unCross.pas 10 KB
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unit unCross;

interface
const
  naMax=10000;
type
  cross_ratio_type=array[1..naMax] of double;

procedure get_cross_ratio(Z1_in,m1_in,Z2_in,m2_in,q_in,E_in,b_max_in : double; nb_in: integer; ep_in : double; theta_initial,dtheta : double; ntheta : integer; var cross_ratio : cross_ratio_type);

implementation

uses math;

const
  nmax = 500000;
type
  re = extended;
var
  Z1,Z2 : re;
  ro    : re;
  rmax  : re;
  Ecm   : re;
  b     : double;
  ep    : double;
  phi   : array[0..nmax] of single;
  a_scr : re;
  Tmax,
  vmax : re;

function V_pot(r : re) : re;
var
  i : integer;
begin
  if r > rmax then
  begin
   V_pot := 0;
  end
  else
  begin
    i := round(r/rmax*nmax);
    if i = nmax then V_pot := 0 else
    V_pot := Z1*Z2/r*exp((r-i*rmax/nmax)/(rmax/nmax) * (ln(phi[i+1])-ln(phi[i])) + ln(phi[i]));
  end;
end;



function V_pot_prime(r : re) : re;
var
  i : integer;
begin
  if r > rmax then
  begin
    V_pot_prime := 0;
  end
  else
  begin
   i := round(r/rmax*nmax);
   V_pot_prime := Z1*Z2*(-phi[i]/sqr(r) +(phi[i+1]-phi[i])/(r*rmax/nmax) );
 end;
end;

procedure get_ro;
var
  F,G   : re;
  F_old : re;
  r, dr : re;
  count : integer;
begin
  r := sqrt(sqr(b) + sqr(Z1*Z2/Ecm));

  count := 0;
  F := 100;
  repeat
    inc(count);
    F_old := F;
    F := sqr(r)*(1 - V_pot(r)/Ecm) - sqr(b);

    G :=  2*r*(1 - V_pot(r)/Ecm)-sqr(r)*V_pot_prime(r)/Ecm;
    dr := F/G;
    r := r - dr;
    if count > 1000 then
    begin

     writeln(count,' ',r,dr/r,F);
    end;
  until ((abs(dr/r) < 1e-14) and (count > 10))or ((count > 1000) and (F_old*F < 0));
//  writeln(count);
  ro := r;
end;

function fun(x: re) : re;
var
 r : re;
begin
  r := ro/(1-sqr(x));
  fun := x/sqrt(abs(1-V_pot(r)/Ecm-sqr(b/r)));
end;



function simp_inner(a1,b,ep : re): re;
type
  arr = array[1..30] of re;
var
  dx,epsp,x2,x3,
  f2,f3,f4,fmp,
  fbp,est2,est3 : arr;
  nrtr          : array[1..30] of integer;
  pval          : array[1..30,1..3] of re;
  sum,eps,fm,f1,
  a,absar, est,da,
  fa,fb,sx,est1 : re;
  lvl,l{,l1} : integer;
label
  1,2,3,4,5,7,11,12,13;
begin
{ nonrecursive adaptive integration }
{ algorithm 182 CACM 6 (1983) 315   }
{ adaptation from : Numerical Integration , Davis-Rabinowitz, pg. 198 }

  a := a1;
  eps := ep;
  lvl := 0;
  absar := 0;
  est := 0;
  da := b-a;
  fa := 0;{fun(a);}
  fm := 4*fun((a+b)*0.5);
  fb := fun(b);
{ 1 = recur }
1: inc(lvl);
   dx[lvl] := da/3;
   sx := dx[lvl]/6;
   f1 := 4*fun(0.5*dx[lvl] + a );
   x2[lvl] := a + dx[lvl];
   f2[lvl] := fun(x2[lvl]);
   x3[lvl] := x2[lvl] + dx[lvl];
   f3[lvl] := fun(x3[lvl]);
   epsp[lvl] := eps;
   f4[lvl] := 4*fun(dx[lvl]*0.5+x3[lvl]);
   fmp[lvl] := fm;
   est1 := sx*(fa+f1+f2[lvl]);
   fbp[lvl] := fb;
   est2[lvl] := sx*(f2[lvl] +f3[lvl] + fm);
   est3[lvl] := sx*(f3[lvl] + f4[lvl] + fb);
   sum := est1+ est2[lvl]+ est3[lvl];
   absar :=absar -  abs(est) + abs(est1) + abs(est2[lvl]) + abs(est3[lvl]);
   if abs(est-sum) - epsp[lvl]*absar  > 0 then goto 3 else goto 2;
3: if lvl < 30 then goto 4 else goto 2;
  { 2=up }
2: dec(lvl);
  l := nrtr[lvl];
  pval[lvl,l] := sum;
  case l of
    1 : goto 11;
    2 : goto 12;
    3 : goto 13;
  else
    goto 4;
  end;
4: nrtr[lvl] := 1;
   est := est1;
   fm := f1;
   fb := f2[lvl];
7: eps := epsp[lvl]/1.7;
   da := dx[lvl];
   goto 1;
11: nrtr[lvl] := 2;
    fa := f2[lvl];
    fm := fmp[lvl];
    fb := f3[lvl];
    est := est2[lvl];
    a := x2[lvl];
    goto 7;
12: nrtr[lvl] := 3;
    fa := f3[lvl];
    fm := f4[lvl];
    fb := fbp[lvl];
    est := est3[lvl];
    a := x3[lvl];
    goto 7;
13: sum := pval[lvl,1] + pval[lvl,2] + pval[lvl,3];
    if lvl>1 then goto 2 else goto 5;
5: simp_inner := sum;
end;


