Second-order quadrupole interaction for rotating crystal.
Contributor: R. Hajjar




Home and Applets > Quadrupole Interaction > Definition > W(4,0) in VAS and MAS NMR

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:

W2-0-VAS

with

W2-VAS

The final expression of W(4,0) in VAS (MAS) NMR is:

W40-VAS-final

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];
 

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