thermal nucleation of the normal state in superheated superconducting tin grains

e

e

e

e

e;

PHYSICAL REVIEW B 1 SEPTEMBER 1998-IIVOLUME 58, NUMBER 10

Thermal nucleation of the normal state in superheated superconducting tin grains

H. DubosGroupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05, Paris, Franc

and Centro de Fı´sica Nuclear, Universidade de Lisboa, 1699 Lisbon, Portugal

T. A. GirardCentro de Fı´sica Nuclear, Universidade de Lisboa, 1699 Lisbon, Portugal

and Groupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05,Paris, France

G. WaysandGroupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05, Paris, Franc

P. Perrier and V. JeudyGroupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05, Paris, Franc

and Centro de Fı´sica Nuclear, Universidade de Lisboa, 1699 Lisbon, Portugal

D. LimagneGroupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05, Paris, Franc

J. SecoCentro de Fı´sica Nuclear, Universidade de Lisboa, 1699 Lisbon, Portugal

J. I. CollarPPE Division/CERN, 1211 Geneva 23, Switzerland;

Groupe de Physique des Solides (UA 17 CNRS), Universite´s Denis Diderot and Pierre et Marie Curie, 75251 Cedex 05, Paris, Francand Centro de Fı´sica Nuclear, Universidade de Lisboa, 1699 Lisbon, Portugal

~Received 5 March 1998!

Suspensions of superheated type-I superconducting Sn microspheres of 16–25 and 10–25mm diameterswere irradiated at temperatures below 1 K and above 2 K by decay electrons from14C and35S, respectively.Measurements of the radiation-induced gaps in the superheating transition curves indicate a full heating of thesmallest grain volume of each suspension, contrary to a partial heating model based on the difference betweenquasiparticle relaxation and diffusion rates in Sn. The phonons are in thermal equilibrium with the quasipar-ticles when the normal state is nucleated.@S0163-1829~98!05834-2#

pesiib

ttouct

Inel-th

tthu

-seford

ta-t

ith

’t iseat-

lax-or

anes-

I. INTRODUCTION

The thermodynamic response of metastable type-I suconducting microspheres to irradiation has received conerable attention in recent years as a result of their possapplications in particle detection.1,2 Thermal nucleation ofthe normal state in the superheated material is induced byinteraction of radiation if the energy deposited is sufficientraise its temperature across the phase boundary. How mof the volume is required to be heated in order to indunucleation of the normal state has, however, been a poinsome dispute.

There are essentially two possibilities: some or all.‘‘local-heating’’ descriptions, only a fraction of the volumis required; in a ‘‘global-heating’’ description, the entire voume is uniformly heated above the phase line beforenormal state is nucleated.

To our knowledge, a complete description of ‘‘local heaing’’ has never been satisfactorily elaborated. Generally,volume fraction has been related to the heating of an eq

PRB 580163-1829/98/58~10!/6468~8!/$15.00

r-d-le

he

cheof

e

-ea-

torial surface area, or nucleation germ~surface defect,boundary, or impurity!. The significance of defects was demonstrated by single-microsphere~hereafter referred to a‘‘grain’’ ! experiments in which a variation of thsuperheated-to-normal transition field was observedvariation of the grain orientation relative to the appliefield.2,3 Similar results were obtained on single-grain rotions under irradiation;3 the effect is more pronounced aincreasingly lower temperatures.

On the basis of a series of irradiation measurements wsingle grains, Franket al.4 recently advanced a ‘‘unified’’heating model that combines both ‘‘local’’ and ‘‘global’heating behaviors by whether or not an equatorial defecheated above the phase line before or with the uniform hing of the grain, which in turn depends on~1! the location ofthe interaction site relative to that of the defect, and~2! thedifference between the materials-dependent thermal reation and diffusion rates of the excited quasiparticles. Fmaterials in which the diffusion time is significantly less ththe phonon relaxation time, no local temperature can be

6468 © 1998 The American Physical Society

olfoandonal

eotiooneb

orld

io

oc

th

iotincats

.sia

ea

incehsader

ndta-III

n in

ownm

on.po-tter

s

de-

hed

he

-

or

PRB 58 6469THERMAL NUCLEATION OF THE NORMAL STATE IN . . .

tablished: the defect is uniformly heated with the grain vume, resulting in a global-heating behavior. In contrast,materials in which the relaxation is sufficiently faster ththe diffusion, such as Sn/In, only a ‘‘warm spot’’ is createwhich if sufficiently near a defect provokes its nucleatibefore the remainder of the volume, resulting in a locheating behavior.

