The Grüneisen parameter for aluminum in the temperature range K

Transcription

1 The Grüneisen parameter for aluminum in the temperature range K Alessandro Marini INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, I Frascati, Italy Available critical field data as a function of temperature and pressure are used to calculate the electronic contribution to the Grüneisen parameter for pure aluminum in the superconducting state. Large negative values for the parameter are found in the range of temperatures of interest starting from the value γ e s(t = T c )= 10.7 ± 0.8. This fact could be taken into account as a possible explanation for the cosmic ray large signals detected at a rate higher than expected by the gravitational wave antenna NAUTILUS when operated in superconducting regime. PACS numbers: q, b. 1. INTRODUCTION Studies 1 3 of the signals due to the interactions of cosmic rays impinging on the gravitational wave antenna NAUTILUS, a massive (.3 ton) bar made of an aluminum alloy, have shown that in a run of the detector at thermodynamic temperature T = 1.5K the results are in agreement with the thermo-acoustic model, while in a run at T = 0.14K large signals are detected at a rate higher than expected. In the thermo-acoustic model mechanical vibrations originate from the local thermal expansion caused by warming up due to the energy lost by a particle crossing the material. More precisely, the relation that accounts for the detectable vibrational energy E in the n th longitudinal mode due to a specific energy loss dw/dx of a particle impinging on a cylindrical bar is: 4 E = 4k γ 9π ρlv (dw dx ) F n (z 0,θ 0,l 0 ) where k is the Boltzmann constant, ρ is the density of the bar material,

2 A.Marini L is the length of the bar, v is the longitudinal speed of sound, γ is the Grüneisen adimensional parameter of the material and F n is a function of the impinging track geometrical parameters, not relevant in this context. The thermo-acoustic model predictions have been recently verified by an experiment, 5 operated only at room temperature, using a small suspended aluminum cylinder exposed to an electron beam. The experiment was also motivated by the applicability of the model in the evaluation of the cosmic ray effects on resonant mass gravitational wave detectors of ultrahigh sensitivity. The NAUTILUS results suggest that a more efficient mechanism for particle energy loss conversion into mechanical energy takes place when the bar is in the superconducting state. The behaviour of the Grüneisen parameter at ultra low temperatures is one of the various possible explanations for the observed effects in NAUTILUS. After a short review of the Grüneisen parameter properties at low temperatures and a summary of experimental results obtained for almost pure aluminum (Section ), a calculation of the electronic Grüneisen parameter based on available experimental data in the temperature range K is here reported (Section 3).. PROPERTIES OF THE GRÜNEISEN PARAMETER AT LOW TEMPERATURE AND AVAILABLE EXPERIMENTAL DATA.1. Properties of the Grüneisen parameter at low temperature The Grüneisen parameter γ is defined as: γ(t,v )= βv χ T C V where β[= ( ln V/ T) p ]isthe volume expansion coefficient, V is the molar volume, χ T [= ( ln V/ p) T ]isthe compressibility at constant temperature and C V is the specific heat at constant volume. For aluminum the thermal expansion at the critical temperature and below is too small to be measured directly by present techniques. For the normal state an extrapolation from higher temperatures is possible, but for the superconducting state a thermodinamic derivation from the experimental pressure and temperature dependence of the critical field H c can be used. Differences between normal (n) and superconducting (s) states are expected in the quantities defining the electronic component of the Grüneisen parameter. According to Shoenberg, 6 the differences are expressed in terms

3 Grüneisen parameter for Al below T =1.1K of the pressure and temperature dependence of the critical field. In particular, for Type I superconductors the differences in the volume expansion coefficient and compressibility are to the first order: and β n β s = 1 H c H c 4π T p + H c H c 4π p T (1) χ s χ n = 1 4π ( H c p ) + H c H c 4π p () respectively. From the definition of γ it follows that the electronic component of the Grüneisen parameter in the superconducting state is: 7 γs e = C e,n γn e V (β n β s ) C e,s χc e,s where C e,n,c e,s are the electronic contributions to the specific heat in the normal and superconducting states, γn e is the electronic component of the Grüneisen parameter in the normal state and χ s χ n is assumed (see discussion below). Using an alternate formulation of (1), that exploits the dependence of β n β s on Hc and includes the contribution of the magnetostriction in the superconducting state, the electronic Grüneisen parameter in the superconducting state can be finally expressed as: 7 γs e = C e,n γn e V 1 C e,s χc e,s 8π ( Hc T p χ H c T ) (3) At T = T c γs e takes the form: 7 γs(t e = T c )= C e,n γn e + C n C s d lnt c C e,s C e,s d lnv (4) where C n C s is the difference of the specific heats at the transition... Experimental values of the Grüneisen parameter for aluminum at low temperature Several authors have in the past measured the electronic and lattice components of the Grüneisen parameter for aluminum down to low temperatures. 8 Results have been obtained using methods based on dilatometry with capacitive readout, 9, 10 on pulsed thermoelastic stresses, 11 and on measurements of the variations of the critical field as a function of the pressure and the temperature The accurate results from the dilatometry, obtained at very low values of temperature but higher than the critical value (T c 1.1K), are in extremely

