Effect Of Duo Fermion Spin On The Specific Heat And Entropy Of A Mixture Of Helium Isotopes

Transcription

1 Eect O Duo Fermion Spin On he Speciic Heat And Entropy O A Mixture O Helium Isotopes * S.M. Lusamama iaii University, P.O ox , ungoma, enya. simonlusamama@kiu.ac.ke *Author or orrespondence.w. Sakwa Masinde Muliro University o Science and echnology, P.O ox , akamega, enya. tsakwa@mmust.ac.ke J.O. Odhiamo, iaii University, P.O ox , ungoma, enya. odhiamo@kiu.ac.ke Astract Spin normally determines the character, rate o collision and susequently the properties o ultracold systems. his study investigated the eect o doule spin degeneracy on the thermodynamic properties o Helium-3 and Helium- isotopes speciically the speciic heat and entropy o a grand canonical ensemle. he approach used was statistical in nature where permutation and exclusion was done on ermions and osons. From thermodynamics, the ression o partition unction which was used to derive other ressions o speciic heat and entropy as temperature dependence. Speciic heat and entropy were ound to increase with temperature. he kink in speciic heat at a temperature o 35 implied that there was a phase transition. eywords hermodynamics, Entropy, Speciic heat IRODUIO degeneracy, Experiments on ultracold trapped atomic gases have opened a window onto the phases o quantum matter [1]. ose gas remains conceptually a good starting point to understand the remarkale ehaviour o superluid Helium. It was in this context that ogoliuov analyzed the eect o weak interactions and Pitaevskii constructed the classical theory or inhomogenous ose-einstein condensates []. A gas o osonic atoms in an optical or magnetic trap had een reversily tuned etween superluid and insulating ground states y varying the strength o a periodic potential produced y standing optical waves. his transition was lained on the asis o the ose-huard model with on-site repulsive interactions and hopping etween nearest neighoring sites o the lattice [3]. Further, theoretical studies o osonic atoms with spin and/or pseudospin had also een undertaken. hese studies have revealed a variety o interesting Mott phases and superluidinsulating transitions in these systems. On the ermionic side, the erimental studies have mainly concentrated on the oservation o paired superluid states and the S-E crossover in such systems near a Feshach resonance []. It has also een possile to generate mixtures o ermionic and osonic atoms in a trap. Initially, the main ocus o such erimental studies were to generate quantum degenerate Fermi gases, through sympathetic cooling with osons. However, a host o theoretical studies ollowed, which estalished such ose-fermi mixtures to e interesting physical systems in their own right, exhiiting exciting Mott phases in the presence o an optical lattice [5]. In all o these works, the spin o the ermions in the mixture is taken to e rozen out due to the presence o the magnetic trap. Research works thereater considered a ose-fermi mixture in an optical trap, where the spins o the ermions were dynamical degrees o reedom [6]. It was shown that the interaction etween the osons and the ermions in such a mixture enhanced the s-wave pairing instaility o the ermions. hermodynamic properties o Helium-3 and Helium- isotopes has also een done where the partition unction relations as a means o studying the thermodynamic properties o Helium-3 and Helium- isotopes was developed [7]. Similarly, thermodynamic properties o normal liquid helium-3 y using the lowest order constrained variational (LO) method was done where the ree energy, pressure, entropy, chemical potential and liquid phase diagram as well as the helium-3 speciic heat were evaluated and discussed [8]. In this work, we considered a ose-fermi mixture o Helium isotopes (Helium- and Helium-3) interacting weekly in pairs. It was assumed that the atoms were conined using an optical trap so that Fermions spins were not rozen out. However, the eect o the harmonic trap potential was ignored. heoretical ramework According to [9], the partition unction o Helium isotopes with duo spin degeneracy was derived as JMES

2 JMES (1) Where is the partition unction, denotes numer o osons, numer o ermions is the chemical potential o osons, is the chemical potential o ermions, is the temperature, is the oltzmann constant, is the dual energy o helium isotopes he ollowing are the conditions or determining the speciic heat ( v ), entropy ( S) and lamda transition temperature ( ) or a grand canonical ensemle () S (3) 0 () ased on equations (), (3) and (), the speciic heat ( v ), entropy (S) and the lamda transition temperature ( ) was ound to e v (5)

