Analytical expression of W(4,0) for rotating crystal in VAS and MAS NMR
The second-order quadrupole interaction is related to W(4,0):
In VAS (MAS) NMR experiments, W(4,0) is defined by:
with
The final expression of W(4,0) in VAS (MAS) NMR is:
The analytical expressions of a40, a4n, and b4n coefficients can be determined as follows:
(1) Select and copy the following green lines; then paste them into a cell of Mathematica, a software for numerical and symbolic calculations.
(2) Press "Ctrl-A" for select all; then
press "Shift-enter" for evaluate cells.
(Or in the menu bar, select Kernel > Evaluation > Evaluate Cells)
Using Mathematica-5 running with a 3-GHz processor, some spelling error messages followed by the analytical expression of coefficients involved in W(4,0) for VAS and MAS NMR are obtained in 80 seconds.
(* D4 is a reduced form (5 rows x 9 columns) of the 4-th order
Wigner active rotation matrix *)
D4 = {
{ (1 + Cos[beta])^4*E^(-I*(4*alpha + 4*gamma))/16,
-Sqrt[2]*(1 + Cos[beta])^3*Sin[beta]*E^(-I*(4*alpha + 3*gamma))/8,
Sqrt[7]*(1 + Cos[beta])^2*Sin[beta]^2*E^(-I*(4*alpha + 2*gamma))/8,
-Sqrt[7/2]*(1 + Cos[beta])*Sin[beta]^3*E^(-I*(4*alpha + gamma))/4,
Sqrt[35/2]*Sin[beta]^4*E^(-4*I*alpha)/8,
-Sqrt[7/2]*(1 - Cos[beta])*Sin[beta]^3*E^(-I*(4*alpha - gamma))/4,
Sqrt[7]*(1 - Cos[beta])^2*Sin[beta]^2*E^(-I*(4*alpha - 2*gamma))/8,
-Sqrt[2]*(1 - Cos[beta])^3*Sin[beta]*E^(-I*(4*alpha - 3*gamma))/8,
(1 - Cos[beta])^4*E^(-I*(4*alpha - 4*gamma))/16},
{Sqrt[7]*(1 + Cos[beta])^2*Sin[beta]^2*E^(-I*(2*alpha + 4*gamma))/8,
-Sqrt[7/2]*(1 - 2*Cos[beta])*(1 + Cos[beta])^2*Sin[beta]
*E^(-I*(2*alpha + 3*gamma))/4,
(1 + Cos[beta])^2*(1 - 7*Cos[beta] + 7*Cos[beta]^2)
*E^(-I*(2*alpha + 2*gamma))/4,
-Sqrt[2]*(1 + Cos[beta])*(-1 - 7*Cos[beta] + 14*Cos[beta]^2)*Sin[beta]
*E^(-I*(2*alpha + gamma))/8,
Sqrt[5/2]*(-1 + 7*Cos[beta]^2)*Sin[beta]^2
*E^(-2*I*alpha)/4,
-Sqrt[2]*(1 - Cos[beta])*(-1 + 7*Cos[beta] + 14*Cos[beta]^2)*Sin[beta]
*E^(-I*(2*alpha - gamma))/8,
(1 - Cos[beta])^2*(1 + 7*Cos[beta] + 7*Cos[beta]^2)
*E^(-I*(2*alpha - 2*gamma))/4,
-Sqrt[7/2]*(1 - Cos[beta])^2*(1 + 2*Cos[beta])*Sin[beta]
*E^(-I*(2*alpha - 3*gamma))/4,
Sqrt[7]*(1 - Cos[beta])^2*Sin[beta]^2*E^(-I*(2*alpha - 4*gamma))/8},
{ Sqrt[35/2]*Sin[beta]^4*E^(-4*I*gamma)/8,
Sqrt[35]*Cos[beta]*Sin[beta]^3*E^(-3*I*gamma)/4,
Sqrt[5/2]*(-1 + 7*Cos[beta]^2)*Sin[beta]^2*E^(-2*I*gamma)/4,
