Nonlinear and Hysteretic Ultrasound Propagation in Solid 4 He: Dynamics of Dislocation Lines and Pinning Impurities
aa r X i v : . [ c ond - m a t . o t h e r] N ov Nonlinear and Hysteretic Ultrasound Propagation in Solid He:Dynamics of Dislocation Lines and Pinning Impurities
Izumi Iwasa a and Harry Kojima b a Faculty of Science, Kanagawa University, Kanagawa, Japan 259-1293 b Serin Physics Laboratory, Rutgers University, Piscataway, New Jersey 08854 (Dated: November 19, 2020)We report on the measurements of 9.6 MHz ultrasound propagation down to 15 mK in polycrys-talline quantum solid He containing 0.3 and 20 ppm He impurities. The attenuation and speedof ultrasound are strongly affected by the dislocation vibration. The observed increase in atten-uation from 1.2 K to a peak near 0.3 K is independent of drive amplitude and reflects crossoverfrom overdamped to underdamped oscillation of dislocations pinned at network nodes. Below 0.3K, amplitude-dependent and hysteretic variations are observed in both attenuation and speed. Theattenuation decreases from the peak at 0.3 K to a very small constant value below 70 mK at suf-ficiently low drive amplitudes of ultrasound, while it remains a high value down to 15mK at thehighest drive amplitude. The behaviors at low drive amplitudes can be well described by the effectsof the thermal pinning and unpinning of dislocations by the impurities. The binding energy betweena dislocation line and a He atom is estimated to be 0.35 K. The nonlinear and hysteretic behaviorsat intermediate drive amplitudes are analyzed in terms of stress-induced unpinning which may occurcatastrophically within a network dislocation segment. The relaxation time for pinning at 15 mKis very short ( < PACS numbers: 67.80.B-, 62.65.+k
I. INTRODUCTION
Dislocation lines are one dimensional defects in crys-tal lattice and are important in understanding phenom-ena such as fracture and fatigue , yield strength, plastic-ity, work hardening, creep, and others. Though materialproperties influencing these phenomena may be variedempirically, a complete physical understanding of dis-locations is still lacking. Dislocations in materials canbe directly observed by chemical etching, electron trans-mission microscopy and X-ray diffraction. The etchingis restricted to surfaces; the electron transmission is in-vasive, local and requires vacuum environment; the X-ray techniques are limited by low resolution . Ultra-sound is a sensitive and convenient nondestructive toolfor probing the global properties of dislocation lines inmaterials including solid helium . Ultrasound isespecially suited for studying dislocations where theirlength scale nearly matches the acoustic resonant length.A network of dislocation lines that are strongly pinnedat their intersections(nodes) is formed in a solid sampleduring its growth process. Dislocation lines may be ad-ditionally pinned weakly by bound impurity atoms. Ac-cording to the Granato-L¨ucke theory (GL) (see Sec.IV A), under the periodic stress of ultrasound the dislo-cation lines execute damped vibration like taut stringspinned at their network nodes (average network pinninglength L nA ) and the impurities (average impurity pin-ning length L iA ), and the vibrating dislocation segmentsmodify the ultrasound propagating characteristics.Solid He is particularly suitable for investigating theimpurity effects on the dislocation dynamics: the onlyrelevant impurity is the He atom (nominal concentrationis 3 × − in commercially available “natural purity” He gas). Though the impurity concentration is very low, theimpurities are crucial in understanding the observed phe-nomena. At low temperatures( T ) below 1 K, the He im-purity atoms are highly mobile owing to the rapid quan-tum diffusion process and uniformly distributed withina solid He sample. He impurity atoms lower their elas-tic energy by “condensing” onto (predominantly edge)dislocations and thereby pin dislocation lines. The dis-location lines, however, can be unpinned from the impu-rity atoms by increasing the amplitude of the ultrasoundstress when the imposed force on the pinning site exceedsa critical force. Once unpinning is initiated on a dislo-cation line, it continues at all other impurity atoms onthe line till this line is only pinned at its two end networknodes. This runaway effect was anticipated by GL as“catastrophic break-away” of dislocation lines from im-mobile impurities. An important difference in our caseis that the impurities are mobile . When unpinning oc-curs in solid He, the mobile impurities rapidly diffuseaway from the dislocation and cannot repin the disloca-tion again within the acoustic cycle.Franck and Hewko reported the first measurementof temperature dependence of the ultrasound speed insingle crystals of hcp He above 0.7 K. They discovereda transition in the speed from a power law expected forthe adiabatic sound at high temperatures to approximatetemperature independence below 1.1 K or 2 K dependingon the growth pressure of crystals. Wanner, et al. madesimilar measurements down to 0.2 K and found that thedeviation from the adiabatic sound speed at tempera-tures below 1 K could be both positive and negative de-pending on samples. They identified the origin of theobserved anomalous temperature dependence below 1 Kas the resonant interaction of dislocation lines with ul-trasound. Since then, the effects of dislocations on thepropagation speed as well as attenuation of ultrasound insingle-crystal solid helium have been confirmed by otherstudies down to 80 mK. The effects of pinning bydislocations in solid He doped with additional He havebeen studied in the higher temperature range.The interaction between the dislocation lines and Heimpurities in hcp He was also revealed in the measure-ments of shear modulus in the kHz range . Thesemeasurements were originally stimulated by the observa-tion of the so-called NCRI (non-classical rotational in-ertia) in the torsional oscillator experiments . Thevariations of the shear modulus and dissipation at tem-pratures below 1 K are found to depend on the frequency,strain amplitude and temperature. The mechanism ofthe interaction is either damping of dislocation motionor pinning of dislocation lines by He impurities. Param-eters related to the dislocation such as the dislocationdensity, the average network pinning length, the dampingconstant of dislocation motion, and the binding energyof a He atom to the dislocation are obtained from thesemeasurements.The present work is the first to study the ultrasoundpropagation in polycrystalline solid He extended downto 15 mK. Typically, the attenuation and speed of ultra-sound increase as the sample is cooled from 1.2 to 0.3K. These responses are similar to those in the single-crystalline samples and can be explained by the transi-tion from overdamped to underdamped resonance of dis-locations. The ultrasound responses become nonlinearand hysteretic as the sample is further cooled. When thedrive level is sufficiently low, the attenuation typicallydecreases by 20 dB from 300 to 15 mK, while the speedshows a minimum at 100 mK. These low drive responseswill be explained as an effect due to pinning of the dislo-cations by He impurities in thermal equilibrium. Whenthe drive is sufficiently high, on the other hand, the atten-uation decreases very little upon cooling from 300 to 15mK. Hysteretic responses are observed depending on thehistory of how the drive level and temperature are varied.For example, the attenuation in the warming run at anintermediate drive level is smaller than the correspond-ing attenuation in the cooling run at the same drive level.The increase in attenuation in the warming run will beshown to originate in the stress-induced “catastrophic”unpinning of dislocation lines from He impurities. Theseobservations of nonlinear and hysteretic responses aremade for the first time in the ultrasound measurementon solid He with 0.3 ppm of He.The paper is organized as follows. The experimentalmethods are described in Sec. II. The data on the mea-sured attenuation and speed on typical as well as atypicalsamples are shown in Sec. III. In Sec. IV the theoreticalbackground on the GL theory is first given. Then the be-havior of the typical sample is analyzed. Discussions onthe atypical samples and comparison with other relatedexperiments are given in Sec. V. The paper is concludedwith a brief summary and questions for further research in Sec. VI. Supplemental information is given in the Ap-pendices.The analysis is rather complicated because of the non-linear nature of the phenomena. There are two indepen-dent external parameters, i. e. temperature and stress,which affect pinning and unpinning of dislocations by He impurities. The temperature of the sample can beregarded as uniform all over the sample. The stress, onthe other hand, is inhomogeneous because the amplitudeof the ultrasound pulse decreases along the path of ul-trasound due to attenuation. Therefore, a position de-pendent analysis carried out is described in detail in thelatter part of Sec. IV.