4 A.Marini good agreement with the theoretical expectations for γ 0 [.5] 8 and γ e [1.63], 16 respectively the lattice and the electronic components extrapolated at T =0. Some results for γ e, obtained from the studies of H c and characterized at T = T c, apparently deviate from those coming from dilatometry by a factor of the order approximately greater than three. This discrepancy could be due to the complexity of the experimental technique of the variations of H c with p and T,ortothe theoretical hypothesis assumed for deriving the result. 3. CALCULATION OF γ e s IN THE TEMPERATURE INTERVAL 0.3K T 1.1K The previously cited article by Harris and Mapother contains a tabulation of critical field data in the temperature range starting from T 0.3K up to T c for four values of pressure (p 0, 11, 367, 490 atm). This tabulation can be used as an input to relation (3) for calculating γs e at p =0ina temperature interval from 0.3K up to 1.1K, below T c. Relation (3) can be written in the form γs e = A B C, with A = Ce,n C e,s γn, e B = V 1 χc e,s 8π and C = Hc T p χ H c T, for evaluating the relative weight of the terms. In order to calculate A and B, the fitting functions: 1.34 Tc/T C e,n =ΓT,C e,s =7.1ΓT c e where R is the gas constant, θ = 47.7K, Γ= J/(moleK ), T c = 1.163K, are taken from Phillips 17 and γn e =1.63 is assumed. The values of A and B for some points of the temperature interval are shown in the table. The H c tabulated data are used for computing the terms Hc T p and H c T in C. The fit of Hc with a polynomial of the fourth order in T at fixed p gives directly H c T. Linear fits vs. p of H c T at fixed T give the values of Hc T p. In this approximation the second derivatives of Hc are independent of the pressure. The first term of C is greater by nearly an order of magnitude than the second, computed at p =0,but the errors on the first term, estimated from the covariance matrices and the χ of the fits, reach the maximum value of 80%. The table shows the obtained values for C for some points of the temperature interval. Relation (3) is valid in the approximation that χ s χ n, and the H c tabulated data can be used for calculating () to verify this assumption. A fitting procedure, similar to the one described above and disregarding the contribution of the second derivatives, is used to calculate Hc p at fixed T. Values of χs χn χ n are obtained ranging from to in the

5 Grüneisen parameter for Al below T =1.1K T[K] A B[10 7 K] C[10 7 K 1 ] temperature interval, with an error of the order of 10% evaluated from the covariance matrices and the χ of the fits. Whit this result it is possible to use the value 18 χ = cm /dine both for the normal and the superconducting states. The final calculation of A B C gives negative values for γs, e due in this context to the prevalence respect to A of B C, whose positive sign is determined by the sign of C. However it must be underlined that C is affected by large errors, mainly due to the inaccuracy of Hc T p. Fig. 1 shows the absolute vales of γs e at p =0in the interval 0.3K T 1.1K. Agreement is found for γs(t e =1.1K) computed from the fitting procedure and γs e at T = T c computed from equation (4) using the specific heat data given by Phillips and dlntc dlnv given by Harris and Mapother: γ e s(t = T c )= 10.7 ± CONCLUSIONS In this macroscopic framework, involving thermoelastic properties of pure aluminum in normal and superconducting states, the electronic Grüneisen parameter for the superconductive state, γs, e seems to have absolute values greater than those assumed in the normal state. This circumstance, possibly due to the fitting procedures and approximations used in the treatment of the available data or to the experimental accuracies, could be related to the discrepancy in the rate of high energy cosmic rays observed in NAUTILUS when operated in superconducting state. However this simplified explanation does not take into account other factors, such as the fact that the bar is not made of pure aluminum, but together with other possible explanations it could be object of investigation by means of a suspended small cylindrical bar made of the same aluminum alloy as NAUTILUS, impinged by a high energy electron beam and operated either in the normal or in the superconducting states.

6 A.Marini Fig. 1. Absolute values of γ e s at p =0;broken lines show the estimated errors from the covariance matrices and χ of the fits.

7 Grüneisen parameter for Al below T =1.1K REFERENCES 1. P. Astone et al., Phys. Rev. Lett. 84, 14(000). P. Astone et al., Phys. Lett. B 499, 16(001) 3. P. Astone et al., Phys. Lett. B 540, 179 (00) 4. See for instance: G. Liu and B. Barish, Phys. Rev. Lett. 61, 71 (1988) and references therein 5. G.D. van Albada et al., Rev. Sci. Instrum. 71, 1345 (000) 6. D. Shoenberg, Superconductivity, Cambridge Univ. Press (1965) pagg. 58, R.W. Munn, Phys. Rev. 178, 677 (1969) 8. A comprehensive survey on this argument can be found in: T.H.K. Barron, J.G. Collins and G.K. White, Adv. in Phys. 9, 609 (1980) 9. J.G. Collins, G.K. White and C.A. Swenson, Journ. Low Temp. Phys. 10, 69 (1973) 10. F.R. Kroeger and C.A. Swenson, Journ. Appl. Phys. 48, 853 (1977) 11. W.B. Gauster, Phys. Rev. B 4, 187 (1971) 1. D. Gross and J.L. Olsen, Cryogenics 1, 91(1960) 13. E.F. Harris and D.E. Mapother, Phys. Rev. 165, 5 (1968) 14. C. Palmy et al., (1970). Reported in R. Griessen and R. Ott, Phys.Lett. 36A, 113 (1971) 16. D.C.Wallace, Journ. Appl. Phys. 41, 5055 (1970) 17. N.E. Phillips, Phys. Rev. 114, 676 (1959) 18. N.B. Brandt and N.I. Ginzburg, Sov.Phys.-Usp 8, 0 (1965)