procedure get_theta(var theta : re);
const
  r_cut : re = 4;
var
  x_cut : re;
begin
  r_cut := rmax;
  get_ro;
  r_cut := r_cut + ro;
  x_cut := sqrt(1-ro/r_cut);
  theta := pi - 4*b/ro*simp_inner(0,x_cut,ep/1000)-2*(pi/2-arccos(b/r_cut));
  {theta := pi - 4*b/ro*romb(0,x_cut,ep/10)-2*(pi/2-arccos(b/r_cut));}
end;

procedure prepare_phi_numerical;
var
  in_dat : text;
  name   : string;
  r,ph   : array[0..100] of double;
  i,n,k  : integer;
  x      : double;
  //dummy1 : double;
  //dummy2 : double;
begin
  write('Enter filename to load : '); // screening function in Angstrom
  readln(name);
  assign(in_dat,name);
  reset(in_dat);
  r[0] := 0;
  ph[0] := 1;
  i := 1;
  repeat
    readln(in_dat,r[i],ph[i]);
    r[i] := r[i]/0.529;
    inc(i);
  until eof(in_dat);
  close(in_dat);
  n := i-1;
  rmax := r[n];
  writeln('rmax = ',rmax:7:1);

  for i := 0 to nmax do
  begin
    x := i*rmax/nmax;
    k := -1;
    repeat
      inc(k);
    until (k>n) or ((x>=r[k]) and (x<=r[k+1]));
    if k>n then halt;
    phi[i] := ((x-r[k])/(r[k+1]-r[k])*((ph[k+1])-(ph[k])) + (ph[k]));
  end;
end;

procedure prepare_phi_scaling(rcut : re; pot_type : byte);
var
  i      : integer;
  x      : double;
begin
  rmax := rcut;
  for i := 0 to nmax do
  begin
    x := i*rmax/nmax/a_scr;
    if pot_type = 1 then
    begin
      {moliere screening function}
      phi[i] :=  0.35*exp(-0.3*x)+0.55*exp(-1.2*x)+0.10*exp(-6.0*x);
    end;
    if pot_type = 2 then
    begin
      {zbl screening function}
      phi[i] :=  0.18175*EXP(-3.1998*x) + 0.50986*EXP(-0.94229*x) + 0.28022*EXP(-0.4029*x) + 0.028171*EXP(-0.20162*x);
    end;
  end;
end;


procedure magic(eps,b:re; var c,r : re); // c = cos(theta/2)   eps = E_cm/(Z1 Z2 e^2/a_scr)   b = p/a_scr  r=r_0/a_scr
var
  rr,v,v1,fr,fr1,
  ex1,ex2,ex3,ex4,
  q,roc,sqe,cc,aa,ff,
  delta :re;

begin
  R:=B;
  RR:=ln(0.02817/(EPS*B))/0.2016;
  IF(RR>B) then
   begin
     RR:=ln(0.02817/(EPS*RR))/0.2016;
     IF(RR>B)then R:=RR;
   end;
  REPEAT
    EX1:=0.18175*EXP(-3.1998*R);
    EX2:=0.50986*EXP(-0.94229*R);
    EX3:=0.28022*EXP(-0.4029*R);
    EX4:=0.028171*EXP(-0.20162*R);
    V:=(EX1+EX2+EX3+EX4)/R;
    V1:=-(V+3.1998*EX1+0.94229*EX2+0.4029*EX3+0.20162*EX4)/R;
    FR:=B*B/R+V*R/EPS-R;
    FR1:=-B*B/(R*R)+(V+V1*R)/EPS-1;
    Q:=FR/FR1;
    R:=R-Q;
  UNTIL (ABS(Q/R)<0.001);
  ROC:=-2.0*(EPS-V)/V1;
  SQE:=SQRT(EPS);
  CC:=(0.01185+SQE)/(0.0068338+SQE);
  AA:=2.0*EPS*(1.0+(0.80061/SQE))*power(B,CC);
  FF:=(SQRT(AA*AA+1.0)-AA)*((10.855+EPS)/(16.883+EPS));
  DELTA:=(R-B)*AA*FF/(FF+1.0);
  C:=(B+DELTA+ROC)/(R+ROC);
end;



var
  U,W : array[1..16*1024] of double;
  N_mehler : integer;

