportional to the energy-weighted sum rule) [135, 136]. For this
collective excitation, the nuclear ($N-Z$) asymmetry and the corresponding
symmetry energy provide the restoring force for the oscillation between the
excess neutrons against the ($N=Z$) nuclear core [137]. Thus only a minor part
of the $E1$ strength is expected at lower excitation energies. The electric
dipole strength distribution at low energies (below $\approx 5$ MeV) is
dominated by multi-phonon excitations originating from the coupling between
$J^{\pi}=2^{+}$ quadrupole and $J^{\pi}=3^{-}$ octupole vibrations of the
nuclear shape, being the lowest single-phonon excitations in even-even nuclei
near closed neutron or proton shells. The two-phonon excitations $2^{+}\otimes
3^{-}$, called quadrupole-octupole coupling, lead to a multiplet consisting of
five states with spins and parities $J^{\pi}=1^{-},...,5^{-}$. In photon-
scattering experiments, the 1- member of the $2^{+}\otimes 3^{-}$ multiplet
can be excited selectively by an $E1$ transition from the ground state. Its
subsequent decay is predominantly back to the ground state via an $E1$
transition [138].

At energies between 5 MeV and a few hundred keV above the neutron-separation
threshold, an enhancement of dipole strength was observed in many semi-magical
and magical nuclei [119]. It was found that the $E1$ dipole strength at
energies below the GDR range is of the order of 1$\%$ of the TRK sum for
stable nuclei, and up to about 5$\%$ for the exotic nuclei studied so far
[121]. Owing to its relatively low strength, this low-lying $E1$ transition is
often denoted as the pygmy dipole resonance (PDR). The first PDR observation
was made by Bartholomew in 1961 who found enhanced gamma-ray emission after
neutron capture [139]. The PDR name was coined in 1969, when its impact on
calculations of neutron-capture cross sections was reported [140]. In 1971,
Mohanet et al. proposed a description of the PDR in a three-fluid
hydrodynamical model [141]. One interpretation is an oscillation of excess
neutrons (the neutron skin) against the $N=Z$ core. The PDR is of great
interest because it provides information on the neutron skin and on the
nuclear symmetry energy as a crucial ingredient to the nuclear matter
equation-of-state [142]. On theoretical grounds, employing the relativistic
quasiparticle random phase approximation (RQRPA) approach, a one-to-one
correlation was found between the pygmy dipole strength and parameters
describing the density dependence of the nuclear symmetry energy, and in turn
with the thicknesses of the neutron skin [143].

As discussed in Ref. [137], the determination of the neutron radius of a heavy
nucleus, i.e. its neutron distribution, is a fundamental problem related to
the equation-of-state of nuclear matter with far-reaching consequences in
areas as diverse as atomic PV [144, 145], nuclear structure [146, 147, 148,
149, 150], heavy-ion collisions [151, 152, 153, 154, 155], and neutron-star
structure [156, 157, 158, 159, 160]. Hence, the far-reaching impact of
understanding the $E1$ strength distribution of the PDR motivated still
ongoing extensive experimental and theoretical studies in a wide mass range
[119, 161, 162, 163, 164, 126, 124, 121, 123, 122, 165, 166]. In a recent work
at GSI [164], PDR were studied in a chain of chromium isotopes, see Table 4.

As photon scattering is a key experimental technique to study the $E1$ dipole
strength in nuclei, and in particular the PDR, an intense and high-quality
(i.e. monoenergetic) photon beam as envisaged for the GF would turn out highly
beneficial for this active research field.

###  4.3 GDR and multiphoton excitation

The high photon flux in the GF secondary beam can be used at photon energies
of $10-20$ MeV to excite the GDR in medium-mass and heavy nuclei. In a shell-
model picture the GDR is a superposition of particle-hole excitations out of
the ground state and is not an eigenstate of the nuclear Hamiltonian. These
particle-hole excitations actually do not all have the same energy, leading to
a spreading of the GDR often referred to as Landau damping [167, 168]. The
residual two-body interaction mixes the particle-hole excitations with each
other and with other shell-model configurations and, thus, spreads the GDR
over the eigenstates of the nuclear Hamiltonian. This leads to a Lorentzian
distribution of the dipole strength with a so-called “spreading” width
$\Gamma^{\downarrow}$ of around 5 MeV [169, 170]. Ground-state deformation in
axially-symmetric deformed nuclei leads to a splitting of the GDR Lorentzian
into two components. The GDR spectrum, with its strength, centroid, and
Lorentzian parametrizations thus offer information on the nuclear shape [171],
and in addition, on the nuclear size, the nuclear symmetry energy (important
for the study of neutron-star structure) and the viscosity of the neutron and
proton fluids [172]. In a time-dependent picture, the spreading of the GDR
over nuclear eigenstates can be viewed as statistical equilibration [173] with
a characteristic time scale $\hbar/\Gamma^{\downarrow}$.

According to the Brink-Axel hypothesis, the GDR is a mode of excitation not
only of the nuclear ground state, but of any excited state as well [116, 117].
GDRs built on excited states are termed “hot GDRs” and have so far been
produced in heavy-ion collisions leading to fusion or incomplete fusion [118].
As a result of the collision, a compound nucleus is formed, i.e., a highly
excited nucleus in which the excitation energy is distributed over several or
many particle-hole excitations [173]. Heavy-ion collisions involve large
angular momenta, leading to high angular momentum states in the compound
nucleus. Completely new prospects would be opened by the possibility to reach
the same excitation energies by multi-photon absorption. Absorption of a
dipole photon transfers one unit of angular momentum. The total angular
momentum in a multiple dipole absorption process typically increases with the
square root of the number of absorbed photons [174]. Multiple dipole
absorption would therefore lead to excitation of the compound nucleus at high
energy and at comparatively low angular momentum, far above the yrast line, as
illustrated in Fig. 9. The spectral region of high excitation energy and small
angular momentum is so far mostly unexplored. Little is known experimentally
in that region about state densities, the widths of the GDRs, decay properties
of highly excited states, and, specifically, about the nuclear equilibration
process.