II. APPARATUS AND MEASUREMENTPROCEDURE
Our ultrasound apparatus (a schematic is shown in Fig.1) for studying solid He has been described earlier. Itwas originally designed to study the ultrasound propa-gation simultaneously with torsional oscillation responseof solid He. This report is focussed on the ultrasoundresults. The interrelationship between the ultrasoundand the torsional oscillation phenomena will be describedelsewhere. Briefly, a BeCu torsion rod (outer diameter =4.0 mm, inner diameter = 0.8 mm, length = 14 mm) con-nected to the cylindrical sample chamber provides boththe thermal contact to the mixing chamber of a dilutionrefrigerator and an inlet conduit for the sample gas. Theinner ends of the sample chamber (diameter( D ) = 8.6mm, length( x m ) = 6.6 mm) are terminated by identi-cal (10 mm diameter, 10 MHz) X-cut quartz transducersacting as driver and detector. The sample length is setby the precision-machined steps on the inner wall of thesample chamber as shown in Fig. 1. Ultrasound is ex-cited by applying 1.2 µ s wide RF voltage pulses of am-plitude V and frequency Ω / π = 9 . A ≡ V /V is normalized to an arbitrary valueof V = 2.53 mV. As described in Appendix A on cali-bration of transduers, A = 1 is estimated to be equiva-lent to an applied compressional stress amplitude of 1.36Pa. The estimation is, however, uncertain by as much asa factor of two due to incomplete impedance matching,acoustic mismatch, and ringing of the transducer in theultrasound experiment.A standard pulse-echo apparatus with a superhetero-dyne system is used to measure the ultrasound prop-agation. The received pulse signal is converted to anintermediate frequency of 5 MHz, amplified, and phase-sensitively detected in the spectrometer. The result-ing quadrature video voltages, V sin and V cos , are thenmeasured by a boxcar integrator. Fractional changesin propagation speed v are evaluated from the phaseshift according to: δv/v = arctan( V sin /V cos ) / (Ω P ) where P = x m /v is the transit time. The relative attenuation !" FIG. 1. Cutaway schematic view of ultrasound cell: quartztransducer(Q), electrical contact plate to transducer(E), in-sulating backing (I), one compression spring(C) of three, elec-trical lead(L) attached to spring-loaded contact, bolt holes(B,bolts not shown). The brass cell body encloses the cylindricalsample chamber between the transducers. Helium is filled viathe hole for the upper electrical lead. The cell is attached tothe mixing chamber of the dilution refrigerator via the upperBeCu (torsion) rod with the sample chamber axis orientedvertically. of sound signal amplitude (in dB) is evaluated accord-ing to α = −
20 log p V + V − α v , where α v is thedrive output attenuator setting (in dB). The zero offsetsof the quadrature signals are determined in the presenceof solid He sample in the cell by extrapolating the mea-sured voltages as the drive amplitude is decreased nearlyto zero. The absolute speed of sound can be in principleestimated from the length of the sample and the transittime of the ultrasound pulse in the pulse-echo method.Unfortunately, we could not accurately measure the ab-solue speed because the rising edge of the received signalwas rounded due to ringing of the transducer.Solid He samples are grown in the cylindrical chamberby the blocked capillary method as follows. The cham-ber is initially loaded at 4.2 K with liquid He at 68 bar,using commercial natural purity He gas originated fromTexas with a nominal He impurity concentration( x ) of0.3 ppm . As cooling of the dilution refrigerator sys-tem is initiated, the section of the fill capillary attachedto the high temperature end of the refrigerator rapidlycools down near to 1.2 K. The liquid He within thatsection of the capillary becomes frozen and forms a plug,which now keeps the He mass below the plug and in thesample chamber fixed. Subsequently, an “as-grown” solidsample with a molar volume of 20.3 cm /mole under afinal pressure of about 35 bar is produced over about twohours during which the sample chamber is cooled below50 mK.Annealing is carried out in some samples by raising andmaintaining a constant temperature near 1.5 K for 20 - 48 hours. The sample pressure is indirectly monitored bythe pressure sensor located on the mixing chamber. Thesample chamber and the pressure sensor is connected bythe fill line tube (inner diameter = 0.75 mm and length15 cm). The sensor pressure is found to decrease mono-tonically during annealing. Annealing is considered com-pleted when the changes in the pressure sensor readingare much reduced from the initial rate.The effects of He impurity concentration on ultra-sound response are studied by increasing x to 20 ppm.The dilution refrigerator system is warmed up to about5 K and the remnant commercial natural purity heliumgas from the previous measurements is pumped out fromthe sample cell over 22 hours. The sample cell is nextloaded with the appropriate amount of He gas to makeup the increased x , filled and finally pressurized withcommercial natural purity He at 4.2 K. The mixturesolid sample is subsequently grown in the usual mannerdescribed above. The in situ He concentration withinthe sample solid itself is not measured.Measurements on the samples grown at the initialstages of the experiment were made as a function of tem-perature at relatively high drive amplitudes,
A > α and δv/v were amplitude dependent and hysteretic at temper-atures below 0.3 K. It was finally found that decreasing A below a critical value was crucial in allowing He impuri-ties to pin the dislocations in the sample and suppressingthe nonlinear effects. In order to study the details of theeffects related to changing A , the measurement proce-dure illustrated in Fig. 2 was adopted for the last andmost extensively studied sample T min ∼
15 mK where the relative drive am-plitude is set at an initial low amplitude A i ( ≤ .
07) (see[1] in Fig. 2). This A i is sufficiently small that all thedislocation segments are pinned by the number of Heimpurities in thermal equilibrium. Keeping the tempera-ture constant, A is then increased in several steps to thefinal relative amplitude A f at [2] in Fig. 2. The temper-ature is then increased in a “warming run” in ∼ → [3] → [4]) to T max ∼ . T min over ∼
15 hours ([4] → [5]). At T min A is decreased downto A i ([5] → [1]) and then increased to a different A f forthe next run. The ultrasound data, δv/v and α , are ac-quired throughout the procedure. Depending on A f , theultrasound data can be hysteretic or reversible during thewarming and successive cooling run as described below. III. RESULTS
Ultrasound propagation was studied in eleven solid Hesamples in hexagonal close-packed (hcp) structure. Nineof them were grown from commercial He gas ( x = 0.3 FIG. 2. Measurement procedure of varying the ultrasound relative drive amplitude A and temperature T for the sample T min ( ∼
15 mK) the drive amplitude is increased from A i ≤ .
07 at [1] to A f = 1 −
15 at[2]. The temperature is subsequently increased in a warming run in ∼ T max ( ∼ ∼
15 hours from [4] to [5] in a cooling run. The procedure is completed by decreasing A down to A i ([5] → [1]). The next run is repeated for a different A f . Characteristic temperatures T and T are described in the text(Sec.III A 1). Cartoons (1) − (5) illustrate pinning/unpinning and string-like deflections of one network segment of dislocation pinnedby impurities. Heavy dashed horizontal lines with arrows indicate the critical length L c as A f is varied. Heavy dashed curvesconnecting the two end network nodes indicate that the network segment is catastrophically unpinned(see Sec. IV D) fromimpurities. ppm). Two samples with x = 20 ppm were grown in or-der to examine the effect of He impurity on ultrasoundresponse. The ultrasound response varied in detail fromsample to sample. Characteristics of the samples, how-ever, could be roughly divided into three categories de-pending on the nature of observed ultrasound response.The samples described in this paper are listed in Table Itogether with fitted parameters to be discussed later.Many of the samples show “typical” ultrasound re-sponse that follows from the interaction of the ultra-sound with dislocation lines and is compatible with theGL theory when pinning at the network nodes and by He impurities is taken into account. The typical samples have an attenuation peak (typically 20 dB) around 0.3 K.The attenuation and the change in ultrasound speed be-come nonlinear and hysteretic below the peak tempara-ture. Samples with “small attenuation” belong to thesecond category. Their attenuation peak is less than 3dB and the variation of the speed of ultrasound is almostproportional to T , that is, the response is almost freeof dislocation effects. In the third rare category, samplesexhibit “anomalous” response where the ultrasound re-sponse around 0.7 K is of a relaxation type. This responsecannot be simply explained by the GL theory. However,the nonlinear and hysteretic effects of dislocations similarto the typical samples can be seen also in the anomaloussamples at temperatures below 100 mK. Taken together,our samples allow a comprehensive study of solid Heexhibiting a wide range of ultrasound propagation phe-nomena produced by dislocations.
A. Typical sample response
1. Temperature and drive amplitude dependence
Temperature dependences of α and δv/v of a typi-cal sample taken during warming and subsequent cool-ing runs with three values of drive amplitude ( A f =1.12,3.98, and 14.1) are shown in Fig. 3. The data weretaken on the sample A i ≤ .
07 and then increased to a new A f according to the measurement procedure in Fig. 2.When the drive amplitude was decreased to A i at T min , α and δv/v became α = 10 dB and c = 0 . × − ,respectively, regardless of the previous value of A f . Weregard α and c as the reference values for α and δv/v .There is a broad attenuation peak around 200 ∼ T >
300 mK, theresponse is linear: both α and δv/v are independent of A f as well as the temperature sweep direction. The valuesof α and δv/v decrease monotonically as the temperatureincreases. It will be shown below (see Sec. IV B) that theinteraction between the ultrasound and dislocation linespinned at network nodes can account for the behavior inthis temperature range.In the low temperature range, T <
200 mK, the re-sponse becomes more intricate. It is strongly nonlinear,i.e. dependent on the value of A f , and hysteretic withrespect to the thermal history. The attenuation increasesmonotonically from T min to 200 mK, whereas δv/v showsa distinct minimum around 100 mK and occasionally amaximum at a lower temperature. On cooling runs, both α and δv/v tend to saturate to A f -dependent values as T min is approached. The low-temperature response willbe analyzed in terms of pinning of dislocations by Heimpurities in Sec. IV C and below.On warming at A f =1.12, α remains equal to α up toa characteristic temperature T ( ≈
70 mK), above whichit increases to reach the broad maximum. On cooling at A f =1.12, α decreases from the broad peak and reaches α at T . There is no hysteresis between the warming andcooling runs. Similar reversible behavior was observed inanother run at A f =1.41 (not shown in Fig. 3). Thevariation of δv/v at A f =1.12 is also thermally reversibleand the minimum around 100 mK is the deepest amongthe runs in Fig. 3 (2).The attenuation at at A f =3.98, on the other hand, ishysteretic. On warming, α remains equal to α up toanother characteristic temperature T ( ≈
50 mK), abovewhich it increases towards the broad maximum with asmall kink at T . On cooling, α decreases from the max- FIG. 3. Temperature dependence of α (panel (1)) and δv/v (panel (2)) of the sample He impurityduring warming(w) and subsequent cooling(c) runs with threedifferent drive amplitudes. A f = 1.12 (black closed(w) andopen(c) circles), 3.98 (blue closed(w) and open(c) triangles),and 14.1 (red closed(w) and open(c) squares). Down(up) ar-rows in (1) indicate characteristic temperatures T ( T ) wherethe dislocation line pinning state changes in warming runs.See text(Sec. III A 1) for explanation of these characteristictemperatures and the “vertical” data at T min for A f = 14 . imum along the warming data down to T , where it de-parts from the warming line and levels off. The variationof δv/v at A f =3.98 is also hysteretic. Note that there isa maximum in δv/v on the warming run around T .At A f =14.1, α already increases at T min above α asshown by “vertical points” in Fig. 3(1) and reaches 21 dBover about 2000 s time interval after A is changed from A i to A f . The temporal variations will be described inSec. III A 3. When the temperature is raised, α remainsconstant at 21 dB up to T and then increases towardsthe broad maximum. On cooling at A f =14.1, α decreasesonly slightly from the maximum value. The behavior of δv/v at A f =14.1 shows similar features; δv/v increasesslowly at T min after A is changed from A i to A f , and thevariations in δv/v between 50 and 200 mK are smallerthan those at lower A f .Figure 4 shows α and δv/v of another typical sample A f = 2.51 of the previous FIG. 4. Attenuation (panel (1)) and change in speed (panel(2)) of the sample He impurity in warm-ing (black dots) and cooling (open circles). The drive levelis high ( A f =7.9) and the measurement procedure deviatesslightly from Fig. 2 (see text). run to A f = 7.94 without reducing to A i . As a result, α and δv/v in the warming run did not start from α and c ,respectively. The variations of α and δv/v in the coolingrun are similar to those of the sample A f = 14.1shown in Fig. 3.
