ORDER PARAMETER SINGULARITIES IN SUPERFLUID 3He CONFINED TO SMALL CYLINDERS

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ORDER PARAMETER SINGULARITIES IN

SUPERFLUID 3He CONFINED TO SMALL

CYLINDERS

C. Gould, D. Lee

To cite this version:

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JOURNAL DE PHYSIQUE CoZZoque C6, supplgment au no 8, Tome 39, aoi2 1978, page C6-65

ORDER PARAMETER SINGULAR I T 1 ES I N SUPERFLUID 3 ~ e CONFINED TO SMALL CYLI NDERSt

C.M. Gould and D.M. Lee

Department of P h y s i c s , Come22 U n i v e r s i t y Ithaca, New Y o r k 14853 U.S.A.

RQsum6.- Nous avons mis de I'~H~-A dans des cylindres trSs 6troits afin de stabiliser des textures avec des singularites topologiques r6sultantes. Nous Btudions des textures singulisres qui en r6sul- tent par RMN.

Abstract.- We have placed superfluid 3 ~ e - ~ into verynarrow cylinders in order to stabilize textures with dependable topological singularities. The resulting singular textures are studied by NMR.

Defects in ordered media is a subject of great current inte:est /I/. We report here the first step in studying singularit'ies in superfluid 3 ~ e by loo- king at 3 ~ e - ~ trapped in very small cylinders by means of NMR in a Pomeranchuk cell. The superfluid is introduced into 1.2xl0~c~linders (2pm diameter) which have a length to diameter ratio of 1500. We then study only liquid in these tiny cylinders by excluding bulk liquid from the experimental region. Owing to the boundary condition that must be perpendicular to walls, in long cylinders there is only one class of textures which does not inherently possess singular structures, the usual example being the Mermin-Ho texture 121. When the diameter is com- parable to or smaller than typical bending lengths

(around 10pm) the Mermin-Ho texture is very expen- sive in terms of bending energies. 'Hence this texture should be excluded from our tiny cylinders, leaving only textures which are singular.

Our probe of the texture is CW NMR. The fre- quency shifts observed in transverse and longitudinal experiments on the superfluid come from the di-

A a

polar coupling of the Q and d vectors proportional to

(2.il2.

Singularities in the vector field do not have corresponding singularities in the d field

A

(because d does not suffer the perpendicular boun-

a a

dary condition) so that in our cylinders d and Q in general will be misaligned. This changes the dipole energy and hence the expected resonance fre- quencies. In our experiments an external magnetic field as well as the RF field may be applied paral- lel or perpendicular to the axis of the cylinders. The different resonance configurations allow us to study the curvature of the dipole energy in diffe-

t Work supported by the National Science Foundation through Grant DMR-76-21669 and through the Cornell Materials Science Center Grant DMR-76-81083

rent directions. Our observations are summarized in figure 1 which shows frequency shifts of transverse and longitudinal NMR in the cylinders with respect to bulk liquid signals.

7 " 0

Frequency

-

Fig. 1 : Summary of data : Absorption vs. frequency for different modes. The small bump labelled "Bulk" is absorption from bulk liquid away from the experimental region. The huge solid 3 ~ e signal is shown as the cross-hatched region. In (b) and (c) a time dependence of the relative size of the two transverse pore resonances is seen.

The clearest way to plot longitudinal frequen- cies is to divide by the bulk frequency so as to re- move its strong temperature dependence. The resulting graph is shown in figure 2. To generate the same type of graph for transverse data we assume that the field and temperature dependence is ana- logous to the bulk behaviour so w2 = ( Y % ~ + ( R ~ Q ~ ) ~ and we then plot the ratio RT. The qualitative shape of the resulting curves follows the longitudinal

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curve in that at low temperatures it levels off but NMR frequency shifts in 2 u cylinders for H paral- near T it drops, though the values are different. lel to the cylinder axes at pressures below melting At the time of this writing it is not known whether pressure /3/. Their results agree qualitatively with the simple sum of squares relationship holds in the the data taken in this configuration in our work,

pores. although numerical factors disagree.

References

084 -

082

-

- /1/ Kleman,M. and Toulouse,G., J. Physique Lett.

<

_-

37 (1976) L-149. Mermin,N.D., in Quantum Fluids

and Solids, edited by S.B. Trickey, E.D. Adams 080- - and J.W. Dufty (Plenum Press, New York, 1977) n' p. 3, and P.W. Anderson and R.G. Palmer ibid.

078-

-

p. 23

/2/ Mermin, N.D.,. and Ho, Tin-Lun, Phys. Rev. Lett. 36 (1976) 596

-

/3/ Saunders,J., Betts,D.S., Brewer,D.F., Swithenby S.J., and Truscott, W.S., preprint.

Fig. 2 : Longitudinal resonance frequency as a fraction of the bulk frequency plotted vs. temperature.

A very simple model to calculate NMR frequen-

cies in these tiny cylinders was suggested by Taka-

..

gi (private communication). Assume an R texture of figure 3 with a perpendicular magnetic field.

Fig. 3 : A model calculation. Given a perpendicular magnetic field, the d vector rests as shown. Assu- ming a random angular distribution of textures of which (a) and (b) are extreme examples, we get the symmetric absorption curve ( c ) .

In (a) the combination of texture and field lets

2

sit in a local minimum of the dipole energy giving rise to a positive frequency shift. In (b) however

2

is forced to sit at a saddle point which gives a positive longitudinal resonance, but a negative transverse shift equal and opposite to the shift in

(a). Assuming a random distribution of pinning si- tes for the singularities a "secant-like" line shape results in (c) which resembles the experimental results. While this model is too crude to produce reliable values a symmetric line shape should be retained in a complete calculation.