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




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

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

The first-order quadrupole interaction is related to V(2,0):


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

V20-VAS

with

V2k-VAS

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

V20-VAS-final

The analytical expressions of a0, an, and bn 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 V(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];

(* V2pas 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 unit *)

V2pas = {{eta/2, Sqrt[3/2], eta/2}};

MatrixForm[V2pas];

(* V2k is a row-matrix with 5 columns *)
V2k = V2pas.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];

(* V20vas is an expression *)
V20vas = V2k.ComplexExpand[D2VAS];

m1 = Expand[V20vas] /. {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];

(* a2VAS = amplitude of Cos[2*gamma]*Sin[2*rot*t] *)
a2VAS = FullSimplify[Coefficient[m22[[size - 3, 2]], Cos[2*gamma]]];
a2MAS = FullSimplify[a2VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* b2VAS = amplitude of Sin[2*gamma]*Sin[2*rot*t] *)
b2VAS = FullSimplify[Coefficient[m22[[size - 3, 2]], Sin[2*gamma]]];
b2MAS = FullSimplify[b2VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* a1VAS = amplitude of Cos[gamma]*Sin[rot*t] *)
a1VAS = FullSimplify[Coefficient[m22[[size - 1, 2]], Cos[gamma]]];
a1MAS = FullSimplify[a1VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(* b1VAS = amplitude of Sin[gamma]*Sin[rot*t] *)
b1VAS = FullSimplify[Coefficient[m22[[size - 1, 2]], Sin[gamma]]];
b1MAS = FullSimplify[b1VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];


(* c2VAS = amplitude of Cos[2*gamma]*Cos[2*rot*t] *)
(* c2VAS = -b2VAS *)
c2VAS = FullSimplify[Coefficient[m22[[size - 2, 2]], Cos[2*gamma]]];

(* d2VAS = amplitude of Sin[2*gamma]*Cos[2*rot*t] *)
(* d2VAS = a2VAS *)
d2VAS = FullSimply[Coefficient[m22[[size - 2, 2]], Sin[2*gamma]]];

(* c1VAS = amplitude of Cos[gamma]*Cos[rot*t] *)
(* c1VAS = -b1VAS *)
c1VAS = FullSimplify[Coefficient[m22[[size, 2]], Cos[gamma]]];

(* d1VAS = amplitude of Sin[gamma]*Cos[rot*t] *)
(* d1VAS = a1VAS *)
d1VAS = FullSimplify[Coefficient[m22[[size, 2]], Sin[gamma]]];


(* a0VAS *)
a0VAS = FullSimplify[Sum[m22[[i]], {i, 1, size - 4}]];
a0MAS = FullSimplify[a0VAS /. thetaVAS -> ArcCos[Sqrt[1/3]]];

(*--------------- Table of aiVAS and biVAS for VAS --------------*)

tableV20VAS = List[{"a0VAS", a0VAS}, 
                   {"a1VAS", a1VAS}, {"b1VAS", b1VAS}, 
                   {"a2VAS", a2VAS}, {"b2VAS", b2VAS}];

Print[TableForm[tableV20VAS,
      TableHeadings -> 
      {None, {"VAS", "expression of aiVAS and biVAS in VAS"}}]];

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

(*--------------- Table of aiMAS and biMAS for MAS ---------------*)

tableV20MAS = List[{"a0MAS", a0MAS}, 
                   {"a1MAS", a1MAS}, {"b1MAS", b1MAS}, 
                   {"a2MAS", a2MAS}, {"b2MAS", b2MAS}];

Print[TableForm[tableV20MAS,
      TableHeadings -> 
      {None, {"MAS", "expression of aiMAS and biMAS in MAS"}}]];

Remove[D2, alpha, beta, gamma, V2pas, eta, V2k, D2VAS, thetaVAS,
       rot, t, V20vas, m1, m2, m22, size, a2VAS, a1VAS, a0VAS,
       b2VAS, b1VAS, c2VAS, c1VAS, d2VAS, d1VAS, a2MAS,
       a1MAS, a0MAS, b2MAS, b1MAS, c2MAS, c1MAS, d2MAS,
       d1MAS, tableV20VAS, tableV20MAS];
 

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