Second-order quadrupole shift for MAS rotating crystal.




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Second-order quadrupole shift for MAS rotating crystal

The second-order quadrupole interaction is related to V(2,1) and V(2,2):

second-order quadrupole interaction

(Eq. 1)


The second-order quadrupole interaction shifts the energy level |m> by an amount:

Energy level shift due to the second-order quadrupole interaction

(Eq. 2)


In the spectrum, the second-order quadrupole shift of the line position associated with the transition (m - 1, m) is

Line position due to the second-order quadrupole interaction

(Eq. 3a)


Line position due to the second-order quadrupole interaction

(Eq. 3b)


In MAS NMR experiment, V(2,p) is defined by

EFG tensor component V(2,p)

(Eq. 4)


and V(2,k)MAS by

EFG tensor component V(2,k)MAS

(Eq. 5)


For MAS rotating crystal, first Eq. 4 is used in Eq. 3b. The second-order quadrupole shift of the central-transition line with m = 1/2 is

Line position due to the second-order quadrupole interaction

(Eq. 6)


Then Eq. 5 is used in Eq. 6 that becomes:

Central line position due to the second-order quadrupole interaction for MAS crystal

(Eq. 7)


with

Coefficients D, E, and F

(Eq. 8)


We also provide Mathematica notebook for calculating V(2,1)V(2,-1) and V(2,2)V(2,-2) in the condition of fast rotating crystal. They allow us to define Eq. 6. This notebook also generate V(2,2)MASV(2,-2)MAS + (V(2,0)MAS)2 used in Eq. 6. Their analytical expressions 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, these analytical expressions are obtained in 13 seconds.


(*The EFG tensor in the reference MAS, V2kMAS is a row - matrix with 5 columns*)

V2kMAS = {V22MAS, V21MAS, V20MAS, V2m1MAS, V2m2MAS};

(*D21MAS and D2m1MAS are reduced forms (5 rows x 1 column) of the 2-nd 
order Wigner active rotation matrix*)

D21MAS = {{-(1/2)(1 + Cos[thetaMAS])Sin[thetaMAS]*E^(-2I rot*t)},
      {(Cos[thetaMAS]^2 - (1/2)(1 - Cos[thetaMAS]))*E^(-I rot*t)},
      {Sqrt[3/8]Sin[2thetaMAS]},
      {( (1/2)(1 + Cos[thetaMAS]) - Cos[thetaMAS]^2)*E^(I rot*t)},
      {(1/2)(1 - Cos[thetaMAS])Sin[thetaMAS]*E^(2I rot*t)}};

D2m1MAS = {{-(1/2)(1 - Cos[thetaMAS])Sin[thetaMAS]*E^(-2I rot*t)},
      {((1/2)(1 + Cos[thetaMAS]) - Cos[thetaMAS]^2)*E^(-I rot*t)},
      {-Sqrt[3/8]Sin[2thetaMAS]},
      {(Cos[thetaMAS]^2 - (1/2)(1 - Cos[thetaMAS]))*E^(I rot*t)},
      {(1/2)(1 + Cos[thetaMAS])Sin[thetaMAS]*E^(2I rot*t)}};

(*D22MAS and D2m2MAS are reduced forms (5 rows x 1 column) 
of the 2-nd order Wigner active rotation matrix*)

D22MAS = {{(1/4)(1 + Cos[thetaMAS])^2*E^(-2I rot*t)},
      {(1/2)(1 + Cos[thetaMAS])Sin[thetaMAS]*E^(-I rot*t)},
      {Sqrt[3/8]Sin[thetaMAS]^2},
      {(1/2)(1 - Cos[thetaMAS])Sin[thetaMAS]*E^(I rot*t)},
      {(1/4)(1 - Cos[thetaMAS])^2*E^(2I rot*t)}};

D2m2MAS = {{(1/4)(1 - Cos[thetaMAS])^2*E^(-2I rot*t)},
      {-(1/2)(1 - Cos[thetaMAS])Sin[thetaMAS]*E^(-I rot*t)},
      {Sqrt[3/8]Sin[thetaMAS]^2},
      {-(1/2)(1 + Cos[thetaMAS])Sin[thetaMAS]*E^(I rot*t)},
      {(1/4)(1 + Cos[thetaMAS])^2*E^(2I rot*t)}};

(*thetaMAS is the magic angle*)

D21MAS = D21MAS /. {thetaMAS -> ArcCos[Sqrt[1/3]]};
D2m1MAS = D2m1MAS /. {thetaMAS -> ArcCos[Sqrt[1/3]]};
D22MAS = D22MAS /. {thetaMAS -> ArcCos[Sqrt[1/3]]};
D2m2MAS = D2m2MAS /. {thetaMAS -> ArcCos[Sqrt[1/3]]};

(*V21mas, V2m1mas, V22mas, and V2m2mas are expressions*)

