hfs.hpp

#pragma once
#include "DiracOperator/TensorOperator.hpp"
#include "IO/InputBlock.hpp"
#include "Physics/NuclearData.hpp"
#include "Potentials/NuclearPotentials.hpp"
#include "Wavefunction/Wavefunction.hpp"
#include "fmt/color.hpp"
#include "qip/Maths.hpp"
#include <functional>
 
namespace DiracOperator {
 
//==============================================================================
//! Auxiliary functions for hyperfine operators; F(r) [nuclear distribution] and similar
//! @details See \ref DiracOperator::hfs for operator
namespace Hyperfine {
 
/*!
  @brief Type for radial function F(r, r_Nuc) -- nuclear magnetisation/charge distribution
  @details
  - F(r,rN) contains only finite nuclear size correction, not HFS radial function.
  - F(r,rN) -> 1 for r > rN
  - r and rN always in atomic units
 
  The full HFS radial operator is t^k(r) = F(r, r_N) / r^{k+1}, formed by
  \ref tk_radial. F(r) itself is defined by the chosen nuclear model;
  see \ref sphericalBall_F, \ref VolotkaSP_F, \ref uSP, \ref doublyOddSP_F,
  \ref generic_F.
*/
using RadialFunction = std::function<double(double r, double r_Nuc)>;
 
//==============================================================================
/*!
  @brief Forms the hyperfine radial operator: F(r, r_N) / r^{k+1}
  @details
  \f[
    t^k_{\rm radial}[r_i] = \frac{F(r_i, r_N)}{r_i^{k+1}}
  \f]
 
  See \ref RadialFunction for the contract on F(r, r_N).
  Angular factors and signs are handled elsewhere.
 
  @param k     Multipole rank (1 = M1, 2 = E2, ...)
  @param rN    Nuclear radius (a.u.); passed to @p hfs_F as r_N
  @param r     Radial grid points (a.u.)
  @param hfs_F Nuclear magnetisation/charge distribution F(r, r_N);
               see \ref RadialFunction and \ref sphericalBall_F
  @return      Radial values of t^k(r)
*/
std::vector<double> tk_radial(int k, double rN, const std::vector<double> &r,
                              const RadialFunction &hfs_F);
 
//==============================================================================
/*!
  @brief Spherical ball model for F(r, r_N) [default]. Uniformly distributed point k-poles.
  @details
  Based on a simple multipole expansion, assuming the nucleus is composed of
  uniformly distributed point k-poles. Note: not everyone defines this the same way.
 
  \f[
    F(r, r_N) =
    \begin{cases}
      (r/r_N)^{2k+1} & r < r_N \\
      1               & r > r_N
    \end{cases}
  \f]
*/
RadialFunction sphericalBall_F(int k);
 
//! Spherical shell F(r, r_N): 0 for r < r_N, 1 for r > r_N
RadialFunction sphericalShell_F();
 
//! Pointlike F(r): F = 1 everywhere
RadialFunction pointlike_F();
 
/*!
  @brief Volotka single-particle nuclear model: F(r, rN)
  @details
  Calculates the Bohr-Weisskopf magnetisation distribution using the Volotka
  single-particle model of Phys. Rev. Lett. 125, 063002 (2020).
  Assumes radial nuclear density is a step function.
  F goes to 1 as r_N -> 0. See \ref RadialFunction for F(r) contract.
 
  @param mu     Nuclear magnetic moment (in nuclear magnetons)
  @param I_nuc  Nuclear spin
  @param l_pn   Orbital angular momentum of valence nucleon
  @param gl     Orbital g-factor (1 = proton, 0 = neutron)
  @param print  Print model details
 
  @return Radial function \f$ F_{\mathrm{BW}}(r, r_N) \f$
*/
RadialFunction VolotkaSP_F(double mu, double I_nuc, double l_pn, int gl,
                           bool print = true);
 
/*!
  @brief Elizarov single-particle magnetisation model [extended Volotka]
  @details
  Returns the radial magnetisation function F(r, r_N) for the model of
  Elizarov, A. A. et al., Opt. Spectrosc. 100, 361 (2006).
  Uses nuclear radial density u(r), with:
 
  \f[
    u(r) =
      \begin{cases}
        u_0 (R-r)^n & \text{u1(r)} \\
        u_0 r^n     & \text{u2(r)}
      \end{cases}
  \f]
 
  n=0 should correspond to the Volotka model (see \ref VolotkaSP_F).
  See \ref RadialFunction for F(r) contract.
 
