DESY 21-026

IFJPAN-IV-2021-4

JLAB-THY-21-3337

TMDlib2 and TMDplotter:
a platform for 3D hadron structure studies

N.A. Abdulov1, A. Bacchetta2, S. Baranov3, A. Bermudez Martinez4, V. Bertone5,
C. Bissolotti2,6, V. Candelise7, L.I. Estevez Banos4, M. Bury8, P.L.S.
Connor4,***Now at University of Hamburg, L. Favart9, F. Guzman10, F.
Hautmann11,12, M. Hentschinski13, H. Jung4, L. Keersmaekers11, A. Kotikov14,
A. Kusina15, K. Kutak15, A. Lelek11, J. Lidrych4, A. Lipatov1, G. Lykasov14,
M. Malyshev1, M. Mendizabal4, S. Prestel16, S. Sadeghi Barzani17,11, S.
Sapeta15, M. Schmitz4, A. Signori2,18, G. Sorrentino7, S. Taheri Monfared4, A.
van Hameren15, A.M. van Kampen11, M. Vanden Bemden9, A. Vladimirov8, Q.
Wang4,19, H. Yang4,19
1SINP, Moscow State University, Russia

2Dipartimento di Fisica, Università di Pavia and INFN, Italy

3Lebedev Physics Institute, Russia

4DESY, Hamburg, Germany

5IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France

6EIC Center, Jefferson Lab, Newport News, USA

7INFN Sezione di Trieste (a), Università di Trieste (b) , Trieste, Italy

8Institut für Theoretische Physik, Universität Regensburg, Germany

9Inter-University Institute For High Energies, Universite Libre de Bruxelles,
Belgium

10InSTEC, Universidad de La Habana, Cuba

11Elementary Particle Physics, University of Antwerp, Belgium

12RAL and University of Oxford, UK

13Departamento de Actuaria, Fisica y Matemáticas, Universidad de las Américas,
Puebla, Mexico

14JINR, Dubna, Russia

15Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland

16Department of Astronomy and Theoretical Physics, Lund University, Sweden

17Department of Physics, Shahid Beheshti University, Iran

18Theory Center, Jefferson Lab, Newport News, USA

19School of Physics, Peking University, China

###### Abstract

A common library, TMDlib2, for Transverse-Momentum-Dependent distributions
(TMDs) and unintegrated parton distributions (uPDFs) is described, which
allows for easy access of commonly used TMDs and uPDFs, providing a three-
dimensional (3D) picture of the partonic structure of hadrons. The tool
TMDplotter allows for web-based plotting of distributions implemented in
TMDlib2, together with collinear pdfs as available in LHAPDF.

## PROGRAM SUMMARY

Computer for which the program is designed and others on which it is operable:
any with standard C++, tested on Linux and Mac OS systems

Programming Language used: C++

High-speed storage required: No

Separate documentation available:  No

Keywords:  QCD, TMD factorization, high-energy factorization, TMD PDFs, TMD
FFs, unintegrated PDFs, small-$x$ physics.

Other programs used: LHAPDF (version 6) for access to collinear parton
distributions, Root (any version $>$ 5.30) for plotting the results

Download of the program: `http://tmdlib.hepforge.org`

Unusual features of the program: None

Contacts: H. Jung (hannes.jung@desy.de), A. Bermudez Martinez
<EMAIL_ADDRESS>

Citation policy: please cite the current version of the manual and the
paper(s) related to the parameterization(s).

##  1 Introduction

The calculation of processes at high energy hadron colliders is based in
general on the calculation of a partonic process (matrix element) convoluted
with the likelihood to find a parton of specific flavor and momentum fraction
at a given scale within the hadrons. If the parton density depends only on the
longitudinal momentum fraction $x$ of the hadron’s momentum carried by a
parton, and the resolution scale $\mu$, the processes are described by
collinear factorization with the appropriate evolution of the parton densities
(PDFs) given by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)
evolution equations [1, 2, 3]. Such descriptions are successful for
sufficiently inclusive processes, like inclusive deep-inelastic lepton-hadron
scattering (DIS).

In several less inclusive processes, also the transverse momentum of the
interacting partons plays an important role, leading to an extension of the
collinear factorization theorem to include transverse degrees of freedom.
Different factorization theorems for the inclusion of transverse momenta to
the parton densities have been developed in the past, leading to so-called
Transverse Momentum Dependent (TMD) parton densities and unintegrated parton
densities (uPDFs) [4]. These densities provide a 3D imaging of hadron
structure, extending the 1D picture given by PDFs. For semi-inclusive
processes, like semi-inclusive DIS (SIDIS), Drell-Yan (DY) production and
$e^{+}e^{-}$ scattering, TMD factorization has been formulated [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17]. The high-energy (small-$x$ limit)
factorization was formulated for heavy flavor and heavy boson production in
Refs. [18, 19, 20, 21, 22, 23, 24] using unintegrated gluon distributions [25,
26, 27, 28, 29, 30, 31, 32, 33]. In Refs. [34, 35] the Parton Branching (PB)
method was formulated as a way to obtain TMD distributions for all flavours
over a wide range of $x$, transverse momentum $k_{t}$, and scale $\mu$
essentially by solving next-to-leading-order (NLO) DGLAP equations through
Sudakov form factors, separating resolvable and non-resolvable branchings via
the notion of soft-gluon resolution scale [36, 37], and keeping track of the
transverse momenta at each branching.

Since the number of available TMD densities increases very rapidly, and
different groups provide different sets, it was necessary to develop a common
platform to access the different TMD sets in a common form. In 2014 the first
version of TMDlib (version 1) and TMDplotter was released [38, 39], which made
several TMD sets available to the community. This library has set a common
standard for accessing TMD sets, similar to what was available for collinear
parton densities in PDFlib [40, 41] and LHAPDF [42]. TMDlib is a C++ library
which provides a framework and an interface to a collection of different uPDF
and TMD parameterizations.

In this report, we describe a new version of the TMDlib library, TMDlib2, as
well as the associated online plotting tool TMDplotter. TMDlib2 covers all the
features present already in the previous version and contains significant new
developments, such as the treatment of TMD uncertainties and a more efficient
method to include new TMD sets. The report is structured as follows. In Sec.
2, we give the main elements of the library framework. In Sec. 3 we emphasize
the new features of TMDlib2 compared to the previous version. In Sec. 4 we
provide the essential documentation. We summarize in Sec. 5.

##  2 The TMDlib framework

The TMDlib library and its new version TMDlib2 consider momentum weighted TMD
parton distributions $x{\cal A}_{j}(x,k_{t},\mu)$ of flavor $j$ as functions
of the parton’s light cone longitudinal momentum fractions $x$ of the hadron’s
momentum, the parton’s transverse momentum $k_{t}$, and the evolution scale
$\mu$ [4]. Besides, the library also contains integrated TMDs obtained from
the integration over $k_{t}$, as follows

$x{\cal A}_{int}(x,\mu)=\int_{k_{t,min}}^{k_{t,max}}dk_{t}^{2}\;x{\cal
A}(x,k_{t},\mu)\,,$ (1)

In Fig. 1 (left), we show an example of integrated TMD obtained with
TMDplotter for the PB-NLO-HERAI+II-2018-set1 [43], in which the integral
between $k_{t,min}=0.01$ and $k_{t,max}=100$ GeV is compared with the
collinear PDF set HERAPDF2.0 [44]. By construction both sets are identical.
However, in general, Eq.(1) does not converge to the collinear pdf, which is
shown in Fig. 1 (right) comparing the integral between $k_{t,min}=0.01$ and
$k_{t,max}=100$ GeV of PV17 [45] with the corresponding collinear distribution
of MMHT2014 [46]. Several aspects of the relationship between integrated TMDs
and collinear PDFs have been investigated in the literature. The matching
coefficient between the integrated gluon TMD and the collinear gluon PDF in
the MSbar scheme was first computed in the small-$x$ limit in Ref. [22], with
small-$x$ resummation of logarithmic accuracy $(\alpha_{s}\ln x)^{m}$ to all
orders $m$ in $\alpha_{s}$. Perturbative calculations of the matching
coefficients at finite order have recently been carried out through N3LO in
Refs. [47, 48]. Other aspects of the relationship between integrated TMDs and
collinear PDFs are studied e.g. in [19, 49, 50, 51, 25, 26, 52]. We refer the
reader to the overview [4], and references therein, for further discussions of
this topic.

In TMDlib2 the densities are defined more generally as momentum weighted
distributions $x{\cal A}(x,\bar{x},k_{t},\mu)$, where $x,\bar{x}$ are the
(positive and negative) light-cone longitudinal momentum fractions [53, 49,
54, 52]. In some of the applications $\bar{x}$ is set explicitly to zero,
while in other cases $\bar{x}=0$ means that it is implicitly integrated over.

Figure 1: Comparison of up-type parton distributions, $xf(x,\mu)=x{\cal
A}_{int}(x,\mu)$ as a function of $x$ at $\mu=100$ GeV (left): comparison of
the integrated distribution PB-NLO-HERAI+II-2018-set1 [43] with HERAPDF2.0
[44]. (right): comparison of integrated distribution PV17 [45] with MMHT2014
[46].

###  2.1 Grids and Interpolation

Since the analytic calculation of TMDs as a function of the longitudinal
momentum fraction $x$ (we neglect $\bar{x}$ in the following), the transverse
momentum $k_{t}$ and the scale $\mu$ is very time consuming and in some cases
even not available, the TMDs are saved as grids, and TMDlib provides
appropriate tools for interpolation between the grid points (where the type of
evolution is indicated):

allFlavPDF |  Multidimensional Linear Interpolation in $x$, $k_{t}$ and $\mu$ is used for PB and CCFM-type TMDs.
---|---
Pavia |  Interpolation based on Lagrange polynomials of degree three, performed through APFEL++[55, 56].
InterpolationKS |  Multidimensional cubic spline interpolation in $x$, $k_{t}$ and $\mu$, based on GSL implementation, is used for KS-type TMDs (see Tab. 1).

The parameterizations of TMDs in TMDlib are explicitly authorized for each
distribution by the corresponding authors. A list of presently available TMD
sets is given in Tab. 1. No explicit QCD evolution code is included: the
parameterizations are as given in the corresponding references.

The grids of each selected TMD set are read into memory once (the I/O time
depends on the size of the grid). Each TMD set is initialized as a separate
instance of the TMD class, which is created for each different TMD set, for
example for uncertainty sets, or if several different TMD sets are needed for
the calculation. The memory consumption of TMDlib is determined by the size of
the TMDgrids. Optionally, TMDgrids can be loaded separately, avoiding large
memory consumption.

