Observation of stimulated nuclear spin echoes from solid hydrogen

ED 1

Physica I07B (1981) 269-270 North.Holland Publishing Company

OBSERVATION OF STIMULATED NUCLEAR SPIN ECHOES FROM SOLID HYDROGEN

Insuk Yu, Scan Washburn and Horst Meyer

Department of Physics, Duke University Durham, North Carolina 27706

Solid H 2 samples were investigated with a phase-coherent pulsed NMR technique. Upon application of a three-pulse sequence, up to four stimulated echoes were observed. These echoes were identified as quasi-quadrupolar ones in their nature, and brought about by the intramoleculear dipolar interactions. The decay of the primary stimulated echo was observed, indicating the presence of cross-relaxation between the spin states. A density matrix analysis using the quasi-quadrupolar model was successful in accounting for the results. Among a number of experimental probes, NM_R is a very frequently used one in the study of the properties of solid hydrogen, for instance in the orlentational order-dlsorder transition and the more recently investigated "quadrupolarglass" phase I. Because of the short inter-proton distance in an ortho-hydrogen molecule, the intra-molecular dlpole-dlpole interaction is large and sometimes it is possible to ignore the weaker intermolecular one or treat it separately. The nuclear spin Hamiltonian of a single hydrogen molecule in a magnetic field H o leads to the triplet states that are given to the first order by 2 = E+I = <+'I~ I÷÷>

y2h2 -ThH o +

(1-3 cos2@) 4r 3 2 2

E°

=

1 <++++÷IMI÷+++÷> = ~(i-3

(1)

cosae)

2 2 E_ 1 = <++I~ I++> = yhH o + ~ 3 -(I-3 c°s2@) 4r where y is the gyromagnetic ratio of the proton, r is the inter-proton distance and 8 is the angle between the inter-proton axis and the magnetic field. Metzger and Gaines 3 noticed that the same set of energy levels are obtained by considering the two spin I-½ particles (protons) as a fictitious spin I-I particle whose Hamiltonian is 22 ~eq

. -yhHoIz + I I~3-(1_3 cos28) (3Iz2_2)

(2)

r We note that this is exactly the same HamiltonJan as for a spin I-I particle with a quadrupole moment eQ in the presence of a magnetic field H o and an axially symmetric electric field gradient (~2V/~Z 2) with the Z axis in the direction of H o 2, Q = -yhHol Z + ~ Q

82V 2 ~Z---~(31Z-2)

(3)

0378-4363/81/0000-43000/$02.50 © North-HollandPublishingCompany

where eQ(~2V/~Z 2) is replacing (y2h2/r3(l-3K cos28)) in Eq. (2). Therefore, the spin system of solid hydrogen is expected to behave in a similar way to a system of spin I-I with nuclear quadrupolar interaction rather than a magnetic system of spin lffi½ 4. We found the properties of the spin-echo signal observed after a twopulse (81-T-82(~)) sequence to be in full agreement with the expectations from a density matrix analysis of a spin I-I system 5 indicating the validity of our quasl-quadrupolar model. Here we report the formation of stimulated nuclear spln-echoes observed after the application of three rf pulses (90°-T-~2-t-83) . Our density matrix analysis using the Hamiltonlan eq of Eq. (2) for spin I=I shows that four stimulatea echoes will form at times t+2T, 2t, 2t+T and 2t+2T 6. Such echoes were observed at the predicted times in samples of solid H 2 at various ortho concentrations and temperatures ranging from X=0.09 to X=0.35 and from T-0.08 K to T=I.3 K. A representative spectrum wlth a normal echo E and four stimulated echoes is shown in Fig. I. The height of the primary stimulated echo (El) was measured as a function of the duration t between the second (82 ) and the third (83)pulses and the decay of the envelope is shown in Fig. 22, together with the decay of the free induction decay (FID) signal after a (90°-t-90 °) pulse sequence that measured the spin lattice relaxation time T I. In Fig. 2b, the FID after a 90 ° pulse (with characteristic time T2*) and the echo decay from spin-spin relaxation (SSR, time constan~ T2) are shown for the sake of comparing the various time constants of the sample. The SSR decay was obtained from the difference between the respective decays of the echo following a (90°-T-90°0o) and a (90°-T-90°90 o) sequence. The Justification for this procedure follows from a density matrix analysis 5 . Because the decay rate of the stimulated echo envelope is faster than T1 -I, this indicates the presence of cross-relaxation processes between the spin states of the o-H 2 molecules