  @param mu       Nuclear magnetic moment (in nuclear magnetons)
  @param I_nuc    Nuclear spin
  @param l_pn     Orbital angular momentum of valence nucleon
  @param gl       Orbital g-factor (1 = proton, 0 = neutron)
  @param n        Power; usually 0, 1, or 2. 0 => Volotka
  @param R        Nuclear radius
  @param u_option Selects radial form: true for u1(r), false for u2(r)
  @param print    Print model details
 
  @return Radial magnetisation function F(r, r_N)
 
  @warning Normalisation is done by numerical integration; may be unstable.
           Check that result returns to \ref VolotkaSP_F as n -> 0.
*/
RadialFunction uSP(double mu, double I_nuc, double l_pn, int gl, double n,
                   double R, bool u_option, bool print = true);
 
//! Volotka single-particle model for doubly-odd nuclei
/*! @details
  From: [Phys. Rev. Lett. 125, 063002 (2020).](http://arxiv.org/abs/2001.01907)
 
  Total magnetisation:
  \f[
    g F(r) = 0.5 \left[ g_1 F_1(r) + g_2 F_2(r) 
                        + (g_1 F_1(r) - g_2 F_2(r)) K \right]
  \f]
 
  with
 
  \f[
    K = \frac{[ I_1(I_1+1) - I_2(I_2+1) ]}{[ I(I+1) ]}
  \f]
 
  Returns F(r) (i.e., divided by total g).
 
  g2 is obtained from
  g = 0.5 [ g1 + g2 + (g1 - g2) K ].
 
  @note F1, F2 are the Volotka single-particle functions (see \ref VolotkaSP_F).
 
  @param mut  Total nuclear magnetic moment (in nuclear magnetons)
  @param It   Total nuclear spin
  @param mu1  Magnetic moment of nucleon 1
  @param I1   Spin of nucleon 1
  @param l1   Orbital angular momentum of nucleon 1
  @param gl1  Orbital g-factor of nucleon 1 (1 = proton, 0 = neutron)
  @param I2   Spin of nucleon 2
  @param l2   Orbital angular momentum of nucleon 2
  @param print Print model details
 
  @return Radial function F(r)
*/
RadialFunction doublyOddSP_F(double mut, double It, double mu1, double I1,
                             double l1, int gl1, double I2, double l2,
                             bool print = true);
 
/*!
  @brief Constructs F(r) from a numerical density rho(r) given on the radial grid.
  @details
  Computes the cumulative radial integral with rank-k weighting:
 
  \f[
    F(r) = N \int_0^r x^{2k} \rho(x)\, \d x, \qquad
    N = \frac{1}{\int_0^\infty x^{2k} \rho(x)\, \d x}
  \f]
 
  so that \f$ F(r) \to 1 \f$ as \f$ r \to \infty \f$.
 
  The x^{2k} weighting matches the radial scaling of the multipole operator of rank k
  (consistent with \ref sphericalBall_F, where F ~ r^{2k+1} inside the nucleus).
  
  The density @p rho need not be pre-normalised.
  The returned function ignores its second argument (r_Nuc), since the
  distribution is already encoded in @p rho. Assumes @p r is always a
  grid point (no interpolation).
 
  @param grid  Radial grid on which @p rho is defined.
  @param rho   Nuclear density values on the grid points.
  @param k     Multipole rank (1 = M1, 2 = E2, ...)
  @return Radial distribution function F(r, r_Nuc).
*/
RadialFunction generic_F(const Grid &grid, const std::vector<double> &rho,
                         int k);
 
//------------------------------------------------------------------------------
//! Converts reduced matrix element to A/B coeficients (takes k, 2J, 2J)
double convert_RME_to_HFSconstant_2J(int k, int tja, int tjb);
 
//! Converts reduced matrix element to A/B coeficients (takes k, kappa, kappa)
double convert_RME_to_HFSconstant(int k, int ka, int kb);
 
} // namespace Hyperfine
 
//==============================================================================
//==============================================================================
//==============================================================================
 
//! Generalised hyperfine-structure operator, including relevant nuclear moment
/*! @details
 
  Implements the nuclear multipole hyperfine operator of rank @p k using a
  specified finite-nucleus radial model.
 
