// ( 3/2 J2 J3 )
// ( M1 M2 M3 )
private static double ThreeJ_ThreeHalvesJ(SpinState s1, SpinState s2, SpinState s3)
{
Debug.Assert(s1.TwoJ == 3);
if (s1.TwoM < 0)
{
// m1 is negative; invert the m's
double r = ThreeJ_ThreeHalvesJ(s1.Invert(), s2.Invert(), s3.Invert());
// fix sign!
return(r);
}
else if (s1.TwoM == 1)
{
// m1 = 1/2
if (s3.TwoJ == (s2.TwoJ + 1))
{
// j3 = j2 + 1/2
double r = (s2.JMinusM - s2.TwoM) * Math.Sqrt(1.0 * (s2.JPlusM + 1) / s2.TwoJ / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3));
if (s2.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
else
{
// j3 = j2 + 3/2
double r = Math.Sqrt(3.0 * (s2.JPlusM + 1) * (s2.JPlusM + 2) * (s2.JMinusM + 1) / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3) / (s2.TwoJ + 4));
if (s2.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
}
else
{
// m1 = 3/2
if (s3.TwoJ == (s2.TwoJ + 1))
{
// j3 = j2 + 1/2
double r = -Math.Sqrt(3.0 * (s2.JPlusM + 1) * (s2.JPlusM + 2) * (s2.JMinusM) / s2.TwoJ / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3));
if (s2.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
else
{
// j3 = j2 + 3/2
double r = -Math.Sqrt(1.0 * (s2.JPlusM + 1) * (s2.JPlusM + 2) * (s2.JPlusM + 3) / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3) / (s2.TwoJ + 4));
if (s2.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
}
}
// ( 1/2 J2 J+1/2 )
// ( M1 M2 -M1-M2 )
private static double ThreeJ_HalfJ(SpinState s1, SpinState s2)
{
Debug.Assert(s1.TwoJ == 1);
if (s1.TwoM < 0)
{
// m1 = -1/2
// invert to turn into m1 = +1/2 case
double r = ThreeJ_HalfJ(s1.Invert(), s2.Invert());
if (s2.TwoJ % 2 == 0)
{
r = -r;
}
return(r);
}
else
{
// m1 = +1/2
double r = -Math.Sqrt((s2.JPlusM + 1.0) / (s2.TwoJ + 1.0) / (s2.TwoJ + 2.0));
if (s2.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
}
// ( J1 J2 J3 )
// ( 0 0 0 )
private static double ThreeJ_ZeroM_Int(SpinState s1, SpinState s2, SpinState s3)
{
Debug.Assert(s1.TwoM == 0);
Debug.Assert(s2.TwoM == 0);
Debug.Assert(s3.TwoM == 0);
// this method depends on the ordering j1 <= j2 <= j3
Debug.Assert(s1.TwoJ <= s2.TwoJ);
Debug.Assert(s2.TwoJ <= s3.TwoJ);
int j = (s1.TwoJ + s2.TwoJ + s3.TwoJ) / 2;
if (j % 2 != 0)
{
return(0.0);
}
else
{
// use the method formula derived in
int A = j - s1.TwoJ;
int B = j - s2.TwoJ;
int C = j - s3.TwoJ;
// note that A >= B >= C, because of the ordering of js; therefore we eliminate factors involving C, which are largest
// note also that A, B, and C are even, because j is even and we subtract even numbers from j to get them
int hA = A / 2;
int hB = B / 2;
int hC = C / 2;
// calculate first factor
double f = 1.0 / (j + 1);
int fp1 = B / 2;
int fp2 = A / 2;
int fq1 = A;
int fq2 = A + B / 2;
for (int i = 1; i <= hB; i++)
{
f = f * (hB + i) * (hA + i) * (hA + i) / (A + i) / (A + hB + i) / i;
}
f = Math.Sqrt(f);
// calculate second factor
double g = 1.0;
for (int i = 1; i <= hC; i++)
{
g = g * (hC + i) * (hA + hB + i) * (hA + hB + i) / (A + B + i) / (A + B + hC + i) / i;
}
g = Math.Sqrt(g);
if ((j / 2) % 2 != 0)
{
f = -f;
}
return(f * g);
}
}
// ( 0 J J )
// ( 0 M -M )
private static double ThreeJ_ZeroJ(SpinState s)
{
double r = 1.0 / Math.Sqrt(s.TwoJ + 1);
if (s.JMinusM % 2 != 0)
{
r = -r;
}
return(r);
}
// ( J1 J2 M1 M2 | J M ) =
// (-1)^{J1-J2+M} * Sqrt(2J+1) *
// ( J1 J2 J )
// ( M1 M1 -M )
/// <summary>
/// Computes a Clebsch-Gordon coefficient.