It is the philosophy of TMDlib that the definition of TMD grids is left free,
but a few examples are given: the grids for the PB, CCFM and KS TMD sets are
stored in form of text tables, the grids of the Pavia type TMDs are stored and
read via the `YAML` frame. The method of interpolation and the corresponding
accuracy of the interpolation is left under control of the authors of the
relevant TMD sets.

Figure 2: Transverse momentum distribution $x{\cal A}(x,k_{t},\mu)$ at
$x=0.08$ and $\mu=100$ GeV obtained with PB-NLO-HERAI+II-2018-set2 [43] (left)
and PV17 [45] (right).

###  2.2 Uncertainty TMD sets

The estimation of theoretical uncertainties is an important ingredient for
phenomenological applications, and uncertainties from PDFs and TMDs play a
central role. The uncertainties of TMDs are estimated usually from the
uncertainties of the input parameters or parameterization. There are two
different methods commonly used: the Hessian method [57] which is applied if
the parameter variations are orthogonal or the Monte Carlo method providing
Monte Carlo replicas [58, 59]. The specific prescriptions on how to calculate
the uncertainties for a given TMD set should be found in the original
publication describing the TMDs.

An example of TMDs with uncertainty band is shown in Fig. 2 for the PB set as
well as for the PV17 set. The parameters of intrinsic $k_{t}$\- distribution
are part of the fit of PV17, while they are not fitted for the PB sets (see
discussion in Ref. [43]).

###  2.3 TMDplotter

TMDlib provides also a web-based application for plotting TMD distributions –
TMDplotter, plotting tools for collinear pdfs are available under e.g. [60] or
[61]. In Fig. 3 (left) a comparison of the transverse momentum distributions
of different TMD sets is shown, and in Fig. 3 (right) the gluon-gluon
luminosity calculation for the integrated TMD sets PB-NLO-HERAI+II-2018-set1
[43] at $\mu=100$ GeV compared with the one obtained from HERAPDF2.0 is shown
(the curves obtained from PB-NLO-HERAI+II-2018-set1 and HERAPDF2.0 overlap).

Figure 3: TMDplotter examples: (left) comparison of the transverse momentum
distributions of different TMD sets, (right) gluon-gluon luminosity
calculation using integrated TMD sets (the curves from PB-NLO-
HERAI+II-2018-set1 and HERAPDF2.0 overlap).

TMDplotter is available at http://tmdplotter.desy.de/.

##  3 New features

Having described in Sec. 2 the general framework of the TMDlib library, we
here stress the main new features of TMDlib2 compared with the previous
version [38] of the library. The most important development concerns the
inclusion of many new TMD sets. This is achieved through a new and more
efficient method to add input files. The method is flexible enough that it
will allow new sets, which may become available in the future, to also be
included in a straightforward manner. Another extremely important development
of TMDlib2, which plays an essential role in paving the way to systematic TMD
phenomenology at colliders and fixed target machines, is that the
uncertainties associated with TMD sets are now accessible through the library.
This was not the case in the first version [38]. It is the first time that TMD
uncertainties become available in a library tool. While uncertainties on
collinear PDFs are nowadays available through several different web-based
resources, TMDlib2 is at present the unique tool which contains the full
existing information on uncertainties on TMD sets, and makes it readily
accessible. As such, we expect it to be an essential tool for phenomenological
studies of TMDs and comparisons with experimental data.

To sum up, the main new features of TMDlib2 compared to the earlier version of
the library are as follows.

  * •

TMDlib2 makes use of C++ classes, and the different sets corresponding to
uncertainty sets or sets corresponding to different parameterizations are read
once and initialized as different instances, allowing to load many sets into
memory;

  * •

information about TMD sets is read via YAML from the TMD info files,
containing all metadata;

  * •

including new TMD sets is simplified with the new structure of the input sets;

  * •

the TMD sets are no longer part of the TMDlib distribution, but can be
downloaded via TMDlib-getdata, distributed with TMDlib2.

##  4 TMDlib documentation

TMDlib is written in C++ , with an interface for access from FORTRAN code. The
source code of TMDlib is available from http://tmdlib.hepforge.org/ and can be
installed using the standard autotools sequence configure, make, make install,
with options to specify the installation path and the location of the LHAPDF
PDF library [42], and the ROOT data analysis framework library [62] (which is
used optionally for plotting). If ROOT is not found via root-config, the
plotting option is disabled. After installation, TMDlib-config gives access to
necessary environment variables.

###  4.1 Description of the program components

#### Initialization in C++

TMDinit(name) |  To initialize the dataset specified by its name name. A complete list of datasets available in the current version of TMDlib with the corresponding name is provided in Tab. 1.
---|---
TMDinit(name,irep) |  To initialize a given irep replica of the dataset name.
TMDinit(iset) |  To initialize the dataset specified by its identifier iset.

#### Initialization in Fortran

TMDinit(iset) |  To initialize the dataset specified by its identifier iset.
---|---
TMDset(iset) |  To switch to the dataset iset.

#### Access to TMDs in C++

TMDpdf(x, xbar, kt, mu) |  Vector double-type function returning an array of $13$ variables for QCD parton densities with the values of $x{\cal A}(x,{\bar{x}},k_{t},\mu)$: at index $0,\dots,5$ is $\bar{t},\dots,\bar{d}$, at index $6$ is the gluon, and at index $7,\dots,12$ is $d,\dots,t$ densities.
---|---
TMDpdf(x, xbar, kt, mu, xpq) |  Void-type function filling an array of $13$ variables, xpq, with the values of $x{\cal A}(x,{\bar{x}},k_{t},\mu)$ : at index $0,\dots,5$ is $\bar{t},\dots,\bar{d}$, at index $6$ is the gluon, and at index $7,\dots,12$ is $d,\dots,t$ densities.
TMDpdf(x, xbar, kt, mu, uval, dval, sea, charm, bottom, gluon, photon) |  Void-type function to return $x{\cal A}(x,\bar{x},k_{t},\mu)$ at $x,\bar{x},k_{t},\mu$ for valence u-quarks uval, valence d-quarks dval, light sea-quarks s, charm-quarks c, bottom-quarks b, gluons glu and gauge boson photon.
TMDpdf(x, xbar, kt, mu, up, ubar, down, dbar, strange, sbar, charm, cbar, bottom, bbar, gluon, photon) |  To return $x{\cal A}(x,\bar{x},k_{t},\mu)$ at $x,\bar{x},k_{t},\mu$ for the partons `up`, `ubar`, `down`, `dbar`, `strange`, `sbar`, `charm`, `cbar`, `bottom`, `bbar`, `gluon` and gauge boson `photon` (if available).
TMDpdf(x, xbar, kt, mu, up, ubar, down, dbar, strange, sbar, charm, cbar, bottom, bbar, gluon, photon, Z0, W+,W-,higgs) |  To return $x{\cal A}(x,\bar{x},k_{t},\mu)$ at $x,\bar{x},k_{t},\mu$ for the partons ` up`, `ubar`, `down`, `dbar`, `strange`, `sbar`, `charm`, `cbar`, `bottom`, `bbar`, `gluon`, the gauge bosons `photon`, `Z0`, `W`+, `W-` and `higgs` (if available).

### Access to TMDs in Fortran

TMDpdf(kf,x, xbar, kt, mu, up, ubar, down, dbar, strange, sbar, charm, cbar, bottom, bbar, gluon) |  To return $x{\cal A}(x,\bar{x},k_{t},\mu)$ at $x,\bar{x},k_{t},\mu$ for the partons `up`, `ubar`, `down`, `dbar`, `strange`, `sbar`, `charm`, `cbar`, `bottom`, `bbar`, `gluon` for the hadron flavor `kf`. (`kf` is no longer used, only kept for backward compatibility with TMDlib1)
---|---
TMDpdfEW(x, xbar, kt, mu, up, ubar, down, dbar, strange, sbar, charm, cbar, bottom, bbar, gluon, photon, Z0, W+,W-, higgs) |  To return $x{\cal A}(x,\bar{x},k_{t},\mu)$ at $x,\bar{x},k_{t},\mu$ for the partons ` up`, `ubar`, `down`, `dbar`, `strange`, `sbar`, `charm`, `cbar`, `bottom`, `bbar`, `gluon`, the gauge bosons `photon`, `Z0`, `W`+, `W-` and `higgs` (if available).

#### Callable program components

The program components listed in this section are accessible with the same
name in C++ as well as in Fortran.

TMDinfo(dataset) |  Accesses information from the `info` file.
---|---
TMDgetDesc() |  Returns data set description from `info` file.
TMDgetIndex() |  Returns index number as a string of data set from `info` file.
TMDgetNumMembers() |  Returns number of members of data sets from `info` file.
TMDgetScheme() |  Returns evolution scheme of dataset from `info` file.
TMDgetNf() |  Returns the number of flavours, $N_{f}$, used for the computation of $\Lambda_{QCD}$.
TMDgetOrderAlphaS() |  Returns the perturbative order of $\alpha_{\mathrm{s}}$ used in the evolution of the dataset.
TMDgetOrderPDF() |  Returns the perturbative order of the evolution of the dataset.
TMDgetXmin() |  Returns the minimum value of the momentum fraction $x$ for which the dataset initialized by TMDinit(name) was determined.
TMDgetXmax() |  Returns the maximum value of the momentum fraction $x$ for which the dataset initialized by TMDinit(name) was determined.
TMDgetQmin() (TMDgetQ2min()) |  Returns the minimum value of the energy scale $\mu$ (in GeV), ($\mu^{2}$ (in GeV2)) for dataset.
TMDgetQmax() (TMDgetQ2max()) |  Returns the maximum value of the energy scale $\mu$ (in GeV) , ($\mu^{2}$ (in GeV2)) for dataset.
TMDgetExtrapolation_Q2() |  Returns the method of extrapolation in scale outside the grid definition as specified in `info` file.
TMDgetExtrapolation_kt() |  Returns the method of extrapolation in $k_{t}$ outside the grid definition as specified in `info` file.
TMDgetExtrapolation_x() |  Returns the method of extrapolation in $x$ outside the grid definition as specified in `info` file.
TMDnumberPDF(name) |  Returns the identifier as a value of the associated `name` of the dataset.
TMDstringPDF(index) |  Returns the name associated with `index` of the dataset.

###  4.2 TMDlib calling sequence

In the following simple examples are given to demonstrate how information from
the TMD parton densities can be obtained in C++ and Fortran.