The identification of $\hbar/\Gamma^{\downarrow}$ with the equilibration time
becomes questionable at high excitation energy well above particle threshold.
For example, hot GDRs [118] are characterized by high neutron-evaporation
rates and have correspondingly short lifetimes [175]. Neutron evaporation may
happen prior to complete equilibration. The increase of the GDR width due to
neutron evaporation then does not correspond to a shorter equilibration time.
Such increase is considered to be one of the possible reasons for the damping
of the hot GDR [175, 176, 118]. Sokolov and Zelevinsky confirmed the
interpretation of Ref. [175] and attributed the disappearance of the
collective GDR strength observed experimentally to the complex interplay
between internal and external dynamics with decay into the continuum [177]. A
different explanation postulating an increase of the spreading width with
temperature was also put forward [178].

Figure 9:  Qualitative illustration of two regimes of nuclear excitation. The
yrast line defines the minimum energy of a nuclear state with a certain
angular momentum. Heavy-ion collisions preferentially excite states close to
the yrast line (region depicted by hatched area). Multiple absorption of
multi-MeV dipole photons involves small transfer of angular momentum and could
lead to compound states several hundred MeV above yrast (red arrow). The inset
shows the angular-momentum distributions for $N_{0}=10,20$ and $30$ absorbed
dipole photons starting from an initial state with $J=0$. Adapted from Ref.
[174].

Due to its large width, the GDR can in many cases be investigated with
broadband gamma sources. In the following we identify special cases for which
the high photon flux and/or monochromaticity of the GF may be of advantage.

First, the high photon flux would be beneficial for the study of GDR
excitations in exotic nuclei far from the valley of stability. Due to their
short lifetime, such targets are bound to produce small excitation yields and
are hard to access experimentally. A notable exception is the PDR and GDR
measurement of the neutron-rich unstable 68Ni nucleus [179]. Nuclear gamma
spectroscopy on the GDR at the GF could bring new insights into the shapes and
properties of exotic nuclei.

Second, the GF will enable photofission studies of fission isomers in
actinides. A nuclear reaction such as (d,p) or fragmentation would produce
highly deformed excited states in the second minimum of the fission barrier.
Photoabsorption would excite the GDR built upon these states. The high photon
flux and in particular the monochromaticity of the GF would allow for
photofission studies of these strongly deformed actinides with a much better
sensitivity than so far available to the fission barrier landscape. This is
the subject of Sec. 4.5.

Finally, one may consider the possibility of multiple absorption of dipole
photons, where for instance a second absorbed photon excites a GDR built upon
the GDR reached by the first photon. As shown in Ref. [180], a comparison of
single and multiple GDRs could probe statistical aspects of the resonance
width, exhibiting either a Lorentzian or Gaussian shape. For the focused laser
pulses envisaged at the Nuclear Pillar of the Extreme Light Infrastructure in
Bucharest [181], with pulse durations of tens of zeptoseconds
($1\,\textrm{zs}=10^{-21}$ s), theoretical studies based upon the Brink-Axel
hypothesis predict multi-photon absorption of up to $\approx$hundred photons.
This would lead to excitation energies in the range of several $100$ MeV, to
multiple neutron evaporation and to the formation of neutron-deficient nuclei
far from the valley of stability [182, 174, 183, 184]. In comparison, focusing
of the secondary beam does not appear to be feasible at the GF and the
envisaged pulse durations are many orders of magnitude longer. Thus, even the
probability of two-photon excitation remains small at the GF.

###  4.4 Photonuclear response above particle threshold

The excitation-energy region around the particle-separation threshold is, on
the one hand, of interest from a theoretical perspective because the coupling
of bound quantum states to the continuum of unbound states requires an
extended formalism, which is particularly intricate for exotic nuclei near the
drip-lines where all structures are weakly bound [185]. On the other hand,
this energy region covers the Gamow-window of thermally driven reactions of
nucleons with heavy nuclei. Its understanding is a prerequisite for modelling
nuclear-reaction cascades in hot cosmic objects and thus for understanding
nucleosynthesis. Below the threshold, all excited resonances decay
predominantly by gamma-ray emission, with rare cases of $E0$ transitions
involving internal conversion or electron-positron pair production.

In this regime, the knowledge of the nuclear level densities and the gamma-ray
strength function is of crucial importance as an ingredient, for example, to
model nuclear-reaction-network calculations of nucleosynthesis. From an
experimental perspective, the so-called “Oslo method” was established as a
reliable tool to derive these properties from measured gamma-ray energy
spectra. The Oslo method comprises a set of analysis techniques designed to
extract the nuclear level density and average gamma-decay strength function
from a set of excitation-energy tagged gamma-ray spectra. The method was first
introduced in Ref. [186] and since then continuously further developed [187].
Recently a new software implementation of the method was provided [188].

Above the threshold, the particle-decay channel opens up. Either no gamma rays
can be observed at all or their intensity cannot be used as a measure of the
total electromagnetic excitation strength of the resonance due to the unknown
particle-decay branching ratio. An intense narrow-energy-band photon beam will
open up new horizons for investigation of nuclear photo-response at and above
the separation threshold [189].

####  4.4.1 Neutron-capture cross sections of s-process branching nuclei

Cosmic nucleosynthesis of heavy elements above the so-called iron peak mainly
proceeds via neutron-capture processes; the r process (r: rapid neutron
capture) is connected to scenarios of high neutron densities well above 1020
cm-3 and temperatures on the order of 2-3$\cdot$109 K as occurring in
explosive scenarios such as, for example, supernovae or neutron-star mergers
[72, 190]. In contrast, average neutron densities during s-process
nucleosynthesis (s: slow neutron capture) are rather small ($\approx$108
cm-3), i.e. the neutron capture rate $\lambda_{n}$ is normally well below the
$\beta$-decay rate $\lambda_{\beta}$ and the reaction path is close to the
valley of $\beta$ stability [191, 192, 193]. However, when the s-process
reactions occur at peak neutron densities, reaction path branchings occur at
unstable isotopes with half-lives as low as several days. The half-lives at
these branching points are normally known with high accuracy, at least under
laboratory conditions. One relies on theory for the extrapolation to stellar
temperatures [194]; the neutron-capture cross sections are accessible to
direct experiments only in special cases. Besides the limited availability of
sufficient amounts of target material, intrinsic activity of the target
hinders experimental access especially in the case of branching points
involving isotopes with short half-lives. Moreover, the predictions in the
Hauser-Feshbach statistical model yield different results depending on the
underlying parameter sets. Additionally, studies of branching points involving
long-lived isotopes (e.g. 147Pm, 151Sm, 155Eu) showed that the recommended
values of neutron-capture cross sections in the Hauser-Feshbach statistical
model [195] differ by up to 50$\%$ from the experimentally determined values
[196]. Therefore, further and precise experimental constraints on the
theoretical predictions of these crucial values are needed. The inverse
($\gamma$,n) reaction could be used at the GF to determine optimized model-
input parameters for improved predictions.