  By default, includes the nuclear moment, and relevant factors.
  Input parameter @p GQ is the g-factor (k=1) or nuclear moment (k>1), 
  in units of nuclear magnetons * barns^power.
  If 'units' set to 'au', usually should set mu=I=Q=1 then to get raw t^k matrix elements.
 
  That is, it calculates:
 
  \f[
    GQ * t^k
  \f]
 
  with:
 
  \f[
      t^k = 
      \frac{-1}{r^{k+1}}\,F(r)
      \begin{cases}
        C^k & \text{electric (even k)} \\
        \sqrt{\frac{k+1}{k}}
          \mathbf{\alpha}\!\cdot\!\mathbf{C}^{(0)}_k 
          & \text{magnetic (odd k)}
      \end{cases}
  \f]
 
  where F(r) is nuclear k-pole distribution (=1 for pointlike).
 
  Convert to hyperfine constant by taking stretched state, and multiply by:
 
  \f[
    M = 
    \begin{cases}
      1/J      &  k = 1 \\
      2        &  k = 2 \\
      -1       &  k = 3 \\
      1        &  k \geq 4 \\
    \end{cases}
  \f]
 
  This factor (including the steched 3j symbol) is returned by \ref Hyperfine::convert_RME_to_HFSconstant()
 
  Units: 
  - Assumes nuclear moments in units of:
  - magnetic moments: @f$\mu_N\, b^{(k-1)/2}@f$
  - electric moments: @f$b^{k/2}@f$
  - Matrix element are in MHz by default, otherwise in atomic units.
 
  Here \f$ \mu_N \f$ is the nuclear magneton and \f$ b \f$ is the barn.
 
  See, e.g., Xiao _et al._, [Phys. Rev. A 102, 022810 (2020).](http://arxiv.org/abs/2007.06798)
 
  The radial part is constructed from \ref Hyperfine::tk_radial().
 
  @note See @ref DiracOperator::Hyperfine for details
*/
class hfs final : public TensorOperator {
  using RadialFunction = std::function<double(double, double)>;
 
public:
  //! @brief Constructs hyperfine operator of multipolarity k.
  /*! @details
    Initialises the hyperfine interaction operator.
 
    @param in_k     Tensor rank k (multipolarity) of the hyperfine interaction 
                    (e.g., 1,2,3,... for M1, E2, M3,... etc.)
    @param GQ       Nuclear g-factor for k=1, general moment for k>1
    @param rN_au    Nuclear radius (a.u.), used for finite-size magnetisation models.
    @param rgrid    Radial grid used to define the radial functions.
    @param hfs_F    Radial magnetisation distribution function
                    (see @ref DiracOperator::Hyperfine).
    @param MHzQ     If true outputs in MHz units (assuming input g etc. in muN.barns); otherwise in raw atomic units (pure t^k matrix elements).
 
    @see See also: \ref DiracOperator::Hyperfine
  */
  hfs(int in_k, double GQ, double rN_au, const Grid &rgrid,
      const RadialFunction &hfs_F = Hyperfine::pointlike_F(), bool MHzQ = true)
    : TensorOperator(in_k, Parity::even, GQ,
                     Hyperfine::tk_radial(in_k, rN_au, rgrid.r(), hfs_F)),
      k(in_k),
      magnetic(k % 2 != 0),
      cfg(magnetic ? 1.0 : 0.0),
      cff(magnetic ? 0.0 : 1.0),
      mMHzQ(MHzQ) {
 
    const auto power = magnetic ? (k - 1) / 2 : k / 2;
 
    // Assumes nuclear moment in muN*b^(k-1)/2 for magnetic,
    // and b^k/2 for electric
    const auto unit_au =
      magnetic ? PhysConst::muN_CGS * std::pow(PhysConst::barn_au, power) :
                 std::pow(PhysConst::barn_au, power);
    m_unit = mMHzQ ? unit_au * PhysConst::Hartree_MHz : 1.0;
  }
 
  std::string name() const override final {
    return "hfs" + std::to_string(k) + "";
  }
  std::string units() const override final { return mMHzQ ? "MHz" : "au"; }
 
  double angularF(const int ka, const int kb) const override final {
    return magnetic ? -double(ka + kb) / double(k) *
                        Angular::Ck_kk(k, ka, -kb) * m_unit :
                      -Angular::Ck_kk(k, ka, kb) * m_unit;
  }
 