3 S 15 (7) RESULS AD DISUSSIO Speciic Heat alues o speciic heat are computed using Equation (5) leading to igure 3 which shows the variation o speciic heat with temperature in the range 5 to 80. (6) Speciic Heat,(e/mol.) emperature,() Figure 1: ariation o Speciic Heat with emperature in the range 5 to 80 he speciic heat is zero at very low temperatures elow 5. In this temperature range, particles are all occupying the lower quantum states. As the temperature is increased, ermions shit quickly to the higher states. he speciic heat rises onentially to a peak value o 8.0 x 10-3 e/mol. at a temperature o 35 and reduces gradually to very low values. At this temperature, a phase-like transition appears to take place in a manner that is characteristic o a second-order phase transition. his type o Gaussian shaped curve relating speciic heat to temperature has een oserved y other scientists while investigating relationship etween speciic heat and temperature or varied materials under varied conditions [10]. his indings concurs with [11]. In this study, a ulk mixture o helium-3 and helium- is considered in which there is ree movement o helium-3 atoms into liquid helium-. Helium-3 atoms are seen as impurities in liquid helium- and weak interactions develop etween the helium-3 and helium- atoms which, consequently, shit the transition temperature o liquid helium- to a higher value. Entropy Using equation (6), the variation o Entropy with temperature is ound to e a curve with a gently decreasing slope nearly saturating at 80. JMES

4 Entropy, S(e/) emperature,() Figure : ariation o Entropy with emperature in the range 5 to 80 Entropy is a measure o disorder in a system. On the graph shown aove, entropy is seen increasing instantaneously with increase in temperature. his shows excitation o the particles in the system. he particles in the system do not spin at low temperatures ut they gain momentum as the temperature increases. his results tally with that o [1, 13]. OLUSIOS Dierent research works on the study o thermodynamic properties o Helium isotopes have looked at two models: the independent particle model in isolated system and the ensemle model consisting o large collection o system. In these model, particles are considered to e interacting weakly in pairs. At low temperatures (elow 5 ), the speciic heat and entropy are diminishingly small, this is maorly ecause o the microscopic occupation o the ground state. his concurs with what [1] oserved or disordered osons. he entropy which is determined statistically, naturally, increases with temperature. AOWLEDGEME he authors would like to appreciate the support rom iaii University. REFEREES [1].. Marios, E.S. Pedro, R. Amilson, A.M. aracanhas and. idrim (016), Quantum turulence in trapped atomic ose-einstein condensates, Physics Reports,ol. 6,Pp.1-5. [] M. Wouters M (013), Quantum Fluids o Exciton-Polaritons and Ultracold Atoms, Springer, erlin, Heidelerg,ol. 177,Pp [3]. Sengupta and. Dupuis (005), Mott insulator to superluid transition o ultracold osons in an optical lattice near a Feshach resonance, European physics Letters,ol.70, Pp [] S. Simonucci, P. Pieri and G.. Strinati (005),road vs. narrow Fano-Feshach resonances in the S-E crossover with trapped Fermi atoms, European physics Letters,ol.69,Pp [5] H. Haiping, G Liming and S hen (016), Strongly interacting ose-fermi mixtures in one dimension, ew Journal o Physics,vol.18, [6].S. Mankoc and H..F. ielsen (015), Degrees o reedom o massless oson and ermion ields in even dimension, wesite: 19 th August 018. [7].M. hanna, J. amwitako,.w. Sakwa, Y.. Ayodo, A. Sarai,G.O. arasa and S.J. Prichani (015), hermodynamics o Incommensurate Quantum rystal, International Journal o Physics and Mathematical Sciences, ol. 5 issue, Pp [8] M. Modarres and H.R. Moshegh (009), he thermodynamic properties o normal liquid Helium-3, Journal o Statistical mechanics and its application, ol.388,issue 17,Pp [9] S.M. Lusamama,.W. Sakwa, J.O Odhiamo,.O Oketch and A.A Ocholla (015), Internal Energy JMES

5 o a Grand anonical Ensemle o a Mixture o Helium Isotopes with Duo-Fermion Spin Degeneracy, International Journal o Physics and Mathematical Sciences, ol. 5,issue, Pp [10] J.O. Odhiamo,.W. Sakwa,.W. Rapando and Y. Ayodo (016), Eect o uo plane on the hermodynamic properties o doule i-o layered cuprate ased on an interaction etween cooper pair and an electron, International ournal o Physics and Mathematical Sciences,ol. 6,issue.Pp [11] S.Rohit, S. Gurpreet and S. uldip (011), heoretical investigation o thermophysical properties in two-temperature argon-helium thermal plasma, Physical Plasmas,ol. 18, [1] A. Gross and A. Marx (010), Quantum Fluids, Physical Review Letters, vol.105, [13].W. Sakwa, Y.. Ayodo, P.O. Saisi, A. Sarai,.W. Rapando and.a.omondi (013), A on- Degenerate inary System o Helium Isotopes, East Arican Journal o Engineering, Science & echnology,ol.1,issue,pp [1].W. Sakwa,.M. hanna, M.Mueni,S.. Rotich and P..orongey (00). Four Level Approximations in Disorderd Medium. Indian Journal o Pure and Applied Physics,ol., Pp JMES