####  4.4.2 ($\gamma$,p), ($\gamma,\alpha$) cross sections for p-process
nucleosynthesis

In the framework of cosmic nucleosynthesis, ‘p-nuclei’ are certain proton-
rich, naturally occurring isotopes of some elements which cannot be produced
in either slow (s-process) or rapid (r-process) neutron-capture processes. In
the seminal paper by Burbidge, Burbidge, Fowler and Hoyle (‘B2FH’) [197],
accompanied by [198], the s- and r-processes for the production of
intermediate and heavy nuclei beyond iron were introduced. It was immediately
realized by the authors of both papers that a number of proton-rich isotopes
can never be synthesized through sequences of only neutron captures and
$\beta^{-}$ decays. This required the postulation of a third process. It was
termed p-process, because it was thought to proceed via proton captures at
high temperature, however, later findings shed doubts on the feasibility to
use proton captures for producing all of the nuclides missing from the s- and
r-process mechanisms. In the literature, ‘p-process’ is sometimes used as a
general term denoting whatever production mechanism(s) is/are found to be
responsible for the p-nuclides. Historically, there were 35 p-nuclides
identified, with 74Se being the lightest and 196Hg the heaviest [199].
Compared to the bulk of natural isotopes, p-nuclei generally show abundances
which are 1-2 orders of magnitude lower. Photodesintegration rates, like
($\gamma$,n), ($\gamma$,p) and ($\gamma$,$\alpha$) play a crucial role in the
nucleosynthesis of these nuclei. As the p-reaction network calculations
comprise several hundred isotopes and the corresponding reaction rates,
theoretical predictions of these rates, normally obtained in the framework of
the Hauser-Feshbach statistical model theory [200], are necessary for the
modelling. The reliability of these calculations should be tested
experimentally for selected isotopes.

While the ($\gamma$,n) cross sections in the energy regime of the Giant Dipole
Resonance around 15 MeV were measured extensively already several decades ago
(see, for example, Ref. [201]), many efforts using continuous bremsstrahlung
spectra have been made [202, 203, 204] to determine the reaction rates without
any assumptions on the shape of the dependence of the cross section on energy
in the astrophysically relevant energy region just above the reaction
threshold. Also laser Compton backscattering to produce monoenergetic photon
beams was used to determine reaction rates by an absolute cross section
measurement [205]. In contrast, experimental knowledge about the ($\gamma$,p)
and ($\gamma$,$\alpha$) reactions in the corresponding astrophysical Gamow
energy window is much more scarce. In fact, the experimental data are based on
the observation of the time reversal (p,$\gamma$) and ($\alpha$,$\gamma$)
cross sections, respectively [206, 207, 208, 209] for the neutron-deficient
nuclei with mass numbers around 100. Therefore, being able to measure these
rates directly would represent a significant progress. It should be noted that
deeper insight into the p-process nucleosynthesis would not emerge from the
measurement of only a few selected reactions, but via the development of a
comprehensive database. This necessitates rather short individual measurement
times for studying the stable p nuclei despite their typically low natural
abundances [210]. This will only become possible using intense $\gamma$ beams
as will be provided by the VEGA facility at ELI-NP and, ultimately, by the
Gamma Factory.

However, the majority of nuclides in the p-process network calculations are
radioactive. Here, the access to stored secondary beams at the GF, see Sec. 6,
will enable a unique possibility to investigate photo-induced reactions on
short-lived species. Furthermore, direct $(p,\gamma)$ and $(\alpha,\gamma)$
reactions can be as well studied by employing in-ring, pure, thin, gaseous H2
and He targets [211, 212]. In this way, direct and reverse reactions can be
measured for the same pairs of nuclides, thus enabling immediate testing of
Hauser-Feshbach calculations.

####  4.4.3 Direct measurement of astrophysical $S$-factors

Helium burning in stars, as has been realized at the dawn of nuclear
astrophysics [197], is the key process to understanding the abundances of
chemical elements in the universe. This process proceeds via the
triple-$\alpha\to^{12}$C reaction, enhanced by the Hoyle resonance [213], and
followed by the $\alpha$ radiative capture on carbon,
12C$(\alpha,\gamma)^{16}$O. The rate of the former process at stellar
temperatures is known to $\sim 10\%$. In turn, a $\approx 100\%$ uncertainty
in the rate of the latter reaction represents the main source of uncertainty
in stellar evolution models. A direct measurement of the rate of the
12C$(\alpha,\gamma)^{16}$O reaction at astrophysical energies is considered
‘the holy grail’ of nuclear astrophysics, but is extremely challenging. The
radiative process is rare due to the weakness of the electromagnetic force,
and may be suppressed by 3-6 orders of magnitude with respect to other
processes (e.g., elastic scattering).

Attempts to measure the inverse process generalized to a virtual-photon-
induced disintegration 16O$(e,e^{\prime}\alpha)^{12}$C showed the possibility
to access this process experimentally [214], albeit no satisfactory agreement
of theoretical predictions with the data was achieved. There is a revived
interest in studying this process at electron-scattering facilities [215].
While one wins in the rates due to high intensity of electron sources, the
price to pay are electromagnetic background processes and the necessity to
extrapolate to the real photon point. At GSI, Coulomb dissociation of a 500
MeV/nucleon 16O beam colliding with a Pb target is used to address this
reaction. This is an extremely challenging experiment. Apart from tiny
reaction rates at low center-of-mass energies, the dissociation products have
nearly identical magnetic rigidities [216], making it hard to separate them.
The Gamma Factory will allow to directly study the 16O$(\gamma,\alpha)^{12}$C
photodisintegration process avoiding the aforementioned complications.