2. Hysteresis with respect to drive amplitude
According to the procedure prescription shown in Fig.2, A is decreased at the end of the previous cooling runat T min from A f to A i and then increased to a new A f for the next warming run. Figure 5 shows the ultrasoundresponse, α − α and δv/v − c , of the sample α and c are plotted for clarity here. Each symbol repre-sents a series of changes taken in A at the end of a coolingrun. It can be seen that different symbols trace out thesame hysteretic dependence on A regardless of the valueof A f of the previous cooling run.In the process of decreasing A down to a critical value A c ( ≈ . A in-dicating nonlinear behavior. In the range A i < A < A c ,both α − α and δv/v − c become zero and the responseis linear. In the process of increasing A but only up to athreshold drive amplitude A t ( ≈ α − α and δv/v − c remain zero and the response is linear. When A is in-creased to greater than A t , the response becomes non-linear and time dependent behavior is observed (see thenext subsection). a - a ( d B ) (1) (2) d v / v - c ( - ) A FIG. 5. Hysteresis of attenuation α − α (panel (1)) andspeed δv/v − c (panel (2)) vs. drive amplitude A at T min in the same sample A f at the endof previous cooling run (Fig. 2[5]) to A i (Fig. 2[1]) andthen increased (solid arrows pointing right) to A f of the nextwarming run (Fig. 2[2]). Different symbols indicate varioussequential changes in A starting from different A f at the endof previous cooling run. The vertical dashed arrows indicatetime dependent response (see Sec. III A 3).
3. Time dependent response after drive amplitude change In all of the decreasing steps of A and in the increasingsteps of A up to A t in Fig. 5, the changes in response oc-cur rapidly. However, when A is increased to that above A t , changes in both α and δv/v occur in a remarkabletransient manner accompanied by long relaxation timesindicated by the vertical dashed lines in Fig. 5.Figure 6 shows examples of the ultrasound response at T min as a function of time( t ) after A is changed at t = 0.The amplitude of the received signal, S = p V + V ,and δv/v − c are plotted. In Fig. 6(1), A is decreased by10 dB from 5.01 to 1.58 at t = 0. No change in S occursat t=0, indicating that the response is nonlinear and thatthe attenuation decreases by 10 dB. The speed of sound,on the other hand, decreases by 6 × − at t = 0. Theresponse occurs rapidly within the data acquisition cycletime of ∼ A is increased by 10 dB from 1.41 to 4.47 (Fig.6(2)), S instantly increases by 9.8 dB (3.7 to 11.5 mV)while the change in δv/v is less than 1 × − . Theseresponses are therefore nearly linear.When the drive is stepped up to one above A t , how-ever, the response is dramatically different. The drive isincreased by 10 dB from 4.47 to 14.1 at t = 0 in Fig. 6(3). S initially increases by 10 dB from 11.5 to 35.7 mV butthen decreases slowly, whereas δv/v − c initially becomesnegative and then gradually increases to become posi-tive. These responses correspond to the vertical dashedlines at A = 14 . T min in Fig. 3. The decrease in S cannot bedescribed by a single exponential function. The fittedcurve to S in Fig. 6(3) is a double exponential func-tion of the form, S = s + s exp( − t/τ ) + s exp( − t/τ ),where s = 11 . s = 15 . s = 11 . τ = 90 s, and τ = 640 s. The value of S after t = 2000s is almost the same as that before t = 0, that is, theattenuation increases by 10 dB. The relaxation in δv/v can be similarly fitted by a double exponential function.The observed time dependence of the ultrasound re-sponse following step changes in A is most likely revealingthe characteristic time scale in the dynamics of the in-teraction between ultrasound and dislocation lines. Thetime scale rapidly decreases from 2000 s at T min to be-low 10 s at 65 mK. Unfortunately, detailed temperaturedependence of the relaxation phenomenon was not mea-sured.We will discuss later (see Sec. V F) the physical mech-anism behind the nonlinear effects observed in Fig.6 (1)and (3) in terms of the dynamic process of pinning andunpinning of dislocation lines by He impurities.
B. Small-attenuation samples
Three as-grown samples show a “small-attenuation”response over the measured temperature range as shownin Fig. 7. The overall change in α is less than 3 dBand the hysteresis is small. The data of δv/v can bewell described by a T dependence as expected from thephonon anharmonicity and there is no hysteresis be-tween warming and cooling. The small-attenuation re-sponse occurred in samples with both natural purity ( x = 0.3 ppm) and doped ( x = 20 ppm) samples. C. Anomalous sample
The ultrasound behavior of the sample was given earlier) is unusual and anoma-lous. The data from this sample taken at two drive levels A f = 1 and 2.82 are shown in Fig. 8. Before starting thewarming run at A f = 1, the drive level was decreased to A i = 0.45. On the other hand, the drive level was notchanged before starting the warming run at A f = 2.82.In the high temperature range T >
200 mK in Fig. 8both α and δv/v are independent of the drive amplitude;the response is linear. However, there is an anomalouspeak in α and a rapid change in δv/v around 700 mK.The behavior around 700 mK is totally unexpected from t (s) S ( m V ) (3) -50 0 50 100 150510 (2)24 (1) -20246 d v / v - c ( - ) -202-4-20246 FIG. 6. Temporal variations of ultrasound amplitude S(left ordinate, red open circles) and speed δv/v − c (rightordinate, black dots) before and after step changes in driveamplitude A (at t = 0) in the sample T min . Panel (1): A = 5 . → .
58, panel (2): A = 1 . → .
47 and panel (3):4 . → .
1. The number of plotted data points is reducedin (3) for clarity. The observed relaxations of S and δv/v − c in (3) can be fitted with double exponential decays as shownby the curves (see Sec. III A 3). the GL theory. It can be fitted as the Debye relaxationresponse (see Sec. V D).In the low temperature range below 200 mK, the re-sponse shown in Fig. 8 is strongly sensitive to the driveamplitude. The hysteresis of α at A f = 1 is similar tothat of the typical sample A f = 3.98 in Fig. 3.(The critical amplitude in the sample A c < A f = 2.82 is dueto a different measurement procedure from Fig. 2 wherethe drive level was not decreased at T min to a low A i . D. Sample with He impurity concentrationincreased to 20 ppm
To see if the anomalies in ultrasound propagation insolid He containing x = 27.5 ppm impurity concentra-tion observed by Ho et al. could be reproduced in ourexperiment, He impurity concentration of 20 ppm waschosen. The ultrasound response of the sample x = 20 ppm is shown in Fig. 9. The drive level was notchanged before starting the warming run in the A f = 1.0data while it was increased from 1.0 to 1.41 before the -12-11-10-9-8 a ( d B ) (1)100 10001.01.52.02.5 d v / v ( - ) T (mK) (2)
FIG. 7. Attenuation (panel (1)) and change in speed (panel(2)) of a small-attenuation sample Heimpurity measured at A f = 2 .
51 (black dots for warming andblack open circles for cooling). The number of plotted data isreduced. The scale of ordinate in panel (1) is expanded com-pared with Fig. 3 in order to show more detailed temperaturedependence. The δv/v data during warming and cooling inpanel (2) are almost indistinguishable and follows a T depen-dence (black curve) expected from the phonon anharmonicity. warming run in the A f = 1.41 data. Probably this changein procedure is the reason why the initial value of α in thewarming run at A f = 1.41 is smaller than that at A f =1.0. The critical amplitude in this sample seems A c < T >
200 mK, the response is linear. Theresponse is nonlinear and hysteretic in the low temper-ature range similar to the typical sample response. Weestimate T = 80 ∼
90 mK and T = 100 ∼
110 mKfrom the warming runs. Both of theses temperatures arehigher than those for the x = 0 . δv/v decreases with increasing tempera-ture around 100 mK in this sample (Fig. 9(2)) while itincreases around 70 mK in the anomalous sample (Fig.8(2)). The difference will be discussed in Sec. V G. IV. ANALYSIS
In this section, the observed ultrasound response ofthe typical sample is analyzed. The theory of dislocationlines by Granato and L¨ucke (GL) is introduced first.Pinning of dislocations by impurities is crucial in under-standing the present experimental results at low tem-peratures. The only significant impurities in our solid a ( d B ) (1)
10 100 1000-10-8-6-4-20 d v / v ( - ) T (mK)(2)
FIG. 8. Attenuation (panel (1)) and change in speed (panel(2)) of an anomalous sample x = 0.3 ppm measuredat A f = 1 (black dots for warming(w) and black open criclesfor cooling(c)) and A f = 2 .
82 (blue filled(w) and open(c)triangles). The drive level was decreased to A i = 0 .
45 beforestarting the warming run at A f = 1, while it was not changedat A f = 2 .
82. Red solid curves represent contributions fromthe Debye relaxation to the attenuation and speed (see Sec.V D). He samples are He atoms whose atomic volume is big-ger than that of a He atom due to larger amplitudeof zero-point vibration. Thus He impurities can pin adislocation line via elastic interaction. The phenomenaof pinning and unpinning in thermal equibrium as wellas stress-induced unpinning are described. Then, spa-tial variation of pinning is considered in order to explainthe amplitude-dependent effects. Based on these ideas,the data of α and δv/v of the sample He impurity concentration will be deferred toSec. V.As the analysis involves nonlinearity, both the atten-uation denoted by α and the attenuation coefficient β are used. In the linear case, they are simply related as α = α + βx m . In the nonlinear case, on the other hand,spatial variation of the attenuation coefficent will be con-sidered. A. Granato-L¨ucke theory
GL considered dislocation lines in the presence of ul-trasound as vibrating strings that are pinned strongly atnetwork nodes and weakly at impurities. When ultra- a ( d B ) (1)10 100 1000-4-3-2-101 (2) d v / v ( - ) T (mK)
FIG. 9. Attenuation (panel (1)) and change in speed (panel(2)) of the sample x = 20 ppm at A f = 1 (blackdots for warming(w) and black open cricles for cooling(c)) and A f = 1 .
