Calculated Changes in the Elastic Properties of MgCNi3 at the Superconducting Transition

We calculated the elastic properties of MgCNi3 at the superconducting transition (TTcc) using various thermodynamic and acoustic data. From the calculations, a step discontinuity of 8 ppm in the bulk modulus, 7 ppm in the Young’s modulus, and 3 ppm in the longitudinal sound velocity (vvll) is expected at TTcc. e step discontinuities at the transition temperature indicated the importance of lattice changes to the superconducting mechanism of MgCNi3. e Debye temperature was calculated to be 460K.e electronphonon coupling constants calculated in the weak and strong coupling limits of the BCS theory and the van Hove scenario showed that MgCNi3 is a moderately strong coupled superconductor.


Introduction
Superconductivity in MgCNi 3 with transition temperature (  ) near 8 K is of interest especially from the pairing mechanism point of view.e electron-phonon coupling constant of MgCNi 3 has been determined using speci�c heat measurements [1].Evidence of strong electron-phonon coupling and the van Hove type singularity in the electronic density of states near the Fermi energy has also been suggested [2].
e nature of superconductivity in this material is still controversial.A step discontinuity in the longitudinal sound velocity indicating lattice soening has been observed in the conventional superconductors.us, information on the lattice properties at   is very useful in understanding the MgCNi 3 superconductor.e acoustic method is a sensitive probe of phonon states and electron-phonon coupling in materials.e elastic properties of MgCNi 3 have also been reported [3,4].e objectives of this work were to estimate the electronphonon coupling constant, changes in the elastic properties, and sound velocity at the transition temperature of MgCNi 3 .e elastic properties of MgCNi 3 including the change in the bulk and Young's moduli and discontinuity in the longitudinal sound velocity at   were calculated using various sound velocity and thermodynamic data.e acoustic Debye temperature was also calculated and compared to reported values determined by other methods.e electron-phonon coupling constant in various scenarios is also reported.

Basic Formulation
e standard isotropic elastic medium approximation can be applied to the polycrystalline samples.In this case, two independent elastic moduli, namely, the shear modulus  can be written as    2   , and the bulk modulus  can be written as    2   − (4/3) 2  , where  is the mass density,   is the shear velocity, and   is the longitudinal velocity [5].e longitudinal modulus can be written as    2   , and the Young's modulus is   (3 − 4)/( − ).
For the superconducting transition in �ero �elds, the thermodynamics of second-order phase transition [6,7] gives the following anomalies: is expected to show a step discontinuity, and  is supposed to show a change in the slope at   from the mean-�eld transition [6].e change in  can be written as

Results and Discussion
Several papers have reported the elastic moduli of MgCNi 3 [3,4,8] with a considerable spread in values.Because this material is usually prepared in the polycrystalline form, in this paper, we have used values for polycrystalline form as reported by Shein et al. [3].Using bulk modulus   171.05 GPa, shear modulus of   1.4, and Young's modulus   1.16GPa as published previously [3], the longitudinal sound velocity was calculated to be    270 m/s and shear sound velocity    3100 m/s.e sound velocities can be used to calculate the acoustic Debye temperature by using the standard formula    (ℎ  )(34) 13   , where ℎ is the Planck's constant,   is the Boltzmann's constant,  is the number of mass-point,  is the atomic volume, and   is the mean velocity given by 3 3    1 3  + 2 3  .Using this expression, the acoustic Debye temperature of MgCNi 3 is calculated to be 460 K. is is near to the   values calculated theoretically [4,8,9] but much higher than the value determined from speci�c heat measurement [10,11].
e BCS theory for weak coupling gives    1.13   −1 , where  is the electron-phonon coupling.e electron-phonon coupling constant for the weak coupling case calculated using the previous acoustic Debye temperature is   0.2.
e relevance of van Hove density of states has been suggested for MgCNi 3 [2].e singularity in the density of states at the Fermi level was taken into account in the van Hove scenario of a two-dimensional system.In this scenario,   is given as    2.72   (−1 √ ) , where       ≈ 10  (  is the Fermi energy) [12].e electron-phonon coupling constant calculated using this formula is   0.01.
e McMillan expression for the strong coupling BCS case is given as [13]    (  1.4) exp[−1.04(1+ )( −  * (1 + 0.2)), where  * (with conventional values of 0.0 − 0.1) is the screened Coulomb repulsion between electrons in a Cooper pair.e electron-phonon coupling constant derived from this expression ranged from 0.49 to 0.59.is is consistent with most measured values for MgCNi 3 [1,10].us, this material is a moderately strong coupled superconductor.
e following is a discussion on the elastic properties of MgCNi 3 at   .Using (1a), Δ      mJ/mol K 2 [10], and     0.1 K/GPa [14], the calculated step discontinuity in the bulk modulus is 8 ppm.Using (2), the discontinuity in the longitudinal velocity is 3 ppm.By assuming that     ≈   , the expected discontinuity in the Young's modulus calculated using (1b) is 7 ppm.ese discontinuities are similar to Pb [15] and LiFeAsOF [16] superconductors but are much smaller than the calculated [17] and observed [18] values in the cuprate-based high   superconductors and MgB 2 [19].Table 1 shows the sound velocity and changes in bulk and Young's modulus at   of MgCNi 3 and other superconductors.
e second-order phase transition does not allow step discontinuity in the shear velocity at   in zero �elds� hence, the shear velocity was assumed continuous at   .In these calculations, the thermodynamic data used are representative of the various reported values, which are of the same order of magnitude.e expected step discontinuity in the longitudinal velocity in MgCNi 3 is similar to the measured value of Pb which is a BCS-type superconductors and are within the resolution range of most ultrasonic apparatus.Our calculations showed that MgCNi 3 is a moderately strong coupled BCS-type superconductor.