To be of direct use for understanding stellar nucleosynthesis, measurements at
the GF should be done close to the energy relevant to the helium-burning
conditions in stars, where the ignition plasma temperature is T$\approx 2\cdot
10^{8}$ K. The kinetic energy of the $\alpha$ particle corresponding to the
astrophysics-relevant Gamow window is $E_{\alpha}\approx$300 keV. The
respective threshold for the (inverse) photodisintegration process
16O($\gamma,\alpha$)12C is $E^{th}_{\gamma}\approx$7.6 MeV. For comparison,
the competing proton-knockout reaction threshold lies at $\approx$11.6 MeV.

####  4.4.4 Alpha clustering in heavy nuclei

Although alpha clustering in light nuclei is well known and the famous Hoyle
state in 12C plays an important role in the evolution of the Universe, there
are many open questions concerning the physics of clusterization in nuclei
[217]. Even the geometrical structure of the most clusterized nuclei, such as
6He in the ground state and 12C in the Hoyle state (and higher similar
states), is still under debate. In the case of 6He, one critical feature is
the correct prediction of the $\beta$-decay to 6Li that is sensitive to the
details of the wavefunction of 6He. The case of 8He is also important as its
size is somewhat smaller than that of 6He due to the pressure of outer
neutrons.

Much less is known concerning the alpha clustering in heavier nuclei. At the
Budker Institute of Nuclear Physics (Novosibirsk), more than 30 years ago, an
experiment with a super-thin nuclear jet target crossing the electron beam in
a storage ring [218] at electron energy $E_{e}=130$ MeV demonstrated a
significant cluster component in the double magic 16O nucleus. There were
observed various cluster-decay channels, with the final states in 12C, 8Be and
four alpha particles. These promising experiments were discontinued and this
storage ring does not exist anymore. Also the electron-ion scattering
experiment ELISe, aiming at reaction studies on stored short-lived nuclides,
is significantly delayed at FAIR [219].

A series of experiments on alpha clustering in heavier nuclei were performed
at Jyväskylä with alpha scattering from heavy nuclei [220]. This reference is
the latest detailed publication on the experiment with 36Ar$+\alpha$ at 150
MeV energy. There is a systematic accumulation of collective states, like
cluster of cluster states with the same spin going from 16O all the way to
40Ca. In the shell-model language, such states are regular combinations of
many selected states from higher shells. The only currently available picture
is that of a quasicrystal structure where the analogs of crystal bands,
probably overlapping, are formed built on alpha constituents. The bands are
split by alpha tunneling as electron bands in normal crystals.

The study with gamma rays of high energy and therefore appropriate wavelength,
a distant relative of X-ray diffraction from crystals, can shed light on this
structure that has no analogs in other nuclear phenomena. The
$(\gamma,\alpha)$ and possibly also $(\gamma,2\alpha)$ reactions [221],
studied as a function of photon energy and polarization can provide unique
information concerning specific features of alpha-clusterization, especially
in even-even $N=Z$ nuclei [222].

The development of nuclear physics away from the stability band brought to the
experimental light such exotic nuclei as those in vicinity of 100Sn, the
heaviest double magic nucleus with $N=Z$. Here, there are indications of alpha
structure continuing up to such heavy systems. This should be a regular part
of the equilibrium nuclear structure in distinction to alpha decay of
superheavy elements. There are various arguments in favor of the existence of
such a substructure including the quadrupole collective states in heavier tin
isotopes, in spite of the absence of valence protons in this shell according
to the primitive shell-model scheme. Especially characteristic are extremely
strong alpha decays of nuclei near 100Sn [223]. The decay of 105Te to 101Sn is
one of the strongest alpha decays in the whole periodic chart, with a half-
life of only 620 ns [224]. This is comparable to the 229 ns half-life of 212Po
which is just a bound state of the alpha particle and the core of 208Pb. The
strongest known $\alpha$ decay of 104Te to 100Sn has an upper limit half-life
of only 18 ns [223]. A detailed study of alpha structures in heavy nuclei
would also be a driving force for the development of corresponding theory
(some directions are outlined in Ref. [222] as the nuclear analog of the
transition between the Bardeen–-Cooper-–Schrieffer (BCS) picture of
superconductivity and boson condensation).

####  4.4.5 Fano effect in nuclear gamma spectroscopy

The Fano effect [225], well known in atomic and molecular physics, is a
characteristic asymmetric spectral lineshape that arises due to the
interference of the resonant and nonresonant transition amplitudes in the
vicinity of a resonance. In nuclear physics, the Fano effect was studied, for
example, in 15N(7Li,7Be)15C [226] and d(9Li,10Li)p [227] reactions. However,
to the best of our knowledge, the Fano effect has not as yet been observed in
nuclear gamma transitions. A loosely related experiment [228] on X-ray
interference with 57Fe Mössbauer nuclei observed Fano interference in the
X-ray reflectivity of thin-film cavities around the nuclear resonance energy.
However, in that case the nonresonant channel was given by the electronic
scattering of the cavity, while only the resonant channel was stemming from
the driving of the nuclear Mössbauer transition.

A study of nuclear Fano resonances via photoexcitation was proposed in the
context of three-body Efimov states [229]. Photoabsorption on the ground state
of 20C ($\tau_{1/2}\approx 16$ ms) should be sensitive to the presence of the
$n+^{19}$C and $n+n+^{18}$C states embedded in the 20C continuum and display
Fano resonances around the respective energies [230].

With tunable narrow-band gamma rays from the GF, one can envision a systematic
study of Fano interference in the vicinity of nuclear gamma transitions, for
example the ones in 13C discussed in Appendix B.