  * •

in C++

              string name ="PB-NLO-HERAI+II-2018-set2";
          double x=0.01, xbar=0, kt=10., mu=100.;
          TMD TMDtest;
          int irep=0;
          TMDtest.TMDinit(name,irep);
          cout << "TMDSet Description: " << TMDtest.TMDgetDesc() << endl;
          cout << "number       = " << TMDtest.TMDnumberPDF(name) << endl;
          TMDtest.TMDpdf(x,xbar,kt,mu, up, ubar, down, dbar, strange, sbar,
                   charm, cbar, bottom, bbar, gluon, photon);

  * •

in Fortran (using multiple replicas of the TMD)

              x = 0.01
          xbar = 0
          kt = 10.
          mu = 100.
          iset = 102200
          call TMDinit(iset)
          write(6,*) ’ iset = ’, iset
          call TMDinit(iset)
          nmem=TMDgetNumMembers()
          write(6,*) ’ Nr of members ’, nmem,’ in Iset = ’, iset
          do i=0,nmem
            isetTMDlib = iset+i
            write(6,*) ’ isetTMDlib = ’, isetTMDlib
            call TMDinit(isetTMDlib)
            call TMDset(isetTMDlib)
            call TMDpdf(kf,x,xbar,kt,mu,up,ubar,dn,dbar,strange,sbar,
        &    charm,cbar,bottom,bbar,glu)
            call TMDpdfew(kf,x,xbar,kt,mu,up,ubar,dn,dbar,strange,sbar,
        &    charm,cbar,bottom,bbar,glu,photon,z0,wplus,wminus,higgs)
          end do

###  4.3 Installation of TMD grids

The TMD grid files are no longer automatically distributed with the code
package, but have to be installed separately. A list of available TMD
parameterizations is given in Tab. 1.

    # get help
    bin/TMDlib-getdata --help

    # install all data sets
    bin/TMDlib-getdata all

    # install a single data (for example: SV19_nnlo)
    bin/TMDlib-getdata SV19_nnlo

###  4.4 Structure of TMD grids

In TMDlib2 the TMDgrids are stored in directories with the name of a given TMD
set which is located in installation_prefix/share/tmdlib/TMDsetName. Every
such directory contains info file and grid file(s), for example for a TMD set
called `test`:

    ~/local/share/tmdlib> ls test
    test.info     test_0000.dat

The `info` file contains general information on the TMDset (inspired by a
similar strategy in LHAPDF), as described below, and the file(s)
`test_0000.dat` contains the TMDgrid. If further replicas are available (for
example for uncertainties), the files are numbered as `test_0000.dat,
test_0001.dat,...`, with the number of files given by `NumMembers` as
described below.

The `info` file must contain all the information to initialize and use the
TMDgrid:

    SetDesc: "Description of the dataset "
    SetIndex: XXXXX
    Authors: XXXX
    Reference: XXXX
    Particle: 2212
    NumMembers: 34
    NumFlavors: 6
    TMDScheme: PB TMD
    Flavors: [-5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 21]
    Extrapolation_x: fixed
    Extrapolation_Q2: fixed
    Extrapolation_kt: fixed
    AlphaS_MZ: 0.118
    AlphaS_OrderQCD: 1
    OrderQCD: 1
    XMin: 9.9e-07
    XMax: 1.
    KtMin: 0.01
    KtMax: 13300.
    QMin: 1.3784
    QMax: 13300
    MZ: 91.1876
    MUp: 0.
    MDown: 0.
    MStrange: 0.
    MCharm: 1.47
    MBottom: 4.5
    MTop: 173

The meaning of most entries is obvious from their name, with `TMDScheme`
different structures for the TMD grids can be selected:

  * PB TMD

used for the PB TMD series

  * PB TMD-EW

used for the PB TMD series including electroweak particles

  * Pavia TMDs

used for the PaviaTMD (or similar TMD) series

iset | uPDF/TMD set | Subsets | Ref.
---|---|---|---
101000 | `ccfm-JS-2001` | 1 | [63]
101010 | `ccfm-setA0` | 4 | [63]
101020 | `ccfm-setB0` | 4 | [63]
101001 | `ccfm-JH-set1` | 1 | [64]
101002 | `ccfm-JH-set2` | 1 | [64]
101003 | `ccfm-JH-set3` | 1 | [64]
101201 | `ccfm-JH-2013-set1` | 13 | [65]
101301 | `ccfm-JH-2013-set2` | 13 | [65]
101401 | `MD-2018` | 1 | [66]
101410 | `KLSZ-2020` | 1 | [67]
102100 | `PB-NLO-HERAI+II-2018-set1` | 35 | [43]
102200 | `PB-NLO-HERAI+II-2018-set2` | 37 | [43]
102139 | `PB-NLO-HERAI+II-2018-set1-q0` | 3 | [43]
102239 | `PB-NLO-HERAI+II-2018-set2-q0` | 3 | [43]
103100 | `PB-NLO+QED-set1-HERAI+II` | 1 | [68]
103200 | `PB-NLO+QED-set2-HERAI+II` | 1 | [68]
10904300 | `PB-NLO_ptoPb208-set1` | 1 | [69]
10904400 | `PB-NLO_ptoPb208-set2` | 1 | [69]
10901300 | `PB-EPPS16nlo_CT14nlo_Pb208-set1` | 1 | [69]
10901400 | `PB-EPPS16nlo_CT14nlo_Pb208-set2` | 1 | [69]
10902300 | `PB-nCTEQ15FullNuc_208_82-set1` | 33 | [69]
10902400 | `PB-nCTEQ15FullNuc_208_82-set2` | 33 | [69]
200001 | `GBWlight` | 1 | [70]
200002 | `GBWcharm` | 1 | [70]
210001 | `Blueml` | 1 | [71]
400001 | `KS-2013-linear` | 1 | [72]
400002 | `KS-2013-non-linear` | 1 | [72]
400003 | `KS-hardscale-linear` | 1 | [73]
400004 | `KS-hardscale-non-linear` | 1 | [73]
400101 | `KS-WeizWill-2017` | 1 | [74]
500001 | `EKMP` | 1 | [75]
410001 | `BHKS` | 1 | [76]
300001 | `SBRS-2013-TMDPDFs` | 1 | [77]
300002 | `SBRS-2013-TMDPDFs-par` | 1 | [77]
601000 | `PV17_grid_pdf` | 201 | [45]
602000 | `PV17_grid_ff_Pim` | 201 | [45]
603000 | `PV17_grid_ff_Pip` | 201 | [45]
604000 | `PV17_grid_FUUT_Pim` | 100 | [45]
605000 | `PV17_grid_FUUT_Pip` | 100 | [45]
606000 | `PV19_grid_pdf` | 216 | [78]
607000 | `PV20_grid_FUTTsin_P_Pim` | 101 | [79]
608000 | `PV20_grid_FUTTsin_P_Pip` | 101 | [79]
701000 | `SV19_nnlo` | 23 | [80]
702000 | `SV19_nnlo_all=0` | 21 | [80]
703000 | `SV19_n3lo` | 23 | [80]
704000 | `SV19_n3lo_all=0` | 21 | [80]
705000 | `SV19_ff_pi_n3lo` | 23 | [80]
706000 | `SV19_ff_pi_n3lo_all=0` | 21 | [80]
707000 | `SV19_ff_K_n3lo` | 23 | [80]
708000 | `SV19_ff_K_n3lo_all=0` | 21 | [80]
709000 | `SV19_pion` | 7 | [81]
710000 | `SV19_pion_all=0` | 7 | [81]
711000 | `BPV20_Sivers` | 25 | [82]
Table 1: Available uPDF/TMD parton sets in TMDlib.

##  5 Summary

The authors of this manual set up a collaboration to develop and maintain
TMDlib and TMDplotter, respectively a C++ library for handling different
parameterizations of uPDFs/ TMDs and a corresponding online plotting tool. The
aim is to update these tools with more uPDF/TMD parton sets and new features,
as they become available and are developed. TMDlib2 improves on the efficiency
of previous versions, allows for simpler C++ interfaces and simplifies the
inclusion of new uPDF/TMD sets.

## Acknowledgments

N. Abdulov was also supported by the RFBR grant 19-32-90096. S. Baranov, A.
Lipatov and M. Malyshev are grateful the DESY Directorate for the support in
the framework of Cooperation Agreement between MSU and DESY on phenomenology
of the LHC processes and TMD parton densities. V. Bertone is supported by the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 824093. C. Bissolotti is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No. 647981, 3DSPIN) and by the U.S. Department of
Energy contract DE-AC05-06OR23177 under which Jefferson Science Associates
operates the Thomas Jefferson National Accelerator Facility. F. Hautmann
thanks DESY, Hamburg and CERN, Theory Group for hospitality and support. M.
Hentschinski gratefully acknowledges support by Consejo Nacional de Ciencia y
Tecnología grant number A1 S-43940 (CONACYT-SEP Ciencias Básicas). A. Lelek
acknowledges funding by Research Foundation-Flanders (FWO) (application
number: 1272421N). M. Malyshev and N. Abdulov were supported by the grant of
the Foundation for the Advancement of Theoretical Physics and Mathematics
(grants 20-1-3-11-1 and 18-1-5-33-1, respectively. A. Signori acknowledges
support from the European Commission through the Marie Skłodowska-Curie Action
SQuHadron (grant agreement ID: 795475). S. Taheri Monfared thanks the Humboldt
Foundation for the Georg Forster research fellowship and gratefully
acknowledges support from IPM. A. van Hameren acknowledges support from the
Polish National Science Centre grant no. 2019/35/ST2/03531. K. Kutak
acknowledges the support by Polish National Science Centre grant no.
DEC-2017/27/B/ST2/01985 Q. Wang and H. Yang acknowledge the support by the
Ministry of Science and Technology under grant No. 2018YFA040390 and by the
National Natural Science Foundation of China under grant No. 11661141008.