The model of Ref. 4 comes closest to including the dtailed microscopic aspects of the involved energy transpAs appealing as it may appear, and independent of quesregarding the applicability of the near-equilibrium relaxatirates of Kaplanet al.5 in such nonequilibrium situations, thexperimental basis given for the model must, however,discountedin toto: neither the Sn grains of Refs. 3 and 4 nthe In grains of Ref. 4 exhibited the full superheating fie(Hsh). A partial compilation3,4,6,7 of recent Hsh measure-ments for Sn is shown in Fig. 1. The fact that the transitfields of Refs. 3 and 4 are near the bulk thermodynamcritical field8 (Hc) suggests that the observed transitionscurred from intermediate states,9 which would both lower thetransition fields and generate the low-energy tail inirradiation-induced transition spectra that are observed.

Nonetheless, the possibility of any local-heating behavfor Sn has a significant impact on the current developmenSn-based devices for particle-detection applications, sonly those materials exhibiting a global-heating behaviorgenerate a well-defined energy threshold. The experimenRef. 4 were performed on relatively large~34–46mm! diam-eter grains using heavily ionizing 4-MeVa particles that aregenerally stopped within about 10mm of the grain surfaceWe here report a series of experiments on suspensionsmaller diameter, fully metastable Sn grains, under irradtion by the decay electrons of14C and 35S in which theenergy depositions are distributed across the grain volumwhich clearly demonstrate a global heating in their normstate nucleation. Although single-grain measurementsprinciple yield more precise results, uncertainties arisfrom size distributions and local-field effects can be reduby working with well-separated suspensions of grains. Tapproach does not require the extreme sensitivity necesin single-grain measurements, and has generally yielmore accurate and reliable determinations of the GinzbuLandau parameter in type-I superconductors.2,7 In Sec. II, we

FIG. 1. Compilation of recent experimental measurementsHsh(T) in Sn microspheres. The solid line represents the contouthe thermodynamic critical field, obtained from Ref. 8.

-r

,

-

-rt.ns

e

nic-

e

rofcenof

of-

s,l-ingdisryd

g-

describe in further detail the distinction between global- alocal-heating models, indicating their different manifestions in single-grain and multigrain experiments. Sectiondescribes our experiments, the results of which are giveSec. IV.

II. THEORETICAL CONSIDERATIONS

A. Single grain

The phase diagram of a single superheated grain is shin Fig. 2. The precise location of a grain within the diagrais ideally at 3

2 the applied field (Ha), corresponding to itsmaximum local surface field as a result of demagnetizatiThe presence of a nucleation germ effectively raises thissition nearer the phase line as a result of its ability to scaquasiparticles.10

The temperature distance (DT) to the phase boundary ifixed by the grain location (Ta ,Ha). Thermal nucleation ofthe normal state depends only on the magnitude of theposited energy (DE):

DE<VheatedETa

Ta1DT

CsdT, ~1!

where Vheated is the heated volume of the grain;Cs is thevolume specific heat of the superconductor, given by

Cs5@agTce2bTc /Ta1aTa

3#, ~2!

where $a,b,a,g% are materials-dependent parameters. Tfirst and second terms of Eq.~2! describe the electron anphonon contributions, respectively. In Fig. 2,Ta1DT is re-lated to the superheated field by

Ta1DT5@Ta21 3

2 Tc2$DH/Hsh~0!%#1/2, ~3!

which is derived from the BCS temperature variation of tsupercritical field,11

Hshlocal~T!5Hsh

local~0!@12~Ta /Tc!2#. ~4!

