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Spin-lattice relaxation of quasi-steadystates in nitrogen-14 N.Q.R. multipulseexperimentsD.Ya. Osokin aa Kazan Physico-Technical Institute of the Academy of Sciences of theUSSR , Sibirski trakt 10/7, Kazan, 420029, U.S.S.R.Published online: 22 Aug 2006.

To cite this article: D.Ya. Osokin (1983) Spin-lattice relaxation of quasi-steady states in nitrogen-14N.Q.R. multipulse experiments, Molecular Physics: An International Journal at the Interface BetweenChemistry and Physics, 48:2, 283-291

To link to this article: http://dx.doi.org/10.1080/00268978300100211

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MOLECULAR PHYSICS, 1983, VOL. 48, No. 2, 283-291

Spin-lattice relaxation of quasi-steady states in nitrogen-14 N.Q.R. multipulse experiments

by D. YA. O S O K I N

Kazan Physico-Technical Institute of the Academy of Sciences of the USSR, Sibirski trakt 10/7, Kazan, 420029 U.S.S.R.

(Received 4 May 1982 ; accepted 18 September 1982)

Us ing average hami l ton ian theory the rate equa t ion descr ib ing the evolut ion of quas i -s teady states to equ i l ib r ium wi th the lattice in n i t rogen-14 N.Q.R. mul t ipu lse expe r imen t s is obta ined . In general, these states are character ized by a non-ze ro t ransverse magnet iza t ion . T h e exper imenta l curves of spin echo envelopes in the phase-a l t e rna ted mul t ipulse sequence are given and conf i rm the theory.

1. INTRODUCTION

In previous papers [1-3] it has been shown that the phase-alternated multipulse sequence tw1°'-(r-tw21S°°-2r-tw2°°-r)n originates a quasi-steady state in a quadrupolar spin-1 system for a time of the order of T 2. This quasi- steady state is characterized by the both longitudinal and transverse magnetiza- tions in the ratio depending on the cycle pulse width (tw2) rather than on the initial conditions (the preparatory pulse width t,,.1). As experiments show [2, 3] in the course of the further time development the system relaxes towards an equilibrium state with a non-zero transverse magnetization. In this paper we discuss the influence of spin-lattice interactions on the time evolution of the quasi-steady state in a quadrupolar spin-1 system irradiated by a phase-alternated multipulse sequence.

2. THEORY

We consider an ensemble of equivalent spins with I = 1 in the representation where the quadrupole hamiltonian .~Q = K[3Iz 2 - 12 + ~(Ix 2 - Iu2)] has diagonal form. The eigenstates and eigenvalues of ~ Q are

[ + ) = [ y ) = ( l + l ) + [ - 1 ) ) / X / 2 ; E + = K ( I + ~ ) ;

[0 ) = ] z ) = [0) ; E o = -2K; I - ? = l x ) = ( l + l ) - I - 1 ) ) / x / 2 ; E _ = K ( 1 - ~ ) .

Here Ix, I v and If are nuclear spin components in the principal axis system of the electric field gradient (EFG) tensor, [ + 1 ) , [0), and [ - 1 ) are the eigen- states of the operator If with eigenvalues + 1, 0, - 1 respectively, K = e 2 qQ/4 is the nuclear quadrupole coupling constant and ~ is the asymmetry parameter of the EFG tensor.

0026-8976183/4802 0283 804.00 ~ 1983 Taylor & Francis Ltd M.P. K

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284 D. Ya. Osokin

In general, the quadrupole spin-lattice relaxation in molecular crystals is governed by the torsional oscillations of molecules. The non-secular part of the hamiltonian of these interactions is usually written in frequency units as follows [4] :

a~'SL = OxOJ+o( I~,I z + I~I~)- OuOJ_o( I~l . + I . I z ) - OzoJ._( IxI~, + Iulx),

where

o~+0=%=E+-E0, OJ_o=OJu=E_-Eo, o ~ . _ = % = E + - E _

are the transition frequencies between quadrupolar energy levels E+, E 0, and E_, and 8x, 0N, O z are the instantaneous values of the angles of torsional oscillations in the EFG tensor principal axis system. Using the fictitious spin-l/2 operator formalism [5] this hamiltonian can be rewritten as