/// </summary>
/// <param name="s1">The first spin state.</param>
/// <param name="s2">The second spin state.</param>
/// <param name="s">The total spin state.</param>
/// <returns>The Clebsch-Gordon coefficient measuring the contribution of the given first and
/// second spin states to the given total spin state.</returns>
public static double ClebschGodron(SpinState s1, SpinState s2, SpinState s)
{
double f = Math.Sqrt(s.TwoJ + 1);
if (s1.JPlusM % 2 != 0)
{
f = -f;
}
if (s2.JMinusM % 2 != 0)
{
f = -f;
}
return(f * ThreeJ(s1, s2, s.Invert()));
}
// ( J1 J2 J1+J2 )
// ( M1 M2 -(M1+M2) )
private static double ThreeJ_MaxJ(SpinState j1, SpinState j2)
{
double r = AdvancedIntegerMath.LogFactorial(j1.TwoJ) +
AdvancedIntegerMath.LogFactorial(j2.TwoJ) +
AdvancedIntegerMath.LogFactorial(j1.JPlusM + j2.JPlusM) +
AdvancedIntegerMath.LogFactorial(j1.JMinusM + j2.JMinusM) -
AdvancedIntegerMath.LogFactorial(j1.TwoJ + j2.TwoJ + 1) -
AdvancedIntegerMath.LogFactorial(j1.JPlusM) -
AdvancedIntegerMath.LogFactorial(j1.JMinusM) -
AdvancedIntegerMath.LogFactorial(j2.JPlusM) -
AdvancedIntegerMath.LogFactorial(j2.JMinusM);
r = Math.Exp(r / 2.0);
return(r);
}
/// <summary>
/// Enumerates all spin states that may be obtained by combining two spin states.
/// </summary>
/// <param name="s1">The first spin state.</param>
/// <param name="s2">The second spin state.</param>
/// <returns>A list of spin states which may be obtained.</returns>
public static SpinState[] Combine(SpinState s1, SpinState s2)
{
int tj_max = s1.TwoJ + s2.TwoJ;
int tj_min = Math.Abs(s1.TwoJ - s2.TwoJ);
int tm = s1.TwoM + s2.TwoM;
if (Math.Abs(tm) > tj_min)
{
tj_min = Math.Abs(tm);
}
SpinState[] states = new SpinState[(tj_max - tj_min) / 2 + 1];
for (int i = 0; i < states.Length; i++)
{
states[i] = new SpinState((tj_min + 2 * i) / 2.0, tm / 2.0);
}
return(states);
}
private static double ThreeJ_ZeroHalfM(SpinState s1, SpinState s2, SpinState s3)
{
Debug.Assert(s1.TwoM == 0);
Debug.Assert(Math.Abs(s2.TwoM) == 0);
int j = (s1.TwoJ + s2.TwoJ + s3.TwoJ) / 2;
// if m2 < 0 invert and re-call
if (j % 2 == 0)
{
double f = -Math.Sqrt(1.0 * (s1.TwoJ - s2.TwoJ + s3.TwoJ + 2) * (s1.TwoJ + s2.TwoJ - s3.TwoJ) / (s2.TwoJ + 1) / (s3.TwoJ + 1)) / 2.0;
double g = ThreeJ_ZeroM(new SpinState(s1.TwoJ / 2.0, 0.0), new SpinState((s2.TwoJ - 1) / 2.0, 0.0), new SpinState((s3.TwoJ + 1) / 2.0, 0.0));
return(f * g);
}
else
{
double f = Math.Sqrt(1.0 * (j + 2) / (s2.TwoJ + 1) / (s3.TwoJ + 1)) / 2.0;
throw new NotImplementedException();
}
}
// ( J1 J2 J3 )
// ( 0 0 0 )
private static double ThreeJ_ZeroM(SpinState s1, SpinState s2, SpinState s3)
{
Debug.Assert(s1.TwoM == 0);
Debug.Assert(s2.TwoM == 0);