The lowestHa at which a grain transition occurs, corresponding to the largest inducedDT, is given by

fof FIG. 2. A phase diagram for a single microsphere;Hl(T)

5(3/2)Ha .

in

a

ha

thgeiu

fth

tae

Sn

e

rmu-xro

oren

ns-bhecafm

ndoflaalcn

hei-

totic-sus-This

be

ctate-

ate;cal

thein-

ll

.

es

Asrum

hoter-the.

6470 PRB 58H. DUBOSet al.

ETa

Ta1DTmaxCsdT5DEmax/Vmin , ~5!

whereDEmax is the maximum energy deposition in the graand Vmin is the minimumVheated. For a sufficiently largeenergy of the incident electron,DE in a grain is approxi-mately linear with the electron range. This implies thDEmax is proportional to the grain radius~R!. In a global-heating model, the grain is uniformly heated, such tVmin[Vgrain, and DEmax/Vgrain5*Ta

Ta1DTmaxCsdT'R22

5const, independent of temperature.In local-heating descriptions, nucleation occurs when

requisite temperature increase reaches a defect near theequator. For energy depositions near this defect, the tempture increase reaches a maximum above the equilibrvalue before the full grain volume; in theextremecase,Vmincorresponds to the defect, and the required depression oorder parameter implies10 a scaling with the coherence leng~j!. Sincej(T)'j0(12T/Tc)

21/2,12

DEmax/V~j!'@3DEmax/4pj03#@12T/Tc#

1.5, ~6!

wherej0 is the Pippard coherence length of the pure me~Sn: 0.2mm!. For energy depositions further away from thdefect site, Vheated'0.5Vgrain1Vdefect'0.5Vgrain.13 In anyevent, the transition occurs at a lowerHa than expected fromDEmax/Vgrain.

In the model of Ref. 4, the local-heating behavior ofresults from the short time scale~'1 ns! for quasiparticlerelaxation to the energy gap by phonon emission, relativthe longer spreading time~'10 ns! of the quasiparticlesacross the grain diameter. In this case,Vheated<Vgrain: thequasiparticles quickly relax to phonons, creating a ‘‘waspot’’ within the grain volume which then spreads diffsively. Although the warm quasiparticles continue to echange energy with cold phonons during the diffusion pcess, the normal-state nucleation in these materialsinitiated prior to a full warming of the phonon system; fthe Sn results of Ref. 4, only 20–50 % of the depositedergy is in the phonon system.

B. Multigrains

The situation for an ensemble of grains isa priori morecomplicated than for a single grain, owing to the inherevariation of the individual local fields resulting from the ditribution of sizes, defects, and diamagnetic interactionstween the grains.14 These yield an apparent spreading of ttransition fields of the ensemble, as shown in the typiunirradiated experimental differential superheating curve otest suspension of Fig. 3, which was recorded during a raof Ha from zero to well aboveHsh at fixed temperature. Thewidth of the curve results from the local-field variations, afor small filling factors corresponds to the distributionlocal magnetic-field states populated by the grains. Thegrains to undergo a transition are those more metallurgicperfect, and they experience virtually no diamagnetic effesince the majority of the previously superconducting esemble has already become normal.

Reduction of the filling factor reduces the effects of tdiamagnetic interactions.15 Nonetheless, given the nonun

,

t

t

erainra-m

the

l

to

--is

-

t

e-

lap

stlyts-

formity of grain metallurgy and size, it seems impossibleavoid the existence of some distribution of local magnefield states within a phase diagram of the ensemble: thepension constitutes homogeneous, disordered media.spread of the local-field maxima at the grain equators canrepresented schematically by the vertical linexy in the phasediagram of Fig. 4, which in principle incorporates any defepresence: there is one and only one phase line for the mrial, given byHsh(T). In this diagram, for a givenHa , somefraction of the suspension has transited to the normal steach grain of the remainder sees a different maximum lofield (Hl), resulting in a distributionDTl5@Tsh(Hl)2Ta#required for normal-state transitions, whereTsh is the super-heated transition temperature.