~ ' S L = 2 ~ mmOmS~ ra. r a=p

In the notation adopted (see Appendix) the lower index of S-operators corre- sponds to that of Pauli matrices and the upper index denotes the pairs of energy levels. In order to preserve the symmetry of notation the transition frequencies are denoted as OJ+o = E+ - E0, o~ 0_ = E o - E_, oJ_. = E_ - E+. In the interaction representation

a~eSL = 2 ~ oJ,~Om(S~ m cos o~mt + S=" sin oJmt ) rn=p

As it was pointed out earlier [1-3] the phase-alternated multipulse sequence conserves its cyclic properties in N.Q.R., that enables one to use the average

• hamiltonian theory [6] in the case concerned. Assuming that the radio-fre- quency field is applied at the transition frequency COp one obtains the zero- order average hamiltonian

,

-~ + w,k, sin + Suq cos

-co~g~ sin ~ ) + $7 ( w ~ , cos ~¢~- cokq..q sin ~ )

+Su,(oj ,g, c o s ~ + % g q s i n ~ - - ~ ) ] c o s ~ } (1)

where te te

gm= I Ore(t) cos mint dt and k,, = I O,~(t) sin wmt dt 0 0

are the real and imaginary parts of the instantaneous spectrum of 0,~ (it should be emphasized that they are random functions), t~ is the duration of a cycle and ~b = yHlcmt w is the rotation angle of nuclear magnetization, where y is the gyromagnetic ratio and c,n are the direction cosines of the radio-frequency field v e c t o r I - I 1 .

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Relaxation in multipulse N.Q.R. 285

The first-order average hamihonian has the form :

.~sL ta) - [2~% 2 - G'(~%)-OJq ~ G ' ( c O q ) - eOr 2 G'(oJ,)] ( S p cos --¢22

- Sfl' sin cos + [OJq 2 G'(wq) - 0,, 2 G (w~)](S~ - S,'),

where

(2)

te

G'(o~.,) = ~ O,,(t)O,,(t + r ' ) sin o~,,,r' dr' 0

is the imaginary spectral density of torsional oscillations. Next we need to proceed to solving the-quantum Liouville equation with

the average hamiltonians (1, 2). The well-known commutator expansion of the formal solution of this equation reduces the problem to the conventional procedure used in obtaining rate equations :

p(ntc) = exp (-i,~s1jnte)p(O ) exp ( i~sLntc)

(ntc) 2 =P(O)--i[:'~SL, P(O)]nt~-- T [J~SL, [~SL, p(0)]] + .... (3)

Now in order to calculate the commutators the initial density matrix p(0) should be determined. It should describe the state of internal equilibrium (quasi-equilibrium) in a spin system in presence of the multipulse sequence. Such a density matrix was obtained neglecting spin-lattice interactions [2, 3]. It was shown that the solution of quantum Liouville equation for the average hamiltonians of interactions broadening N.Q.R. lines contains a constant term :

where

~ = 1-~ f~p~(l_cosq~2)2+4col~(l+cos¢2) kT (5)

is the average spin temperature in a quasi-steady state. Here ~2p is a com- ponent of the tensor of homonuclear dipole-dipole interactions and o~ 1 is the sum of the constants of interactions in the average hamihonians of inhomo- geneous shifts (resonance offset), heteronuclear dipole-dipole interactions and the secular part of the hamiltonian of interactions with the torsional oscillations of the molecules [1].