Debug.Assert(s3.TwoM == 0);
int j = (s1.TwoJ + s2.TwoJ + s3.TwoJ) / 2;
if (j % 2 != 0)
{
// zero by symmetry
return(0.0);
}
else
{
double f = -AdvancedIntegerMath.LogFactorial(j + 1)
+ AdvancedIntegerMath.LogFactorial(j - s1.TwoJ)
+ AdvancedIntegerMath.LogFactorial(j - s2.TwoJ)
+ AdvancedIntegerMath.LogFactorial(j - s3.TwoJ);
f = Math.Exp(f / 2.0);
int k = j / 2;
double g = AdvancedIntegerMath.LogFactorial(k)
- AdvancedIntegerMath.LogFactorial(k - s1.TwoJ / 2)
- AdvancedIntegerMath.LogFactorial(k - s2.TwoJ / 2)
- AdvancedIntegerMath.LogFactorial(k - s3.TwoJ / 2);
g = Math.Exp(g);
if (k % 2 != 0)
{
return(-f * g);
}
else
{
return(f * g);
}
}
}
public void SpinStateInvalid1()
{
// spin state indices are smaller than spin indices
SpinState s = new SpinState(1.0, -2.0);
}
private static double ThreeJ_Any(SpinState j1, SpinState j2, SpinState j3)
{
// compute prefactor
double f1 = 0.0
+ AdvancedIntegerMath.LogFactorial((j1.TwoJ + j2.TwoJ - j3.TwoJ) / 2)
+ AdvancedIntegerMath.LogFactorial((j1.TwoJ - j2.TwoJ + j3.TwoJ) / 2)
+ AdvancedIntegerMath.LogFactorial((-j1.TwoJ + j2.TwoJ + j3.TwoJ) / 2)
- AdvancedIntegerMath.LogFactorial((j1.TwoJ + j2.TwoJ + j3.TwoJ) / 2 + 1);
double f2 = 0.0
+ AdvancedIntegerMath.LogFactorial(j1.JPlusM)
+ AdvancedIntegerMath.LogFactorial(j1.JMinusM)
+ AdvancedIntegerMath.LogFactorial(j2.JPlusM)
+ AdvancedIntegerMath.LogFactorial(j2.JMinusM)
+ AdvancedIntegerMath.LogFactorial(j3.JPlusM)
+ AdvancedIntegerMath.LogFactorial(j3.JMinusM);
double f = Math.Exp((f1 + f2) / 2.0);
if ((j1.JPlusM - j2.JMinusM) % 2 != 0)
{
f = -f;
}
Console.WriteLine("f={0}", f);
// determine maximum and minimum values of k in sum
int kmin = 0;
int k23 = j2.JPlusM - j3.JMinusM;
if (kmin < k23)
{
kmin = k23;
}
int k13 = j1.JMinusM - j3.JPlusM;
if (kmin < k13)
{
kmin = k13;
}
int k123 = (j1.TwoJ + j2.TwoJ - j3.TwoJ) / 2;
int kmax = k123;
int k1 = j1.JMinusM;
if (k1 < kmax)
{
kmax = k1;
}
int k2 = j2.JPlusM;
if (k2 < kmax)
{
kmax = k2;
}
Console.WriteLine("{0} <= k <= {1}", kmin, kmax);
Debug.Assert(kmin <= kmax);
// compute the sum
double g = 0.0;
for (int k = kmin; k <= kmax; k++)
{
double gt = AdvancedIntegerMath.LogFactorial(k)
+ AdvancedIntegerMath.LogFactorial(k123 - k)
+ AdvancedIntegerMath.LogFactorial(k1 - k)
+ AdvancedIntegerMath.LogFactorial(k2 - k)
+ AdvancedIntegerMath.LogFactorial(k - k13)
+ AdvancedIntegerMath.LogFactorial(k - k23);
gt = Math.Exp(-gt);
if (k % 2 != 0)
{
gt = -gt;
}
g += gt;
}
Console.WriteLine("g={0}", g);
return(f * g);
}
public void SpinStateInvalid2()
{
// spin states indices match the parity of spin indices
SpinState s = new SpinState(1.0, 0.5);
}
public void SixJThreeJRelation()
{
SixJSymbol[] symbols = GenerateRandomSixJSymbols(50.0, 5);