Irradiating the suspension results in energy loss ofincident radiation to the material, generating temperaturecreases in accordance with Eq.~1!: in the simplest descrip-tion, with sufficiently long irradiation times or intensity, agrains within the regionDHmax5Hsh(Ta)2Hr of Fig. 4, cor-responding toDEmax of Eq. ~1!, undergo a phase transitionThe fieldHr corresponds to theTa1DT of Eq. ~3! for which

FIG. 3. Superposition of differential superheating curv(dN/dH) at 480 mK, with~d! and without irradiation~j!.

FIG. 4. An effective phase diagram of a grain’s suspension.indicated, electrons from the lower part of the energy-loss spectare capable of flipping only grains with smallDT. Sufficiently longirradiation times or intense radiation fields establish an effectiveborder atHr ; grains introduced into the region between the supheated phase boundary and the ‘‘hot border’’ via increase ofapplied magnetic field can transition only by thermal nucleation

herra

eo

i

of

s,g

-

ha-

t

,ls

teis

sersmeeete

rre

mbyfil

ska

e0

su

dil,r

in

d0of

Kent,zeroasis-

no

natan

edhis

aan

x-


Eq. ~5! is satisfied. This is observed in Fig. 3: the gap in tdifferential superheating curve of the suspension under idiation results from the transiting of grains to the normstate during a pause atHa5300 G inserted into the fieldramping, which has been recorded separately. The slopthe high-field region of the gap results from the transitiongrains introduced into the hot border zone by increasingHafollowing the pause.

The return of the suspension’s superheating curve tounirradiated behavior occurs whenHr has been raised toHsh(Ta). The maximum gap width (DHmax) thus provides ameasure ofDTmax induced by the maximum energy lossthe irradiation, which is defined by the minimumVheatedofthe suspension atHr . Despite the many-body complicationthe energy loss remains proportional to the electron ranandDEmax/Vgrain varies asRmin

22 . In a global-heating description, DHmax⇔DH5Hsh2Hstepof Ref. 4,Vmin corresponds tothe smallest grain of the suspension andDEmax/Vmin5constfor all temperature. For local-heating models,DHmax@DH:Vmin continues to correspond to that of a defect, so tDHmax scales withV(j) and must therefore exhibit a temperature dependence.

III. EXPERIMENT

Measurements ofDHmax were conducted for two differensuspensions of Sn grains, of 10–25mm diameters forTa.2 K and 16–25mm diameters forTa,1 K. In each casedata acquisition was accomplished using fast-puelectronics.16 The normal-state nucleation of a grain creadiscontinuities in the flux cut by a detecting loop, whichconnected via two transformers to a LeCroy HQV810-bapreamplifier. Only irreversible flux entry is detected: reveible flux changes are outside of the bandwidth of the prealifier. The input signal is shaped with a fast LeCroy amplifiand discriminated with a LeCroy MVL407 ultrafast voltagcomparator. Computer control synchronizes the magnstep rise with the opening of a gate during which flux pulsare detected. Owing to the transformer presence, the cutiming resolution of the system is limited to about 10ms.

A. Temperatures >2 K

The grains were of spherical geometry, varying in diaeter from 10–25mm, and were part of a sample madesonic dispersion of the molten metal in an oil bath, sizetration using calibrated sieves, and uniform folding17 into aparaffin dielectric. The suspension consisted of three dieach 5 mm in diameter and 0.3 mm thick. Each disk westimated to contain 1.43105 tin grains based on a volumfilling factor of 20%. The disks were spaced along a 10mm-diameter U-shaped copper wire loop of 225mm widthimprinted on an epoxy board.

Each disk was covered with one equal-diameter, tispaper foil containing 5mCi of evaporated35S, a pure beta-decay source with an end-point energy of 167 keV, anhalf-life of 82 days. Owing to the width of the sense coonly about 7% of the total grains were sensitive to measument.

The sensor was operated in a pumped4He refrigerator;the temperature was pressure regulated, providing a nom

a-l

off

ts

e,

t

es

d-p-r

icsnt

-

-

s,s

-

e

a

e-

al

stability of 1023 K. The applied magnetic field, generateperpendicular to the loop by a niobium Helmholtz coil of 1cm diameter and 1 cm separation, provided a field stability0.01 G and homogeneity of 3% over the detector region.