The physical meaning of p~.~ may be clearly seen in the particular case of one-particle interactions. The solution of the quantum Liouville equation for the average hamihonian [1 ]

( ~ 1 = c°x SzP cos 2 u sin + co2(S f l - S~,)

and the initial density matrix after the preparatory pulse tw~

P ° = l [1] - k-T ( Szp c°s ~a - S 'p sin q3k T- "( S f l - Sz')

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286 D. Ya. Osokin

is as follows

cos ~2 sin - ~ ) cos ( q , 1 - - ~ )

I This solution is identical with that for a free induction decay signal, which

may be obtained from the above by setting ~ = 0

Pl =~ t [ 1 ] -"wv(S~v cos ~b 1 - Svv sin ~b 1 cos wi t+ Sx p sin sin COlt )

if one rotates the reference frame by the angle - ~ J 2 around x-axis in the fictitious subspace ' p '. So the constant term represents the longitudinal magnetization in a new reference frame (rotated by the angle -4,2/2). After averaging over the distribution of interactions (~ol) the oscillating terms vanish and a spin-1 system may be characterized by an average spin temperature. The solution for the total hamiltonian is more complicated and will be given elsewhere.

Thus there are two quasi-integrals of motion (average spin temperatures in a multipulse sequence representation) ~ and • in a spin-1 system, ~ describing the evolution of the population difference of p-levels (irradiated by the multi- pulse sequence) and B describing that between the half of the sum of p-levels populations and the population of the third level.

Now we may proceed to evaluating the commutators in (3) assuming that P(0) -Pss- The first-order average hamiltonian (2) commutes with p(0). The commutator of the zero-order average hamiltonian (1) with p(0) vanishes after an ensemble averaging. The double commutator is :

n 2 tc2[~SL (0), [~SL (°), p(0)]]

( cos ¢2 sin~-~){~c%[4wv2(ko2+g,2cos2~-~) = n 2 \ Szv -~ - S~,v

+½%~(kq2+gq2)(l+cos~)+½oo.2(k.2+g,2)(l+cos~)]

× [tOq2(kq2 +gq~)+w,2(kr2 +g,~)]}( l +cos~) .

Let us consider equation (3) with the explicit form of the right-hand side. Taking n = 1, transposing p(0), and dividing both sides of (3) by t~ one obtains,

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Relaxation in multipulse N.Q.R. 287

on the left-hand side, the density matrix derivative with respect to the effective discrete t ime nte. On the right-hand side the term (k,, 2 +g,,2) is the square of the modulus of an instantaneous power spectrum or, in other words, the square of the modulus of a current power spectrum increment. This increment, divided by to, gives the derivative of a current power spectrum with respect to ntc, which after ensemble averaging is equal to the spectral density [7] :

d

We also assume that km ~=gm ~ which causes O,,(t) to be a symmetric or anti- symmetric time function with equal probabilities.

Introducing the transition probabilities

w., =

and equating the coefficients of the same operators on both sides of equation (3) one obtains the coupled differential equations for a and/3 :

d , d-t = - oc[2 W'p + ~( W' a + W'.)] + 3fl(% - w.)( W'q - W'.)/2o%

d--] = - ~fl( W'q + W'r) + a~%( W'q - W'r)/Z(o,q - w,),

where

are reduced transition probabilities. The solution of these equations for e

= C,. exp ( - t/Tl.,) + CI exp ( - t/Tlt),

where

1

TI.~, t

and

= W'pAW Wtq--~ Wtr-Jv-~ [ ( W ' p - W'q)2-Je(W'q - Wtr)2--~(W'r - W'p)2] 112

W'q- W',) TI.~

,/(, ,) Tflo3(W'q- W . ) ~Fu T,,

gives the infinite temperature approximation due to the classical t reatment of the lattice. In order to avoid this, p(t) should be replaced by p ( t ) - Peq, where Peq is the equilibrium value of p. In the absence of a multipulse sequence Peq is known to be given by Boltzmann distribution. When a multipulse sequence is applied Peq is assumed to be the Boltzmann density matrix transformed by the sequence, which may be obtained from equation (5) by setting 41=0. This assumption will be checked experimentally, but the quantum-mechanical treat- ment of the lattice in obtaining the rate equation is also desirable. With this

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288 D. Ya. Osokin

assumption the final expression for ~ is as follows

1 [ 8mlz(m + cos 4e) ]{1 [cos (41 ~ ) 1 + f~p2(1 _ cos 4~) + 4o~x2(1 + cos 42) 2

- c o s 1+ l[Tx~-l/T11 /t exp -

+ 1 ] e x p - +cos . (6)

Thus, the phase-alternated multipulse sequence changes the transition probabilities by the factors

according to the levels irradiated, and the equilibrium state becomes dependent on the intensity of the radio-frequency field. Except for these changes, (6) corresponds to the equation of free relaxation in a spin-1 system [4] and turns into it when 42 = O.