## References

  * [1] V. N. Gribov and L. N. Lipatov,  Sov. J. Nucl. Phys. 15, 438 (1972).  [Yad. Fiz.15,781(1972)]
  * [2] G. Altarelli and G. Parisi,  Nucl. Phys. B 126, 298 (1977)
  * [3] Y. L. Dokshitzer,  Sov. Phys. JETP 46, 641 (1977).  [Zh. Eksp. Teor. Fiz.73,1216(1977)]
  * [4] R. Angeles-Martinez et al.,  Acta Phys. Polon. B 46, 2501 (2015).  1507.05267
  * [5] J. C. Collins and D. E. Soper,  Nucl. Phys. B 193, 381 (1981).  [Erratum: Nucl. Phys.B213,545(1983)]
  * [6] J. C. Collins and D. E. Soper,  Nucl. Phys. B 194, 445 (1982)
  * [7] J. C. Collins, D. E. Soper, and G. F. Sterman,  Nucl. Phys. B 223, 381 (1983)
  * [8] J. C. Collins, D. E. Soper, and G. F. Sterman,  Phys. Lett. B 109, 388 (1982)
  * [9] J. C. Collins, D. E. Soper, and G. F. Sterman,  Nucl. Phys. B 250, 199 (1985)
  * [10] J. Collins,  Foundations of perturbative QCD, Vol. 32.  Cambridge monographs on particle physics, nuclear physics and cosmology., 2011
  * [11] R. Meng, F. I. Olness, and D. E. Soper,  Phys. Rev. D 54, 1919 (1996).  hep-ph/9511311
  * [12] P. M. Nadolsky, D. R. Stump, and C. P. Yuan,  Phys. Rev. D 61, 014003 (2000).  [Erratum: Phys.Rev.D 64, 059903 (2001)], hep-ph/9906280
  * [13] P. M. Nadolsky, D. R. Stump, and C. P. Yuan,  Phys. Rev. D 64, 114011 (2001).  hep-ph/0012261
  * [14] X.-D. Ji, J.-P. Ma, and F. Yuan,  Phys. Rev. D 71, 034005 (2005).  hep-ph/0404183
  * [15] X.-D. Ji, J.-P. Ma, and F. Yuan,  Phys. Lett. B 597, 299 (2004).  hep-ph/0405085
  * [16] M. G. Echevarria, A. Idilbi, and I. Scimemi,  JHEP 07, 002 (2012).  1111.4996
  * [17] J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein,  Phys. Rev. Lett. 108, 151601 (2012).  1104.0881
  * [18] S. Catani, M. Ciafaloni, and F. Hautmann,  Phys. Lett. B 242, 97 (1990)
  * [19] S. Catani, M. Ciafaloni, and F. Hautmann,  Nucl. Phys. B 366, 135 (1991)
  * [20] E. M. Levin, M. G. Ryskin, Y. M. Shabelski, and A. G. Shuvaev,  Sov. J. Nucl. Phys. 53, 657 (1991)
  * [21] J. C. Collins and R. K. Ellis,  Nucl. Phys. B 360, 3 (1991)
  * [22] S. Catani, M. Ciafaloni, and F. Hautmann,  Phys. Lett. B307, 147 (1993)
  * [23] S. Catani and F. Hautmann,  Nucl. Phys. B427, 475 (1994).  hep-ph/9405388
  * [24] F. Hautmann,  Phys. Lett. B 535, 159 (2002).  hep-ph/0203140
  * [25] E. Avsar (2012).  1203.1916
  * [26] E. Avsar,  Int. J. Mod. Phys. Conf. Ser. 04, 74 (2011).  1108.1181
  * [27] S. Jadach and M. Skrzypek,  Acta Phys. Polon. B 40, 2071 (2009).  0905.1399
  * [28] F. Dominguez,  Unintegrated Gluon Distributions at Small-x.  Ph.D. Thesis, Columbia U., 2011
  * [29] F. Dominguez, J.-W. Qiu, B.-W. Xiao, and F. Yuan,  Phys. Rev. D 85, 045003 (2012).  1109.6293
  * [30] F. Dominguez, A. Mueller, S. Munier, and B.-W. Xiao,  Phys. Lett. B 705, 106 (2011).  1108.1752
  * [31] F. Hautmann,  Acta Phys.Polon. B 40, 2139 (2009)
  * [32] F. Hautmann, M. Hentschinski, and H. Jung (2012).  1205.6358
  * [33] F. Hautmann and H. Jung,  Nucl. Phys. Proc. Suppl. 184, 64 (2008).  0712.0568
  * [34] F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik,  JHEP 01, 070 (2018).  1708.03279
  * [35] F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik,  Phys. Lett. B 772, 446 (2017).  1704.01757
  * [36] B. R. Webber,  Ann. Rev. Nucl. Part. Sci. 36, 253 (1986)
  * [37] R. K. Ellis, W. J. Stirling, and B. R. Webber,  Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8, 1 (1996)
  * [38] F. Hautmann, H. Jung, M. Krämer, P. Mulders, E. Nocera, et al.,  Eur. Phys. J. C 74, 3220 (2014).  1408.3015
  * [39] P. Connor, H. Jung, F. Hautmann, and J. Scheller,  PoS DIS2016, 039 (2016)
  * [40] H. Plothow-Besch,  Comput. Phys. Commun. 75, 396 (1993)
  * [41] H. Plothow-Besch,  Int. J. Mod. Phys. A 10, 2901 (1995)
  * [42] A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt,  Eur. Phys. J. C 75, 132 (2015).  1412.7420
  * [43] A. Bermudez Martinez, P. Connor, F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik,  Phys. Rev. D 99, 074008 (2019).  1804.11152
  * [44] ZEUS, H1 Collaboration, H. Abramowicz et al.,  Eur. Phys. J. C 75, 580 (2015).  1506.06042
  * [45] A. Bacchetta, F. Delcarro, C. Pisano, M. Radici, and A. Signori,  JHEP 06, 081 (2017).  1703.10157
  * [46] L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. Thorne,  Eur. Phys. J. C 75, 204 (2015).  1412.3989
  * [47] M. A. Ebert, B. Mistlberger, and G. Vita,  JHEP 09, 146 (2020).  2006.05329
  * [48] M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu,  Phys. Rev. Lett. 124, 092001 (2020).  1912.05778
  * [49] G. Watt, A. D. Martin, and M. G. Ryskin,  Eur. Phys. J. C31, 73 (2003).  hep-ph/0306169
  * [50] F. Hautmann,  Phys. Lett. B 655, 26 (2007).  hep-ph/0702196
  * [51] F. Hautmann,  Phys. Lett. B 643, 171 (2006).  hep-ph/0610078
  * [52] J. C. Collins, T. C. Rogers, and A. M. Stasto,  Phys. Rev. D 77, 085009 (2008).  0708.2833
  * [53] G. Watt, A. D. Martin, and M. G. Ryskin,  Phys. Rev. D70, 014012 (2004).  hep-ph/0309096
  * [54] J. Collins and H. Jung (2005).  hep-ph/0508280
  * [55] V. Bertone,  PoS DIS2017, 201 (2018).  1708.00911
  * [56] V. Bertone, S. Carrazza, and J. Rojo,  Comput. Phys. Commun. 185, 1647 (2014).  1310.1394
  * [57] J. Pumplin, D. Stump, J. Huston, H. Lai, P. M. Nadolsky, et al.,  JHEP 0207, 012 (2002).  hep-ph/0201195
  * [58] W. T. Giele and S. Keller,  Phys. Rev. D 58, 094023 (1998).  hep-ph/9803393
  * [59] W. T. Giele, S. A. Keller, and D. A. Kosower (2001).  hep-ph/0104052
  * [60] ApfelWeb.  https://apfel.mi.infn.it/
  * [61] Durham PDFplotter.  http://hepdata.cedar.ac.uk/pdf/pdf3.html
  * [62] R. Brun and F. Rademakers,  Nucl. Instrum. Meth. A 389, 81 (1997)
  * [63] H. Jung (2004).  hep-ph/0411287
  * [64] M. Hansson and H. Jung (2003).  hep-ph/0309009
  * [65] F. Hautmann and H. Jung,  Nuclear Physics B 883, 1 (2014).  1312.7875
  * [66] N. A. Abdulov, H. Jung, A. V. Lipatov, G. I. Lykasov, and M. A. Malyshev,  Phys. Rev. D 98, 054010 (2018).  1806.06739
  * [67] A. V. Kotikov, A. V. Lipatov, B. G. Shaikhatdenov, and P. Zhang,  JHEP 02, 028 (2020).  1911.01445
  * [68] H. Jung, S. T. Monfared, and T. Wening,  Physics Letters B 817, 136299 (2021).  2102.01494
  * [69] E. Blanco, A. van Hameren, H. Jung, A. Kusina, and K. Kutak,  Phys. Rev. D 100, 054023 (2019).  1905.07331
  * [70] K. J. Golec-Biernat and M. Wusthoff,  Phys. Rev. D 59, 014017 (1998).  hep-ph/9807513
  * [71] J. Blumlein,  On the $k_{T}$ dependent gluon density in hadrons and in the photon,  in ’95 QCD and high-energy hadronic interactions. Proceedings, 30th Rencontres de Moriond, Moriond Particle Physics Meetings, Hadronic Session, Le Arcs, France, March 19-25, 1995, pp. 191–197.  1995\.  Also in preprint hep-ph/9506446
  * [72] K. Kutak and S. Sapeta,  Phys. Rev. D 86, 094043 (2012).  1205.5035
  * [73] K. Kutak,  Phys. Rev. D 91, 034021 (2015).  1409.3822
  * [74] P. Kotko, K. Kutak, S. Sapeta, A. M. Stasto, and M. Strikman,  Eur. Phys. J. C 77, 353 (2017).  1702.03063
  * [75] M. G. Echevarria, T. Kasemets, P. J. Mulders, and C. Pisano,  JHEP 07, 158 (2015).  1502.05354
  * [76] M. Bury, A. van Hameren, H. Jung, K. Kutak, S. Sapeta, and M. Serino,  Eur. Phys. J. C 78, 137 (2018).  1712.05932
  * [77] A. Signori, A. Bacchetta, M. Radici, and G. Schnell,  JHEP 1311, 194 (2013).  1309.3507
  * [78] A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, F. Delcarro, F. Piacenza, and M. Radici,  JHEP 07, 117 (2020).  1912.07550
  * [79] A. Bacchetta, F. Delcarro, C. Pisano, and M. Radici (2020).  2004.14278
  * [80] I. Scimemi and A. Vladimirov (2019).  1912.06532
  * [81] A. Vladimirov,  JHEP 10, 090 (2019).  1907.10356
  * [82] M. Bury, A. Prokudin, and A. Vladimirov,  Phys. Rev. Lett. 126, 112002 (2021).  2012.05135.


# Nuclear fission chain reaction in cooling white dwarf stars

C. J. Horowitz<EMAIL_ADDRESS>Center for Exploration of Energy and
Matter and Department of Physics, Indiana University, Bloomington, IN 47405,
USA  M. E. Caplan<EMAIL_ADDRESS>Illinois State University, Department of
Physics, Normal, IL 61790

###### Abstract

The first solids that form as a white dwarf (WD) starts to crystallize are
expected to be greatly enriched in actinides. Previously [PRL 126, 1311010] we
found that these solids might support a nuclear fission chain reaction that
could ignite carbon burning and provide a new Type Ia supernova (SN Ia)
mechanism involving an isolated WD. Here we explore this fission mechanism in
more detail and calculate the final temperature and density after the chain
reaction and discuss a number of open physics questions.