269

270

1.0

X --0.32 T =0.5K 9MHz

l

== :'=

'

I

'

%... ~

"..

I

' I J X =0.32 ] T = 0 5K "1

t.C

~. 0.."

o.,

!

E,L,~L'

L-

.-

.~

o

io

ioo ffsec'

I

I

T+t 21Y

2{*27-

t+2T time

2t,T

due to inter-molecular dipolar spin-flip interaction, and which we characterize by a time TI2. From an analysis for the spin I=I system, made along the lines of Ref. 6, the normalized height of the primary stimulated echo (E I) at the time t+2T after a three-pulse sequence is found to be S(t+2T) = -sin~2sin~3exp(-t/T1)ext(-t/Tl2 )

(4)

Therefore, TI2 can be obtained from the measured decay El(t) and from T I. The average value of TI2 can also be calculated approximately using the method in Appendix 2 of Ref. 6. We obtain

8

g T 2 exp

1

30

i

I

i

I

iO0 200 2 "E ( ~sec )

i

I"

300

dered and in the glass phase of solid H 2.

Fig. I. Spectrum of echoes from a (90o-41 psec60o-131 ~sec-60 o) three-pulse sequence. E is the ordinary echo and E 1 through E 4 are the stimulated echoes.

TI2 = ~

~'o t (msec}

Fig. 2. The primary stimulated echo E 1 vs. t. SLR and SSR are respectively the spinlattice and the spin-spin relaxation measured as described in the text. The FID represents the free induction decay after a 90 ° pulse of width 3 usec and measures T 2.

90°,82 E © T

boo,,

~T21J

(5)

where g is an adjustable parameter which takes into account the correlations in three-particle interactions. Using the relaxation times T 2 and T~ in Fig. 2b and using the value g=2.31 which was chosen in the work of Ref. 6, the cross-relaxatlon time TI2 was found to be T12=6.6 maec. This is rather in good agreement with what was obtained from the measured E l decay T12=9.2 msec ± 0.2 msec. In conclusion, the formation of the stimulated echoes we found in the solid hydrogen samples confirms the validity of our quasl-quadrupolar model, and a density matrix analysis permits the determination of the cross-relaxatlon processes, which we intend to study in the dlsor-

One of the authors (IY) is grateful to Professor A.A.V. Givson for his mentioning of the Ref. 7. The research was supported by a grant from the National Science Foundation.

REFERENCES: [I] l.P. Silvera, Rev. Mod. Phys. 52, 393 (1980) and references therein. [2] A. Abragam, The Principles of Nuclear Magnetism, (Oxford University Press, 1961), Ch. VII. ]3] D.S. Metzger and J.R. Gaines, Phys. Rev. 147, 644 (1966). [4] This may not be applicable when the intramolecular dlpole-dipole interaction is effectively reduced by the fast modulation of the intermolecular electric quadrupolequadrupole (EQQ) interaction. [5] I. Yu et al., to be published. [6] Such an analysis has been made for half integer spins by P. Mansfield, D.E. McLaughlln and J. Butterworth, J. Phys. C, Solid State Phys. 3, 1071 (1970). [7] A similar mDdel has been used for isolated methyl groups (=3/2). See P.S. Allen, W. Harding and P. Mansfield, J. Phys. C, Solid State Phys. ~, 189 (1972).