###  4.5 Photofission with monochromatic gamma beams

Photofission measurements enable selective studies of extremely deformed
nuclear states in light actinides and can be utilized to better understand the
landscape of the multiple-humped potential energy surface in these nuclei. The
selectivity of these measurements originates from the low amount of angular
momentum transferred during the photoabsorption process. Studies of fission
isomerism in the actinide mass region provided evidence already in the 1960s
for the existence of a second, minimum of the potential energy surface [231]
corresponding to superdeformation (SD), with a ratio of 2:1 between the long
and short principal axes of the strongly deformed nuclear shape, parameterized
as a rotational ellipsoid. This corresponds to a value of $\beta_{2}\sim 0.7$
of the quadrupole-deformation parameter in the nuclear surface
parameterization in terms of a spherical harmonic multipole expansion. Three
decades later also the existence of a hyperdeformed (HD) third minimum of the
fission barrier (axis ratio 3:1, $\beta_{2}\approx$ 0.9, octupole deformation
$\beta_{3}\approx$ 0.3) could be established (see review articles [232, 233]).
The lower part of Fig. 10 illustrates such a triple-humped fission barrier
(expressed by the potential energy as a function of the nuclear quadrupole
deformation $\beta_{2}$).

Figure 10: Schematic overview of the multiple-humped fission barrier in light
actinide isotopes together with the corresponding nuclear shapes. Lower part:
cut through the potential energy surface along the fission path, revealing -
besides the normal deformed (ND) first minimum with the nuclear ground state
(GS) - a superdeformed (SD) second minimum at an axis ratio of 2:1 (with a
fission isomer (FI) as its ground state) and a hyperdeformed (HD) third
minimum at an axis ratio of 3:1. In the upper part the corresponding nuclear
shapes are displayed as a function of the quadrupole and octupole degrees of
freedom. Figure adapted from [233].

Spectroscopic information on the properties of these extremely deformed
nuclear states was obtained for the superdeformed second minimum via direct
decay studies (conversion electron and $\gamma$ spectroscopy) and isomeric
(delayed) fission, while for the third minimum, transmission-resonance
spectroscopy was performed, analyzing resonances in the prompt fission
probability. Figure 11 displays the fission barrier of 236U (left) and, in the
right part, the related prompt fission cross section with transmission
resonances [234].

Figure 11: Left part (adapted from [234]): The triple-humped potential-energy
surface of 236U. Also damped compound nuclear states in the normal deformed
(ND) first, superdeformed (SD) second and hyperdeformed (HD) third potential
minima are shown as hatched areas. For strongly mixed ND and SD states,
transmission resonances of HD states can occur as visible in the prompt
fission probability shown in the right part.

While direct decay spectroscopy is limited to the second minimum (due to the
thin outer barrier of the third well and thus too short decay lifetimes
compared to fission) and to an excitation-energy range up to about 1.5 MeV
above the SD ground state, transmission resonance spectroscopy in the third
minimum probes the energy region of about 1-1.5 MeV below the barrier top.
Thus the intermediate range of excitation energies between 3.5 MeV and 4.5 MeV
(representing the region of the second and third vibrational phonon) remained
so far inaccessible to high-resolution studies that could have provided deeper
insight, for example, into the harmonicity of the nuclear potential at these
extreme deformations. Instead, early photofission studies in actinides
performed with bremsstrahlung photons with an effective energy bandwidth of
about $\Delta E=$200-300 keV revealed the existence of a so-called ‘isomeric
shelf’ in the photofission cross section of 238U in this energy region, as
visible below the sharp bend at ca. 4.5 MeV in Fig. 12a) [235]. This isomeric
shelf is interpreted as the result of a competition of prompt and delayed
(isomeric) fission, following the $\gamma$ decay to the isomeric ground state.
Due to the high selectivity of the ($\gamma$,f) reaction in terms of spin and
parity, only 1- and 2+ states are formed following the absorption of $E1$ and
$E2$ multipolarity, respectively. Therefore, Fig. 12a shows an analysis of the
experimental data in terms of quadrupole and dipole contributions, exhibiting
a dominant 1- component for the isomeric shelf.

Figure 12: a) Photofission yield for 238U as a function of the excitation
energy: experimental data (full symbols) and 2+ and 1- contributions from
model calculations (labelled solid lines) [235]. b) Photofission yield data
from a) (solid line to guide the eye) as accessible with bremsstrahlung
photons of an effective bandwidth $\Delta E\sim$ 300 keV. c) Expected
photofission yield of 238U when using a $\gamma$ beam of $\Delta E/E\sim$
10-6, based on resonances tentatively reported in an early photofission
experiment with limited resolution [236]. Figure adapted from [235].

While the photon bandwidth achievable with bremsstrahlung limits the spectral
resolution as shown for the existing data in Fig. 12b, using narrow-bandwidth
photons of the monochromatized beam from the GF could allow resolving a
resonance structure underlying the isomeric shelf, as illustrated in Fig. 12c.
The resonance positions and strengths are based on tentative findings of an
early photofission experiment with limited resolution [236], that could
conclusively be performed with the envisaged high-resolution photon beam of
the GF.

Candidates for photofission studies at the GF are illustrated in Fig. 13. For
232Th and 238U these plots show an expected triple-humped fission barrier
structure as a function of the quadrupole deformation $\beta_{2}$. For a long
time only a shallow third potential minimum was assumed to exist, until it
could be demonstrated for 234U [237], 236U [234] and 232U [238] that the outer
third minimum is in fact as deep as the second minimum. This also leads to a
new interpretation of early photofission data, where a fine structure in the
6.0 MeV resonance of 232Th was measured by Zhang et al. [239, 240]. In view of
our improved knowledge on the triple-humped barrier we have to conclude that
in these measurements in fact the depth of the third well instead of the
second minimum was determined. For 238U also the potential landscape based on
a triple-humped fission barrier is drawn. For a long time, photofission in
238U was only studied via measurements using bremsstrahlung, where resonances
could not be resolved, and with a weak tagged-photon beam of 100 keV bandwidth
resulting in low-statistics resonances reported by Dickey and Axel [241]. The
early 236U(t,pf) (pf here indicates that a proton is emitted before the
residual nucleus undergoes fission) measurements by Back et al. [242] show
several pronounced transmission resonances between 5 and 6 MeV, while a whole
sequence of further (yet unresolved) transmission resonances at lower energies
is expected to explain the isomeric shelf [235].