41 (blue filled(w) and open(c) triangles). The drivelevel was not changed before the warming run at A f = 1,while it was increased from 1.0 to 1.41 at A f = 1 .
41. Thedown and up arrows in panel (1) indicate T and T at A f =1 .
41, respectively. sound impinges on the dislocation lines, their displace-ments contribute an extra strain in addition to the elas-tic strain. This interaction leads to changes in the prop-agation speed and the attenuation of the ultrasound .Ultrasound in the MHz range excites resonant vibrationin the dislocation lines. We apply the GL theory to solid He from the overdamped vibration near 1.2 K down tounderdamped vibration near 0.3 K. Below 0.3 K, the im-purities induce profound modifications on the ultrasoundpropagation.The effect of dislocation lines on ultrasound responsemay be expressed by writing the net fractional changein the longitudinal sound speed v and the attenuation α ,respectively, as: δvv = δv d v + δv a v + c, (1)and α = α d + α , (2)where δv a /v = aT accounts for the phononanharmonicity( a is a fitting parameter), δv d /v and α d represent the effects of the dislocation line motion on thespeed and attenuation, respectively. The constants c and α are included as reference points which are to be deter-mined in the data fitting procedure. In the linear region,we can write α d = β d x m , (3) where β d is the attenuation coefficient due to disloca-tions.According to GL, δv d /v and β d in the absence of impu-rity pinning are given by sums of the effects of all disloca-tion segments with a distribution in the network pinninglength L : δv d v = 4 Rv t π Ω Z ∞ LN ( L )[1 − ( L L ) ] dL [1 − ( L L ) ] + g , (4)and β d = 4 Rv t π v Ω Z ∞ LN ( L ) gdL [1 − ( L L ) ] + g . (5)Here R is the “orientation factor” of the dislocationline relative to the sound propagation direction, v t thespeed of transverse sound, and g a dimensionless damp-ing parameter. The dislocation segments are assumeddistributed in length by N ( L ) dL , where the distributionfunction N ( L ) is the number of dislocation segments perunit volume per unit length. The “resonant length” L is defined by: L = r − ν v t Ω , (6)where ν ( ≈ .
3) is the Poisson ratio. Taking the speed oftransverse sound at 35 bar as v t = 267 m/s , we estimate L = 7 . µ m at the applied ultrasound frequency.The distribution function is not a priori known.Typically , the distribution of the network dislocationsegments L n ( L ) is assumed to be a temperature indepen-dent exponential: N n ( L ) = Λ L nA exp( − LL nA ) , (7)where Λ is the dislocation density, i. e. the total “mo-bile” dislocation line length per unit volume and L nA is the average network pinning length. Where the net-work dislocation segments dominate, N ( L ) in Eq. (4)and (5) is replaced by N n ( L ). Where temperature de-pendent impurity-pinned dislocation segments play cru-cial role in the ultrasound response, a new distributionfunction N A ( L, T ) will be introduced (see Eq. (11)).The damping mechanism of dislocation line vibra-tion in solid He has been identified experimentally atlow temperatures as the “fluttering” interaction withphonons so that the temperature dependence of g is givenby g = g T , where g is a constant. The dislocation linemotion in our experiment varies over a wide range fromthe overdamped ( g >
1) regime at high temperatures tounderdamped ( g <
1) regime at low temperatures. Thecrossover from the overdamped to the underdamped vi-bration occurs around 560 mK where g = 1.The major focus of this report is the underdampedregime of the dislocation line vibration at low tempera-tures, where the motion of those line segments of length0close to L results in “resonant ultrasound response.”The integrand in Eq. (5) shows that β d is characteris-tically dependent on the magnitude of g . In the over-damped dislocation vibration regime, the denominatorbecomes approximately g and β d is proportional to g − .In the underdamped dislocation vibration regime, the in-tegrand shows the resonant character around L = L .In the limit of g <<
1, the height of the integrand is L N ( L ) g − and the width is approximately gL . Thus, β d in the underdamped regime at low temperatures be-comes a constant independent of T : β d ( g << ≈ Rv t L N n ( L ) π v Ω = 2 R Λ v t L π vL nA Ω exp (cid:18) − L L nA (cid:19) . (8)The integrand in Eq. (4) shows that δv d /v is deter-mined by the balance between the negative change fromthose line segments with L < L and positive changefrom those line segments with L > L . The “anti-resonance” character of the effects of dislocations on thepropagation speed of ultrasound introduces some intri-cate dependence on temperature. B. Analysis of amplitude independent data
At high temperatures,
T >
300 mK, the impuritiesare thermally driven off of dislocations and pinning byimpurities becomes negligible. So the dislocations arepinned only at the network nodes. The ultrasound re-sponse would then be independent of the drive ampli-tude as in fact observed above 300 mK in the typicalsample α data with A f = 14 . < T < α , L nA , R Λ and g as adjustable parameters. The bestfit parameter values for the sample α d down to 10 mK with these fitparameters is shown as α in Fig. 10(1). Note that α is essentially constant below 300 mK as expected fromthe underdamped motion of dislocation lines in the lowtemperature limit as given by Eq. (8).The δv/v data with A f = 14 . < T < a and c as adjustable while fixing all other parameters, L nA , R Λ and g , listed in Table I for the sample a and c are also listed in Table I. The calculated δv d /v + δv a /v down to 10 mK with the parameters for the sample δv/v ) in Fig. 10(2).The “network-pinned” response, α and ( δv/v ) , re-produces the amplitude-independent data at T >
T <
300 mK in the limit of highamplitude.
FIG. 10. Calculated attenuation α d (panel (1)) and changein speed δv d /v + δv a /v (panel (2)) in the two limiting states ofthe sample α and ( δv/v ) , and the purely impurity-pinnedstate(red dashed curves) are α and ( δv/v ) . C. Pinning and unpinning of dislocations byimpurities in thermal equilibirum
The process of pinning and unpinning of dislocationlines by the He impurity atoms is central to understand-ing the amplitude-dependent and hysteretic ultrasoundresponse observed at
T <
300 mK. In thermal equilib-rium, the number of impurities on the dislocation linesis set by the balance between the temperature indepen-dent pinning rate and the thermally activated unpinningrate. .The average impurity pinning length in thermal equi-librium L iA ( T ) is expected to follow the Arrhenius law: L iA ( T ) = L i exp (cid:18) − E b T (cid:19) , (9)where E b is the binding energy and L i a constant. Thetemperature-dependent distribution function of the im-purity pinning length is assumed to be exponential: N i ( L, T ) = Λ L iA ( T ) exp (cid:18) − LL iA ( T ) (cid:19) . (10)Then the effective temperature-dependent distributionfunction of the combined impurity-network pinninglength is also exponential: N A ( L, T ) = Λ L A ( T ) exp (cid:18) − LL A ( T ) (cid:19) , (11)1 TABLE I. Parameters derived from the experimental data; the He concentration ( x ), the reference points for attenuationand speed of sound ( α , c ), the anharmonic parameter ( a ), and the parameters related to dislocations ( g , R Λ, L nA , L i , and E b ). The last column indicates the type of sample behavior. The blank entries indicate that sufficient experimental data wasnot taken to enable extracting those parameters.Sample x (ppm) α (dB) c (10 − ) a (10 − K − ) g (K − ) R Λ(10 m − ) L nA ( µ m) L i ( µ m) E b (K) type ∗ ∗ where the effective average pinning length L A ( T ) is takenas the parallel combination of L nA and L iA ( T ): L A ( T ) = L nA L iA ( T ) L nA + L iA ( T ) . (12)When the ultrasound drive amplitude is sufficientlysmall, the dislocations are pinned at network nodes andby He impurities in thermal equilibrium. This pinningstate will be simply called as impurity-pinned state. The α and δv/v data at A f = 1 .
12 shown in Fig. 3(1) and (2),respectively, are established as the impurity-pinned stateof dislocations based on the observations that the datafrom warming and cooling runs are identical and α is thelowest among the runs. In this case, we assign N A ( L, T )as the distribution function N ( L ) in Eq. (4) and (5). Theattenuation data of the cooling run with A f = 1 .
12 in theentire temperature range is fitted with Eq. (5) (togetherwith Eq. (9), (11), and (12)) with two fitting parameters L i and E b while keeping α , g , R Λ, and L nA fixed tothose found in Sec. IV B and listed in Table I for the sam-ple L i = 101 µ mand E b = 0 .
35 K as shown in Table I. With these param-eters, α d and δv d /v + δv a /v are calculated and shown inFig. 10 as α and ( δv/v ) , respectively. They representthe response of the impurity-pinned state in the limit oflow amplitude. Figure 11 shows the measured attanua-tion and speed in sample A = 1 .
12 and the calculated curves for the impurity-pinned state along with other curves for E b = 0 .
67 K(seeSec. V B) and δv a /v + c .The Arrhenius law, Eq. (9), becomes unphysical at lowtemperatures around T min because the minimum valueof the impurity pinning length is limited to the nearest-neighbor atomic distance, b = 0 .
36 nm. A more realistictemperature dependence would be L iA ( T ) = L i exp (cid:18) − E b T (cid:19) + γb, (13)where γ is a numerical factor. Setting γ = 0, Eq. (13)is equivalent to Eq. (9). We expect γ > a ( db ) (1)10 100 1000-1.0-0.50.00.51.01.5 d v / v ( - ) T (mK) (2)
FIG. 11. Cooling data of the sample A f = 1.12 (blackclosed circles) compared with calculated curves. Panels (1)and (2) show attenuation and change in speed, respectively.Black solid curves represent calculations with E b = 0 .