  double angularCff(int, int) const override final { return cff; }
  double angularCgg(int, int) const override final { return cff; }
  double angularCfg(int, int) const override final { return cfg; }
  double angularCgf(int, int) const override final { return cfg; }
 
  static std::unique_ptr<TensorOperator> generate(const IO::InputBlock &input,
                                                  const Wavefunction &wf) {
 
    input.check(
      {{"", "Most following will be taken from the default nucleus if "
            "not explicitely given"},
       {"mu", "Magnetic moment in mu_N"},
       {"Q", "Nuclear quadrupole moment, in barns. Also used as overall "
             "constant for any higher-order moments [1.0]"},
       {"k", "Multipolarity. 1=mag. dipole, 2=elec. quad, etc. [1]"},
       {"rrms",
        "nuclear (magnetic) rms radius, in Fermi (fm) (defult is charge rms)"},
       {"units",
        "Units for output (only for k=1,k=2). MHz or au. For MHz, "
        "assumes nuclear moments g/Q are in mu_N*barns^p. For atomic "
        "units, best to set g=Q=1 to get raw t^k matrix elements [MHz]"},
       {"F", "F(r): Nuclear moment distribution: ball, point, shell, "
             "SingleParticle, doublyOddSP, uSP, Fermi, or Gaussian [ball]"},
       {"", "The following options are all for F(r) details. See "
            "https://ampsci.dev for full details (search `hyperfine`)"},
       {"printF", "Writes F(r) to a text file [false]"},
       {"print", "Write F(r) info to screen [true]"},
       {"", "The following are only for F=SingleParticle or doublyOddSP"},
       {"I", "Nuclear spin. Taken from nucleus"},
       {"parity", "Nulcear parity: +/-1"},
       {"l", "l for unpaired nucleon (automatically derived from I and "
             "parity; best to leave as default)"},
       {"gl", "=1 for proton, =0 for neutron"},
       {"", "The following are only used if F=doublyOddSP"},
       {"mu1", "mag moment of 'first' unpaired nucleon"},
       {"gl1", "gl of 'first' unpaired nucleon"},
       {"l1", "l of 'first' unpaired nucleon"},
       {"l2", "l of 'second' unpaired nucleon"},
       {"I1", "total spin (J) of 'first' unpaired nucleon"},
       {"I2", "total spin (J) of 'second' unpaired nucleon"},
       {"", "The following are only for u(r) function (F=uSP)"},
       {"u", "u1 or u2 : u1(r) = (R-r)^n, u2(r) = r^n [u1]"},
       {"n", "n that appears above. Should be between 0 and 2 [0]"},
       {"", "The following are only for Fermi function (F=Fermi)"},
       {"t",
        "skin thickness (normally 2.3). note: c taken from wavefunction, or "
        "from rrms above [2.3]"}});
    if (input.has_option("help")) {
      return nullptr;
    }
 
    const auto nuc = wf.nucleus();
    const auto isotope = Nuclear::findIsotopeData(nuc.z(), nuc.a());
 
    auto mu = input.get("mu", isotope.mu ? *isotope.mu : 1.0);
    auto I_nuc = input.get("I", isotope.I_N ? *isotope.I_N : 1.0);
    const auto print = input.get("print", true);
    const auto k = input.get("k", 1);
 
    const auto use_MHz =
      qip::ci_compare(input.get<std::string>("units", "MHz"), "MHz");
 
    if (k <= 0) {
      fmt2::styled_print(fg(fmt::color::red), "\nError 246:\n");
      std::cout << "In hyperfine: invalid K=" << k << "! meaningless results\n";
    }
    if (I_nuc <= 0 && k == 1) {
      fmt2::styled_print(fg(fmt::color::orange), "\nWarning 253:\n");
      std::cout << "In hyperfine: invalid I_nuc=" << I_nuc
                << "! meaningless results\nSetting I=1\n";
      I_nuc = 1;
    }
    if (mu == 0.0 && k == 1) {
      fmt2::styled_print(fg(fmt::color::orange), "\nWarning 352:\n");
      std::cout << "Setting mu=1\n";
      mu = 1;
    }
 
    const auto g_or_Q =
      (k == 1) ? (mu / I_nuc) :
      (k == 2) ? input.get("Q", isotope.q ? *isotope.q : 1.0) :
                 input.get("Q", 1.0);
 
    enum class DistroType {
      point,
      ball,
      shell,
      SingleParticle,
      doublyOddSP,
      uSP,
      Fermi,
      Gaussian,
      Error
    };
 
    const std::string default_distribution = "ball";
 
