IsotopeResponses

Data characterizing stable isotopes in Earth, and their response functions for scattering with dark matter. Ancillary data for arXiv: 2312.08478

This repository contains two files:

1. Isotope_Data.dat contains information characterizing the properties and distribution of nuclear isotopes in the bulk of the Earth.

   We include only elements which are either (1) stable or (2) metastable with decay lifetime longer than the age of the Earth.

   There are 281 total entries; the lightest isotope is hydrogen (Z=1, A=1) and the heaviest is lead (Z=82, A=208).

   For each isotope, we include:

   • the element symbol (one or two letters) specifying the identity of the atom (e.g. helium is He, carbon is C).
   • the atomic number Z (e.g. helium has Z=2, carbon has Z=6).
   • the mass number A of the specific isotope (e.g. helium has A=3 or A=4, carbon has A=12 or A=13).
   • the spin (integer or half-integer) and parity (+ or -) of the isotope (e.g. 4He has 0+, 13C has 1/2-).

   • the isotopic fraction (between 0 and 1) of a given element which is found in the specified isotope (e.g. 0.9893 of all carbon in the Earth is in the form of 12C).

     Note that the isotopic fractions, summed over all isotopes for a given element, should equal unity.

     Note that the entries are rounded to four decimal places, so a value of 0 in the table implies an isotopic fraction less than 10^{-5}.

- the **magnetic moment** of each isotope, measured in units of the Bohr magneton _e/(2mp)_ with _mp_ the proton rest mass and _e_ the elementary charge.

    In natural units, _e/(2mp) = 0.16 GeV^{-1}_. In SI units, _e/(2mp) = 5.05078*10^{-27} Joule/Tesla_ .

- the **number density** of each _element_ in Earth's core in units of _km^{-3}_.
- the **number density** of each _element_ in Earth's mantle in units of _km^{-3}_.
- the **number density** of each _element_ in Earth's crust in units of _km^{-3}_.

    The number density of a given isotope can be estimated by multiplying the number density of an element by the relevant isotopic fraction.

1. NuclearResponseData.dat contains the coefficients of the polynomial characterizing dark-matter--nucleus scattering through a variety of non-relativistic operators. These are calcluated using the nuclear shell code BigStick, available online at this link via GitHub; see this paper for information.

- The 20 isotopes included are: 12C, 14N, 24Mg, 25Mg, 27Al, 28Si, 29Si, 30Si, 54Fe, 56Fe, 57Fe, 58Fe, 58Ni, 60Ni, 70Ge, 72Ge, 73Ge, 74Ge, 87Sr, 88Sr. These are labeled by their atomic and mass numbers (_Z,A_).

- The 8 operators included are: M, Σ'' (Sigmapp), Σ' (Sigmap), Φ'' (Phipp), \tilde{Φ}' (Phitildep), Δ (Delta), MΦ'' (MPhipp), Σ'Δ (SigmapDelta).

- For each isotope and each operator, we specify all isospin combinations, (τ,τ')=(00,01,10,11), separately.
 For example, the entry beginning with _[WSigmapp10      12  25         ...]_ is the response function for 25Mg scattering through the Σ'' operator with (τ,τ')=(1,0).

- The remaining entries are the coefficients ηi of the polynomial characterizing the response function, which take the form

    W_A^{ττ'} = e^{-2y}(η0 + η1 * y + η2 * y^2 + η3 * y^3 + η4 * y^4 + η5 * y^5 + η6 * y^6 + η7 * y^7 + η8 * y^8)

    where _y=(b*q/2)^2_ with _q_ the momentum transfer and _b=sqrt[41.467/(45*A^{-1/3} - 25*A^{-2/3})] fm_ the nuclear oscillation parameter. We truncate the polynomial expansion at eighth order.

 Note that due to nuclear physics uncertainty and limitations of the <tt>BigStick</tt> codebase, we expect errors to be sub-percent for light elements, _Z<20_, and approaching _10-50%_ for elements as heavy as iron (and heavier).

Final note: if you become aware of any error in this repository, please inform the owner immediately.