35 Kand other parameters shown in Table I for sample α and ( δv/v ) in Fig. 10 shiftedby α = 10 .
31 dB and c = 0 . × − . Red broken curvesrepresent calculations with E b = 0 .
67 K, L i = 2440 µ m andother parameters in Table I for sample the repulsive force between neighboring He atoms on adislocation line. Figure 12 shows the temperature vari-ation of L iA for γ = 0, 1, 10, and 100. The lines of α and ( δv/v ) in Fig. 10 are calculated with γ = 0, buteven when γ = 100, the deviations from these lines areinsignificant. Thus the value of γ cannot be determinedfrom the present measurements.Let us consider the variation of L iA with x . For sim-2
10 100 10001E-40.0010.010.1110100
T (mK) L i A ( m m ) FIG. 12. Temperature dependences of the average impuritypinning length calculated from Eq. (13) for x = 0.3 ppmwith L i = 101 µ m and E b = 0.35 K for γ = 0 (solid line), 1(dotted line), 10 (dashed line), and 100 (dash-dotted line). plicity, we assume that the number of He atoms con-tributing to pinning the dislocation lines is proportionalto x . Then we expect L iA ( x , T ) = Kx exp (cid:18) − E b T (cid:19) , (14)from Eq. (9) where K/x = L i . We obtain K = 30 µ m · ppm from assuming x =0.3 ppm and L i = 101 µ m. D. Stress-induced unpinning
As noted in Sec. IV A, the dislocation lines are stronglypinned at network nodes but also weakly pinned by im-purities. As a He atom has a larger atomic volume thana He atom, the difference in the atomic size causes impu-rity pinning. The interaction between a dislocation anda He atom can be described by the elastic potential en-ergy (see Appendix C, Eq. (C1)). If the applied force F on a pinning impurity is sufficiently large, the dislocationline can be unpinned there. The compressional stress ofthe longitudinal ultrasound induces shear stress compo-nent σ s on the glide plane of a dislocation line and σ s inturn induces a force on the dislocation line. When thedislocation segments with lengths L and L adjoining apinning site bow out, the unpinning force on the pinningsite is given by F = bσ s L + L , (15)where b is the magnitude of Burgers vector which is equalto the nearest neighbor atomic distance. A critical force, F c , is required to produce unpinning. The condition of unpinning, F > F c , can be transformed to L + L > L c , (16)where L c is the critical length L c = 2 F c bσ s . (17)For our sample L c is estimated to be (see AppendixB, Eq. (B8)) L c [ µ m] = 12 A . (18)The critical force is estimated (see Appendix B) to be F c = 1 . × − N. For comparison, F c has beenevaluated from a torsional-oscillator experiment tobe 1 . × − N and from a shear-modulus experiment to be 6 . × − N.There are two types of stress-induced unpinning inour experiment. One is the “catastrophic” unpinningwhich occurs when the stress is increased at low temper-atures. The other is unpinning of a single impurity atomon a network segment which occurs when the sample iscooled at constant stress amplitude. The catastrophicunpinning is discussed first.Consider a network segment, i. e. a dislocation linepinned by two adjacent network nodes, which is addi-tionally pinned by several impurities distributed betweenthe two ends under a small external stress. When the ex-ternal shear stress increases as the ultrasound drive am-plitude is increased, L c decreases according to Eq. (18).Eventually, L c becomes shorter than the sum of some ad-jacent two impurity segments along the dislocation lineand the dislocation is unpinned from the impurity. Onceunpinning is initiated, it continues catastrophically to thenext adjacent impurity segments till this network seg-ment is pinned only at its two end network nodes.In the original model of the catastrophic unpinning ,the pinning impurities are assumed to be immobile sothat the dislocation line is pinned by the impurities againas soon as the stress is decreased. In the case of solid he-lium, on the other hand, the He impurities drift awayfrom the dislocation line once unpinned. Therefore, asthe catastrophic unpinning is initiated, the network seg-ment becomes free from impurity pinning.Consider next the situation when a network segmentof length L N is pinned by a single impurity atom, whichdivides the network segment into two impurity segmentsof lenghts L and L where L + L = L N . When anultrasound pulse at a drive level A corresponding to thecritical length L c is applied, the impurity atom is un-pinned if L N > L c , while it remains pinned if L N < L c .In a cooling run, pinning by impurities becomes appre-ciable at temperatures below E B ( ≈
350 mK). Accordingto the calculation in Fig. 12, L iA at 300, 200, and 100mK are 31.2, 17.4, and 3.0 µ m, respectively. For exam-ple, a network segment with L N ≈ µ m is pinned byone He impurity atom on average when the sample is3 s s ( P a ) x L c a nd L ( m m ) FIG. 13. Schematic spatial variations of shear stress am-plitude and critical length at T min . The driver transducer islocated at x = 0 and the receiver transducer at x m = 6 . A f = 4, which leads to σ s = 2 .
72 Pa from the transducer calibration constant(seeEq. (A4)). The shear stress amplitude (solid black line)is assumed to vary near the drive transducer according to σ s ( x ) = σ s exp( − βx ), where β = 380 m − is the attenuationcoefficient in the network-pinned region. The sample dividesinto two regions separated by the border at x = 2 . x < x ) where L c (dashed blue line)is shorter than the resonance length L (dash-dotted red line)and the impurity-pinned region ( x > x ) where L c ≥ L . Thecritical force F c = 1 . × − N is assumed. cooled down to 100 mK. If the amplitude of ultrasoundis A = 4 corresponding to L c = 3 µ m, longer networksegments ( L N > µ m) pinned by an impurity are un-pinned because L N > L c , while shorter ones ( L N < µ m) pinned by an impurity remain pinned. When thesample is further cooled, the dynamical aspects of pin-ning and unpinning must be taken into account. Thetemperature-independent pinning rate on a network seg-ment of length L N = 3 µ m is about 1 atom/s (see Eq.(25)), which is much smaller than the pulse repetitionfrequency (1 kHz). When a He atom happens to ap-proach a network segment of length L N > L c and pins itdown just after an ultrasound pulse has traversed, thenext ultrasound pulse comes within 1 ms and unpinsthe network segment from the impurity atom. As a re-sult, the network segments longer than L c stay unpinneddown to T min . Those shorter than L c , on the other hand,are pinned by impurities and the average pinning lengthdecreases with decreasing temperature according to Eq.(12). E. Spatial variation of pinning
In the previous section, the ultrasound is assumed totravel unattenuated throughout the sample. We intro- duce in this section a more realistic model that the am-plitude of the ultrasound pulse is attenuated along thepath through the sample. The local amplitude is writ-ten as A ( x ) where x is the position in the sample fromthe driver transducer end ( x = 0 mm) towards the re-ceiver end ( x m = 6 . σ s ( x ) and L c ( x ), respectively. In order to analyzethe amplitude dependent attenuation below 300 mK, thespatial variation of the response of dislocation lines to ul-trasound field must be considered. In the underdampedregime of dislocation vibration, the network segments oflengths around L mainly contribute to α d and δv d /v owing to their resonant behavior. As the attenuation inthe sample T < A ( x ) and σ s ( x ) decrease by a fac-tor of 10 while L c ( x ) increases by a factor of 10 withinthe sample from x = 0 to x m . An example of spatialvariations of σ s ( x ) and L c ( x ) with the drive amplitudeinitially set to A f = 4 .
0, corresponding to L c = 3 . µ mat x = 0 mm, is illustrated in Fig. 13.The local L c ( x ) increases with x and eventually ex-ceeds L = 7 . µ m. In Fig. 13, x is the position where L c becomes equal to L . The sample can be dividedinto two regions of distinct ultrasound response to dislo-cations: (1)“network-pinned” state region nearer to thedriver, 0 < x < x , and (2)“impurity-pinned” state re-gion nearer to the receiver, x < x < x m . We define acharacteristic fraction r as r = x /x m . ( r = 0 .
36 in Fig.13).In the network-pinned region, the network segmentswith lengths near L are not at all pinned by impurities.This region corresponds to the high drive amplitude limitconsidered in Sec. IV B. The ultrasound response is thusdescribed by α and ( δv/v ) shown in Fig. 10.In the impurity-pinned region, L c is longer than L so that the network segments with lengths near L arepinned by He impurities. This region corresponds to thelow drive amplitude limit considered in Sec. IV C. Theimpurity pinning length given by Eq. (9) is strongly tem-perature dependent and the ultrasound response is thusapproximately described by α and ( δv/v ) shown in Fig.10. The ultrasound attenuation α becomes vanishinglysmall at low temperatures below 70 mK, so that σ s and L c are spatially independent at T min in the impurity-pinned region ( x > x ) in Fig. 13.The net ultrasound response of attenuation and changein speed to the dislocation motion is written as: α d = rα + (1 − r ) α (19) δv d v = r (cid:18) δvv (cid:19) + (1 − r ) (cid:18) δvv (cid:19) . (20)The distinction of network-pinned and impuritiy-pinnedregions becomes meaningless at T >
300 mK as L iA becomes longer than L . Nevertheless, Eq. (19) and(20) can be applied in the whole temperature range forconvenience since α d and δv d /v are independent of r at T >
300 mK.4
F. Analysis of cooling runs at various A f To facilitate the analysis of data with Eqs. (19) and(20), the reference points α and c are subtracted fromthe experimental data of the sample A f = 10 . r is assumed to be atemperature-independent constant in each cooling runbecause α is almost constant below 300 mK. As α = 0at T min , the value of r ( A f ) is determined from Eq. (19)as r ( A f ) = α ( A f ) − α α ( T min ) , (21)where α ( A f ) − α is the attenuation of a cooling run at T = T min in Fig. 14.The empirical values of r ( A f ) for the cooling runs with A f = 1.12, 3.98, 10.0, and 14.1 are 0.036, 0.373, 0.767,and 0.868, respectively, as shown in Fig. 16 and in Fig.17(2). Fitting of the data except for A f = 1.12 results in r ( A f ) = ( ≤ A f < . . ln A f . (1 . ≤ A f ) (22)which is also shown in Fig. 16. The functional form ofEq. (22) is derived in Appendix B.The values of r ( A f ) in cooling runs are entered in Eq.(19) to calculate α d . The resulting fits are drawn as blacksolid lines in Fig. 14. The observed temperature depen-dence of attenuation is generally well reproduced by thefits. The same numerical values are used in Eq. (20)together with the phonon anharmonicity term, δv a /v , tocalculate the fit curves to the δv/v − c data of the coolingruns shown in Fig. 15 as black solid lines. The agree-ment between the experimental and calculated changesin speed at T min is not as good as that of attenuationlikely because r is determined from the attenuation dataalone. G. Analysis of warming runs
According to the measurement procedure in Fig. 2, thedrive amplitude is decreased to A i ( < .
