Interaction of two magnetic resonance modes in polar phase of superfluid 3He
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Interaction of two magnetic resonance modes in polar phase ofsuperfluid He V. V. Dmitriev +1) , A. A. Soldatov + ∗ , A. N. Yudin + + Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia ∗ Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
Submitted September 7, 2018
We report results of low frequency nuclear magnetic resonance (NMR) experiments in the superfluid polarphase of He which is stabilized by a new type of “nematic” aerogel – nafen. We have found that an interactionbetween transverse and longitudinal NMR modes may essentially influence the spin dynamics. Theoreticalformulas for NMR resonant frequencies are derived and applied for interpretation of the experimental results.
1. INTRODUCTION “Nematic” aerogels (N-aerogels) consist of strandswhich are oriented along the same direction. Diametersof the strands ( ∼
10 nm) are less than the superfluid co-herence length of He, so in the case of liquid He con-fined in N-aerogel these strands play a role of anisotropicimpurities, and theory predicts that superfluid phaseswhich do not exist in bulk He may become favorable,namely the polar-distorted A phase, the polar-distortedB phase and the pure polar phase [1, 2, 3, 4]. There aretwo types of N-aerogel: “Obninsk aerogel” and nafen[5]. Polar-distorted A and B phases have been observedin He confined in Obninsk aerogel [6, 7, 8], while thepure polar phase is realized in He in nafen [9]. As wellas in other superfluid phases of He, in the polar phasetwo NMR modes may be observed. Frequencies of thesemodes depend on temperature, on a magnetic field ( H )and on the orientation of H with respect to the orbitalpart of the order parameter. Here we present resultsof continuous wave (CW) NMR experiments in the po-lar phase of He in nafen which demonstrate a stronginteraction between these two resonant modes at cer-tain conditions. Exact expressions for NMR frequenciesin the polar phase in the limit of small excitations arederived and used to explain experimental data. e-mail: [email protected]
2. THEORY
The order parameter of the polar phase is A jk = ∆ e iφ d j m k , (1)where ∆ is the gap parameter, φ is the phase factor, d is the unit spin vector and m is the unit vector in theorbital space which direction is fixed along the direc-tion of strands of N-aerogel [1]. We choose H = H ˆz and m y = 0, so that m z = cos ϕ and m x = sin ϕ ,where ϕ is the angle between H and m . In the equilib-rium state orientation of d is determined by minimiza-tion of the sum of dipole ( U D ∝ ( dm ) ) and magnetic( U H ∝ ( dH ) ) energies, that is d k ˆy .Spin dynamics of the polar phase is described byLeggett equations [10]:˙ M = γ M × H − Ω P ω L ( d × m ) ( dm ) , ˙ d = d × ( γ H − ω L M ) , (2)where γ ≈ · Oe is the gyromagnetic ratio of He, ω L = γH is the Larmor frequency, M is the mag-netization normalized to its equilibrium value ( χH ), χ is the magnetic susceptibility and Ω P is the Leggett fre-quency of the polar phase, which is zero at the super-fluid transition temperature and grows up to ∼
100 kHzon cooling. Introducing f M z = M z − M x = ω L M y − Ω P ω L ( dm ) d y m z , ˙ M y = − ω L M x − Ω P ω L ( dm ) ( d z m x − d x m z ) , ˙ f M z = Ω P ω L ( dm ) d y m x , ˙ d x = ω L (cid:16) d z M y − d y f M z (cid:17) , ˙ d y = ω L (cid:16) d x f M z − d z M x (cid:17) , ˙ d z = ω L ( d y M x − d x M y ) . (3)1 V. V. Dmitriev, A. A. Soldatov, A. N. Yudin
We consider small deviations of M and d from the equi-librium state, that is: f M z ≪ M x,y ≪ d x,z ≪ d y ≈
1. Then from Eqs.(3) it follows:¨ M x = − (cid:0) ω L + Ω P cos ϕ (cid:1) M x + Ω P f M z sin ϕ cos ϕ, ¨ f M z = Ω P M x sin ϕ cos ϕ − Ω P f M z sin ϕ, (4)that results in the equation for NMR frequencies ω : ω − (cid:0) ω L + Ω P (cid:1) ω + Ω P ω L sin ϕ = 0 . (5)Eq.(5) has two solutions: ω ± = 12 (cid:0) ω L + Ω P ±± q ( ω L + Ω P ) − ω L Ω P sin ϕ (cid:19) . (6)We note that there is a frequency gap between ω + and ω − modes: there are no solutions of Eq.(5) forΩ P > ω > Ω P sin ϕ . In general, both modes representcoupled oscillations of transverse and longitudinal com-ponents of M , and the ratio of amplitudes of transverseand longitudinal oscillations for a given ω is R = | ω − Ω P sin ϕ | Ω P sin ϕ cos ϕ . (7)Expressions for NMR frequencies in the polar phasederived in previous papers [1, 6, 9, 11] were obtainedin the assumption that equations for M z ( t ) and M x,y ( t )are decoupled. In this case we get two noninteractingNMR modes: longitudinal and transverse. Frequenciesof these modes for small amplitudes are ω k = Ω P sin ϕ,ω ⊥ = p ω L + Ω P cos ϕ. (8)Eqs.(8) are valid only for ϕ = 0 and ϕ = 90 ◦ , or if ω L ≫ Ω P . In other cases, as it follows from Eqs.(4),the coupling of longitudinal and transverse modes be-comes essential. This is illustrated by Fig.1a, where wepresent calculated field dependencies of ω + , ω − , ω k , and ω ⊥ for realistic experimental conditions. It is seen thatthe coupling between resonant modes results in their“repulsion” and transformation into two nonintersect-ing branches with frequencies ω + and ω − . We note thatat ϕ = 90 ◦ we get two intersecting modes, but even atsmall deflections of ϕ from 90 ◦ a qualitative change ofthe NMR spectrum occurs (Fig.1b).