Material at white dwarf (WD) densities is ionized and crystallizes to form a
Coulomb solid where the melting temperature of different chemical elements
scales with atomic number $Z^{5/3}$ Caplan and Horowitz (2017). Phase
separation and crystallization can play important roles during the cooling of
WD stars. The Gaia space observatory has determined parallax distances to
large numbers of galactic stars Gaia Collaboration (2018), which allow for
unprecedented modeling of WD and their evolution. Core crystallization, long
predicted, is now resolved van Horn (1968); Tremblay  _et al._ (2019). Phase
separation or sedimentation of the neutron rich isotope 22Ne, in a C/O WD,
could release significant gravitational energy and delay cooling Caplan  _et
al._ (2020); Bildsten and Hall (2001); Hughto  _et al._ (2010); Blouin  _et
al._ (2020); Camisassa  _et al._ (2020); Cheng  _et al._ (2019); Bauer  _et
al._ (2020).

Pure uranium ($Z=92$) has a melting temperature 95 times higher than the
melting temperature of C. When a WD starts to crystallize, the first solids
may be very strongly enriched in actinides, because these have the highest
$Z$.

In a previous paper Horowitz and Caplan (2021) we found that these first
solids could be so enriched that they support a fission chain reaction.
Fission could possibly ignite carbon burning and produce a thermonuclear
supernova (SN Ia). These stellar explosions are important distance indicators
in cosmology Abbott  _et al._ (2019); Howell (2011); Sullivan (2010). The
exact SN Ia explosion mechanism is poorly understood but is thought to involve
a WD interacting with a binary companion that is itself either a WD or a
conventional star Wang and Han (2012); Hillebrandt  _et al._ (2013); Ruiz-
Lapuente (2014). Presently there is tension between Hubble constant values
determined from SN and in other ways Riess  _et al._ (2021); di valentino  _et
al._ (2021). This tension provides additional motivation to study SN Ia
mechanisms and to try and understand how thermonuclear supernovae explode.

In the present paper we explore further the SN mechanism proposed in ref.
Horowitz and Caplan (2021) where fission in an isolated WD ignites carbon
burning. This mechanism involves the steps shown in Fig. 1. To begin, U must
be produced in an r-process event and be incorporated into a pre-stellar
nebula. During the main sequence the U remains mixed through the star, and
following the giant phases and expulsion of a planetary nebula the resulting
WD retains some of this U. As the WD cools some U may sink and enhance the
core U number abundance. When the core cools to the actinide freezing
temperature the actinides begin to crystallize, effectively forming
precipitates, which may be so enhanced in U that they undergo a fission chain
reaction which triggers a supernova.

Figure 1:  Sequence from uranium production in an r-process nucleosynthesis
event (top) to a fission chain reaction (bottom), see text.

We discuss the composition of the first solids when uranium crystallizes, the
criticality of this composition for a fission chain reaction, the
configuration of the system and the size of the central region, the critical
mass, energy transport during fission including losses from heat conduction,
and the final temperature and density that will be reached. We end by
identifying and discussing a number of possibly open issues with this new SN
mechanism as listed in Table 1.

Table 1:  Selected physics issues discussed in text. Issue | Impacts
---|---
Enrichment of 235U | Criticality
Delay time | Enrichment
Mass of WD | Delay time
Amount of C/O in crystal |  $n$ moderation, spectra
Amount of Pb in crystal | Heat capacity, final $T$
Mass of central region (pit) | Total E released
T dependence of $\sigma_{f}(^{238}$U) | Fission E released
Heat conduction losses | Final $T$ and C ignition
CO vs ONe WD | Carbon ignition

Initial enrichment of 235U during nucleosynthesis: We start with the initial
enrichment $f^{0}_{5}$. This is the fraction of U that is 235U following
nucleosynthesis. Actinides are very likely produced during the r-process where
seed nuclei rapidly capture many neutrons Horowitz  _et al._ (2019); Cowan
_et al._ (1991); Goriely and Arnould (2001); Truran (1998). Recently,
observations of the kilonova associated with the gravitational wave event
GW170817 strongly suggest that neutron star mergers are an important r-process
site. Modern simulations of r-process nucleosynthesis in neutron rich ejecta
during a merger find a ratio of 235U (and mother nuclei that will later decay
into 235U) to 238U (and mother nuclei for 238U) near one. This implies an
initial enrichment $f_{5}^{0}\approx 50\%$. There are nuclear physics
uncertainties for $f_{5}^{0}$ from for example the nuclear mass model used to
describe heavy very neutron rich nuclei. These uncertainties may be relatively
modest because 235U and 238U are very near in mass. Therefore changes in mass
model may change the total U production more than the isotopic ratio.
Furthermore, new radioactive beam accelerators such as the Facility for Rare
Isotope Beams (FRIB) can produce and study many of the neutron rich nuclei
involved in the r-process and FRIB experiments may further reduce nuclear
physics uncertainties.

In addition to nuclear physics uncertainties, there are astrophysical
uncertainties associated with conditions in the r-process site. Many of these
uncertainties are associated with the electron fraction $Y_{e}$ that governs
the ratio of neutrons to protons in the r-process events ejecta. One likely
needs a moderately small $Y_{e}$ in order to significantly produce actinides.
Given this, the dependence of the isotopic ratio on $Y_{e}$ appears to be
small 111Erika Holman private communication. Therefore the astrophysical
uncertainties in $f_{5}^{0}$ may also be modest, see for example Korobkin  _et
al._ (2012). These uncertainties should be studied further in future work. For
now we assume r-process events synthesizes U with $f_{5}^{0}\approx 0.50$.

This fresh r-process material may then mix with older U that has a lower
$f_{5}$ because some of the older U has had time to decay. The solar system
provides an example of how this might work. Radioactive dating suggests the
last event that contributed significant r-process material to the solar system
occurred about 100 Myr before solar system formation Lugaro  _et al._ (2014).
Material from this event and earlier production contributed to the total U
inventory for the solar system. The total enrichment at formation must have
been $f_{5}\approx 0.25$ so that radioactive decay over the 4.6 Gyr lifetime
of the solar system could result in the observed $f_{5}=0.7\%$ today. Thus
$f_{5}^{0}\approx 50\%$ was diluted to $f_{5}\approx 25\%$ by mixing with
material from earlier r-process events.

Delay time: The delay time is the total time for all of the steps shown in
Fig. 1 to occur and is important because U is radioactive. As 235U decays
$f_{5}(t)$ will decrease with time $t$ as,

$f_{5}(t)=\Bigl{[}\bigl{(}\frac{1}{f_{5}^{0}}-1\bigr{)}\exp(\frac{t}{\tau})+1\Bigr{]}^{-1}\,$
(1)

with $\tau\approx 1.20$ Gyr. If $f_{5}^{0}\approx 0.50$ after nucleosynthesis
and one needs $f_{5}\approx 0.12$ or larger for criticality (see below), then
$t$ must be less than 2.4 Gyr or one will have $f_{5}<0.12$ when U
crystallizes so that a chain reaction is no longer possible. If the average
enrichment from multiple r-process events, shortly after the last event, is
only 0.25 then $t$ must be less than 1.1 Gyr in order for $f_{5}>0.12$ at the
time of U crystallization.

Some contributions to the delay time include (1) the time between
nucleosynthesis and star formation, (2) the main sequence lifetime of the
star, and (3) the cooling time of the WD before U crystallization. Both the
main sequence lifetime and the WD cooling time depend strongly on mass and
decrease rapidly for larger masses. The main sequence lifetime of a star of
mass $M_{ms}$ is $\approx 10(M_{\odot}/M_{ms})^{2.5}$ Gyr. The star may then
lose considerable mass before becoming a WD with mass $M_{wd}<M_{ms}$. The
cooling time of the WD depends on the crystallization temperature $T_{c}$
which we characterize with the Coulomb parameter $\Gamma\propto 1/T_{c}$, see
after Eq. 13. A 50/50 mixture of C/O crystallizes near $\Gamma\approx 200$
Horowitz  _et al._ (2010). Previously we found that U is expected to
crystallize out at about twice the C/O temperature Horowitz and Caplan (2021).
Therefore, we calculate the cooling time of the WD for U crystallization
$t_{U}$ as the time until $\Gamma\approx 100$ at the center of the star. We
note that this time is much less than the time until C/O crystallize
$t_{C/O}$. This is because the cooling rate depends on the star’s luminosity
and this decreases significantly as the star cools. Therefore, it takes a WD
more than three times as long to cool to $\Gamma=200$ as it does to
$\Gamma=100$.

We list some of these times in Table 2 and plot them in Fig. 2 for stars of
different masses. We use the initial final mass relation (IFMR) from ref.
Cummings  _et al._ (2018) to relate $M_{wd}$ and $M_{ms}$ and we use the H
poor evolutionary sequences from ref. Camisassa  _et al._ (2017) to determine
cooling times. White dwarfs significantly less massive than 0.84 M⊙ will have
$t_{ms}+t_{U}\gg 860$ Myr and $f_{5}$ could be too small for the system to be
critical. White dwarfs more massive than 0.93 M⊙ will have $t_{ms}+t_{U}<610$
Myr. Therefore, we are primarily interested in massive WD with $M_{wd}$ larger
than $\approx 0.84$ M⊙. Lower mass stars may have too long a time delay for a
fission chain reaction.

Figure 2:  Time delay $t_{d}$ from star formation until U crystallization (solid black line) versus WD mass. The red shaded region shows the main sequence lifetime and the red numbers indicate the original main sequence mass in M⊙. The black shaded region shows the WD cooling time until U crystallization.  Table 2:  Main sequence mass $M_{ms}$ and lifetime $t_{ms}$ and WD mass $M_{wd}$ and WD cooling time for U crystallization $t_{U}$, see text. $M_{ms}$ | $t_{ms}$ | $M_{wd}$ | $t_{U}$ | $t_{ms}+t_{U}$
---|---|---|---|---
M⊙ | Myr | M⊙ | Myr | Myr
2.79 | 770 | 0.705 | 730 | 1500
3.12 | 580 | 0.767 | 530 | 1100
3.50 | 440 | 0.837 | 420 | 860
4.33 | 260 | 0.934 | 350 | 610
5.41 | 150 | 1.050 | $\approx 200$ | $\approx 350$

Progenitor Masses: The upper limit on $M_{wd}$ may be set by electron capture
or by the composition of very massive WD. Above a density of about $9.2\times
10^{7}$ g/cm3 electron capture may remove 235U via 235U$(e,\nu_{e})^{235}$Pa
and likely prevent the actinide crystal from becoming critical. This central
density corresponds to a C/O WD with a mass $\approx 1.15$ M⊙. Interestingly,
above $1.98\times 10^{8}$ g/cm3 Pa may capture another electron to produce
235Th. This nucleus with even $Z$ and odd $N$ may be fissile and could support
a chain reaction. Thus a very massive WD with $M_{wd}\approx 1.25$ M⊙ may
undergo 235Th fission.