3. EXPERIMENTAL RESULTS AND DISCUSSION The experiment was carried out with a coherent pulsed N.Q.R. spectro-

meter on a NaNO 2 monocrystalline sample at 77 K. A brief description of the spectrometer has been given elsewhere [1]. In order to examine the theory the time dependence of the spin echo intensity given by equation (6) was measured experimentally for characteristic values of 42- It follows from (6) that the transverse magnetization decays to zero with a double-exponential time de- pendence when 41=90 ° and 4~=180 °. On the other hand, when 41=42 the time dependent term in (6) equals zero and a non-zero equilibrium state is established for a time of the order of T z. The experimental curves of the spin echo envelopes for these two cases are shown in figure 1 and confirm the theory. Setting 41 = 180° and 42 = 90° one obtains another important result of equation (6). It is predicted that the spin system relaxes from the inverted populations of the levels p, to the equilibrium values realized for the case of 41 = 42 = 90°, with the sign of the transverse magnetizations inverted. The experimental spin echo envelope is shown in figure 2 and also agrees with the theory. Another check of the theory arises from the comparison of the relaxation rates in figures 1 and 2. The reduced transition probabilities W'p and W'q+ W'r for the case of 42=180 ° are 0.5 Wp and 0.5 (Wq+ Wr) whereas for the case of 42=90 ° they are 0.75 Wp and 0.85 (W q+ Wr) which agrees with the more rapid relaxation rate in figure 2.

In conclusion we intend to discuss the application of the multipulse tech- nique for spin-lattice relaxation measurements in nitrogen-14 N.Q.R. It is seen that multipulse sequences can considerably shorten the time of relaxation measurements, but there are some circumstances restricting the utilization of equation (6).

First, the dependence of the reduced transition probabilities on the rotation angle of nuclear magnetization, 42, requires averaging of (6) over 42 when

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Relaxation in multipulse N.O.R. 289

J f0

Figure 1. Nitrogen-14 N.Q.R. spin echo envelopes in the phase-alternated multipulse sequence at the transition frequency f+0=4-93125 MHz for the cases: ~1=9~ = 90°-1 and2~l=~.a=180°-2 ; r=0.5ms.

/ I I

to t~1 =ntJs)

Figure 2. The same as in figure 1 for the case of ,61 = 2 d'~ = 180 ~.

polycrystalline samples are used. Th e possibility of obtaining the analytical expression for such an integral seems to be doubtful so computer calculation of the three transition probabilities characterizing the tensor of molecular motion is likely to be required. In this connection, a s tudy of the spin-latt ice relaxation of quasi-steady states for other multipulse sequences, in an at tempt to find a simpler analogue of (6) appears to be of interest.

Second, the average hamiltonian theory holds only for the case of t c~ T2 and, as a result, all the expressions obtained using this theory are independent of to. However there is a pronounced dependence on T of T2~ in both N.M.R. and N.Q.R. [6, 8]. Investigation of such a dependence in the pulsed spin locking sequence t,,l°°-(~--tw~9°°-r), [3] shows that T ~ is independent on ~" when ~-< 0-7 ms, then there is a sharp T ~ decrease up to the T 2 value when ~" increases to 2-3 ms. Similar T2~ behaviour can be observed in the phase- alternated multipulse sequence also. Th e spin-spin relaxation time T~ in N a N O 2 at the transition f requency [+o is about 7 ms. Taking this into account one may draw the approximate conclusion that the spin echo envelope decay is governed by molecular motions when ~ < 0.1 T 2.

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290 D. Ya. Osokin

Another contribution to the spin echo envelope decay can arise from the so-called multispin resonances predicted [9] and observed [10] in N.M.R. pulsed spin locking. In short, they are caused by the spin quantization in a mean radio-frequency field and the absorption of modulation frequency quanta. Although the phase-alternated multipulse sequence has zero mean field the possibility of such phenomenon should be considered.