procedure MEHLER(N : integer);
const
  HALF = 0.5;
var
  W0,Y,Z    : double;
  J      : integer;
begin
  N_mehler := N;
  W0 := PI / (N + N);
  Y := COS( W0 );
  Z := HALF * Y;
  U[1] := SQRT( HALF + Z );
  U[N] := SQRT( HALF - Z );
  Z := U[1] * U[N] + U[1] * U[N];
  W[1] := W0 * U[N];
  W[N] := W0 * U[1];
  for J := 2 to (N+1) div 2 do
  begin
    U[J] := Y * U[J-1] - Z * U[N-J+2];
    U[N-J+1] := Z * U[J-1] + Y * U[N-J+2];
    W[J] := W0 * U[N-J+1];
    W[N-J+1] := W0 * U[J];
  end;
end;

procedure scattering(var theta : re);
var
  i  : integer;
  q,
  qa,qb,
  qx,
  qq,
  qz : double;
begin
  get_ro;
  qz := 0;
  qx := ro/Ecm;
  for i := 1 to N_mehler do
  begin
    QA := ro/ U[I];
    QQ := b* U[I];
    QB := qx * U[I] * V_pot(QA)*qa;
    QQ := (ro + QQ) * (ro - QQ);
    Q := W[I] / SQRT(QQ - QB);
    QZ := QZ + Q
  end;
  QA := b * QZ;
  theta := pi-2*qa;
end;

procedure test_mehler;
var
  theta1,theta2,theta3,c,r0 : re;
begin
  MEHLER(16);
  b := 1;
  repeat
    write('b = ');readln(b);
    get_theta(theta1);
    writeln;
    write('simp = ',theta1*180/pi:8:6);
    scattering(theta2);
    write('        mehler =  ', theta2*180/pi:8:6);
    magic(Ecm/(z1*z2/a_scr),b/a_scr,c,r0);
    theta3 := 2*arccos(c);
    writeln('        magic(zbl) =  ', theta3*180/pi:8:6);


  until b > 10;
end;


procedure scan_b(bmax : re; nb : longint; theta_initial,dtheta : double; ntheta : integer ; var cross_ratio : cross_ratio_type);
var
  theta    : re;
  i        : longint;
  db,b_old,
  b_mean,
  Rutherford,
  theta_mean_std,
  theta_mean_rec,
  theta_mean,
  theta_old : re;
  cross_ratio1:re;
  itheta : integer;

begin
  MEHLER(16*1024);
  theta := pi;
  b     := 0;
 // for itheta := 1 to ntheta do cross_ratio[itheta] := 1e100;    <======
  
  for i := 1 to nb do
  begin
    theta_old := theta;
    b_old     := b;
    b         := bmax*power(i/nb,3);
    db        := b - b_old;
    b_mean    := 0.5*(b+b_old);

//    ep := 1e-4;  get_theta(theta); // simpson
    ep := 1e-3;  scattering(theta); // mehler
//   theta := 2*arctan(Z1*Z2/Ecm/b/2);


    theta_mean_std := 0.5*(theta_old+theta);
    theta_mean_rec := 1/(0.5*(1/theta_old+1/theta));
    theta_mean     := 0.6*theta_mean_rec + 0.4*theta_mean_std;
    Rutherford := sqr(Z1*Z2/4/Ecm) / sqr(sqr(sin(theta_mean/2)));
    cross_ratio1 :=  abs(b_mean/sin(theta_mean) * db/(theta_old-theta)) / Rutherford;

    for itheta := 1 to ntheta do
    if (theta_initial+(itheta-1)*dtheta <= theta_old) and (theta_initial+(itheta-1)*dtheta >= theta) then
    begin
      cross_ratio[itheta] := cross_ratio1;      // <====
    end;
  end;
end;

procedure get_cross_ratio(Z1_in,m1_in,Z2_in,m2_in,q_in,E_in,b_max_in : double; nb_in: integer; ep_in : double; theta_initial,dtheta : double; ntheta : integer; var cross_ratio : cross_ratio_type);

var
  b_max,
  r_max,
  m1,m2,E,q : double;



begin
  z1 := Z1_in {6} {79}  {1, 2, 14, 92};  // H, He, Si, U
  q  := q_in {3} {46};
  z2 := z2_in {13};     // Si 14, Hf 72
  m1 := m1_in {12} {197} {1, 4, 28, 238};
  m2 := m2_in {27};     // Si 28, Hf 178
  ep := ep_in {1e-5};
//  prepare_phi_numerical;

  b_max := b_max_in {1};
  r_max := 10*b_max;
  a_scr := 0.8854/sqrt(power(Z1-q,2/3)+power(Z2,2/3));
  prepare_phi_scaling(r_max,1);

  E := E_in*m1; // 25, 100, 1000  5000 keV/u
  E := E*1000/27.2;
  Ecm := E*m2/(m1+m2);
  Tmax := 4*m1*m2/sqr(m1+m2)*E;
  vmax := sqrt(2*Tmax/(m2*1836));
//  test_mehler;
  scan_b(b_max,nb_in,theta_initial, dtheta,ntheta,cross_ratio);
end;
end.