Figure 13: Triple-humped potential energy landscape for 238U (top) and 232Th
(bottom) as a function of the quadrupole deformation $\beta_{2}$. The solid
lines show a parameterization in the harmonic model, where potential barriers
and minima are described by joint parabolas. The dashed curves represent the
spin-dependent barriers for dipole (1-) and quadrupole (2+) excitations,
differing at the inner but degenerate at the middle barrier [243]. Expected
transmission resonances in various potential wells are indicated with their
energies, spins and cross sections. For 232Th extrapolated partial lifetimes
for delayed fission and ($\gamma$) back decay are also indicated.

Using a brilliant photon beam for photofission studies on strongly deformed
actinides will provide many advantages compared to the experimental approach
pursued over decades and lead to a renaissance of photonuclear physics in
general and photofission in particular. So far the potential energy surface of
actinides was mapped via particle induced reactions like (d,pf), (d,tf) or
(3He, df), resulting in a statistical population of states [200] in the second
and third potential well. In these cases, the typical population probability
amounts to $10^{-4}-10^{-5}$, equivalent in typical experimental scenarios to
about one isomeric fission event per second. Therefore only strong resonances
with resonance strengths $\sigma\Gamma\sim$ 10 eVb could be studied, in the
vicinity of a strong background from prompt fission. When using a
monochromatic $\gamma$ beam with a spectral density of about
$10^{4}-10^{6}\gamma$/eV/s, the spin selectivity of the photonuclear reaction
together with the narrow energy bandwidth of $<10^{-3}$ would result in a
strongly increased isomeric fission rate of $10^{2}-10^{6}$ isomeric fission
events per second. Moreover, due to the almost complete absence of background
from prompt fission, clean spectra would emerge, granting access also to weak
resonances with $\sigma\Gamma\sim$ 0.1 eVb and widths from about a few 100 eV
to 100 keV.

Using a monochromatized $\gamma$ beam with $\Delta E/E\approx 10^{-6}$ would
provide a $\gamma$-beam linewidth approaching the thermal Doppler broadening
limit of about 5 eV at E${}_{\gamma}=$5 MeV for heavy nuclei of $A\approx$240
at 300 K, thus enabling a new era of photofission studies with the GF:
exploring the fission barrier landscape of actinides would enable tests of
macroscopic-microscopic nuclear models with unprecedented sensitivity,
allowing, for example, to improve the input to astrophysical nucleosynthesis
reaction-network calculations. High-resolution $\gamma$ spectroscopy of
nuclear configurations at large deformations could be used to study Nilsson
orbitals with energies decreasing as a function of nuclear deformation. In the
Nilsson-orbital framework, such studies will provide information on the
structure of (less deformed) superheavy elements. Dynamical aspects of the
fission process could be revisited with much improved sensitivity to angular,
mass and charge distributions. Finally, also applications of the fission
process would benefit from such high-quality photofission studies, for
example, from the understanding of strong $E1$ resonances as doorway states to
fission in order to better understand and optimize the mechanisms of minor
actinide transmutation in nuclear waste treatment (see also Sec.14.2).

Another application of nuclear photofission, considered also for the ELI-NP
facility [244], is measurement of magnetic moments of isomeric states of
neutron-rich nuclei.

###  4.6 Odd harmonics in angular distribution of fission fragments

Odd harmonics in angular distribution of fission fragments appear due to
interference of the fission amplitudes from nuclear states of opposite parity.
They may appear due to mixing of opposite-parity nuclear states by the weak
interaction or, in the case of neutron-induced fission, due to population of
opposite-parity states by capture of neutrons in s- and p-waves. In the case
of photofission, opposite parity states may be populated by the capture of
$E1$ and $M1$ or $E2$ photons. Observation of odd harmonics requires
separation of light and heavy fission fragments ($A_{fragment}<A/2$ and
$A_{fragment}>A/2$).

For many years it was believed that observation of odd harmonics is impossible
because of the large number of final states of the fragments (fragments are
formed in excited internal states) so that any interference effect should
average out, see for example, the classic nuclear physics book by A. Bohr and
B. Mottelson [245]. However, pioneering work by the Danilyan group and others
[246, 247, 248, 249, 107] discovered that the parity violating correlation
${\vec{\sigma}\cdot\vec{p}_{l}}$ is not suppressed in the neutron-induced
fission (here $\vec{\sigma}$ is the neutron spin and $\vec{p}_{l}$ is the
light-fragment momentum), it is actually about $10^{-4}$, i.e. it is enhanced
by three orders of magnitude relative to the ratio $10^{-7}$ of the weak to
strong interactions in nuclei.

The theory of this phenomenon was developed in Refs. [107, 250, 251].
Interference between opposite-parity amplitudes is not suppressed since the
orientation of the strongly deformed nucleus is produced before the separation
of the fragments due to mixing of the doublet of the opposite-parity
rotational states. The enhancement of PV happens at the initial stage due to
mixing of very close opposite-parity states in the spectrum of compound
resonances which are formed after neutron capture. Earlier it had been assumed
that this mixing is “forgotten” during the complicated fission process and
does not appear at the final fission stage. Indeed, it is the case where
temporal description of the fission process is applicable, with different
stages of fission separated in time. However, there is an uncertainty relation
between the energy resolution and time resolution, $\delta E\delta
t\gtrsim\hbar$. To temporally separate different stages of fission, we need a
large energy spread $\delta E$. For the case of nearly monochromatic thermal
neutrons ($\delta E\sim T\ll D$ where $T$ is neutron temperature and $D$ is
the distance between the compound resonances), we have all components of the
nuclear wavefunction present simultaneously, the time dependence is given by a
common factor $\exp(-iEt/\hbar)$. As a result, the mixed-parity compound state
contains also the mixed rotation doublet in the strongly deformed nucleus
before the separation of the fragments. The details of the corresponding
calculations may be found in Refs. [107, 251].

Odd harmonics in fission may also appear due to the interference of $s$\- and
$p$-wave neutron capture resulting in the correlations
${\vec{p}_{n}\cdot\vec{p}_{l}}$ and
${\vec{\sigma}\cdot(\vec{p}_{n}\times\vec{p}_{l})}$, where $\vec{p}_{n}$ is
the neutron momentum [107]. The correlation
${\vec{\sigma}\cdot(\vec{p}_{n}\times\vec{p}_{l})}$ has been observed with the
predicted magnitude.