Third, in obtaining equation (6) we do not take into account ultra-slow molecular motions having correlation times of the order of t e [11]. In order to do this further development of the theory is required.

The author wishes to thank Professor B. N. Provotorov for critically reading the manuscript and helpful discussions.

APPENDIX

In this Appendix we collect the notations and definitions of fictitious spin-1/2 operators and their commutation relations to facilitate reading the paper.

In the representation where the quadrupole hamiltonian is diagonal, fictitious spin-l/2 operators can be written in terms of components of a nuclear spin as follows :

& l - o _ ~- I + o _ x l d ~ , ), & +o _ - 2 x, Su - ~ ( l u I , + =1(/2 luZ),

s~o--~-~/~, s o - = i ( i d . + U = ) , & o - = ½ ( / ~ - U ) ,

S.-+=½I. , S~,-4-=1(I.~I~,+I~I,:), S.-+=½(IuZ-I,fl)

and in matrix representation have the form :

-~o h h S~' =~ 0 , S+O=

0

o o h[ao ] i 0 , S~ + = ~ 0 - 1 0 ,

0 0 0 0 0

& o - = -2 0 ,

- i

h S O - = 0 0 , Sfl- = ~ 0 t ,

0 1 0 0 1

h h S~-+ = ~ 0 , S ~ - + =

0

00 ] lo0] o o , & - + = ~ o 0 .

- i 0 0 1

This last table clearly shows the physical meaning of the notation adopted.

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Relaxation in multipulse N . Q.R. 291

T h e fol lowing c o m m u t a t i o n re la t ions are val id :

[Sap, Sbp ] = iScP , [Szp , Szq ] = [(Szp - S q), Sar ] = O,

i i s ~ [sxp ' s ~ ] = - I S / , s p] =_-~ s~,, [ s ~ , s ~ ] : - [ s x ~ , s x~ ] = - ~ ~ ,

i i [ s 2 , so~] = [ s ~ , s t ] = ~ s ~, [s~p, s ~ ] = [ s ~, s~,] = - ~ s ~

where a, b, c deno te x, y , z or cyclic p e r m u t a t i o n , and p, q, r deno te the pa i rs of levels + 0 , 0 - , - + or cyclic p e r m u t a t i o n .

REFERENCES

[1] OSOKIN, D. YA., 1980, Phys. Stat. Sol. (b), 102, 681. [2] OSOKIN, D. YA., 1982, Phys. Stat. Sol. (b), 109, K7. [3] OSOKIN, D. Ya., 1981, Coherent Multipulse Sequences in Nitrogen-14 NQR, Report

on 6th International Symposium on N.Q.R. Spectroscopy, Moscow; 1982, ft. molec. Struct., 83, 243.

[4] SAFIN, I. A., and OSOKIN, D. Y^., 1977, Nuclear Quadrupole Resonance in Nitrogen- containing Compounds (in Russian) (Nauka) ; and references therein.

[5] VEGA, S., 1975, ff. chem. Phys., 63, 3769 ; 1978, Ibid., 68, 5518. V~.CA, S., and PINES, A., 1977, ft. chem. Phys., 66, 5624.

[6] WAt:GH, J., 1978, New N M R Methods in Solid States Physics (in Russian) (Mir). [7] KHhaKEVlCH, A. A., 1967, Spectra and Analysis (in Russian) ( G I T T L ) . PACE, C. H.,

1952, ft. appl. Phys., 23, 103. LAMPARD, D. G., 1954, ft. appl. Phys., 25, 802. [8] MARtNO, R., and KLAINEa, S., 1977, ~. chem. Phys., 67, 3388. [9] IVANOV, Yu. N., PnOVOTOaOV, B. N., and FELDMAN, E. B., 1978, Zh. dhsp. teor. Fiz.,

75, 1847. [10] EROr~EV, L. H., SHUMM, B. A., and MANELIS, G. B., 1978, Zh. ~ksp. teor. Fiz., 75,

1837. [11] MErmlNC, M., 1976, High Resolution N M R Spectroscopy in Solids (Springer-Verlag).

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