Measurements18 were conducted at 2.30, 3.10, and 3.44with the radiation source installed. For each measuremthe detector was tuned by first cooling the suspension atfield. The applied field was then raised until the last grain htransited to the normal state; the threshold level of the dcriminator was adjusted to the minimum level at whichnoise effect was observed in the curve.

Following electronic tuning, the applied field was theramped in increments of 0.29 G in time intervals of 0.02 s2.30 K, 0.04 s at 3.10 K, and 0.09 s at 3.44 K to a mevalue in the linear part of the superheating curve~well abovethe region in which intermediate states may exist! and a 3-s‘‘pause’’ effected before the ramping was again continuuntil the last grain had transited to the normal state. Tprotocol, shown schematically in Fig. 5~a!, was followed bya return of the field to zero; it was repeated many times.

B. Temperatures<1 K

Sn grains of 16–25mm diameter were also part ofsample made by sonic dispersion of the molten metal in

FIG. 5. Protocols of magnetic-field variation in the various eperiments: ~a! T.2 K, ~b! T,1 K, ~c! systematic test forT,1 K, which was used to establish the time delay in~b!.

n

y0p

peeon

8

G-sis

itio-n

tld

e

asthelly

allthe

sna-

e asng,

achto-

r,

ne

ral

nture

in

the

by


oil bath, size filtration using calibrated sieves; they were uformly suspended17 in paraffin to a 20% volume filling fac-tor. The source activity, 144 kBq of14C, a pure beta decawith an end-point energy of 157 keV and half-life of 573yr, was evaporated onto a thin absorbent paper. The sussion, 2.531.530.02 cm3, was installed on the mixingchamber of a3He-4He dilution refrigerator without 1 K pot~DILUETTE!, sandwiched between a U-shaped coppickup loop and the source paper. The applied magnetic fiwas effected by a Helmholtz coil mounted on the barrelthe vacuum shield orthogonal to the refrigerator axis, aprovided a field stability of 231022 G with homogeneity of1%.

Measurements were performed at 100, 200, 480, andmK. The measurement protocol used in these tests wassentially that of previous researchers,3 rather than that of theT.2 K tests: the applied field was first raised to 3001DH in steps of 18 G s21, then lowered to 300 G for measurement, as shown in Fig. 5~b!. This measurement field iwell above the region in which intermediate states may exGrains that undergo a magnetically induced phase transbetween 300 G and 300 G1DH remain normal after decreasing the field to 300 G because the phase transitioirreversible. With this protocol, a zone of depthDH from thesuperheating boundary is magnetically depopulated, andparameterDH can be considered as an energy threshoDuring the measurement time (tmeas), detected transitions ar

FIG. 6. A typical rate curve of Sn grains transitions inducedirradiation, obtained during a pause period atTa5850 mK.

i-

en-

rldfd

50es-

t.n

is

he.

only due to irradiation. The measurement period, 50 s, wselected on the basis of the observed transition rate ofsuperconducting grains, which as seen in Fig. 6 is typicaexponential.

Following measurement, the field was raised untilgrains were in the normal state, then recycled to zero andprocedure repeated many times. The variation inDH was 1G.

The pause atHp1DH resulted from systematic testwithout irradiation in which the field was raised by aamountdH just after reaching 300 G and before the mesurement, as shown in Fig. 5~c!. In this case, fordH,DH,no transitions were recorded; fordH.DH, transitions oc-curred because of the delay in the magnetic-field responsa result of the field coil damping. Because of this dampian additional 1-s delay at 300 G1DH was required in orderfor the field to reach its command value.

IV. RESULTS AND DISCUSSION

The maximum magnetic gap widths measured in etemperature range are presented in column 2 of Table I,gether with the correspondingDTmax in column 3. For theT.2 K measurements,$DTmax% was obtained via Eq.~3!with Hsh

local~0!5570 G.6 TheT,1 K measurements, howeverequired a different treatment since Eq.~4! is not well estab-lished forTa!Tc . Moreover, because of the slow variatioof Hsh with temperature for this region, a small error in thdetermination ofHsh(T) induces a large error inDT. In thiscase,$DTmax% was determined from theDH measurementand from a separately measuredHsh(T) curve.