foreach (SixJSymbol symbol in symbols) {
Spin a = symbol.J1;
Spin b = symbol.J2;
Spin c = symbol.J3;
Spin d = symbol.J4;
Spin e = symbol.J5;
Spin f = symbol.J6;
SpinState[] ams = GenerateRandomSpinStates(a, 2);
SpinState[] bms = GenerateRandomSpinStates(b, 2);
foreach (SpinState am in ams) {
foreach (SpinState bm in bms) {
if (Math.Abs(am.M + bm.M) > c.J) continue;
SpinState cm = new SpinState(c, -(am.M + bm.M));
double g1 = SpinMath.ThreeJ(am, bm, cm);
double g2 = SpinMath.SixJ(a, b, c, d, e, f);
double p = g1 * g2;
double q = 0.0;
List<double> ts = new List<double>();
SpinState[] dms = d.States();
foreach (SpinState dm in dms) {
if (Math.Abs(dm.M + cm.M) > e.J) continue;
SpinState em = new SpinState(e, dm.M + cm.M);
SpinState mem = new SpinState(e, -em.M);
double f1 = SpinMath.ThreeJ(dm, mem, cm);
if (Math.Abs(em.M + am.M) > f.J) continue;
SpinState fm = new SpinState(f, em.M + am.M);
SpinState mfm = new SpinState(f, -fm.M);
double f2 = SpinMath.ThreeJ(em, mfm, am);
SpinState mdm = new SpinState(d, -dm.M);
double f3 = SpinMath.ThreeJ(fm, mdm, bm);
double t = f1 * f2 * f3;
int s = (int) Math.Round(dm.J + dm.M + em.J + em.M + fm.J + fm.M);
if (s % 2 != 0) t = -t;
q += t;
ts.Add(t);
}
Console.WriteLine("{0} v. {1}", p, q);
//Assert.IsTrue(TestUtilities.IsNearlyEqual(p, q));
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(ts, p));
}
}
}
}
// equality
private static bool Equals(SpinState s1, SpinState s2)
{
return((s1.spin == s2.spin) && (s1.twoM == s2.twoM));
}
// equality
private static bool Equals(SpinState s1, SpinState s2)
{
return ((s1.spin == s2.spin) && (s1.twoM == s2.twoM));
}
private static SpinState[] GenerateRandomSpinStates(Spin j, int n)
{
int tj = (int) (2.0 * j.J);
SpinState[] result = new SpinState[n];
Random rng = new Random(1);
for (int i = 0; i < n; i++) {
int tm = -tj + 2 * rng.Next(tj + 1);
result[i] = new SpinState(j, tm / 2.0);
}
return (result);
}
public void ThreeJRacahSymmetry()
{
foreach (ThreeJSymbol symbol in GenerateRandomThreeJSymbols(50.0, 10)) {
// compute the 3j symbol
SpinState s1 = symbol.Column1;
SpinState s2 = symbol.Column2;
SpinState s3 = symbol.Column3;
double C = SpinMath.ThreeJ(s1, s2, s3);
// form other 3j symbols related by Racah symmetry
double k1, k2, k3, n1, n2, n3;
SpinState t1, t2, t3;
// rows 1->2->3
k1 = (s2.J + s2.M + s3.J + s3.M) / 2.0;
k2 = (s1.J + s1.M + s3.J + s3.M) / 2.0;
k3 = (s1.J + s1.M + s2.J + s2.M) / 2.0;
n1 = s1.J - s1.M - k1;
n2 = s2.J - s2.M - k2;
n3 = s3.J - s3.M - k3;
t1 = new SpinState(k1, n1);
t2 = new SpinState(k2, n2);
t3 = new SpinState(k3, n3);
double CC = SpinMath.ThreeJ(t1, t2, t3);
Assert.IsTrue(TestUtilities.IsNearlyEqual(C, CC));