However, before these large masses are reached for electron capture, the main
sequence star may have burned some carbon in its core to produce an O/Ne WD
instead of a C/O WD. It may be significantly harder for fission to ignite O
burning instead of C burning, although this should be checked. Therefore, the
upper limit on the WD mass could be set by its composition. White dwarfs more
massive than about 1.05 M⊙ may have O/Ne cores. We conclude that our fission
mechanism may be most straight froward for C/O WD with masses between $\approx
0.85$ M⊙ and $\approx 1.05$ M⊙. These WD masses may correspond to main
sequence masses between 3.5 M⊙ and 5.4 M⊙. For an initial mass function
$\propto M^{-2.3}$ we note that there are about 125% as many stars between 3.5
and 5.4 $M_{\odot}$ compared to stars with masses above 8 M⊙. If a large
fraction of these massive stars produce core collapse supernovae then there
are very likely plenty of lower mass stars to produce SN Ia. Note it could be
that only a fraction of WD in this mass range have $f_{5}$ large enough for
fission to ignite a SN.

Composition of first solids: We are interested in the very first solids to
form as a WD just starts to crystallize. In the dense interior of a WD
electrons form a degenerate Fermi gas. As a result, conventional chemistry
based on closed electron shells is replaced by the chemistry of a Coulomb
plasma where the melting temperature of different elements scales with atomic
number $Z^{5/3}$. As such, we expect them to be actinide-rich because these
elements have the largest $Z$.

Ref. Horowitz and Caplan (2021) predicted the composition of the first solid
to include significant Pb, U and Th and also some C and O. A simple linear
mixing model for the free energies was used. More recently we have performed
molecular dynamics simulations that find two possible solid phases. One
contains a mixture of Pb, U and Th and no C and O and the other contains the
heavy elements (Pb, U and Th) in a one-to-one ratio by number with a mixture
of C and O. We will present these MD simulations in a later publication. For
now we adopt the simpler composition, without any light C and O, as a model
starting point, see Table 3. Although Pb has a significantly smaller $Z=82$
than $Z=92$ for U, we still expect significant Pb in the first solids because
the solar abundance of Pb is $\approx 100$ times larger than that for U. We
emphasize that uncertainties in the model composition of Table 3 should be
explored in future work.

Table 3:  Composition of central pit (abundance by mass), electron fraction $Y_{e}^{pit}$, average charge $\bar{Z}$, and average mass $\bar{A}$. $x_{U}$ | $x_{Th}$ | $x_{Pb}$ | $Y_{e}^{pit}$ | $\bar{Z}$ | $\bar{A}$
---|---|---|---|---|---
0.287 | 0.210 | 0.503 | 0.391 | 86.26 | 220.78

Criticality: Following ref. Horowitz and Caplan (2021) we calculate the 235U
abundance for criticality of an infinite system with the composition in Table
3. We refer to the crystal as the pit (in analogy with a nuclear weapon). The
multiplication factor $k_{\infty}$ is equal to the number of fissions in one
neutron generation over the fissions in the proceeding generation. We write
$k_{\infty}$ as the probability for fission of 235U times the number of
neutrons released per fission $\nu$ over the probability of neutron absorption
on various U and Th isotopes,

$k_{\infty}=\frac{\nu f_{5}\sigma_{f}(^{235}{\rm
U})}{f_{5}\sigma_{a}(^{235}{\rm U})+(1-f_{5})\sigma_{n\gamma}(^{238}{\rm
U})+N_{\rm Th}\sigma_{n\gamma}(^{232}{\rm Th})}\,.$ (2)

This must be $\geq 1$ for a chain reaction. Here $\sigma_{f}(^{235}{\rm U})$
is the fission cross section for 235U and $\sigma_{a}(^{235}{\rm U})$ is the
sum of $\sigma_{f}(^{235}{\rm U})$ and $\sigma_{n,\gamma}$ for the $n,\gamma$
reaction. Likewise $\sigma_{n\gamma}(^{238}$U) and
$\sigma_{n\gamma}(^{232}$Th) are the $n,\gamma$ cross sections for 238U and
232Th. Finally $N_{\rm Th}=238x_{\rm Th}/(232x_{\rm U})\approx 0.75$ is the
number density of Th over the number density of U. For simplicity we evaluate
these cross sections for 1 MeV neutrons using the ENDF data set from Dec. 2011
that is available from the National Nuclear Data Center NND , see Table 4. We
find $k_{\infty}\geq 1$ for $f_{5}\geq 0.12$. Thus the 235U enrichment must be
at least 12% for the system to be critical.

Table 4:  Cross sections and fission neutron multiplicity $\nu$ for a neutron energy of one MeV NND . $\sigma_{f}(^{235}$U) |  $\sigma_{n\gamma}(^{235}$U) |  $\nu(^{235}$U) |  $\sigma_{n\gamma}(^{238}$U) |  $\sigma_{n\gamma}(^{232}$Th)
---|---|---|---|---
1.25 b | 0.105 b | 2.52 | 0.129 b | 0.143 b
Figure 3:  Composition of central region or pit (left) and bulk liquid (right)
vs distance $r$ from center of pit.

Configuration of system and size of central region or pit: We expect a small
seed crystal to grow by diffusion until a background neutron initiates a chain
reaction. The rate of crystal growth and the background neutron rate from
spontaneous fission were discussed in ref. Horowitz and Caplan (2021) where it
was estimated that the crystal will grow to about 5 mg before a chain reaction
is initiated. We consider a C/O WD with a central density of about $2\times
10^{8}$ g/cm3 and a spherical crystal of mass 5 mg and radius
$r_{p}=1.67\times 10^{-4}$ cm. Note the density of this crystal is slightly
higher (by a factor of $0.5/Y_{e}^{pit}$), $2.56\times 10^{8}$ g/cm3, because
$Y_{e}^{pit}$ is less than 0.5. The crystal is surrounded by a bulk liquid
that is assumed to be a 50/50 mixture of C and O. We expect fission, once
initiated, to quickly melt the crystal. This configuration is illustrated in
Fig. 3.

Critical mass: The critical radius $R_{*}$ of the system is the minimum size
necessary for the system to be critical. This is of order the neutron mean
free path $l$. If the system size is much smaller than $l$ too many neutrons
will likely escape preventing a chain reaction. A diffusion calculation
predicts that $R_{*}$, using $k_{\infty}$ from Eq. 2, is approximately Serber
(1992); Reed (2015),

$R_{*}=x\Bigl{(}\frac{l_{t}l_{f}/3}{k_{\infty}-1}\Bigr{)}^{1/2}\,.$ (3)

Here the value $x$ depends on properties of the “tamper” or bulk region in
Fig. 3, see below, and the critical mass is $M_{*}=4\pi R_{*}^{3}\rho/3$. The
mean free path for fission is $l_{f}^{-1}=n_{U}f_{5}\sigma_{f}(^{235}$U) and
the transport mean free path is
$l_{t}^{-1}\approx(n_{U}+n_{Th}+n_{Pb})\sigma_{t}$ where $n_{i}$ are the
number densities of the heavy elements and we assume equal transport cross
sections of $\sigma_{t}\approx 4$ b. If the transport mean free path in the
tamper region of Fig. 3 is $l_{t}^{\rm tam}$, $x$ satisfies Reed (2015),

$x\cot x=1-\frac{l_{t}^{\rm tam}}{l_{t}}\,.$ (4)

The C/O liquid surrounding the crystal provides a very good tamper that
efficiently reflects neutrons back into the pit and significantly reduces the
critical mass. The transport cross section for scattering from C or O is
comparable to that for scattering from the heavy nuclei however their number
density is much higher. As a simple approximation we assume the transport mean
free path scales with one over the ion density and the ion density is the
electron density over the average charge $Z$. Therefore $l_{t}^{\rm
tam}/l_{t}\approx 7/\bar{Z}$ where 7 is the average $Z$ of C/O and $\bar{Z}$
is from Table 3 so that $x\approx 0.5$ from Eq. 4. If $k_{\infty}\approx 1.1$
then $R_{*}\approx 3\times 10^{-6}$ cm and $M_{*}\approx 3\times 10^{-8}$ g
222In ref. Horowitz and Caplan (2021) we did not include the full effect of
the tamper and got a larger $M_{*}$.. We note that $M_{*}$ is much less than
the 5 mg mass of the pit and $R_{*}\ll r_{p}$. Therefore we expect most
neutrons to be absorbed in the pit and very few will escape out the surface.

Fission chain reaction: We now consider a fission chain reaction in the pit
that is initiated by a neutron from spontaneous fission. We expect diffusion
to be slow and viscosity to be small so that hydrodynamic evolution can be
well approximated by a single equation describing energy conservation Timmes
and Woosley (1992),

$\frac{\partial E(T)}{\partial t}+P\frac{\partial(1/\rho_{b})}{\partial
t}=\frac{1}{\rho_{b}}\frac{\partial}{\partial x}\bigl{(}\sigma\frac{\partial
T}{\partial x}\bigr{)}+\frac{\partial S}{\partial t}\,.$ (5)

Here $E(T)$ is the energy per baryon of the system. The temperature is $T$,
$P$ is the pressure, $\rho_{b}$ is the baryon density, and $\sigma$ is the
thermal conductivity. Finally $S$ is the energy per baryon generated from
fission. Below we will argue that the thermal conductivity term is small
compared to $\partial S/\partial t$. If one neglects this term, Eq. 5 can be
integrated over time to yield,

$E(T_{f})-E(T_{i})+P\bigl{(}\frac{1}{{\rho_{b}}_{f}}-\frac{1}{{\rho_{b}}_{i}}\bigr{)}\approx
S\,.$ (6)

Here $T_{f}$ is the final temperature after the fission chain reaction and
$T_{i}\approx 0$ is the initial temperature. Likewise ${\rho_{b}}_{f}$ is the
final and ${\rho_{b}}_{i}$ the initial baryon density and we note that the
system is at constant pressure.