07) correspondingto L c > µ m at T min as estimated from Eq. (18) beforestarting each warming run. Practically all the networksegments are densely pinned by He impurity atoms sothat each warming run starts with the totally impurity-pinned state ( r = 0). The drive level is then increased toa new A f (see [5] → [1] → [2] in Fig. 2).When A f = 1 .
12, the critical length is estimated tobe L c = 10 . µ m so that stress-induced unpinning doesnot occur for the network segments with lengths near L = 7 . µ m (see Fig. 2(2a) and (3a)). Therefore weexpect r = 0 throughout the warming run as shown in Fig. 17(1). The calculated curves of α d and δv d /v + δv a /v for the warming run at A f = 1 .
12 (dotted magentalines in Fig. 14 and 15, respectively) coincide with thosefor the cooling run. The increase in α d at temperaturesabove T is due to the thermal unpinning effect describedby Eq. (9).When A f = 3 . L c is decreased to 3.0 µ m. Stress-induced unpinning, however, does not occur at T min be-cause the average impurity pinning length, L iA ( T min ),is much shorter than this L c (see Fig. 2(2b)). As thetemperature is increased, L iA ( T ) becomes longer. At T = T ( ≈
50 mK), the condition of unpinning, Eq.(16), is evidently satisfied for some inpurity-pinned seg-ments and the catastrophic unpinning is initiated (seeFig. 2(3b)) so that r starts to increase as shown in Fig.17(1). More network segments are catastrophically un-pinned and r continues to increase as the temperature isfurther increased. At the same time, however, the rel-ative ultrasound amplitude at the border between thenetwork- and impurity-pinned regions, A ( x ), decreasesdue to the attenuation in the extended network-pinnedregion. At T = T ( ≈
70 mK), A ( x ) is expected to be-come smaller than 1.54 and L c becomes longer than L .Indeed the value of r at 70 mK is estimated to be 0.5,from which we obtain A ( x ) = 1.26. A closer inspectionshows that there is a kink in the attenuation data at T .We assume that r stays constant at T > T as shown inFig. 17(1).When the drive amplitude is increased to A f = 10 . T min (see Fig. 2(2c)). As a result, α increases (withaccompanied time dependence, see Fig. 6(3)) at T min asshown in Fig.14(c) and (d). In terms of r , it increasesfrom r = 0 to 0.17 and 0.44 for A f = 10 . A ( x ) =6.53 and 4.79, respectively. When the sample iswarmed, thermally-assisted stress-induced catastrophicunpinning occurs between T and T similar to the warm-ing run at A f = 3 .
98. The experimental values of r at T are 0.83 for both A f = 10 . A ( x ) decreaseto 1.25 and 1.81 for A f = 10 . H. Extraction of parameters and consistentdescription of ultrasound response
The calculated attenuation and speed using Eq. (19),(20) and the temperature dependence of r shown in Fig.17 are drawn as dotted lines for warming runs and solidlines for cooling runs in Fig. 14 and 15. The observed andcalculated attenuation agree well except in the coolingruns from 350 mK to T . The cause of dicrepancy may bethe oversimplified assumption of a constant value of r foreach cooling run. The agreement between the observedand the calculated speed of sound is worse than that ofattenuation likely because r is determined only from theattenuation data. Nevertheless, the local “peaks” and5 A f ( a r b ) a - a ( d B ) T ( m K ) (d)(c)(b)(a) FIG. 14. Temperature dependence of α − α in the sample A f = 1.12(a), 3.98(b), 10.0(c) and 14.1(d). Data and symbols are identical to those in Fig. 3(1) except the addedwarming data (closed orange stars) and cooling data (open orange stars) for (c). Curves(black solid for cooling and dottedmagenta for warming) are α d calculated from Eq. (19) with the values of r in Fig. 17. Down(up) arrows indicate characteristictemperatures T ( T ) where the dislocation line pinning state changes. Vertical data at T min in the warming runs of (c) and (d)indicate time dependent response as in Fig.3. “valleys” found in the observed temperature dependenceof δv/v are reproduced by the calculations.The hysteretic ultrasound response versus A at T min shown in Fig. 5 can be interpreted in terms of the changesin r . At sufficiently low temperatures ( T <
50 mK),Eq. (19) reduces to α d = rα because α = 0. Sim-ilarly, Eq. (20) reduces to δv d /v = r ( δv d /v ) . As A is decreased from A f of the previous cooling run, r de-creases according to Eq. (22) with A f replaced by A .When A < A c ( ≈ . r becomes zero so that α d = 0and δv d /v = 0. As A is now increased in the range A < A t ( ≈ r remains zero. When A exceeds A t , catas-trophic unpinning of dislocations occurs and r increases.It is this unpinning process that shows long relaxation.Thus the origin of the hysteretic behavior at T min shownin Fig. 5 can be traced to the hysteresis in r . V. DISCUSSIONA. Relation and brief comparison with earlierultrasound experiments on solid He The present ultrasound results in the higher temper-ature range (
T >
200 mK) are fairly well understoodas a transition from the overdamped to underdampedresonance of dislocations by the Granato-L¨ucke theory.These results are similar to the previous ultrasound mea-surements on single crystalline samples of He , al-though our samples are likely polycrystalline. The simi-larity comes about owing probably to the size of crystalgrains (typically 1 mm) in our samples being muchlarger than the pinning length of dislocation segments(typically 10 µ m).Our observations show that the dislocations are pinned6 - . - . . . . A f ( a r b ) d v / v - c ( - ) T ( m K ) (d)(c)(b)(a) FIG. 15. Temperature dependence of δv/v − c in the sample A f = 1.12(a), 3.98(b), 10.0(c) and 14.1(d). Themeanings of symbols are the same as in Fig. 14. Curves are δv d /v + δv a /v calculated from Eq. (20) with the values of r inFig. 17. Vertical data at T min in the warming runs of (c) and (d) indicate time dependent response. by He impurities in the lower temperature range (
T <
200 mK). Such observations were not reported in theprevious ultrasound studies on single-crystalline solid He grown from commercial He gas . Probably, thetemperature was not sufficiently lowered or the drive levelwas too high in the early measurements. The pinning ef-fect is most likely present also in single-crystalline solid He grown from commercial He gas when the tempera-ture is lowered down to 15 mK with sufficiently low drivelevel.In one of the early measurements , the temperaturewas lowered below 100 mK but no dislocation effects, es-pecially no pinning effects were reported because eitherthe dislocation contributions to the ultrasound speed andattenuation were too small or the He concentration wastoo low (the authors used a higher purity gas with x =5 ppb). They observed contributions to the speed andattenuation from thermally activated elementary excita-tions and an additional resonant attenuation. Our obser- vations did not show such effects.Ho, Bindloss and Goodkind reported ultrasoundmeasurements on solid He containing 27.5 ppm of He.They observed a new anomaly at about T p = 165 mKcharacterized by a sharp attenuation peak and an in-crease in the speed of sound with decreasing tempera-ture. They analyzed the attenuation peak and the ac-companied speed change as a relaxation mechanism phe-nomenon, and explained the anomaly as due to a con-tinuous phase transition (second order phase transition),suggesting supersolidity . In our measurements on sam-ples with x = 20 ppm (see Fig. 9), a somewhat similarincrease in speed of sound was observed at temperaturesbelow 150 mK, but the attenuation did not show a peakin contrast to their data. Our observations below 150 mKcan be described in terms of pinning of dislocations by He impurities just like in our other samples with lower He concentration. The characteristic temperatures T and T would be shifted higher in the x = 20 ppm sam-7 FIG. 16. Empirical and fitted values of r for the cooling runsof the sample A f as red filled circlesand a black line, respectively. ple since the average impurity pinning length at a giventemperature becomes shorter as x is increased. If theanomaly observed by Ho et al. is related to the disloca-tion pinning, amplitude dependence should be observedon the lower side of T p . They, on the contrary, observedamplitude dependence on the higher side between T p and500 mK. Hence, the anomaly does not seem to be causedby the dislocation pinning.Iwasa and Suzuki reported ultrasound measurementson single-crystalline He containing various concentra-tions of He; x =30 ppm, 300 ppm, and 1 %. They ob-served nonlinear (amplitude-dependent) attenuation andspeed of sound between 120 and 900 mK in the samplewith x =30 ppm. They argued that the nonlinearity wasrelated to pinning and unpinning of dislocations by Heimpurities. This is consistent with our present results.The higher onset temperature of nonlinearity (900 mK)in their measuerements at x =30 ppm compared with theonset temperature (150 mK) in our x =20 ppm samplepossibly indicates that the actual He concentration inone or both of the samples may be substantially differentfrom the given values.