// transpose rows and columns
/*
k1 = s1.J;
k2 = (s2.J + s3.J + s1.M) / 2.0;
k3 = (s2.J + s3.J - s1.M) / 2.0;
n1 = s2.J - s3.J;
n2 = s3.J - s3.M - k2;
n3 = s3.J + s3.M - k2;
t1 = new SpinState(k1, n1);
t2 = new SpinState(k2, n2);
t3 = new SpinState(k3, n3);
double CT = SpinMath.ThreeJ(t1, t2, t3);
Assert.IsTrue(TestUtilities.IsNearlyEqual(C, CT));
*/
}
}
public void SpinStateInvalid3()
{
// spin state values are integer or half-integer
SpinState s = new SpinState(3.0, 0.75);
}
// method described by Shulten & Gordon in J. Math. Phys. 16 (1975) 1961
// uses the recurrsion
// j3 A_(j3+1) ( j1 j2 j3+1 ) + B_j3 ( j1 j2 j3 ) + (j3+1) A_j3 ( j1 j2 j3-1 )
// ( m1 m2 m3 ) ( m1 m2 m3 ) ( m1 m2 m3 )
// plus the normalization condition
// \sum_j3 (2 j3 + 1) ( j1 j2 j3 )^2 = 1
// ( m1 m2 m3 )
// we recurse on j3 because, for j1 <= j2 <= j3, the typical number of recursion steps
// will be 2 * j1, the lowest of the three possible spins to recurse on
private static double ThreeJ_ShultenGordon_RecurseJ(SpinState s1, SpinState s2, SpinState s3)
{
// find the range of j
int tj_min = Math.Abs(s1.TwoJ - s2.TwoJ);
if (tj_min < Math.Abs(s3.TwoM))
{
tj_min = Math.Abs(s3.TwoM);
}
int tj_max = s1.TwoJ + s2.TwoJ;
// find the midpoint of the range
int tj_mid = (tj_min + tj_max) / 2;
if ((tj_mid % 2) != (tj_max % 2))
{
tj_mid++;
}
// variables to store recurrence coefficients and (unscaled) 3j values
double ap, a0, b0;
double cm, c0, cp;
int tj;
// a variable to store the sought value
double c = 0.0;
// iterate in from the right
double NR = 0.0;
cp = 0.0;
ap = 0.0;
c0 = 0.5 / tj_max;
if (((s1.TwoJ - s2.TwoJ - s3.TwoM) / 2) % 2 != 0)
{
c0 = -c0; // sign convention
}
tj = tj_max;
while (true)
{
// keep track of the normalization constant
NR += (tj + 1) * c0 * c0;
// remember the (unscaled) value of the desired 3j symbol
if (tj == s3.TwoJ)
{
c = c0;
}
// check whether we have reached the center
if (tj <= tj_mid)
{
if (Math.Abs(c0) <= Math.Pow(2.0, -40.0) * Math.Abs(cp))
{
// if the middle value is zero, the left and right values cannot be matached
// (acutally, the numbers will be two random tiny numbers, and their ratio will
// be a random number of order one); in this case, go down one more step
tj_mid -= 2;
}
else
{
break;
}
}
// compute required coefficients
b0 = ShultenGodron_B(s1.TwoJ, s2.TwoJ, tj, s1.TwoM, s2.TwoM, s3.TwoM);
a0 = ShultenGordon_A(s1.TwoJ, s2.TwoJ, tj, s3.TwoM);
// use recursion to compute the tj-2 symbol
cm = (-b0 * c0 - (tj / 2.0) * ap * cp) / ((tj + 2) / 2.0) / a0;
// prepare for the next cycle
cp = c0;
c0 = cm;
ap = a0;
tj -= 2;
}
// remember the (unscaled) middle value
double c1 = c0;
// iterate in from the left
double NL = 0.0;