3. DETAILS OF EXPERIMENT
In the present work we use the same experimentalchamber as in experiments described in [12]. The cham-ber is made of Stycast-1266 epoxy resin and has two ||L + H (Oe) / ( k H z ) P (b)(a) H (Oe) / ( k H z ) Fig. 1: CW NMR frequencies in the polar phase versusH at Ω P / π = 82 kHz. (a) From Eq.(6) (thick solidlines) and from Eqs.(8) (thin solid lines). Dashed linecorresponds to the Larmor frequency. ϕ = 68 ◦ . (b)From Eq.(6) for ϕ = 89 ◦ (solid lines) and for ϕ = 90 ◦ (short-dashed lines). cells with nafen samples which were produced by ANFTechnology Ltd (Tallinn, Estonia). In the experimentsdescribed below we use the cell with nafen with overalldensity of 243 mg/cm (nafen-243). The sample has acuboid shape with sizes of 4 mm and is placed freely inthe cell. It consists of Al O strands with diameters of ∼ He-B. To avoid a param-agnetic signal from surface solid He, the sample waspreplated by ∼ . He.Experiments were performed using transverse CW nteraction of two magnetic resonance modes in polar phase of superfluid He ÷
111 Oe (correspondingNMR frequencies were 82 ÷
360 kHz) and at a pressure of29.3 bar. The superfluid phase diagram of He in nafen-243 is presented in [9]. At 29.3 bar the superfluid tran-sition temperature of He in nafen-243 is suppressed by ∼
2% with respect to the superfluid transition temper-ature ( T c ) in bulk He, and down to the lowest reachedtemperature ( ∼ . T c ) the only observed superfluidphase is the polar phase.Two solenoids were used in order to apply the ex-ternal magnetic field in directions parallel ( H k , longi-tudinal field) and perpendicular ( H ⊥ , transverse field)to nafen strands. So, the resultant field H = H k + H ⊥ could be rotated by an arbitrary angle ϕ with respectto the anisotropy axis of nafen. CW NMR measure-ments in the normal phase of He show that an anglebetween axes of longitudinal and transverse solenoids is90 ◦ ± . ◦ . However, we estimate an accuracy of set-ting ϕ as ± ◦ due to a possible misalignment betweenthe axis of the longitudinal solenoid and the anisotropyaxis of the nafen sample. For H ⊥ ∼
25 Oe it limits anaccuracy in determining of H k to 0.4 Oe.
4. RESULTS
Experiments were carried out using transverse CWNMR at a fixed frequency ω for ϕ ≈ ◦ and for ϕ closeto 90 ◦ . We applied fixed field H k and swept H ⊥ torecord the NMR line. In terms of H k and H ⊥ Eq.(5)can be rewritten as follows:( γH ⊥ ) = ω h − (cid:0) γH k (cid:1) / (cid:0) ω − Ω P (cid:1)i . (9)In this case H ⊥ /H k = tan ϕ = const , but the changeof H ⊥ during the sweep through the NMR line is small,and the corresponding change of ϕ is less than 0 . ◦ .The measured dependence of the resonant field H ⊥ on temperature (and Ω P ) at ϕ ≈ ◦ is shown in Fig.2.It is seen that the experimental data are in a good agree-ment with Eq.(9) (solid curve) which fits the data muchbetter than Eq.(8) (dashed curve). We note that smallvariations of H k in the equations result only in verti-cal shift of the theoretical curves (by ∼
100 kHz for δH k = 0 . H k is due to the inaccuracyin setting of H k due to the above mentioned misalign-ment.In experiments presented in Fig.2 only the modewith frequency ω + was excited because ω was alwaysgreater than Ω P . The transition between ω + and ω − modes can be observed if at a fixed ω the temperature ( H / ) ( k H z ) ( P /2 ) (10 kHz ) T/T c Fig. 2: CW NMR transverse magnetic field in the po-lar phase as a function of temperature (upper scale) andΩ P (lower scale). Open circles are experimental data at ω/ π = 146 . H k ≈ . H ⊥ is in the range of40 . ÷ . ϕ is in the range of 67 . ◦ ÷ . ◦ .Solid curve is best fit by Eq.(9) with a single fit pa-rameter of H k which was found to be equal to 16.53 Oe.Dashed curve corresponds to Eq.(8) rewritten in termsof H k and H ⊥ with H k = 16 .