B. Comparison with shear modulus measurements
It is interesting to compare our ultrasound results withthose of shear modulus. Haziot et al. find that theirshear modulus measurements and analysis of polycrys-talline He samples show a dislocation density of 5 . × m − with an average length of 59 µ m. Our R Λ is consis-tent with their dislocation density, considering that theorientation factor of a randomly oriented polycrystallinesample is 1/4 for the shear wave and 1/16 for the longitu-dinal ultrasound. Our L nA , on the other hand, is smaller FIG. 17. Dependence of r on A f and T during warm-ing(1) (Sec. IV G) and cooling(2) (Sec. IV F) runs: A f = 1.12(continuous(black line)), 3.98(dashed(red)), 10.0(dot-ted(magenta)) and 14.1(dash-dotted(blue)). In the warmingruns, the drive amplitude is increased from A i to A f at T min ,thermally-assisted stress-induced unpinning of impurity sitesoccurs where T < T < T , and only thermal unpinning oc-curs where T > T . In the cooling runs, the values of r do notvary with T . The slanted hash marks indicate the tempera-ture range where the calculated ultrasound response does notdepend on r . by one order of magnitude than their length. This dis-crepancy may arise from the number of pinning pointsaffecting the ultrasound propagation being much greaterthan in the shear modulus experiment. Since the networknodes act as pinning points in both shear modulus andultrasound experiments, there must be additional pin-ning points in the ultrasound experiment. A candidatefor the additional pinning points is jogs, but we do nothave sufficient information on the microscopic structureof basal dislocations and jogs for further analysis.Fefferman et al. reported the binding energy of Heatoms on dislocations as E b = 0 .
67 K and the criticalforce F c = 6 . × − N. Their binding energy is about afactor of two greater than ours. The discrepancy may bedue to the different models in the analysis. The bindingenergy in the present work is obtained from the tempera-ture dependences of the sound speed and attenuation as-suming a T-dependent impurity pinning length (pinningmodel). The binding energy in the shear modulus mea-surements, on the other hand, was obtained from the fre-quency dependence of the dissipation-peak temperatureassuming that the damping force on dislocation motionat low temperature was proportional to the concentra-tion of He bound to the dislocations (damping model).If E b = 0 .
67 K is assumed in the analysis of the attenu-8ation and speed of sample A f = 1 .
12, the resultsof fitting are shown as red broken curves in Fig. 11. Ascan be seen, the variation of the curves between 60 and300 mK differs from the measurement because L iA with E b = 0 .
67 K varies faster than that with E b = 0 .
35 K.Since E b is the depth of the spatially dependent bind-ing potential and F c is its maximum slope, E b and F c are simply related to each other as shown in AppendixB. Entering Fefferman et al.’s value of F c into Eq. (B4)gives E b = 0 .
23 K. This binding energy is different fromtheir own estimate but close to ours.Kang et al. reported observations on the hysteresis ofshear modulus while scanning the temperature and theapplied stress. Their measurement procedure did not al-low observations of thermal hysteresis like in ours. On theother hand, they did observe stress-dependent hysteresisand explained it in terms of the pinning/unpinning of dis-locations by He impurities similar to our analysis. Theyassumed distributions in the impurity pinning length aswell as in the network pinning length. Although dislocation effects can be observed in bothshear modulus and ultrasound measurements, significantdifferences should be noted. The ultrasound is selectivelysensitive to those dislocation segments with lengths closeto L . This is advantageous since L nA tends to be closein range to L in many samples. The shear modulus issensitive to those with much broader range of lengths.Spatial variation in the stress amplitude plays an impor-tant role in ultrasound, while the stress amplitude in theshear modulus experiment can be assumed to be uniformowing to the much longer wavelength. C. Small-attenuation response
There are several possible effects that would producethe small-attenuation response shown in Fig. 7; (1) lowdislocation density, (2) small orientation factor, (3) shortnetwork pinning length due to high dislocation density,(4) suppression of dislocation motion due to high im-purity concentration, etc. A small orientation factor ispossible for a single crystal when the angle between thec-axis and the sound propagation direction, θ , is equal to0 or 90 ◦ (the orientation factor of longitudinal ultrasoundin a single-crystal hcp He is given by R = (1 /
8) sin θ ). The small-attenuation response of a single crystal isreported in Fig. 3 in Calder and Franck which maybe due to the effect (2) above. The small-attenuationresponse due to the effect (4) is observed for a singlecrystal doped with 1 % He. However, the real originof the small-attenuation response of the present likelypolycrystalline samples is not clear.
D. Anomalous response
The sample showing anomalous response (see Fig. 8)accompanied by an unexpected peak in attenuation and rapid change in speed near 700 mK is now discussed. Theresponse is suggestive of a Debye relaxation process. Thechange in speed and attenuation according to the Debyerelaxation model can be written as δv r v = − φ τ r ) , (23)and α r x m = − φv Ω τ r τ r ) , (24)where φ is a constant and τ r is a characteristic relax-ation time. Eq. (23) is fitted to the δv/v data ofthe sample τ r =2 . × − exp( − . /T ). The activation energy in τ r islarger than E b and smaller than the formation energy ofa vacancy (more than 10 K). The position and height of α r calculated from Eq. (24) are in good agreement withthe data as shown in Fig. 8(1). Although the anomalyis well described with the Debye relaxation model, itsphysical origin is not clear yet. E. Dependence on He impurity concentration
Consider the variation in the onset temperatures T and T for the thermally-assisted stress-induced unpin-ning as x is increased from 0.3 to 20 ppm. Qualitatively,the average impurity length L iA decreases with increas-ing x so that unpinning would become more difficult,and T and T would become higher. Indeed T doesshift up from 50 mK to 80 ∼
90 mK and T from 70mK to 100 ∼
110 mK. Quantitatively, it is expected thatthe onset occurs at the temperatures where the averageimpurity length L iA is the same in the two impurity con-centrations. In the x = 0 . E b = 0.35 K gives L iA = 0 . µ m at the observed T =50 mK. In the case of x = 20 ppm, the same L iA occursat 125 mK, which is fairly close to the observed valueof T for x = 20 ppm. Similar calculation for T gives437 mK for x = 20 ppm, which is much higher thanthe observed value. Note there are uncertainties in thisestimation. (1) L iA may be proportional to x − / in-stead of Eq. (14). In this case T and T at x = 20 ppmare expected to be 83 mK and 159 mK, respectively, inbetter agreement with the observation. (2) The in-situ value of x may be different from 20 ppm. More study isrequired for quantitative analysis. F. Relaxation phenomena
The relaxation phenomena as shown in Fig. 5 and6 are, as stated earlier, likely related to the dynam-ics of “capture/release” of He impurities during pin-ning/unpinning of dislocations. The processes of decreas-ing and increasing of A involve distinct mechanisms andare discussed in order below.9The process of decreasing A in Fig. 5 leads to a de-crease in r according to Eq. (22) in which A f is re-placed by A . As A is decreased, pinning of dislocationsby impurities occurs in the network-pinned region near x = x , where the stress amplitude due to ultrasound isthe smallest. The response of ultrasound signal to thestepwise decreases in the drive amplitude is completedwithin less than a few seconds at T min (see Fig. 6(1)).This indicates that impurities are captured by disloca-tions on this short time scale ( ∼ α and δv/v would occur when A is decreased. According to Iwasa , the impurity pinningrate R is independent of T and given by R = Lx Q (25)where L is the length of dislocation segment and Q =1 . × (m · s) − is a constant. For a resonant dislo-cation segment, L = L = 7 . µ m, and with x = 0 . R = 3 s − and the relaxationtime, 1 /R = 0 . , on the other hand, theo-retically considered the interaction between He impurityand screw dislocation and found that the impurity cap-ture relaxation time would be in the order of hours anddays. Such a long relaxation time is not observed.The process of increasing A in Fig. 5 is very different.Initially, the drive level is set to the small value A i andthe dislocations in the entire sample are pinned to themaximum extent possible by impurities. When A is in-creased to a value less than A t , the dislocations remainpinned and the response is linear, i.e. α and δv/v donot change and the signal S increases in proportion to A with a short response time (see Fig. 6(2)). When A is in-creased, however, beyond A t , unpinning is initiated andthe catastrophic unpinning follows. The long relaxationtime in Fig. 6(3) indicates the unpinning as a stochasticand dynamical process.Immediately after the increase in A but before anysignificant unpinning has occurred, the stress amplitudeis uniform throughout the sample since the attenuationcoefficient in the purely pinned state is negligibly small.Unpinning can then be induced by ultrasound stress any-where in the sample. As the number of unpinned seg-ments increases, the stress amplitude decreases along x and the probability of unpinning also decreases.The stochastic and dynamical unpinning process maybe pictured as follows. The critical length for A = 14 . L c = 0 . µ m from Eq. (18). Ac-cording to Eq. (13), the average impurity pinning lengthbecomes L iA = γb at low temperature. If γ = 1, all thelattice sites along the dislocation line are occupied by He atoms and unpinning is impossible. We believe thata repulsive force between He atoms results in γ muchbigger than unity. Nevertheless, L iA at 15 mK is smaller than L c even with γ = 100 as shown in Fig. 12. Accord-ing to the distribution of the impurity pinning lengths,Eq. (11), some of the impurity segments may be muchlonger than L iA . In addition, individual impurity pin-ning length may fluctuate because He impurity atomscan move along the dislocation line at low tempereturesdue to quantum tunneling. The condition of unpinning,Eq. (16), can be occasionally satisfied due to the fluctu-ation and the dislocation segment can be unpinned. Allthese processes are likely involved in the observed slowrelaxation shown in Fig. 6(3). The relaxation time de-creases to less than 10 s at 65 mK, but no systematicstudy on the temperature dependence of the relaxationtime was made.More work at temperatures between 10 and 100 mK isclearly needed in elucidating the relaxation phenomena inthe unpinning process. Possible experiments include (1)increasing A f systematically at T min to see the changein the unpinning time, (2) applying a sudden DC stressand observing the dislocation unpinning, and (3) a kindof pump-probe method in which a large amplitude RFpulse is applied on the transmitter and the subsequentultrasound response is measured. G. Thermal effects at low temperatures
At temperatures below 200 mK in the underdampedregime, the damping of dislocation vibration due to thefluttering mechanism can be neglected. The reversibletemperature dependences of α and δv/v at low drive levelsuch as the data at A = 1 .