Sqrt[5]*Cos[beta]*(-3 + 7*Cos[beta]^2)*Sin[beta]*E^( -I*gamma)/4,
(3 - 30*Cos[beta]^2 + 35*Cos[beta]^4)/8,
-Sqrt[5]*Cos[beta]*(-3 + 7*Cos[beta]^2)*Sin[beta]*E^( I*gamma)/4,
Sqrt[5/2]*(-1 + 7*Cos[beta]^2)*Sin[beta]^2*E^(2*I*gamma)/4,
-Sqrt[35]*Cos[beta]*Sin[beta]^3*E^(3*I*gamma)/4,
Sqrt[35/2]*Sin[beta]^4*E^(4*I*gamma)/8},
{ Sqrt[7]*(1 - Cos[beta])^2*Sin[beta]^2*E^(-I*(-2*alpha + 4*gamma))/8,
Sqrt[7/2]*(1 - Cos[beta])^2*(1 + 2*Cos[beta])*Sin[beta]
*E^(-I*(-2*alpha + 3*gamma))/4,
(1 - Cos[beta])^2*(1 + 7*Cos[beta] + 7*Cos[beta]^2)
*E^(-I*(-2*alpha + 2*gamma))/4,
Sqrt[2]*(1 - Cos[beta])*(-1 + 7*Cos[beta] + 14*Cos[beta]^2)*Sin[beta]
*E^(-I*(-2*alpha + gamma))/8,
Sqrt[5/2]*(-1 + 7*Cos[beta]^2)*Sin[beta]^2
*E^(2*I*alpha)/4,
Sqrt[2]*(1 + Cos[beta])*(-1 - 7*Cos[beta] + 14*Cos[beta]^2)*Sin[beta]
*E^(-I*(-2*alpha - gamma))/8,
(1 + Cos[beta])^2*(1 - 7*Cos[beta] + 7*Cos[beta]^2)
*E^(-I*(-2*alpha - 2*gamma))/4,
Sqrt[7/2]*(1 - 2*Cos[beta])*(1 + Cos[beta])^2*Sin[beta]
*E^(-I*(-2*alpha - 3*gamma))/4,
Sqrt[7]*(1 + Cos[beta])^2*Sin[beta]^2*E^(-I*(-2*alpha - 4*gamma))/8},
{ (1 - Cos[beta])^4*E^(-I*(-4*alpha + 4*gamma))/16,
Sqrt[2]*(1 - Cos[beta])^3*Sin[beta]*E^(-I*(-4*alpha + 3*gamma))/8,
Sqrt[7]*(1 - Cos[beta])^2*Sin[beta]^2*E^(-I*(-4*alpha + 2*gamma))/8,
Sqrt[7/2]*(1 - Cos[beta])*Sin[beta]^3*E^(-I*(-4*alpha + gamma))/4,
Sqrt[35/2]*Sin[beta]^4*E^(4*I*alpha)/8,
Sqrt[7/2]*(1 + Cos[beta])*Sin[beta]^3*E^(-I*(-4*alpha - gamma))/4,
Sqrt[7]*(1 + Cos[beta])^2*Sin[beta]^2*E^(-I*(-4*alpha - 2*gamma))/8,
Sqrt[2]*(1 + Cos[beta])^3*Sin[beta]*E^(-I*(-4*alpha - 3*gamma))/8,
(1 + Cos[beta])^4*E^(-I*(-4*alpha - 4*gamma))/16}
};
MatrixForm[D4];
(* W4pas is a row-matrix with 5 columns
containing the 5 nonzero eigenvalues of the EFG
expressed as a 4-th rang spherical tensor, in (eq)(eq) unit *)
W4pas = {{eta*eta/4, 3*eta/(2*Sqrt[7]), (9 + eta*eta/2)/Sqrt[70],
3*eta/(2*Sqrt[7]), eta*eta/4}};
MatrixForm[W4pas];
(* W4k is a row-matrix with 9 columns *)
W4k = W4pas.ComplexExpand[D4];
(* D4VAS is a reduced form (9 rows x 1 column) of the 4-th order
Wigner active rotation matrix *)
D4VAS = {
{Sqrt[35/2]*Sin[thetaVAS]^4
*E^(-4*I*rot*t)/8},
{-Sqrt[35]*Cos[thetaVAS]*Sin[thetaVAS]^3
*E^(-3*I*rot*t)/4},
{Sqrt[5/2]*(-1 +7*Cos[thetaVAS]^2)*Sin[thetaVAS]^2
*E^(-2*I*rot*t)/4},
{-Sqrt[5]*Cos[thetaVAS]*(-3 +7*Cos[thetaVAS]^2)*Sin[thetaVAS]