The energy per baryon of a low temperature very relativistic electron gas and
ion system is

$E(T)\approx
Y_{e}^{pit}\bigl{[}\frac{3}{4}\epsilon_{F}^{0}+\frac{\pi^{2}}{2}\frac{(k_{b}T)^{2}}{\epsilon_{F}^{0}}\bigr{]}+\frac{3}{2\bar{A}}k_{b}T+O(k_{b}T)^{4}\,,$
(7)

with Fermi energy $\epsilon_{F}^{0}=(3\pi^{2}n_{e})^{1/3}$ for initial
electron density $n_{e}=Y_{e}^{pit}{\rho_{b}}_{i}$. We neglect the electron
mass in $\epsilon_{F}^{0}$. Finally $\bar{A}$ is the average mass number of
the heavy nuclei. This term in Eq. 7 describes the heat capacity of the ions
and is negligible compared to the first term 333In Ref. Horowitz and Caplan
(2021) we assumed the heat capacity was dominated by ions and neglected the
electron gas contribution. This oversight leads to an over estimation of the
final temperature.. The pressure at low temperatures is,

$P\approx\frac{{\epsilon_{F}}^{4}}{12\pi^{2}}+\frac{{\epsilon_{F}}^{2}(k_{b}T)^{2}}{18}+O(k_{b}T)^{4}\,.$
(8)

Setting $P$ to a constant allows one to solve for $\epsilon_{F}$ and the
density $\rho$ as a function of temperature. The result is

$\rho(T_{f})\approx\frac{\rho(T_{i}\approx
0)}{1+\frac{\pi^{2}}{2}\bigl{(}\frac{k_{b}T_{f}}{\epsilon_{F}^{0}}\bigr{)}^{2}}\,.$
(9)

For typical values $\epsilon_{F}^{0}\approx 2$ MeV and $k_{b}T_{f}\approx 0.5$
MeV one has $\rho(T_{f})\approx 0.75\rho(T_{i}=0)$. This modest 25% decrease
in density is much too small to stop the fission chain reaction by allowing
more neutrons to escape. Given the small critical mass, it would take a
decrease in density by a factor of $\approx 400$ to stop the chain reaction.
This is an important result that implies the fission reaction, once started,
will likely continue to burn until so much U is fissioned that the system is
no longer critical.

Multiple neutron capture: We now discuss the fraction of U and Th nuclei that
may fission. Our system may behave in ways unlike conventional nuclear
reactors or terrestrial atomic bombs. In a reactor there is time for beta
decay so that a reactor can breed 239Pu by neutron capture on 238U. A
conventional atomic bomb disassembles so rapidly that there may be limited
time for heavy nuclei to capture two or more neutrons. In our system 236U,
239U, and 233Th nuclei, formed by neutron capture, can capture additional
neutrons. Each of these nuclei has a significant fission cross section,
although not as large as for 235U, see Table 5. Fission of these nuclei will
contribute further heating and neutrons to the chain reaction.

Table 5:  Fission cross sections for additional neutron capture nuclei (at 1 MeV) NND . 236U |  239U |  233Th
---|---|---
0.36 b | 0.39 b | 0.095 b

Thermal effects: We expect fission cross sections for 238U and 232Th to
significantly increase, as the temperature increases, because of thermal
effects. Zhu and Pei calculate that the spontaneous fission half life of 240Pu
decreases by 12 orders of magnitude as the temperature is increased from 0 to
0.1 MeV Zhu and Pei (2016). We expect significant increases in neutron induced
fission cross sections with similar modest increases in temperature. This
should be verified with further theoretical work. We expect a large fraction
of both U and Th nuclei to eventually fission because of the chance to capture
multiple neutrons and because of the increase in fission cross section with
increasing temperature.

We write the total energy released in fission (per baryon) as,

$S\approx 200{\rm MeV}(\frac{x_{U}}{238}b_{U}+\frac{x_{Th}}{232}b_{Th})\,.$
(10)

Here $b_{U}$ is the fraction of all U nuclei (235U, 239U, etc.) that
eventually fission and $b_{Th}$ is the fraction of Th nuclei that fission.
Using Eqs. 7, 8, 9 in Eq. 6, we have the simple result that the integral of
the heat capacity at constant pressure is equal to the energy released from
fission, $5Y_{e}^{pit}\pi^{2}(k_{b}T_{f})^{2}/(8\epsilon_{F}^{0})=S$. Solving
for $T_{f}$ gives,

$k_{b}T_{f}=\Bigl{(}\frac{8\epsilon_{F}^{0}S}{5\pi^{2}Y_{e}^{pit}}\Bigr{)}^{1/2}\,.$
(11)

This is plotted in Fig. 4. For $\epsilon_{F}^{0}\approx 2$ MeV and the
composition in Table 3, the final temperature is $T_{f}=6.9\times 10^{9}$ K if
$b_{U}=b_{Th}=1$. If only the U fissions we have $T_{f}=5.2\times 10^{9}$ K
for $b_{U}=1$, $b_{Th}=0$.

Figure 4:  Final temperature versus fraction of uranium $b_{U}$ or thorium
$b_{Th}$ that fissions, see Eq. 11. The factor of 0.75 reflects the lower Th
number density.

Heat conduction losses: We neglected the heat conduction term involving
$\partial/\partial x(\sigma\partial T/\partial x)$ compared to $\partial
S/\partial t$ in Eq. 5. We obtain a first estimate of heat conduction by
assuming the temperature varies over a length scale equal to the size of the
system in Fig. 3 so that $\partial T/\partial r$ is of order $T/r_{p}$. The
thermal conductivity $\sigma$ is dominated by conduction from the degenerate
electrons and the electron mean free path is limited by electron ion
scattering. We use the simple form in ref. Yakovlev and Urpin (1980) where,

$\sigma=\frac{\pi^{3}k_{b}^{2}Tn_{e}}{4\alpha^{2}\epsilon_{F}^{2}\bar{Z}\Lambda_{ei}}\,.$
(12)

Here the Coulomb logarithm describing electron ion scattering is

$\Lambda_{ei}\approx\ln\Bigl{[}\bigl{(}\frac{2}{3}\pi\bar{Z}\bigr{)}^{\frac{1}{3}}\bigl{(}1.5+\frac{3}{\Gamma}\bigr{)}^{\frac{1}{2}}\Bigr{]}-\frac{1}{2}\,.$
(13)

The Coulomb parameter for our multicomponent plasma is $\Gamma=\langle
Z^{\frac{5}{3}}\rangle e^{2}/(a_{e}k_{b}T)\approx 30.6$ at $k_{b}T=0.5$ MeV.
Here the electron sphere radius is $a_{e}=[3/(4\pi n_{e})]^{1/3}$. We note
that the thermal conductivity is reduced by the high average charge $\bar{Z}$
of our system, see Table 3. We have at $k_{b}T=0.5$ MeV and $r_{p}=1.7\times
10^{-4}$ cm,

$\frac{1}{\rho_{b}}\frac{\partial}{\partial x}\sigma\frac{\partial T}{\partial
x}\approx\frac{\sigma T}{\rho_{b}r_{p}^{2}}\approx 6.0\times 10^{8}{\ \rm
MeV/s}\,.$ (14)

We compare this to the rate of fission heating that we approximate as

$\frac{dS}{dt}\approx\bigl{(}\frac{200{\rm
MeV}}{238}\bigr{)}\frac{b_{1}}{\tau}\approx 3.5\times 10^{11}{\ \rm MeV/s}\,.$
(15)

Here the neutron fission mean free path over velocity is $\tau=l_{f}/v\approx
2.4\times 10^{-14}$ s and $b_{1}$ is the maximum fraction of U that fissions
in any one neutron generation. The fission rate rises exponentially during the
chain reaction as the number of neutrons increases very rapidly. What is most
important is the comparison of the fission rate to the thermal conductivity
loses when the fission rate is very high and most of the energy is being
released. We expect the fission rate to be very high near the end of the chain
reaction, just as the system starts to run out fuel (nuclei to fission). We
somewhat arbitrarily consider $b_{1}\approx 0.01$. With this choice the
fission energy release in Eq. 15 is 1,000 times larger than the heat
conduction rate in Eq. 14. This justifies our neglect of heat conduction loses
in Eq. 11.

Carbon ignition: Timmes and Woosley have explored the conditions necessary to
ignite C burning via a deflagration Timmes and Woosley (1992). According to
their Fig. 6, a trigger mass of $M^{*}=5$ mg needs to be heated above $\approx
5\times 10^{9}$ K for carbon ignition. If a significant fraction of the U and
or Th fissions then this temperature may be reached. Note that the pit is
small and there is only a small amount of U in the star. Therefore fission
must ignite carbon fusion in order to provide enough energy for a SN. After
carbon ignition, the deflagration could possibly turn into a detonation
Poludnenko  _et al._ (2019). We conclude, it is possible that a fission chain
reaction could ignite a thermonuclear supernova (SN). However there are a
number of possibly open issues and we discuss some of them now.

Importance of the composition of the first solids: We emphasize that the
composition of the first solids is very important. If more Pb is present the
heat capacity would increase and the final temperature decrease. This could
possibly prevent C ignition. If significant C and O are present, they would
act as a moderator and reduce neutron energies. This could prevent the system
from becoming critical or slow down the fission chain reaction and allow more
losses from heat conduction. Finally C, if present in the crystal, could
provide additional fusion energy. This C should be relatively easy to fuse
because of very strong screening from the high $Z$ elements.

Future work should be carried out in many areas. For example, one should study
further the phase diagram of multicomponent plasmas that include very high $Z$
elements to better constrain the composition of the first solids. The
temperature dependence of the neutron induced fission cross section on 238U
should be calculated and one should study the total energy released from a
fission chain reaction. Hydrodynamic simulations should be performed to see if
this fission reaction can ignite carbon burning, and if so will this lead to a
SN Ia. Finally one should study U enrichment in r-process nucleosynthesis and
if this enrichment will be high enough for criticality.

Multiple SN Ia mechanisms and cosmological bias: There may be more than one
way to produce a thermonuclear supernova. For example some SN with shorter
delay times could be due to fission in a sub Chandrasekhar mass WD while SN
with longer delay times could be from more traditional single degenerate or
double degenerate mechanisms. Furthermore, the relative contribution of
different mechanisms could change with metallicity. For example earlier in the
history of a galaxy there may be less “old” 238U from earlier r-process
events. As a result the average U enrichment of a pre-stellar cloud could be
higher and closer to the $\approx 50$% enrichment of the latest r-process
event. The higher enrichment may allow lower mass stars, with longer delay
times, to explode via fission ignition. These lower mass stars will likely
produce SN with less nickel and lower luminosities. If this change in
luminosity is not perfectly corrected by an empirical correlation with for
example a light curve decay time, a cosmological bias could result.

Presently there is tension between Hubble constant values determined from SN
and in other ways Riess  _et al._ (2021); di valentino  _et al._ (2021). It
may be premature to speculate now on the size of a possible cosmological bias
from our fission mechanism. Nevertheless, the Hubble tension provides strong
motivation to explore this and other SN Ia mechanisms and to try and better
understand how thermonuclear supernovae explode.

Conclusions: The first solids that form as a white dwarf (WD) starts to
crystallize are greatly enriched in actinides because of their large charges.
We estimate that these first solids could be so enriched in actinides that
they may support a fission chain reaction. This reaction could ignite carbon
burning and lead to the explosion of an isolated WD in a thermonuclear
supernova.

