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




Home and Applets > Quadrupole Interaction > Definition > W(2,0) in VAS and MAS NMR

Analytical expression of W(2,0) for rotating crystal in VAS and MAS NMR

The second-order quadrupole interaction is related to W(2,0):


In VAS (MAS) NMR experiments, W(2,0) is defined by:

W2-0-VAS

with

W2-VAS

The final expression of W(2,0) in VAS (MAS) NMR is:

W20-VAS-final

The analytical expressions of a20, a2n, and b2n 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(2,0) for VAS and MAS NMR are obtained in 15 seconds.


(* D2 is a reduced form (5 rows x 3 columns) of the 2-nd order 
   Wigner active rotation matrix *)

D2 = {

      {         (1 + Cos[beta])^2*E^(-I*(2*alpha + 2*gamma))/4,

       -(1 + Cos[beta])*Sin[beta]*E^(-I*(2*alpha +   gamma))/2,

            Sqrt[3/8]*Sin[beta]^2*E^(-2*I*alpha),

       -(1 - Cos[beta])*Sin[beta]*E^(-I*(2*alpha -   gamma))/2,

                (1 - Cos[beta])^2*E^(-I*(2*alpha - 2*gamma))/4},


       {    Sqrt[3/8]*Sin[beta]^2*E^(-2*I*gamma),

            Sqrt[3/8]*Sin[2*beta]*E^(-I*gamma),

           (-1 + 3*Cos[beta]^2)/2,

           -Sqrt[3/8]*Sin[2*beta]*E^(I*gamma),

            Sqrt[3/8]*Sin[beta]^2*E^(2*I*gamma)},


      {         (1 - Cos[beta])^2*E^(I*(2*alpha - 2*gamma))/4,

        (1 - Cos[beta])*Sin[beta]*E^(I*(2*alpha -   gamma))/2,

            Sqrt[3/8]*Sin[beta]^2*E^(2*I*alpha),

        (1 + Cos[beta])*Sin[beta]*E^(I*(2*alpha +   gamma))/2,

                (1 + Cos[beta])^2*E^(I*(2*alpha + 2*gamma))/4}

     };

MatrixForm[D2];

(* W2pas is a row-matrix with 3 columns 
   containing the 3 nonzero eigenvalues of the EFG 
   expressed as a 2-nd rang spherical tensor, in (eq)(eq) unit *)

W2pas = {{Sqrt[3/7]*eta, (eta*eta - 3)/Sqrt[14], Sqrt[3/7]*eta}};

MatrixForm[W2pas];

(* W2k is a row-matrix with 5 columns *)
W2k = W2pas.ComplexExpand[D2];

(* D2VAS is a reduced form (5 rows x 1 column) of the 2-nd order 
   Wigner active rotation matrix *)

D2VAS = {

          {Sqrt[3/8]*Sin[thetaVAS]^2*E^(-2*I*rot*t)},

         {-Sqrt[3/8]*Sin[2*thetaVAS]*E^(-I*rot*t)},

         {(-1 + 3*Cos[thetaVAS]^2)/2},

          {Sqrt[3/8]*Sin[2*thetaVAS]*E^(I*rot*t)},

          {Sqrt[3/8]*Sin[thetaVAS]^2*E^(2*I*rot*t)}

        };

MatrixForm[D2VAS];

(* W20vas is an expression *)
W20vas = W2k.ComplexExpand[D2VAS];

m1 = Expand[W20vas] /. {Cos[a_ + b_] -> Cos[a] Cos[b] - Sin[a] Sin[b],
                        Sin[a_ + b_] -> Cos[b] Sin[a] + Cos[a] Sin[b]};

m2 = Collect[m1, {Cos[x_ *rot*t], Sin[y_ *rot*t],
                  Cos[rot*t],     Sin[rot*t]    }];

(* suppression of the double curve brackets {{}} of m2 *)
m22 =m2[[1,1]];

size = Length[m22];

(* a22VAS = amplitude of Cos[2*gamma]*Sin[2*rot*t] *)
a22VAS = FullSimplify[Coefficient[m22[[size - 3, 2]], Cos[2*gamma]]];
a22MAS = FullSimplify[a22VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* b22VAS = amplitude of Sin[2*gamma]*Sin[2*rot*t] *)
b22VAS = FullSimplify[Coefficient[m22[[size - 3, 2]], Sin[2*gamma]]];
b22MAS = FullSimplify[b22VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* a21VAS = amplitude of Cos[gamma]*Sin[rot*t] *)
a21VAS = FullSimplify[Coefficient[m22[[size - 1, 2]], Cos[gamma]]];
a21MAS = FullSimplify[a21VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* b21VAS = amplitude of Sin[gamma]*Sin[rot*t] *)
b21VAS = FullSimplify[Coefficient[m22[[size - 1, 2]], Sin[gamma]]];
b21MAS = FullSimplify[b21VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];


(* c22VAS = amplitude of Cos[2*gamma]*Cos[2*rot*t]*) 
(* c22VAS = -b22VAS *)
c22VAS = FullSimplify[Coefficient[m22[[size - 2, 2]], Cos[2*gamma]]];

(* d22VAS = amplitude of Sin[2*gamma]*Cos[2*rot*t] *)
(* d22VAS = a22VAS *)
d22VAS = FullSimply[Coefficient[m22[[size - 2, 2]], Sin[2*gamma]]];

(* c21VAS = amplitude of Cos[gamma]*Cos[rot*t] *)
(* c21VAS = -b21VAS *)
c21VAS = FullSimplify[Coefficient[m22[[size, 2]], Cos[gamma]]];

(* d21VAS = amplitude of Sin[gamma]*Cos[rot*t] *)
(* d21VAS = a21VAS *)
d21VAS = FullSimplify[Coefficient[m22[[size, 2]], Sin[gamma]]];


(* a20VAS *)
a20VAS = FullSimplify[Sum[m22[[i]], {i, 1, size - 4}]];
a20MAS = FullSimplify[a20VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(*--------------- Table of a2jVAS and b2jVAS for VAS --------------*)

tableW20VAS = List[{"a20VAS", a20VAS},
                   {"a21VAS", a21VAS}, {"b21VAS", b21VAS},
                   {"a22VAS", a22VAS}, {"b22VAS", b22VAS}];

Print[TableForm[tableW20VAS,
      TableHeadings -> 
      {None, {"VAS", "expression of a2jVAS and b2jVAS in VAS"}}]];

Print["******************************************"];

(*--------------- Table of a2jMAS and b2jMAS for MAS --------------*)

tableW20MAS = List[{"a20MAS", a20MAS},
                   {"a21MAS", a21MAS}, {"b21MAS", b21MAS},
                   {"a22MAS", a22MAS}, {"b22MAS", b22MAS}];

Print[TableForm[tableW20MAS,
      TableHeadings -> 
      {None, {"MAS", "expression of a2jMAS and b2jMAS in MAS"}}]];

Remove[D2, alpha, beta, gamma, W2pas, eta, W2k, D2VAS, thetaVAS,
       rot, t, W20vas, m1, m2, m22, size, a22VAS, a21VAS, a20VAS,
       b22VAS, b21VAS, c22VAS, c21VAS, d22VAS, d21VAS, a22MAS,
       a21MAS, a20MAS, b22MAS, b21MAS, c22MAS, c21MAS, d22MAS,
       d21MAS, tableW20VAS, tableW20MAS];
 

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