The last column of Table I is calculated from the integof Eq. ~1! with the$DTmax% column entries using Eq.~2! with$a,b,a,g%5$7.85, 1.42, 0.101 keV/K4 mm3, 0.6718 keV/K2

mm3% for Sn.19 As evident, the integral is constant withierrors for each suspension over its respective temperarange.

A. Variation of DEmax /V with temperature

The last column of Table I indicates no variationDEmax/V for either temperature range, especially forT.2 K, where the effects of aVheated(j) would be expected toappear. We show in Fig. 7 these results together with

TABLE I. Experimental maximum gap widths and associated parameters.

T ~K! DHmax ~G! DTmax ~K! *Ta

Ta1DTmaxCs dT ~keV/mm3!

T,1 K ~16–25-mm B suspension,14C irradiation!0.8560.04 10.060.12 0.1860.10 0.02760.0080.4860.02 14.060.12 0.5160.08 0.03460.0070.2060.02 16.060.12 0.7560.08 0.02960.0050.1060.01 17.060.12 0.8660.08 0.03060.007

Mean 0.03060.013T.2 K ~10–25-mm B suspension,35S irradiation!3.4460.04 3.260.1 1.0260.12 0.08360.0043.1060.02 3.660.1 1.2460.17 0.08160.0052.3060.02 6.360.1 2.9360.23 0.09360.003

Mean 0.08660.004

orto

-

te

tohed

nts

s

a-ate

ssbu-ffin

och

of

y,sion

ngbyx-

ldandyo

n-ult


temperature variation ofDEmax/V(j) anticipated from thelowest temperature result in each temperature range. FT,1 K, wherej'j0 , the results are insufficiently sensitivebe conclusive; however, the normalization impliesRheated'35j0 suggesting aVheatedmuch larger than would be expected by any local-heating description. For theT.2 K re-sults, the sensitivity is significantly better than the predictemperature variation.

FIG. 8. Calculation of the maximum field gaps as a functiontemperature, with~—! and without~---! the phonon contribution tothe specific heat:~a! T,1 K, ~b! T.2 K.

FIG. 7. Comparison of the experimentalDEmax/Vheatedwith thepredicted temperature dependence of the coherence length;V(j)has been normalized to reproduce the lowest temperature reseach temperature range.

d

B. Heating of the phonon system

Analysis of the phonon system warming, correspondingFigs. 4 and 5 of Ref. 4, is shown in Fig. 8, together with tmeasured$DHmax% from Table I. The contours are obtaineby computing theT1DTmax in the integral of Eq.~1!, whichis necessary to provide the same value ofDEmax/Vmin towithin 1%. The area defined within the solid lines represethe results for Cs5Celectron1Cphonon, with Rmin

5Rmeasurement61 mm; the area within the dotted lines,Cs

5Celectrononly. For theT.2 K data, the theoretical contouroverlap and the data supports either hypothesis; theT,1 Kresults, however, lie well within the area anticipated fromfull specific heat (electron1phonon), in contrast to the corresponding Fig. 5 of Ref. 4: nucleation of the normal stproceeds with a fully warmed phonon system.

C. Heated volumes

The DEmax of each decay for different grain radii wacomputed using a simulation20 that randomly emits electronfrom a distributed source consistent with the energy distrition of the decay, and tracks them through both the paraand grains in accordance with Moliere scattering;21 the en-ergy loss in each medium is computed using the Bethe-Blequation.22 Figure 9 shows the results for the14C decay elec-trons. For the minimum radius of theT.2 K and T,1 Kensembles,DEmax54561 and 6561 keV, respectively. Di-vided by the respective mean values of the last columnTable I, these yieldRheatedof 8.062.1 and 5.060.086mmfor the T,1 K and T.2 K measurements, respectivelequivalent to the smallest grain radius in each suspenand well abovej0 .

