Definition

• EFG tensor
• EFG tensor
  (mathML)
• EFG tensor
  (mathML + SVG)

  ---

• Hamiltonian in high field
• Hamiltonian in high field
  (mathML)

  ---

• Wigner rotation matrices
• Static crystal
• Rotating crystal

  ---

• V20 static
• Second-order quadrupole shift, V21 and V22 static
• W20 static
• W40 static

  ---

• Second-order quadrupole shift, V20 and V22 MAS
• V20 VAS (MAS)
• W20 VAS (MAS)
• W40 VAS (MAS)

One-pulse MAS

MQ-MAS

Reference

Analytical expression of V_(2,0) for static crystal in NMR

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

In static NMR experiment, V_(2,0) is defined by:

Its analytical expression 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, the analytical expression of V_(2,0) is obtained in 5 seconds.

---

(* 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}};

(* D2 is a reduced form (3 rows x 1 column) of the 2-nd order 
   Wigner active rotation matrix *)

D2 = {

        {Sqrt[3/8]*Sin[beta1]^2*E^(-2*I*alpha1)},

       {(-1 + 3*Cos[beta1]^2)/2},

        {Sqrt[3/8]*Sin[beta1]^2*E^(2*I*alpha1)}

     };

(* V20static is an expression *)
V20static = FullSimplify[V2pas.ComplexExpand[D2]];

(* suppression of the double curve brackets {{}} of V20static *)
Print[V20static[[1, 1]]];

Remove[V2pas, eta, D2, alpha1, beta1, V20static];