The GF allows us to investigate both parity violating and parity conserving
odd harmonics in photofission. The corresponding theory is presented in Ref.
[252]. The variable spread of the photon energies will allow investigation of
the transition from small $\delta E$ regime, where odd harmonics are not
suppressed, to the “lost-memory” regime where there is a suppression by a
factor of $\sqrt{\delta E/D}$. The explanation of this suppression factor is
simple: the number of compound states captured in the interval $\delta E$ is
$N\approx\delta E/D$, the amplitudes of the fission from different compound
states have random signs, so the effect is suppressed by $\sqrt{N}$.

###  4.7 Photoinduced processes on the proton and light nuclei

Low-energy Compton scattering off the proton p$(\gamma,\gamma^{\prime})$p and
light nuclei (viz. deuteron or helium), is traditionally used to measure the
nucleon polarizabilities, see the recent reviews [253, 254, 255, 256]. The
present status of the dipole electric ($\alpha_{E1}$) and magnetic
($\beta_{M1}$) polarizabilities of the proton and neutron is shown in Fig. 14.
The GF could improve upon the latest measurements at the operating photon-beam
facilities (HI$\gamma$S and MAMI) given its broader energy range, high energy
resolution and photon flux. Nucleon polarizabilities enter as input to
precision atomic physics, most notably, spectroscopy of muonic atoms [257].
Better precision of the polarizability measurements would reduce the current
uncertainties in atomic calculations [258, 259, 260].

Figure 14: Plot of $\alpha_{E1}$ versus $\beta_{M1}$ for the proton (upper
panel) and neutron (lower panel). The orange band represents the latest Baldin
sum-rule evaluation [261]. ‘PDG’ denotes the latest Particle Data Group
summary. Other references can be found in Ref. [255].

Compton scattering above the pion-production threshold provides better
sensitivity to polarizabilities compared to the case of lower energies. In
addition, a strong correlation between the values of nucleon polarizabilities
and the properties of nucleon excited states, in particular, the ratio of the
electric quadrupole to magnetic dipole $E2/M1$ strengths at the $\Delta(1232)$
resonance has been observed [253]. The delta resonance topic is discussed
further in Sec. 4.10.

The optical theorem, in conjunction with other general principles such as
micro-causality, fully determines the amplitude of forward Compton scattering
in terms of integrals of photoabsorption cross sections [262]. For example,
the spin-independent forward Compton scattering amplitude, $f(\nu)$, as
function of the photon energy $\nu$, is given in terms of total
photoabsorption cross section, $\sigma(\nu)$, as:

$\displaystyle f(\nu)$ $\displaystyle=$
$\displaystyle-\frac{(Ze)^{2}}{M}+\frac{2\nu^{2}}{\pi}\int_{\nu_{\pi}}^{\infty}d\nu^{\prime}\frac{\sigma(\nu^{\prime})}{\nu^{\prime\,2}-\nu^{2}},$
(15)

where the first term is the classical Thomson amplitude characterized by the
ratio of the target charge and mass, and the lower integration limit reads
$\nu_{\pi}=m_{\pi}(1+m_{\pi}/2M)$, with $m_{\pi}$ the pion mass. Expanding
this expression in powers of $\nu/m_{\pi}$ yields a number of useful sum
rules; first of all, the Baldin sum rule for the sum of dipole
polarizabilities [263]:

$\displaystyle\alpha_{E1}+\beta_{M1}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\int_{0}^{\infty}d\nu\frac{\sigma(\nu)}{\nu^{2}}.$
(16)

Our current knowledge of the empirical cross sections, by means of this sum
rule, provides stringent constraints on the nucleon polarizabilities, as shown
by the yellow band in Fig. 14.

Similar considerations for spin-dependent Compton scattering lead to the
Gerasimov-Drell-Hearn (GDH) sum rule [264, 265] which relates the anomalous
magnetic moment $\kappa$ of a particle of spin $S$ and mass $M$ to an energy-
weighted integral over the spin-dependent photoabsorption cross section via,

$I^{\rm
GDH}\equiv\int_{0}^{\infty}\frac{d\nu}{\nu}\left[\sigma^{P}(\nu)-\sigma^{A}(\nu)\right]=4\pi^{2}\kappa^{2}\frac{\alpha}{M^{2}}S,$
(17)

where $\sigma^{P}$ ($\sigma^{A}$) stands for the photoabsorption cross section
with circularly polarized photons and the target polarization parallel
(antiparallel) to the photon momentum, respectively. The GDH sum rule was
shown to hold exactly for the electron, order by order in QED [266, 267]. For
the nucleon, a perturbative QCD calculation is not helpful, but the comparison
has been done based on the measured helicity-dependent cross section
$\sigma^{P}-\sigma^{A}$ [268, 261, 269]. For the proton, the result for the
r.h.s is $I^{\rm GDH}_{p}\simeq 204\,\mu$b (1 b=10-24 cm2), to be compared to
the l.h.s. value $205(21)\,\mu$b based on the empirical cross section.
Likewise, for the neutron $I^{\rm GDH}_{p}=233\,\mu$b to be compared to the
l.h.s. value $225(\dots)(\dots)\,\mu$b.

For the case of the deuteron the empirical verification of the GDH sum rule is
much more delicate. The deuteron anomalous magnetic moment is small,
$\kappa_{d}\simeq-0.143$ (compared to proton $\kappa_{p}\approx 1.79$ and
neutron $\kappa_{n}\approx-1.91$), leading to a tiny sum-rule value, $I^{\rm
GDH}_{d}=0.65\,\mu$b. This small sum rule value implies an almost complete
cancellation of the contributions from the hadronic range $\sim I^{\rm
GDH}_{p}+I^{\rm GDH}_{n}$ vs. the near-threshold photodisintegration $\gamma
d\to pn$. At present, an account for all channels leads to $I^{\rm
r.h.s.}_{d}=27.3\,\mu$b, deviating from the sum-rule expectations. Recent
experimental data at photon energies between the breakup threshold and 10 MeV
from HI$\gamma$S [270], while compatible with the sum rule, feature
significant a uncertainty.