According to the discussion of Sec. II A, the local-heatibehavior observed in Ref. 4 for Sn and In is characterizeda low-energy tail in the differential superheating curves etending 30–50 G belowHstep ~see Fig. 3 therein!; in prin-ciple, the lowest recorded transition field of this tail shoucorrespond to the nearest-defect energy deposition,Vheated'V(j). In fact, these lowest transition fields implRheated'8.5mm, well abovej0 but sufficiently less thanRgrain to further suggest the intermediate-state presence.

f

FIG. 9. The results of simulation of the maximum electron eergy loss from the decay of14C in Sn microspheres of varyingdiameter, as described in the text.

in

eioon

so

taen

th

thtinxe

actai

s

u-

urno

laths

utailinionInn-edtheofis,of

ainsn inter-

ffer-thisch

eat-

on-i-ntal

o.ma

s ofg


V. CONCLUSIONS

In order to investigate the local-heating conclusion of R4 for Sn, we have performed a series of electron irradiatexperiments on two different, fully metastable, suspensiof Sn grains in two different temperature ranges belowTc .

Measurements forT,1 K with a suspension of Sn grain16–25mm in diameter irradiated by the decay electrons14C clearly demonstrate that nucleation of the normal sproceeds with a fully warmed phonon system; measuremfor T.2 K with a suspension of Sn grains 10–25mm indiameter irradiated by35S decay electrons clearly deny bothe temperature variation ofDEmax/V(j) and Vheated'0.5Vmin . In each case, the heated volume derived frommaximum gap width induced in the respective superheacurves by the irradiation, combined with the calculated mamum energy deposition in the material, is in good agreemwith the smallest grain of its suspension.

We thus conclude that the electron-induced thermnucleation of a suspension of superheated superconduSn grains is consistent with a global heating of the grvolume, contrary to the model of Ref. 4.

The energy of thea irradiation of Ref. 4 experiments igenerally deposited within 10mm of the grain surface. Al-though the electrons of these experiments have a maximrange of some 70mm in Sn, about 40% of the incident activity is also stopped within 10mm, with energy sufficient toraise a defect volume by several K throughout the measment range. However, the distinction between local- aglobal-heating behavior of the Ref. 4 model is basedwhether or not the material-dependent quasiparticle reation rate is less than the diffusion rate, independent ofenergy deposition mechanism. The defect presence is

d

, A

v.

Ja

cl

s

s

f.ns

ftets

eg

i-nt

lingn

m

e-dnx-e

ig-

naled by the step atHstep in the measurements of Ref. 4, bthe local-heating behavior is manifested by the low-field tbelowHstep. According to Ref. 4, the presence of this tailSn is explained by the more rapid quasiparticle relaxatrather than diffusion rate. This tail is also observed in theirradiations, and a local-heating behavior is similarly cocluded for the same reason; for Al and Zn, no tail is observand these transitions are designated as ‘‘global’’ becauserelaxation rate is slower than the diffusion. Observationthe local-heating tail in the Sn/In experiments of Ref. 4however, most likely due only to the significant presenceintermediate states, evidenced by the fact that the grnever reached the theoretical limit of superheating as seethe experimental superheating fields. Since this can be demined in advance by measurement of the associated diential superheating curve, we strongly recommend thatcurve be examined prior to the future reporting of any suexperiments.

Curiously, the model of Ref. 4 would seem to ‘‘maksense’’ physically. Since it constitutes a reasonable firsttempt at explaining the heat transport in a type-I supercducting material, why it fails to manifest itself in our experments is perhaps a greater question. Further experimeinvestigations are clearly needed.

ACKNOWLEDGMENTS

This work was supported in part by JNICT Grants NC/FAE/1051/95 and No. C/FAE/1110/96 under PrograCERN of Portugal. Two of us~T.A.G. and J.I.C.! againthank the CNRS and Groupe de Physique des SolideUniversites Paris 7/6 for their hospitality and support durinthis work.

ition-

rein

itude-d

M.r

1G. Waysand, inSuperconductivity and Particle Detection, editedby T. A. Girard, A. Morales, and G. Waysand~World Scientific,Singapore, 1995!, p. 1.