12 in Fig. 3 are due to pinningand unpinning of dislocations by He impurities whichoccurs uniformly in the whole sample irrespective of themagnitude of L N . These effects of temperature are quitedifferent from those of stress that causes unpinning pref-erentially in the region at higher amplitude and for thedislocation segments with longer pinning length.Thus the temperature dependences of α and δv/v atlow drive level originate from that of L A ( T ). Numericalcalculations of α d and L A ( T ) using the parameters for thesample α d = 0 . L A ( T ) = 0 . µ mat 74 mK; α d = 1 dB and L A ( T ) = 0 . µ m at 80 mK;and α d = 10 dB and L A ( T ) = 1 . µ m at 105 mK. Similarcalculation of δv d /v shows that a minimum of δv d /v = − . × occurs at 98 mK where L A ( T ) = 1 . µ m. Itindicates that the velocity minimum occurs at L A /L =0 . δv d /v in the network-pinned statecalculated with Eq. (4) and (7) shows that δv d /v < L nA < . µ m corresponding to the case of the sample δv d /v > L nA > . µ m corresponding to the case ofthe sample VI. CONCLUSION
A systematic study of 9.6 MHz ultrasound propaga-tion in solid He was made between 1.2 K and 15 mK.The attenuation and the changes in the speed of propaga-tion were measured as functions of temperature and thedrive amplitude. Depending on the drive amplitude, thepropagation characteristics showed linear, nonlinear, re-versible and hysteretic behaviors. Most of the observedbehaviors in the typical samples could be explained interms of the interaction between the ultrasound and thestring-like vibration of dislocation lines that are stronglypinned at network nodes and weakly pinned by He im-purity. The small impurity concentration at the level of0.3 ppm in the typical samples plays a crucial role in theultrasound propagation at low temperature below 200mK.There are two extreme states; the network-pinned stateat sufficiently high A f and the impurity-pinned state atsufficiently low A f . The temperature dependences of α and δv/v in the network-pinned state can be described bythe GL theory with the T -independent distribution of thedislocation pinning lengths similar to those in the single-crystalline He. Those in the impurity-pinned state aredescribed by the GL theory including the T -dependentpinning effects of dislocations by He impurity atoms.In our simplified model, the sample is divided in tworegions; the network-pinned region on the transmitterside and the impurity-pinned region on the receiver side,and the state of the sample is described with a single pa-rameter r which gives the T -, A f - and history-dependentfraction of network-pinned region of the sample.When the sample is cooled from 1.2 K, the value of r depends on the drive amplitude A f but not on T . Whenthe drive level is decreased from A f to A i at T min , r becomes 0. When A is increased at T min , r remains 0 upto A f = A t ( ≈ A f > A t , r starts to increasewith a long relaxation time.In warming runs when A f < A c ( ≈ . r remains0 throughout and the attenuation increases at T > T where thermal unpinning becomes appreciable. Whenthe value of A f is intermediate between A c and A t , forexample A f = 4, r remains 0 up to T . Then thermally-assisted stress-induced unpinning starts at T > T and r starts to increase at the same time. At temperatures T > T thermal unpinning becomes dominant so that α continues to increase but r stays constant.The present work has raised questions about pinningand unpinning processes of dislocations by He impuri-ties. The origins of the short relaxation time for pin-ning as well as the long relaxation time for unpinningare yet to be clarified. Measurement on the tempera-ture dependence of the relaxation time may be helpful.In order to get a more comprehensive picture, studies inthe frequency range between 100 kHz and 1 MHz wouldbe desirable. Ultrasound measurements on single crys-talline He samples at temperatures below 100 mK arealso desirable in order to study the pinning mechanism of dislocations by He impurities. Possible effects of grainboundaries in polycrystalline samples can be eliminatedin single crystals. Finally, a systematic study of the de-pendence on He impurity concentration including ultrahigh purity samples down to 10 mK and lower would beinteresting to explore the extent of the validity of ourinterpretations.
ACKNOWLEDGMENTS
We are most grateful to John Goodkind for providingus with the ultrasound instrumentation system utilized inthe experiments. We thank Elizabeth Eibling and ParthJariwala for help in designing of apparatus, Bettina Hein,Rebecca Cebulka, Michelle Goffreda and Chirag Soni fordata analysis, and Michael Keiderling for data acquisi-tion. We are grateful to the support and encouragementfrom Bob Bartynski. This research was supported byNSF DMR-1005325.
Appendix A: Calibration of transducers
The relation between the ultrasound drive amplitudeand the stress exerted onto the solid He samples isroughly calibrated by measuring the losses and gains inthe measurement system components as shown in Fig.A1. The present estimation is, however, uncertain by asmuch as a factor of two due to incomplete impedancematching, acoustic mismatch, and ringing of the trans-ducer in the ultrasound experiment. When the amplitudeof the transmitter output is V out = 80 mV and the vari-able step attenuator is arbitrarily set to α v = 41 dB,corresponding to the relative drive amplitude A = 1, thevoltage on the driver transducer is V = 2 .
53 mV. Thespectrometer signal through the solid He sample cooledto near 15 mK is V s = 2 . − X + 42 [dB] =20 log( V s /V ) = − . X = 21 dB, where X is theconversion loss at the transducer/solid He interfaces.Consider a voltage pulse of amplitude V and of timeinterval τ . The corresponding electrical pulse energy is E = ( V / Z ) τ where Z = 50 Ω is taken as the cableimpedance. Using V /V = A , we can write E = V Z τ A . (A1)The mechanical pulsed energy E transmitted from thequartz transducer into solid He is reduced by X = 21dB, E = E − / = 0 . E . (A2)The elastic energy is also written as E = C l ǫ πD vτ, (A3)1 FIG. A1. Energy diagram of the ultrasound system. Lossesand gains in the components in the measurement system areindicated. The loss in the ultra-miniature coaxial cable wascalculated to be 12 dB by measuring the input (80 mV peak-to-peak RF pulse) and output (20 mVp-p) through the cable. X is the transmission loss at each of the interfaces from drivequartz transducer to solid He and from solid He to receiverquartz transducer. When the variable step attenuator is setto α v = 41 dB corresponding to A = 1, the applied voltageon the drive quartz transducer is V = 2 .
53 mV. The outputsignal amplitude of the spectrometer is V s = 2 . He sample is at our minimum temperature( ∼
15 mK). where C l = 5 . × Pa is the longitudinal modulus ofhcp He, ǫ is the strain amplitude, D = 8 . v = 534 m/s is the longitudinalspeed of sound. From Eqs. (A1) through (A3) it is foundthat ǫ = 2 . × − A and the corresponding compres-sional stress amplitude is σ = C l ǫ = 1 . A Pa. Sincethe sample is probably polycrystalline and each grain israndomly oriented, the relative angle between the soundpropagation axis and the c -axis on average is arbitrarilytaken as θ = 45 ◦ . Then, the shear stress amplitude( σ s )becomes, σ s = σ sin θ cos θ = 0 . A Pa , (A4)and the corresponding shear strain amplitude( ǫ s ) is ǫ s = σ s C s = 4 . × − A, (A5)where C s = 1 . × Pa is the shear modulus. Theshear stress amplitude calibration is used in estimatingthe critical force for unpinning of dislocation lines(seeAppendix B).
Appendix B: Calculation of r ( A f ) We assume at first that the sample is a uniformmedium with the attenuation coefficient β . Ultrasound pulses are excited at x = 0 and propagate in the x -direction. The local amplitude of the pulses are givenby A ( x ) = A f exp( − βx ) , (B1)where A f is the relative drive amplitude at x = 0. Thelocal shear stress σ s ( x ) is proportional to A ( x ) (see Eq.(A4)) and the local critical length L c ( x ) is inversely pro-portional to σ s ( x ) (see Eq. (17)), so that L c ( x ) is in-versely proportional to A ( x ); L c ( x ) = qA ( x ) = qA f exp( βx ) , (B2)where q = 2 F c bσ s ( x ) A ( x ) = 2 F c . b . (B3)Eq. (A4) is used in the second equality.We next assume that the position, x , of the boundarybetween network- and impurity-pinned regions is deter-mined by the condition L c ( x ) = L . (B4)From Eq. (B2) and (B4), x and r are determined to be x = 1 β ln A f L q (B5)and r ( A f ) = x x m = 1 βx m ln A f L q . (B6)Note the range of r is between 0 and 1, so that r = 0 for A f ≤ q/L and r = 1 for A f ≥ ( q/L ) exp( βx m ).When we fit the data of r except for A f = 1 .
12 shownin Fig. 16 with Eq. (B6), we obtain r ( A f ) = 12 . A f . . (B7)Comparing Eq. (B7) with Eq. (B6), we obtain q =1 . · L = 1 . × − m, L c ( x )[m] = 1 . × − A ( x ) (B8)from Eq. (B2), and F c = 1 . × − N from Eq. (B3).We also obtain βx m = 2 . β and length x m to be20 log A f A ( x m ) = 21 . . (B9)This value is consistent with the attenuation in thenetwork-pinned state shown in Fig. 10, α ( T min ) = 23 . F c , on the other hand,is likely accurate only to an order of magnitude owing tovarious uncertainties in deriving Eq. (A4).2 Appendix C: Relation between E b and F c The interaction energy between an edge dislocation ly-ing along the z -axis at (0,0) and a He impurity atom at( x, y ) is given by W ( x, y ) = 4(1 + ν )3(1 − ν ) µb e r a δ yx + y , (C1)where x and y are the coordinates of the impurity paralleland perpendicular to the slip plane, respectively, ν is thePoisson’s ratio, µ is the shear modulus, b e is the edgecomponent of the Burgers vector, r a is the atomic radiusof a He atom, r a (1 + δ ) is that of a He atom, and δ isthe misfit parameter. We note b e = b for a perfect edgedislocation on the basal plane and r a = b/
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