*E^(-I*rot*t)/4},
{(3 - 30*Cos[thetaVAS]^2 + 35*Cos[thetaVAS]^4)/8},
{Sqrt[5]*Cos[thetaVAS]*(-3 +7*Cos[thetaVAS]^2)*Sin[thetaVAS]
*E^(I*rot*t)/4},
{Sqrt[5/2]*(-1 +7*Cos[thetaVAS]^2)*Sin[thetaVAS]^2
*E^(2*I*rot*t)/4},
{Sqrt[35]*Cos[thetaVAS]*Sin[thetaVAS]^3
*E^(3*I*rot*t)/4},
{Sqrt[35/2]*Sin[thetaVAS]^4
*E^(4*I*rot*t)/8}
};
MatrixForm[D4VAS];
(* W40vas is an expression *)
W40vas = W4k.ComplexExpand[D4VAS];
v1 = Expand[W40vas] /. {Cos[a_ + b_] -> Cos[a] Cos[b] - Sin[a] Sin[b],
Sin[a_ + b_] -> Cos[b] Sin[a] + Cos[a] Sin[b]};
v2 = Collect[Expand[v1], {Cos[x_ *rot*t], Sin[y_ *rot*t],
Cos[rot*t], Sin[rot*t] }];
(* suppression of the double curve brackets {{}} of v2 *)
v22 =v2[[1,1]];
size = Length[v22];
(* a44VAS = amplitude of Cos[4*gamma]*Sin[4*rot*t] *)
a44VAS = FullSimplify[Coefficient[v22[[size, 2]], Cos[4*gamma]]];
a44MAS = a44VAS /. thetaVAS -> ArcCos[Sqrt[1/3]];
(* b44VAS = amplitude of Sin[4*gamma]*Sin[4*rot*t] *)
b44VAS = FullSimplify[Coefficient[v22[[size, 2]], Sin[4*gamma]]];
b44MAS = b44VAS /. thetaVAS -> ArcCos[Sqrt[1/3]];
(* c44VAS = amplitude of Cos[4*gamma]*Cos[4*rot*t] *)
(* c44VAS = -b44VAS *)
c44VAS = FullSimplify[Coefficient[v22[[size - 1, 2]], Cos[4*gamma]]];
(* d44VAS = amplitude of Sin[4*gamma]*Cos[4*rot*t] *)
(* d44VAS = a44VAS *)
d44VAS = FullSimplify[Coefficient[v22[[size - 1, 2]], Sin[4*gamma]]];
(* a43VAS = amplitude of Cos[3*gamma]*Sin[3*rot*t] *)
a43VAS = FullSimplify[Coefficient[v22[[size - 2, 2]], Cos[3*gamma]]];
a43MAS = a43VAS /. thetaVAS -> ArcCos[Sqrt[1/3]];
(* b43VAS = amplitude of Sin[3*gamma]*Sin[3*rot*t] *)
b43VAS = FullSimplify[Coefficient[v22[[size - 2, 2]], Sin[3*gamma]]];
b43MAS = b43VAS /. thetaVAS -> ArcCos[Sqrt[1/3]];
(* c43VAS = amplitude of Cos[3*gamma]*Cos[3*rot*t] *)
(* c43VAS = -b43VAS *)
c43VAS = FullSimplify[Coefficient[v22[[size - 3, 2]], Cos[3*gamma]]];
(* d43VAS = amplitude of Sin[3*gamma]*Cos[3*rot*t] *)
(* d43VAS = a43VAS *)
d43VAS = FullSimplify[Coefficient[v22[[size - 3, 2]], Sin[3*gamma]]];
(* a42VAS = amplitude of Cos[2*gamma]*Sin[2*rot*t] *)
a42VAS = FullSimplify[Coefficient[v22[[size - 5, 2]], Cos[2*gamma]]];
a42MAS = FullSimplify[a42VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];
(* b42VAS = amplitude of Sin[2*gamma]*Sin[2*rot*t] *)
b42VAS = FullSimplify[Coefficient[v22[[size - 5, 2]], Sin[2*gamma]]];
b42MAS = FullSimplify[b42VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];