2J. Feder and D. McLachlin, Phys. Rev.177, 763 ~1969!.3M. Frank, P. Freund, J. Gebauer, K. Pretzl, A. Singsaas, an

Stodolsky, Nucl. Instrum. Methods Phys. Res. A287, 583~1990!; J. Fent, P. Freund, G. Gebauer, K. Pretzl, N. SchmitzSingsaas, L. Stodolsky, and G. Vesztergombi, inLow Tempera-ture Detectors for Neutrinos and Dark Matter, edited by K.Pretzl, N. Schmitz, and L. Stodolsky~Springer-Verlag, Berlin,1987!, p. 30.

4M. Frank, A. Singsaas, L. Stodolsky, and S. Cooper, Phys. Re43, 5321~1991!.

5S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S.farey, and D. J. Scalapino, Phys. Rev. B14, 4854~1976!.

6A. Larrea, A. Morales, G. Waysand, and J. Bartolome, NuInstrum. Methods Phys. Res. A317, 541 ~1992!.

7F. W. Smith, A. Baratoff, and M. Cardona, Phys. KondenMater.12, 145 ~1979!.

8D. K. Finnemore and D. E. Mapother, Phys. Rev.140, A507~1965!.

9P. G. de Gennes,Superconductivity of Metals and Alloy~Addison-Wesley, New York, 1966!, p. 40.

10E. V. Thuneberg, J. Low Temp. Phys.62, 27 ~1986!.

L.

.

B

-

.

.

11M. Tinkham, Introduction to Superconductivity~McGraw-Hill,New York, 1996!, p. 119.

12D. Hueber, C. Valette, and G. Waysand, Physica B & C 108,1229 ~1981!.

13This expression is customarily stated as valid only forT'Tc ; amore general validity over the phase space near the transboundary follows from the work of Gork’ov and others. Moreover, high-field calculations ofHc2 ~treated in, for example,Refs. 9 or 11! yield Hc2'j22(t); measurements of this fieldover 0,t,1 ~for example, Refs. 2, 6, and 7! yield Hc3 /Hc

'(12t)1/6→j(t)'(12t)21/2.14U. Geigenmueller, inSuperconducting and Low-Temperatu

Particle Detectors, edited by G. Waysand and G. Chard~Elsevier, Amsterdam, 1989!, p. 191; J. Phys.~France! 49, 405~1988!.

15Several groups have reduced this spreading an order of magnby resorting to ordered arrays@for microspheres, see, for example, M. Le Gros, A. da Silva, B. G. Turrell, A. Kotlicki, anA. K. Drukier, Appl. Phys. Lett.56, 2234~1990!; for microdots,see, for example, R. Leoni, G. Shirripa Spagnola, P. Carelli,G. Castellano, and M. Cirillo, inLow Temperature Detectors foNeutrinos and Dark Matter, edited by N. Booth and G. L.Salmon ~Editions Frontieres, Paris, 1992!, p. 57#. None have,however, been able to eliminate it.

,

oc

n-c

ys.

. B

ta


16A. de Bellefon, D. Broszkiewicz, R. Brue`re-Dawson, P. Espigatand D. Sotiras, Nucl. Instrum. Methods Phys. Res. A277, 211~1989!.

17V. Jeudy, D. Limagne, and G. Waysand, Nucl. Phys. B, PrSuppl.28A, 482 ~1992!.

18T. A. Girard, R. P. Henriques, J. I. Collar, M. Godinho, G. Bofait, G. Waysand, D. Limagne, V. Jeudy, and V. Pagesy, NuInstrum. Methods Phys. Res. A334, 645 ~1993!.

.

l.

19H. R. O’Neal and N. E. Phillips, Phys. Rev.137, A748 ~1965!.20J. I. Collar, T. A. Girard, and G. Waysand, J. Low Temp. Ph

93, 497 ~1993!.21G. R. Lynch and O. I. Dahl, Nucl. Instrum. Methods Phys. Res

58, 6 ~1991!; Phys. Lett. B239, 13 ~1990!.22R. M. Sternheimer, M. J. Berger, and S. M. Seltzer, At. Da

Nucl. Data Tables30, 261 ~1984!.

thermal nucleation of the normal state in superheated superconducting tin grains

Documents