53 Oe. Temperature de-pendence of Ω P was measured independently by CWNMR at ϕ = 0. (and consequently Ω P ) is changed, so that we cross thefrequency gap between modes. Maximal value of Ω P which we could obtain at T ≈ . T c was 107 kHz.Therefore, in order to observe the transition betweenmodes we used the NMR frequency of 82 kHz, althoughour NMR setup was not optimal for such a low fre-quency, and the signal-to-noise ratio was rather poor. Itis also worth noting that an absolute value of ∂ω/∂H ⊥ decreases if ω is approaching Ω P (or Ω P sin ϕ ) result-ing in a broadening of the NMR line and in an ad-ditional decrease of the signal-to-noise ratio near thetransition region. For these reasons we were able to seethe transition between the modes only if ϕ is close to90 ◦ where the frequency gap is small enough. Trans-verse CW NMR absorption lines recorded at ϕ ≈ ◦ during slow warming in the polar phase are shown inFig.3. The transition between ω + and ω − modes occursin a narrow temperature region near ∼ . T c where theNMR signal practically disappears. At higher temper-atures we excite ω + -mode while at lower temperatures ω − -mode is observed.The dependence of the resonant field H ⊥ on Ω P (cal-culated from the first moment of the NMR line) is shown V. V. Dmitriev, A. A. Soldatov, A. N. Yudin c = 0.658 I ( a . u . ) H /2 (kHz)
Fig. 3: CW NMR absorption lines recorded duringwarming in the polar phase. The lines at differenttemperatures are shifted in vertical direction for a bet-ter view. H k was set to 0.29 Oe, but further analysis(see Fig.4) pointed out that it corresponds to 0.54 Oe( ϕ ≈ . ◦ ) that may be due to misalignment betweenthe axis of the longitudinal solenoid and the directionof nafen strands. ω/ π = 82 kHz, H ≈
25 Oe. in Fig.4 by filled circles. It agrees well with the depen-dence following from Eq.(9) if we assume that the realvalue of H k equals 0.54 Oe (solid curve).We also have done a similar experiment where we set H k to be equal to 0.17 Oe (open circles in Fig.4). In thiscase the NMR signal was absent in essentially smallerrange of temperatures (between 0 . T c and 0 . T c ) andat temperatures 0 . T c < T < . T c the experimentaldata cannot be fitted by Eq.(9). We assume that thisis explained by variations of ϕ inside the sample aboutthe mean value. If we do not take into account the datain this temperature range then best fit by Eq.(9) cor- ( H / ) ( k H z ) ( P /2 ) (10 kHz )T/T c Fig. 4: The dependence of H ⊥ on temperature (up-per scale) and Ω P (lower scale). H k was set to 0.29 Oe(filled circles) and to 0.17 Oe (open circles). Lines arebest fits of the data by Eq.(9) with H k = 0 .
54 Oe (solidline) and H k = 0 .
27 Oe (dashed line). ω/ π = 82 kHz, H ≈
25 Oe. responds to H k = 0 .
27 Oe (i.e. ϕ ≈ . ◦ ), and thevariations of ϕ may be estimated as ∼ . ◦ .To sum up, our results of low frequency transverseCW NMR experiments in the polar phase of He innafen are in a good agreement with the developed theo-retical model. Experiments at ϕ ≈ ◦ (Fig.2) confirmthe validity of Eqs.(6,9), while experiments at angles ϕ close to 90 ◦ (Fig.3 and Fig.4) prove the existence of twonon-intersecting branches of the NMR spectrum.
5. CONCLUSIONS
It was observed for the first time that in the polarphase of He at even small deviations of ϕ from 90 ◦ thelongitudinal and transverse NMR modes become cou-pled that results in their repulsion and transformationinto two non-intersecting branches of the NMR spec-trum. The coupling between these modes is possible dueto N-aerogel which fixes m along the aerogel strands.In contrast, in equilibrium homogeneous state in bulksuperfluid He the coupling of longitudinal and trans-verse NMR modes cannot be observed because the mag-netic field orients order parameters of A and B phases sothat the modes do not interact. However, the couplingmay appear in the presence of inhomogeneities of theorder parameter, caused by boundaries or textural de-fects (solitons or vortices). So, a decay of the transverse nteraction of two magnetic resonance modes in polar phase of superfluid He [14] where field dependencies of two AFM res-onant modes were measured for different orientations ofa magnetic field.
6. ACKNOWLEDGEMENTS
We are grateful to to G.E. Volovik for useful com-ments and I.M. Grodnensky for providing samples ofnafen. This work was supported in part by RFBR grant16-02-00349 and the Basic Research Program of the Pre-sidium of Russian Academy of Sciences.
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