The spin asymmetry of deuteron photodisintegration [271] as well as pion
photoproduction on the deuteron is of great interest not only in view of the
GDH sum rule but also in view of a sensitive test of present day theoretical
models, although we are not aware of any measurements of the beam-target spin
asymmetry in the d($\gamma$,p)n reaction.

A polarized deuteron (spin-1) contains polarized proton and neutron.
Absorption of a circularly-polarized photon is only possible in an
antiparallel helicity configuration, whereas only scattering (a Compton
process suppressed by one power of $\alpha$) is possible in the parallel
configuration. The threshold for photodesintegration of the deuteron is 2.22
MeV, the maximum is reached around 5 MeV. A 100% asymmetry is expected.
Nuclear effects are not expected to play a significant role; in addition, they
are calculable in chiral effective field theories (EFT) [272].

In general, polarization observables offer sensitive tests of the theoretical
understanding [273]. For example, the polarization of the outgoing neutron
Py(n) in d($\gamma$,n)p at low energies shows a discrepancy between the theory
and existing data.

###  4.8 Pion photoproduction

Several main directions in contemporary nuclear science are addressing
problems related to astrophysics and investigating weakly bound nuclei at the
limit of nuclear stability. Such nuclei usually have a neutron halo or other
weakly bound external nucleons [274, 275, 276]. Loosely bound nuclei have been
studied by electroinduced two-nucleon knockout [277]. Several new approaches
can be envisaged for the future GF. One approach is to use photoexcitation
with variable gamma energy at wavelength comparable to the size of the exotic
orbitals. Ideally, one would perform a series of ($\gamma$,NN) experiments on
a sequence of isotopes from stable to exotic that could provide a “photo
image” of the evolution of valence orbitals as a function of the mass number.
Another related image can be derived from pion photoproduction with the
registration of the pion and emitted nucleons on a series of such isotopes.
One can expect that pion production on such nuclei will have a component
similar to this process on a free nucleon.

Investigation of ($\gamma,\pi$) reactions is one of the rare techniques to
probe the wavefunction of loosely bound nucleons, for example by the
difference of pion production from the halo and from normal strongly bound
isotopes. The idea is to check how the $\pi$ production depends on the isospin
asymmetry of the nucleus. The $(\gamma,\pi)$ cross section at several hundred
MeV scales with the atomic weight $A$ approximately as $A^{\alpha}$ dependence
with $\alpha\approx 2/3$ [278], close to what one might expect for an
incoherent surface effect. $\pi$ production on the proton and deuteron is well
studied, and this reaction on a free neutron is known with a minimal model
dependence, providing reference for measurements on other systems. The targets
of interest are the long chains of stable isotopes in Sn, Yb, Sm, and Dy,
Table 4. We note that the isotopic dependence of atomic PV was recently
measured in the Yb isotopic chain [279].

ZX |  Isotopes A: abundance and/or $T_{1/2}$
---|---
24Cr |  48: 21.6 h, 50: 4.34% $\geq 1.3\times 10^{18}$ y, 51: 27.7 d, 52: 83.8%, 53: 9.50%, 54: 2.36%
50Sn |  110: 4.11 h, 112: 0.97% $\leq 1.3\times 10^{21}$ y, 113: 115 d, 114: 0.66%, 115: 0.34%, 116: 14.5%, 117: 7.68%, 118: 24.2%, 119: 8.59%, 120: 32.6%, 121: 27.0 h, 122: 4.63%, 123: 129 d, 124: 5.79% $\geq 1.2\times 10^{21}$ y, 125: 9.64 d, 126: $2.30\times 10^{5}$ y, 127: 2.10 h
62Sm |  142: 72.5 m, 144: 3.07%, 145: 340 d, 146: 10.3$\times 10^{7}$ y, 147: 15.0% $1.06\times 10^{11}$ y, 148: 11.2% $7\times 10^{15}$ y, 149: 13.8%, 150: 7.38%, 151: 90 y, 152: 26.8%, 153: 46.3 h, 154: 22.8%, 156: 9.4 h
66Dy |  152:2.38 h, 153: 6.4 h, 154: $3.0\times 10^{6}$ y, 155: 9.9 h, 156: 0.056%, 157: 8.14 h, 158: 0.095%, 159: 144 d, 160: 2.33%, 161: 18.9%, 162: 25.5%, 163: 24.9%, 164: 28.3%, 165: 2.33 h, 166: 81.6 h
70Yb |  164: 75.8 m, 166: 56.7 h, 168: 0.123%, 169: 32.0 d, 170: 2.98%, 171: 14.1%, 172: 21.7%, 173: 16.1%, 174: 32.0%, 175: 4.18 d, 176: 13.0%, 177: 1.91 h, 178: 74 m
Table 4: Examples of isotope chains. Isotopes in the table have half-lives
longer than 1 hr [55].

####  4.8.1 Photoproduction of bound $\pi^{-}$

Secondary photons at the GF which have highly tunable energy and narrow width
obtained by proper collimation provide a unique opportunity to realize
resonance photoproduction of pionic atoms. It is realized through the reaction
$\gamma+n\rightarrow p+\pi^{-}$ inside a nucleus, i.e.,

$\gamma+{}^{A}_{Z_{i}}\textrm{X}\rightarrow(^{A}_{Z}\textrm{X}^{\prime}+\pi^{-})_{nl},$
(18)

where ${}^{A}_{Z_{i}}$X and ${}^{A}_{Z}\textrm{X}^{\prime}$ are the initial
and final nucleus, respectively (both in their nuclear ground state 141414In
principle, pionic atoms with final nuclei in excited states can also be
resonantly produced with higher photon energies.), $Z_{i}$ and $Z=Z_{i}+1$ are
the corresponding atomic numbers, $A$ is the number of nucleons which is the
same for the initial and final nucleus, and $l$ is the orbital angular
momentum of the bound pion.

Estimates of total widths of bound $ns$ states in pionic atoms, the effective
photoproduction cross section of the initial nucleus, dominant contributions
to photon-attenuation background, maximal photoproduction rate $p_{max}$ of