(* c42VAS = amplitude of Cos[2*gamma]*Cos[2*rot*t] *)
(* c42VAS = -b42VAS *)
c42VAS = FullSimplify[Coefficient[v22[[size - 4, 2]], Cos[2*gamma]]];
(* d42VAS = amplitude of Sin[2*gamma]*Cos[2*rot*t] *)
(* d42VAS = a42VAS *)
d42VAS = FullSimply[Coefficient[v22[[size - 4, 2]], Sin[2*gamma]]];
(* a41VAS = amplitude of Cos[gamma]*Sin[rot*t] *)
a41VAS = FullSimplify[Coefficient[v22[[size - 7, 2]], Cos[gamma]]];
a41MAS = FullSimplify[a41VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];
(* b41VAS = amplitude of Sin[gamma]*Sin[rot*t] *)
b41VAS = FullSimplify[Coefficient[v22[[size - 7, 2]], Sin[gamma]]];
b41MAS = FullSimplify[b41VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];
(* c41VAS = amplitude of Cos[gamma]*Cos[rot*t] *)
(* c41VAS = -b41VAS *)
c41VAS = FullSimplify[Coefficient[v22[[size - 6, 2]], Cos[gamma]]];
(* d41VAS = amplitude of Sin[gamma]*Cos[rot*t] *)
(* d41VAS = a41VAS *)
d41VAS = FullSimplify[Coefficient[v22[[size - 6, 2]], Sin[gamma]]];
(* a40VAS *)
a40VAS = FullSimplify[Sum[v22[[i]], {i, 1, size - 8}]];
a40MAS = FullSimplify[a40VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];
(*----------- Table of a4jVAS and b4jVAS for VAS ----------*)
tableW40VAS = List[{"a40VAS", a40VAS},
{"a41VAS", a41VAS}, {"b41VAS", b41VAS},
{"a42VAS", a42VAS}, {"b42VAS", b42VAS},
{"a43VAS", a43VAS}, {"b43VAS", b43VAS},
{"a44VAS", a44VAS}, {"b44VAS", b44VAS}];
Print[TableForm[tableW40VAS,
TableHeadings ->
{None, {"VAS", "expression of a4jVAS and b4jVAS in VAS"}}]];
Print["******************************************"];
(*----------- Table of a4jMAS and b4jMAS for MAS ----------*)
tableW40MAS = List[{"a40MAS", a40MAS},
{"a41MAS", a41MAS}, {"b41MAS", b41MAS},
{"a42MAS", a42MAS}, {"b42MAS", b42MAS},
{"a43MAS", a43MAS}, {"b43MAS", b43MAS},
{"a44MAS", a44MAS}, {"b44MAS", b44MAS}];
Print[TableForm[tableW40MAS,
TableHeadings ->
{None, {"MAS", "expression of a4jMAS and b4jMAS in MAS"}}]];
Remove[D4, alpha, beta, gamma, W4pas, eta, W4k, D4VAS, thetaVAS,
rot, t, W40vas, v1, v2, v22, size, a44VAS, a43VAS, a42VAS,
a41VAS, a40VAS, b44VAS, b43VAS, b42VAS, b41VAS,
c44VAS, c43VAS, c42VAS, c41VAS, d44VAS, d43VAS, d42VAS, d41VAS,
a44MAS, a43MAS, a42MAS, a41MAS, a40MAS,
b44MAS, b43MAS, b42MAS, b41MAS, c44MAS, c43MAS, c42MAS, c41MAS,
d44MAS, d43MAS, d42MAS, d41MAS, tableW40VAS, tableW40MAS];