    // For compatability with old notation of 'F(r)' input option
    const auto Fr_str =
      input.has_option("F") ?
        input.get<std::string>("F", default_distribution) :
      input.has_option("nuc_mag") ?
        input.get<std::string>("nuc_mag", default_distribution) :
        input.get<std::string>("F(r)", default_distribution);
 
    const auto distro_type =
      (qip::ci_wc_compare(Fr_str, "point*") || qip::ci_compare(Fr_str, "1")) ?
        DistroType::point :
      qip::ci_compare(Fr_str, "ball")          ? DistroType::ball :
      qip::ci_compare(Fr_str, "shell")         ? DistroType::shell :
      qip::ci_wc_compare(Fr_str, "Single*")    ? DistroType::SingleParticle :
      qip::ci_wc_compare(Fr_str, "doublyOdd*") ? DistroType::doublyOddSP :
      (qip::ci_compare(Fr_str, "spu") || qip::ci_compare(Fr_str, "uSP")) ?
                                                 DistroType::uSP :
      qip::ci_wc_compare(Fr_str, "Fermi*") ? DistroType::Fermi :
      qip::ci_wc_compare(Fr_str, "Gauss*") ? DistroType::Gaussian :
                                             DistroType::Error;
    if (distro_type == DistroType::Error) {
      fmt2::styled_print(fg(fmt::color::red), "\nError 271:\n");
      std::cout << "\nIn hyperfine. Unkown F(r) - " << Fr_str << "\n";
      std::cout << "Defaulting to pointlike!\n";
    }
 
    const auto r_rmsfm =
      distro_type == DistroType::point ? 0.0 : input.get("rrms", nuc.r_rms());
    const auto r_nucfm = std::sqrt(5.0 / 3) * r_rmsfm;
    const auto r_nucau = r_nucfm / PhysConst::aB_fm;
 
    if (print) {
      std::cout << "\nHyperfine structure: " << wf.atom() << "\n";
      std::cout << "K=" << k << " ("
                << (k == 1     ? "magnetic dipole" :
                    k == 2     ? "electric quadrupole" :
                    k % 2 == 0 ? "electric multipole" :
                                 "magnetic multipole")
                << ")\n";
      std::cout << "Using " << Fr_str << " nuclear distro for F(r)\n"
                << "w/ r_N = " << r_nucfm << "fm = " << r_nucau
                << "au  (r_rms=" << r_rmsfm << "fm)\n";
      std::cout << "Points inside nucleus: " << wf.grid().getIndex(r_nucau)
                << "\n";
      if (k == 1) {
        std::cout << "mu = " << mu << ", I = " << I_nuc << ", g = " << g_or_Q
                  << "\n";
      } else {
        std::cout << "Q = " << g_or_Q << "\n";
      }
    }
 
    // default is BALL:
    auto Fr = Hyperfine::sphericalBall_F(k); // should never be used
 
    if (distro_type == DistroType::ball) {
      Fr = Hyperfine::sphericalBall_F(k);
 
    } else if (distro_type == DistroType::shell) {
      Fr = Hyperfine::sphericalShell_F();
 
    } else if (distro_type == DistroType::SingleParticle) {
      const auto pi = input.get("parity", isotope.parity ? *isotope.parity : 0);
      const auto l_tmp = int(I_nuc + 0.5 + 0.0001);
      auto l = ((l_tmp % 2 == 0) == (pi == 1)) ? l_tmp : l_tmp - 1;
      l = input.get("l", l); // can override derived 'l' (not recommended)
      const auto gl_default = wf.Znuc() % 2 == 0 ? 0 : 1; // unparied proton?
      const auto gl = input.get<int>("gl", gl_default);
      if (print) {
        std::cout << "Single-Particle (Volotka formula) for unpaired";
        if (gl == 1)
          std::cout << " proton ";
        else if (gl == 0)
          std::cout << " neturon ";
        else
          std::cout << " gl=" << gl
                    << "??? program will run, but prob wrong!\n";
        std::cout << "with l=" << l << " (pi=" << pi << ")\n";
      }
      Fr = Hyperfine::VolotkaSP_F(mu, I_nuc, l, gl, print);
 
