The calculation of NMR spectra of rotating dipolar solids by the canonical transformation technique

Volume 88A, number 1

PHYSICS LETTERS

15 February 1982

THE CALCULATION OF NMR SPECTRA OF ROTATING DIPOLAR SOLIDS BY THE CANONICAL TRANSFORMATION TECHNIQUE V.V. LAIKO and B.N. PROVOTOROV Institute of Chemical Physics, Academy of Sciences of the USSR, Chernogolovka, USSR Received 11 November 1981

23Na in a single crystal of sodium chloride rotating at the magic angle are calculated by the canonical NMR spectratechnique. of transformation

Magic angle sample spinning (MASS) is now one of the most effective line narrowing techniques in NMR of solids. The first MASS experiments in solids [1,2] showed that sample rotation produces a binary effect. First, there is a significant line narrowing if the rotational frequency wr exceeds the local field frequency ~“loc’ and secondly, sample rotation generates the rotational side bands. The first phenomenon can be successfully accounted for by the average hamiltonian theory [3]. The side band intensities were calculated earlier either by solution of the model two-spin problem [4] or by introduction of the simplified dipolar hamiltonian [5]. The construction of general theory is connected with the problem of separation of slow nuclear spin motions, due to the dephasing of the spins in time-averaged local fields, from the lowamplitude fast motions accounting for the rotational side bands. The same problem in the pulsed NMR has been successfully solved recently by the canonical transformation technique [6]; this technique will be used here to calculate the NIMR absorption spectra in MASS experiments, The hamiltonian of a sample under MASS conditions in the rotating frame can be written as -

W1~x+~C~(t),

(1)

where ~ is the deviation of radio frequency w1 from the resonance value and ~C~(t)is the secular part of the dipole—dipole interaction:

0 031-9163/82/0000—0000/S 02.75 © 1982 North-Holland

~Z(t)

=

y2h

~iI~ Zik~bik(t), i>k

-

Ii sin2O~kcos2(wrt + ctjk)12

Zik =

=

sin

+

Here

0ik

31iz’kz

20ik

C05(Wrt + ~ik)12”2.

(2)

is the angle between r~kand H

0, a~kis the azimuth angle of rik at t = 0. In the interaction representation the hamiltonian (1) becomes: =

—w1F~exp(i~t)/2

+ ~C~(t),

—

J±=

w1I~ exp(—i~t)/2

±Iz

(3)

Now it is convenient to perform the canonical transformation of the density matrix ~ = exp[—ii~(t)}~ exp[iâ(t)]

,

â(t) =

—

f~c~(t) dt (4)

This transformation does not change the form of the density matrix equation. The main part of the hamiltonian in the transformed frame [6] is = ~w1I~ exp(i~t)/2 w11 exp(—h~t)/2 —

_i[f~c~(t)dt,~]/2.

(5)

Taking into account eq. (2) we can express eq. (5) as

51

Volume 88A, number 1

=

~eff

+

2

PHYSICS LETTERS

and 125 Hz, respectively. The theoretical ratios of 2Wr are I the side0.024 bandsatintensities at 0, wr and 0.36 = 800 Hz (experimental values are

—w1I~exp(i&)/2

n1 i>k aZ~k, 1~Jexp(i&

—

iflwrt)~+ {c.cj.

(6)

Here ~eff and dik are defined as ~eff

=

i 7h

~II~[Zik,Zil]~IklI(wI1~~kril)

=

2~ik a~ 5in 1sin 1)/16 20ik sin 2011 sin(~~k — ct~i) ,

+ sin a~k= 2~2y2hw 20ik expOaik)/(8wrr~k), 1 sin exp(2ictik)/(l6wrr?k). a~k= yhw 2O~k 1 sin

broadening interaction which is independent of spec(7)

As follows from the form of the harniltonian (6) in the transformed frame, it is ~eff that determines slow dephasing of spins. So the second moment M 2 of the spectrum lines can be written as 2}ISP{1~. (8) M2 SP{[Weff,Ix] Taking into account eq. (7) we find that the second moment in eq. (8) coincides with that obtained by

the average hamiltonian theory. From the other hand, squares of the absolute values of harmonic coeffidents in eq. (6) determine the intensities of respec-

tive side bands. A comparison of theory and experiment has been performed for the 23Na absorption spectrum of a single crystal of sodium chloride in MASS experi-

ments [7]. The theoretical values of the line halfwidths from eq. (8) are 114 Hz at = 800 Hz and 57 Hz at = 1600 Hz; experimental values are 245

52

IHz: 0.55 : 0.036) and 1: are 0.092 0.006 at Hence = 1600 (experimental values I : :0.21 : 0). there is a qualitative agreement between the theory and the experiment. For the quantitative discrepancies there are, probably, two reasons. The present theory is the first step in the approximation procedure of the canonical transformation technique [6]. The small parameter of this theory Wloc/Wr is not really in the experiment (1 at = 800 Hz and 0.5 at small = 1600 Hz). Second, according to [8], some

,

i>k>l 20~ksin2O~

~ik1

15 February 1982

imen rotation is presented in sodium chloride ciystals. This interaction, which is probably the quadrupolar interaction of the 23Na nucleus, due to the imperfections of crystal structure, can obscure the

whole picture.

Referr’nces [1] E.R. Andrew, A. Bradbury and R.G. Fades, Nature 182 (1958) 1659. [2] l.J. Lowe, Phys. Rev. Lett. 2 (1959) 285. [3] M. Matti3300. Maricq and J.S. Waugh, J. Chem. Phys. 70 (1979) [4] E.R. Andrew and V.T. Wynn, Proc. R. Soc. 291A (1966) 257. [5] S. Clough and l.R. Gray, Proc. Phys. Soc. 79 (1962) 457. [6] B.N. Provotorov and E.B. Feldman, Zh. Eksp. Teor. Fiz. 79(1980)2206. [7] E.R. Andrew, Prog. Nuci. Magn. Reson. Spectrosc. 8 (1971) 13. 181 F.R. Andrew, A. Bradbury and RG. Eades, Nature 183 (1959) 1802.