cm = 0.0;
a0 = 0.0;
c0 = 0.5 / tj_max;
tj = tj_min;
// j_min = 0 iff j1 = j2 and m3 = 0
// in this case for j3=0, b0 = 0 and all three coefficients vanish, so the recursion is indeterminate
// in this case we do the first recursion step "by hand" using relationships
if (tj == 0)
{
NL += c0 * c0;
cm = c0;
c0 = s1.TwoM / Math.Sqrt(s1.TwoJ * (s1.TwoJ + 2)) * cm;
a0 = ShultenGordon_A(s1.TwoJ, s2.TwoJ, 2, 0);
tj = 2;
}
while (true)
{
// check whether we have reached the center
if (tj >= tj_mid)
{
break;
}
// notice that we terminate before checking whether we are at the desired value,
// or adding to the normalization constant; this is different from the previous loop
// we do this so that the middle element does not double contribute to the normalization
// and because if the middle element is desired, it has already been stored from the previous loop
// remember the desired (unscaled) value
if (tj == s3.TwoJ)
{
c = c0;
}
// add current value to normalization
NL += (tj + 1) * c0 * c0;
// check for end
if (tj >= tj_max)
{
break;
}
// compute required coeficients
b0 = ShultenGodron_B(s1.TwoJ, s2.TwoJ, tj, s1.TwoM, s2.TwoM, s3.TwoM);
ap = ShultenGordon_A(s1.TwoJ, s2.TwoJ, tj + 2, s3.TwoM);
// use recursion relation to compute tj+2
cp = (-b0 * c0 - ((tj + 2) / 2.0) * a0 * cm) / (tj / 2.0) / ap;
// prepare for next iteration
cm = c0;
c0 = cp;
a0 = ap;
tj += 2;
}
// match in the middle
double r = c0 / c1;
NL = NL / (r * r);
if (s3.TwoJ < tj_mid)
{
c = c / r;
}
// normalize
double N = NL + NR;
c = c / Math.Sqrt(N);
return(c);
}
// ( 1 J2 J3 )
// ( M1 M2 M3 )
private static double ThreeJ_OneJ(SpinState s1, SpinState s2, SpinState s3)
{
Debug.Assert(s1.TwoJ == 2);
if (s1.TwoM < 0)
{
// m1 = -1
double r = ThreeJ_OneJ(s1.Invert(), s2.Invert(), s3.Invert());
if ((s1.TwoJ + s2.TwoJ + s3.TwoJ) / 2 % 2 != 0)
{
r = -r;
}
return(r);
}
else if (s1.TwoM == 0)
{
// m1 = 0
if (s3.TwoJ == s2.TwoJ)
{
// j3 = j2
double r = s2.TwoM / Math.Sqrt(s2.TwoJ * (s2.TwoJ + 1) * (s2.TwoJ + 2));
if ((s2.JMinusM % 2) != 0)
{
r = -r;
}
return(r);
}
else
{
// j3 = j2 + 1
double r = -Math.Sqrt(2.0 * (s2.JPlusM + 1) * (s2.JMinusM + 1) / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3));
if ((s2.JMinusM % 2) != 0)
{
r = -r;
}
return(r);
}
}
else
{
// m1 = 1
if (s3.TwoJ == s2.TwoJ)
{
// j3 = j2
double r = Math.Sqrt(2.0 * s2.JMinusM * (s2.JPlusM + 1) / s2.TwoJ / (s2.TwoJ + 1) / (s2.TwoJ + 2));
if ((s2.JMinusM % 2) != 0)
{
r = -r;
}
return(r);
}
else
{
// j3 = j2 + 1
double r = Math.Sqrt(1.0 * (s2.JPlusM + 1) * (s2.JPlusM + 2) / (s2.TwoJ + 1) / (s2.TwoJ + 2) / (s2.TwoJ + 3));
if ((s2.JMinusM % 2) != 0)
{
r = -r;
}
return(r);
}
}
}
/// <summary>
/// Computes a 3j symbol.
