CollisionDB: DataSet D120167

\[\mathrm{He}^{2+} + \mathrm{He} \rightarrow \mathrm{He}^{+} + \mathrm{He}^{+}\]

Process	HCX: Charge Transfer
Data type	cross section \| uploaded on 2023-02-09
Comment	Data entry from page A-98 of the ORNL Redbook on the atomic and molecular collisional data for fusion.
From ALADDIN	Yes

Method	semi-empirical
Frame	Target
Columns	E /eV u^-1 sigma /cm²
Uncertainty	25 %
Ref	B173: C. F. Barnett, "Collisions of H, H₂, He and Li Atoms and Ions with Atoms and Molecules", Oak Ridge National Laboratory Report ORNL-6086, Atomic Data for Fusion 1 (1990). [https://amdis.iaea.org/media/db/ORNL/ORNL-Red_Book_-_1.pdf]
Data	Download (data from fit)

Fitted Data

Fit Function
Details

\[\begin{align*} \text{pfit} (pet) &= \exp\left(\frac{1}{2} \text{pcf}(1) + \sum_{i=1}^{8} \text{pcf}(i+1) \cdot T_i(X)\right) \\ \text{where} \quad X &= \frac{\Big(\ln(pet) - \ln(\text{pcf}(10))\Big) - \Big(\ln(\text{pcf}(11)) - \ln(pet)\Big)}{{\ln(\text{pcf}(11)) - \ln(\text{pcf}(10))}}\\ T_0(X) &=1; \; T_1(X) = X; T_{i+1} = 2XT_i(X) - T_{i-1}(X) \\ \end{align*}\]

Python

def cheb(pet, pcf):
    """
    This function calculates cross sections in cm^2 versus energy in eV/amu
    or rate coefficients in cm^3/s versus electron temperature in eV
    using Chebyshev polynomial fitting coefficients.

    pe: collision energy in eV/amu or electron temperature in eV
    pcf: parameter data array
        pcf[0:9]: parameters for fit to the cross section
        pcf[9]: emin
        pcf[10]: emax, the fit is valid between the limits emin and emax

    """
    emin = pcf[9]
    emax = pcf[10]
    if not (emin <= pet <= emax):
        raise ValueError('Energy outside range of validity of fit in alcheb')

    k = 9
    cheb = pcf[k]
    eminl = np.log(emin)
    emaxl = np.log(emax)
    enl = np.log(pet)
    k -= 1
    xnorm = (enl - eminl - (emaxl - enl)) / (emaxl - eminl)
    twox = 2.0 * xnorm
    prev2 = 0.0
    while k != 0:
        prev = cheb
        cheb = pcf[k] + twox * prev - prev2
        prev2 = prev
        k -= 1
    cheb = 0.5 * pcf[0] + xnorm * prev - prev2
    pfit = np.exp(cheb)
    
    return pfit

Fortran

c
c######################################################################
c
      subroutine alcheb(pet, pcf, kncf, pfit, kermsg)
c
c     this is an ornl:cfadc subroutine to calculate cross sections in
c     (cm[2]) versus energy in (ev/amu) or rate coefficients in
c     (cm[3]/s) versus maxwellian temperature in (ev) from chebyshev
c     polynomial fitting coefficients
c
c     these fits are valid only between the limits emin and emax,
c     which are coefficients pcf(10) and pcf(11) in the entry data field
c
c     pet = collision energy in ev/amu or maxwellian temperature in ev
c
c     kermsg = blank if no errors
c
c     pfit = cross section in cm[2] or rate coefficient in cm[3]/s
c
c     written by h. hunter, cfadc oak ridge national laboratory
c     (modified to aladdin calling structure 4/21/88 r.a. hulse)
c
c------------------------------------------------------------------------
c
      double precision pet, pcf, pfit
      double precision emin, emax, cheb, eminl, emaxl, enl, xnorm
      double precision twox, prev, prev2
      dimension pcf(11)
      character*(*) kermsg
      emin = pcf(10)
      emax = pcf(11)
      if(pet .ge. emin .and. pet .le. emax) then
        kermsg = ' '
      else
        kermsg = 'outside range of fit in alcheb'
        return
      endif
c
c     calculate polynomial using recursion relation
c
      k = 9
      cheb = pcf(k)
      eminl = dlog(emin)
      emaxl = dlog(emax)
      enl= dlog(pet)
      k = k-1
      xnorm = (enl-eminl-(emaxl-enl)) / (emaxl-eminl)
      twox = 2.0d0 *  xnorm
      prev2 = 0.0d+00
   10 prev = cheb
      if(k .ne. 1) then
        cheb = pcf(k) + twox*prev - prev2
        prev2 = prev
        k = k-1
        go to 10
      endif
      cheb = 0.5d0*pcf(1) + xnorm*prev - prev2
      pfit = dexp(cheb)
  100 return
c
      end

Fit Coefficients

pcf(1)	-8.354e+01
pcf(2)	7.584e-01
pcf(3)	-6.738e-01
pcf(4)	-3.681e+00
pcf(5)	-2.027e+00
pcf(6)	-1.482e-01
pcf(7)	2.380e-02
pcf(8)	-7.384e-02
pcf(9)	2.440e-01
pcf(10)	2.400e-04
pcf(11)	2.000e+06
kncf	1.100e+01

x-range

0.00024 – 2000000.0