We thank Erika Holmbeck for sharing preliminary results on uranium synthesis
in the r-process and for many comments on the manuscript. We thank Rolfe
Petschek for discussions on the structure of high $Z$ crystals. We thank
Cameron Reed for pointing out that the C/O liquid is an excellent tamper. We
thank Ezra Booker, Alex Deibel, Witek Nazarewicz, Tomasz Plewa, and Rebecca
Surman for helpful discussions. This research was supported in part by the US
Department of Energy Office of Science Office of Nuclear Physics grants DE-
FG02-87ER40365 and DE-SC0018083 (NUCLEI SCIDAC).

## References

  * Caplan and Horowitz (2017) M. E. Caplan and C. J. Horowitz, Rev. Mod. Phys. 89, 041002 (2017).
  * Gaia Collaboration (2018) Gaia Collaboration, Astronomy & Astrophysics 616, A10 (2018), arXiv:1804.09378 [astro-ph.SR] .
  * van Horn (1968) H. M. van Horn, Astrophys. J. 151, 227 (1968).
  * Tremblay  _et al._ (2019) P.-E. Tremblay, G. Fontaine, N. P. G. Fusillo, B. H. Dunlap, B. T. Gänsicke, M. A. Hollands, J. J. Hermes, T. R. Marsh, E. Cukanovaite, and T. Cunningham, Nature (London) 565, 202 (2019), arXiv:1908.00370 [astro-ph.SR] .
  * Caplan  _et al._ (2020) M. E. Caplan, C. J. Horowitz, and A. Cumming, The Astrophysical Journal 902, L44 (2020).
  * Bildsten and Hall (2001) L. Bildsten and D. M. Hall, The Astrophysical Journal 549, L219 (2001).
  * Hughto  _et al._ (2010) J. Hughto, A. S. Schneider, C. J. Horowitz, and D. K. Berry, Phys. Rev. E 82, 066401 (2010).
  * Blouin  _et al._ (2020) S. Blouin, J. Daligault, D. Saumon, A. Bédard, and P. Brassard, arXiv e-prints , arXiv:2007.13669 (2020), arXiv:2007.13669 [astro-ph.SR] .
  * Camisassa  _et al._ (2020) M. E. Camisassa, L. G. Althaus, S. Torres, A. H. Córsico, S. Cheng, and A. Rebassa-Mansergas, arXiv e-prints , arXiv:2008.03028 (2020), arXiv:2008.03028 [astro-ph.SR] .
  * Cheng  _et al._ (2019) S. Cheng, J. D. Cummings, and B. Ménard, The Astrophysical Journal 886, 100 (2019).
  * Bauer  _et al._ (2020) E. B. Bauer, J. Schwab, L. Bildsten, and S. Cheng, “Multi-gigayear white dwarf cooling delays from clustering-enhanced gravitational sedimentation,” (2020), arXiv:2009.04025 [astro-ph.SR] .
  * Horowitz and Caplan (2021) C. J. Horowitz and M. E. Caplan, Phys. Rev. Lett. 126, 131101 (2021).
  * Abbott  _et al._ (2019) T. M. C. Abbott, S. Allam, P. Andersen, C. Angus, J. Asorey, A. Avelino, S. Avila, B. A. Bassett, K. Bechtol, G. M. Bernstein, E. Bertin, D. Brooks, D. Brout, P. Brown, D. L. Burke, J. Calcino, A. C. Rosell, D. Carollo, M. C. Kind, J. Carretero, R. Casas, F. J. Castander, R. Cawthon, P. Challis, M. Childress, A. Clocchiatti, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, C. Davis, T. M. Davis, J. D. Vicente, D. L. DePoy, S. Desai, H. T. Diehl, P. Doel, A. Drlica-Wagner, T. F. Eifler, A. E. Evrard, E. Fernandez, A. V. Filippenko, D. A. Finley, B. Flaugher, R. J. Foley, P. Fosalba, J. Frieman, L. Galbany, J. García-Bellido, E. Gaztanaga, T. Giannantonio, K. Glazebrook, D. A. Goldstein, S. González-Gaitán, D. Gruen, R. A. Gruendl, J. Gschwend, R. R. Gupta, G. Gutierrez, W. G. Hartley, S. R. Hinton, D. L. Hollowood, K. Honscheid, J. K. Hoormann, B. Hoyle, D. J. James, T. Jeltema, M. W. G. Johnson, M. D. Johnson, E. Kasai, S. Kent, R. Kessler, A. G. Kim, R. P. Kirshner, E. Kovacs, E. Krause, R. Kron, K. Kuehn, S. Kuhlmann, N. Kuropatkin, O. Lahav, J. Lasker, G. F. Lewis, T. S. Li, C. Lidman, M. Lima, H. Lin, E. Macaulay, M. A. G. Maia, K. S. Mandel, M. March, J. Marriner, J. L. Marshall, P. Martini, F. Menanteau, C. J. Miller, R. Miquel, V. Miranda, J. J. Mohr, E. Morganson, D. Muthukrishna, A. Möller, E. Neilsen, R. C. Nichol, B. Nord, P. Nugent, R. L. C. Ogando, A. Palmese, Y.-C. Pan, A. A. Plazas, M. Pursiainen, A. K. Romer, A. Roodman, E. Rozo, E. S. Rykoff, M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, D. Scolnic, S. Serrano, I. Sevilla-Noarbe, R. Sharp, M. Smith, M. Soares-Santos, F. Sobreira, N. E. Sommer, H. Spinka, E. Suchyta, M. Sullivan, E. Swann, G. Tarle, D. Thomas, R. C. Thomas, M. A. Troxel, B. E. Tucker, S. A. Uddin, A. R. Walker, W. Wester, P. Wiseman, R. C. Wolf, B. Yanny, B. Zhang, and Y. Z. and, The Astrophysical Journal 872, L30 (2019).
  * Howell (2011) D. A. Howell, Nature Communications 2, 350 (2011).
  * Sullivan (2010) M. Sullivan, “Type ia supernovae and cosmology,” in  _Lectures on Cosmology: Accelerated Expansion of the Universe_, edited by G. Wolschin (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) pp. 59–97.
  * Wang and Han (2012) B. Wang and Z. Han, New Astronomy Reviews 56, 122 (2012), arXiv:1204.1155 [astro-ph.SR] .
  * Hillebrandt  _et al._ (2013) W. Hillebrandt, M. Kromer, F. K. Röpke, and A. J. Ruiter, “Towards an understanding of type ia supernovae from a synthesis of theory and observations,” (2013), arXiv:1302.6420 [astro-ph.CO] .
  * Ruiz-Lapuente (2014) P. Ruiz-Lapuente, New Astronomy Reviews 62-63, 15 (2014).
  * Riess  _et al._ (2021) A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn, and D. Scolnic, The Astrophysical Journal 908, L6 (2021).
  * di valentino  _et al._ (2021) e. di valentino, o. mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Classical and Quantum Gravity (2021), 10.1088/1361-6382/ac086d.
  * Horowitz  _et al._ (2019) C. J. Horowitz, A. Arcones, B. Côté, I. Dillmann, W. Nazarewicz, I. U. Roederer, H. Schatz, A. Aprahamian, D. Atanasov, A. Bauswein, T. C. Beers, J. Bliss, M. Brodeur, J. A. Clark, A. Frebel, F. Foucart, C. J. Hansen, O. Just, A. Kankainen, G. C. McLaughlin, J. M. Kelly, S. N. Liddick, D. M. Lee, J. Lippuner, D. Martin, J. Mendoza-Temis, B. D. Metzger, M. R. Mumpower, G. Perdikakis, J. Pereira, B. W. O’Shea, R. Reifarth, A. M. Rogers, D. M. Siegel, A. Spyrou, R. Surman, X. Tang, T. Uesaka, and M. Wang, Journal of Physics G: Nuclear and Particle Physics 46, 083001 (2019).
  * Cowan  _et al._ (1991) J. J. Cowan, F.-K. Thielemann, and J. W. Truran, Physics Reports 208, 267 (1991).
  * Goriely and Arnould (2001) S. Goriely and M. Arnould, Astronomy & Astrophysics 379, 1113 (2001), arXiv:astro-ph/0111300 [astro-ph] .
  * Truran (1998) J. W. Truran, Proceedings of the National Academy of Sciences of the United States of America 95, 18 (1998).
  * Note (1) Erika Holman private communication.
  * Korobkin  _et al._ (2012) O. Korobkin, S. Rosswog, A. Arcones, and C. Winteler, Monthly Notices of the Royal Astronomical Society 426, 1940 (2012), https://academic.oup.com/mnras/article-pdf/426/3/1940/3105409/426-3-1940.pdf .
  * Lugaro  _et al._ (2014) M. Lugaro, A. Heger, D. Osrin, S. Goriely, K. Zuber, A. I. Karakas, B. K. Gibson, C. L. Doherty, J. C. Lattanzio, and U. Ott, Science 345, 650 (2014), https://science.sciencemag.org/content/345/6197/650.full.pdf .
  * Horowitz  _et al._ (2010) C. J. Horowitz, A. S. Schneider, and D. K. Berry, Phys. Rev. Lett. 104, 231101 (2010).
  * Cummings  _et al._ (2018) J. D. Cummings, J. S. Kalirai, P.-E. Tremblay, E. Ramirez-Ruiz, and J. Choi, The Astrophysical Journal 866, 21 (2018).
  * Camisassa  _et al._ (2017) M. E. Camisassa, L. G. Althaus, R. D. Rohrmann, E. García-Berro, S. Torres, A. H. Córsico, and F. C. Wachlin, The Astrophysical Journal 839, 11 (2017).
  * (31) National Nuclear Data Center at http://www.nndc.bnl.gov .
  * Serber (1992) R. Serber,  _Los Alamos Primer_ (Univ. of Cal. Press, 1992).
  * Reed (2015) B. C. Reed,  _The Physics of the Manhattan Project_ (Springer, 2015).
  * Note (2) In ref. Horowitz and Caplan (2021) we did not include the full effect of the tamper and got a larger $M_{*}$.
  * Timmes and Woosley (1992) F. X. Timmes and S. E. Woosley, Astrophys. J. 396, 649 (1992).
  * Note (3) In Ref. Horowitz and Caplan (2021) we assumed the heat capacity was dominated by ions and neglected the electron gas contribution. This oversight leads to an over estimation of the final temperature.
  * Zhu and Pei (2016) Y. Zhu and J. C. Pei, Phys. Rev. C 94, 024329 (2016).
  * Yakovlev and Urpin (1980) D. G. Yakovlev and V. A. Urpin, Soviet Astronomy 24, 303 (1980).
  * Poludnenko  _et al._ (2019) A. Y. Poludnenko, J. Chambers, K. Ahmed, V. N. Gamezo, and B. D. Taylor, Science 366, eaau7365 (2019).