/// </summary>
/// <param name="s1">The first column spin state.</param>
/// <param name="s2">The second column spin state.</param>
/// <param name="s3">The third column spin state.</param>
/// <returns>The 3j symbol.</returns>
public static double ThreeJ(SpinState s1, SpinState s2, SpinState s3)
{
// require that Js satisfy triangle inequality
if (s3.TwoJ < Math.Abs(s1.TwoJ - s2.TwoJ))
{
return(0.0);
}
if (s3.TwoJ > (s1.TwoJ + s2.TwoJ))
{
return(0.0);
}
// require that the parity of the spins agree
// for example, spin-1 and spin-2 cannot combine to spin 3/2, even though 1 < 3/2 < 3
if ((s1.TwoJ + s2.TwoJ + s3.TwoJ) % 2 != 0)
{
return(0.0);
}
// require that Ms add to zero
if ((s1.TwoM + s2.TwoM + s3.TwoM) != 0)
{
return(0.0);
}
// determine permutation factor for row switching or inverting
int P;
if (((s1.TwoJ + s2.TwoJ + s3.TwoJ) / 2) % 2 == 0)
{
P = 1;
}
else
{
P = -1;
}
// shuffle so J1 <= J2 <= J3
// keeping track of the sign induced by the permutation
int s = 1;
if (s1.TwoJ > s3.TwoJ)
{
Swap(ref s1, ref s3);
s = s * P;
}
if (s1.TwoJ > s2.TwoJ)
{
Swap(ref s1, ref s2);
s = s * P;
}
if (s2.TwoJ > s3.TwoJ)
{
Swap(ref s2, ref s3);
s = s * P;
}
// now we can test for special cases,
// knowing where the Js will fall in each case
if (s1.TwoJ == 0)
{
// J1 = 0
return(s * ThreeJ_ZeroJ(s2));
}
else if (s1.TwoJ == 1)
{
// J1 = 1/2
return(s * ThreeJ_HalfJ(s1, s2));
}
else if (s1.TwoJ == 2)
{
// J1 = 1
return(s * ThreeJ_OneJ(s1, s2, s3));
}
else
{
// arbitrary Js
if ((s1.TwoM == 0) && (s2.TwoM == 0) && (s3.TwoM == 0))
{
// special case of all M's zero
return(s * ThreeJ_ZeroM_Int(s1, s2, s3));
}
else
{
// general case
return(s * ThreeJ_ShultenGordon_RecurseJ(s1, s2, s3));
}
}
// other possible special cases include:
// J1 = 3/2, J1 = 2
// Ms = 1/2 -1/2 0, 1/2 1/2 1, 1 -1 0 and permutations
// maximum coupling J3 = J1 + J2, minimum coupling J3 = |J1 - J2|
}
// generates only spinors of the same parity as j_min
private static SpinState[] GenerateRandomSpinStates(double j_min, double j_max, int n)
{
int tj_min = (int) Math.Truncate(2.0 * j_min);
int tj_max = (int) Math.Truncate(2.0 * j_max);
SpinState[] states = new SpinState[n];
Random rng = new Random(1);
for (int i = 0; i < n; i++) {
int tj = tj_min + 2 * rng.Next((tj_max - tj_min) / 2);
int tm = -tj + 2 * rng.Next(tj);
states[i] = new SpinState(tj / 2.0, tm / 2.0);
}
return (states);
}