    } else if (distro_type == DistroType::uSP) {
      const auto pi = input.get("parity", isotope.parity ? *isotope.parity : 0);
      const auto u_func =
        input.get("u", std::string{"u1"}); // u1=(R-r)^n, u2=r^n
      const bool u_option = u_func == std::string{"u1"};
      const auto n = input.get("n", 1.0); // u1=(R-r)^n, u2=r^n
      const auto l_tmp = int(I_nuc + 0.5 + 0.0001);
      auto l = ((l_tmp % 2 == 0) == (pi == 1)) ? l_tmp : l_tmp - 1;
      l = input.get("l", l); // can override derived 'l' (not recommended)
      const auto gl_default = wf.Znuc() % 2 == 0 ? 0 : 1; // unparied proton?
      const auto gl = input.get<int>("gl", gl_default);
      if (print) {
        std::cout << "Single-Particle (Volotka formula) with u(r) for unpaired";
        if (gl == 1)
          std::cout << " proton ";
        else if (gl == 0)
          std::cout << " neturon ";
        else
          std::cout << " gl=" << gl
                    << "??? program will run, but prob wrong!\n";
        std::cout << "with l=" << l << " (pi=" << pi << ")\n";
      }
      if (print) {
        std::cout << "u(r) = " << u_func << ": "
                  << (u_option ? "(R-r)^n" : "r^n") << ", n=" << n << "\n";
      }
      Fr = Hyperfine::uSP(mu, I_nuc, l, gl, n, r_nucau, u_option, print);
 
    } else if (distro_type == DistroType::doublyOddSP) {
      const auto mu1 = input.get<double>("mu1", 1.0);
      const auto gl1 = input.get<int>("gl1", -1); // 1 or 0 (p or n)
      if (gl1 != 0 && gl1 != 1) {
        fmt2::styled_print(fg(fmt::color::red), "\nError 324:\n");
        std::cout << "In " << input.name() << " " << Fr_str
                  << "; have gl1=" << gl1 << " but need 1 or 0\n";
        return std::make_unique<NullOperator>(NullOperator());
      }
      const auto l1 = input.get<double>("l1", -1.0);
      const auto l2 = input.get<double>("l2", -1.0);
      const auto I1 = input.get<double>("I1", -1.0);
      const auto I2 = input.get<double>("I2", -1.0);
 
      Fr = Hyperfine::doublyOddSP_F(mu, I_nuc, mu1, I1, l1, gl1, I2, l2, print);
 
    } else if (distro_type == DistroType::Fermi) {
      const auto t_fm = input.get("t", nuc.t());
      const auto c_fm = Nuclear::c_hdr_formula_rrms_t(r_rmsfm, t_fm);
      if (print)
        std::cout << "Fermi distro: c=" << c_fm << " fm, t=" << t_fm << " fm\n";
      const auto rho =
        Nuclear::fermiNuclearDensity_tcN(t_fm, c_fm, 1.0, wf.grid());
      Fr = Hyperfine::generic_F(wf.grid(), rho, k);
 
    } else if (distro_type == DistroType::Gaussian) {
      const auto rN_au = r_rmsfm / PhysConst::aB_fm;
      if (print)
        std::cout << "Gaussian distro: r_rms=" << r_rmsfm << " fm\n";
      std::vector<double> rho;
      rho.reserve(wf.grid().num_points());
      for (const auto ri : wf.grid().r()) {
        rho.push_back(std::exp(-1.5 * ri * ri / (rN_au * rN_au)));
      }
      Fr = Hyperfine::generic_F(wf.grid(), rho, k);
    } else if (distro_type == DistroType::Error) {
      std::cout << "Using pointlike F(r):\n";
      Fr = Hyperfine::pointlike_F();
    }
 
    // Optionally print F(r) function to file
    if (input.get<bool>("printF", false)) {
      std::ofstream of(wf.identity() + "_" + Fr_str + ".txt");
      of << "r/fm  F(r)\n";
      for (auto r : wf.grid()) {
        of << r * PhysConst::aB_fm << " "
           << Fr(r * PhysConst::aB_fm, r_nucau * PhysConst::aB_fm) << "\n";
      }
    }
 
    return std::make_unique<hfs>(k, g_or_Q, r_nucau, wf.grid(), Fr, use_MHz);
  }
 
private:
  int k;
  bool magnetic;
  double cfg;
  double cff;
  bool mMHzQ;
  double m_unit{1.0};
};
 
} // namespace DiracOperator