V21mas = FullSimplify[ComplexExpand[V2kMAS.ComplexExpand[D21MAS]]];
V2m1mas = FullSimplify[ComplexExpand[V2kMAS.ComplexExpand[D2m1MAS]]];
V22mas = FullSimplify[V2kMAS.ComplexExpand[D22MAS]];
V2m2mas = FullSimplify[V2kMAS.ComplexExpand[D2m2MAS]];

z21mas = Expand[V21mas*V2m1mas] ;
z22mas = Expand[V22mas*V2m2mas] ;

(*In rapid rotation of the sample*)

z21mas = z21mas /. {E^(x_) -> 0};
z22mas = z22mas /. {E^(x_) -> 0};

Vterm = FullSimplify[z21mas(24m(m - 1) - 4S(S + 1) + 9) + (1/2)z22mas (12m(
    m - 1) - 4S(S + 1) + 6)];

VtermMAS = Collect[Collect[Collect[Vterm, 
    V20MAS^2], V21MAS V2m1MAS], V22MAS V2m2MAS];

VtermMAS = FullSimplify[
    Expand[VtermMAS[[1, 1]]]] + FullSimplify[Expand[VtermMAS[[1, 
              2]]]] + FullSimplify[Expand[VtermMAS[[1, 3]]]];
VtermMAS = VtermMAS /. {m -> 1/2};
VtermMAS = Simplify[VtermMAS];

coef1 = "omega(2)MAS(-1/2,1/2) = (-2/omegaLarmor)(eQ/(4S(2S-1)hBar))^2 {";
Print[coef1, VtermMAS, "}"];

(**************************************************************************)

(*In the reference PAS, 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*)

D20 = {{Sqrt[3/8]*Sin[beta1]^2*E^(-2I alpha1)},
      {(-1 + 3*Cos[beta1]^2)/2},
      {Sqrt[3/8]*Sin[beta1]^2*E^(2I alpha1)}};

(*V20MAS is an expression*)

V20MAS = FullSimplify[V2pas.ComplexExpand[D20]];
V0MAS = Expand[V20MAS*V20MAS];

(*************************************************)

(*D22 and D2m2 are reduced forms (3 rows x 1 column) of the 2-nd order 
Wigner active rotation matrix*)

D22 = {{(1/4)(1 + Cos[beta1])^2*E^(-2I(alpha1 + gamma1))},
      {Sqrt[3/8]Sin[beta1]^2*E^(-2I gamma1)},
      {(1/4)(1 - Cos[beta1])^2*E^(2I(alpha1 - gamma1))}};

D2m2 = {{(1/4)(1 - Cos[beta1])^2*E^(-2I(alpha1 - gamma1))},
      {Sqrt[3/8]Sin[beta1]^2*E^(2I gamma1)},
      {(1/4)(1 + Cos[beta1])^2*E^(2I(alpha1 + gamma1))}};

(*V22MAS and V2m2MAS are expressions in eq units*)

V22MAS = FullSimplify[V2pas.ComplexExpand[D22]];
V2m2MAS = FullSimplify[V2pas.ComplexExpand[D2m2]];

(*V2MAS is an expression in eq.eq units*)
V2MAS = TrigExpand[V22MAS V2m2MAS];

(*************************************************)

Vterm = V2MAS + V0MAS;

Vterm = Expand[Vterm /. {Sin[beta1]^2 -> 1 - Cos[beta1]^2,
        Sin[beta1]^4 -> 1 - 2 Cos[beta1]^2 + Cos[beta1]^4,
        Sin[alpha1]^4 -> 1 - 2 Cos[alpha1]^2 + Cos[alpha1]^4,
        Sin[2alpha1]^2 -> 1 - Cos[2alpha1]^2,
        Sin[alpha1]^2 -> 1 - Cos[alpha1]^2}];

Vterm = Expand[Vterm /. {Cos[alpha1]^2 -> (1 + Cos[2alpha1])/2,
          Cos[alpha1]^4 -> (1 + Cos[2alpha1])^2/4}];

Vterm = Collect[Collect[Vterm, Cos[beta1]^4], Cos[beta1]^2];

(*Suppression of the double curve brackets {{}} of Vterm*)
ome = Vterm[[1, 1]];
long = Length[ome];

Vterm = Factor[Take[ome, long - 
        2]] + Factor[ome[[long - 1]]] + Factor[ome[[long]]];

coef2 = "omega(2)MAS(-1/2,1/2) = (-2/omegaLarmor)(eQeq/(4S(2S-1)hBar))^2 ((-3/4)+(S+1)S){";
Print[coef2, Vterm, "}"];

Remove[D21MAS, D2m1MAS, D22MAS, D2m2MAS, V2kMAS, V21mas, V2m1mas, V22mas, 
       V2m2mas, z21mas, z22mas, ome, V2pas, D20, D22, D2m2, V22MAS, 
       V2m2MAS, V20MAS, V2MAS, V0MAS, VtermMAS, Vterm, coef1, coef2];
 

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