Quantum Monte Carlo simulation of spin-polarized tritium

Transcription

1 Higher-order actions and their applications in many-body, few-body, classical problems Quantum Monte Carlo simulation of spin-polarized tritium I. Bešlić, L. Vranješ Markić, University of Split, Croatia J. Boronat, J. Casulleras Universitat Politècnica de Catalunya, Spain 26. March CHIN 09 1

2 Outline Introduction Method Interatomic potential Spin-polarized hydrogen (H ) Liquid spin-polarized tritium (T ) Solid T Liquid-Solid Transition in T Pure and mixed T clusters Summary 26. March CHIN 09 2

3 Introduction Spin-polarized hydrogen has 3 isotopes: 1 H =H - boson, 2 H =D - fermion and 3 H =T - boson BEC for H suggested by Stwaley and Nosanow in 1976, realized by Fried et al. in 1998 In Blume et al. suggested using T to achieve BEC with the advantage of nearly exact knowledge o interatomic potential bulk H has been studied from shown to be gas (Etters, Miller and Nosanow, Lanto and Nieminen, Entel and Anlauf,..) 26. March CHIN 09 3

4 Introduction D have three versions involving one (D 1 ), two (D 2 ) and three (D 3 ) nuclear spin states studied in 80ties by Kortscheck et al., Panoff and Clark, Flynn et al.: (D 2 ) and (D 3 ) are both liquids at zero pressure; confirmed in 99 by Skjetne and Ostgaard T is predicted to be liquid Blume et al. did DMC calculations of T clusters; (T ) 3 is Borromean or halo state with energy of only -4.2(7)mK (confirmed by Salci et al. ) 26. March CHIN 09 4

5 Introduction there where no studies of the mixed clusters similary to 3 He it is expected that it ll take more than 30 atoms to form a bound state of (D ) N cluster best candidates for small mixed clusters are T - D and T -H clusters new experimental development: Magnetic trapping of hydrogen after multistage Zeeman deceleration Hogan et al. PRL 101, (2008). starting from a supersonic beam H cooled to about 100mK expected to work well with D and T atoms 26. March CHIN 09 5

6 Method Variational and diffusion Monte Carlo Imaginary-time Schrödinger equation N-particle Hamiltonian Ψ( R, t) = ( H Et ) Ψ( R, t) t H h = N N 2 i m i = 1 i < j V( r ) ij (*) E t is a constant acting as a reference energy, R ( r1, r2,... rn ) is a walker in MC therminology V(r) is the interaction potential DMC solves stochastically the equation (*) 26. March CHIN 09 6

7 use of importance sampling; Ψ( R, t) Φ ( R, t) =Ψ( R, t) ψ ( R) importance samplin wave function in the limit t only the lowest energy state, not ortogonal to ψ(r) survives for liquid ground state and clusters Jastrow approximation N ψ J( R) = f( rij) i< j liquid : f( r ) = exp( b exp( b r )) ij 1 2 ij 26. March CHIN 09 7

8 for clusters Method for solid, simulations of crystalline hcp, fcc and bcc phase Nosanow-Jastrow model for trial wave function ψ b Ν < 70, f r = s r ( R) ψ ( R) hr ( ) NJ J ii i= 1 h(r) is a Gaussian function linking every particle i to a fixed point r I of the lattice periodic boundary conditions 26. March CHIN 09 8 N = 5 ( ij ) exp( ( ) ij ) rij b f r = s r 5 2 N 70, ( ij ) exp( ( ) ij ) rij

9 Method possible bias reduced to the level of statistical noise by: - summing proper tail corrections to the potential and kinetic energy - calculating the energies using different number of particles (from N= ) - checking the dependence with mean population od walkers and with the time step in the second order method 26. March CHIN 09 9

10 Interatomic potential triplet potential b 3 Σ u + (W. Kolos and L. Wolniewicz, Chem. Phys. Lett. 24, 457 (1974). recalculated to larger distances by Jamieson, Dalgarno, and Wolniewicz, PRA 61, (2000 We use spline fit to JDW dana and connect to long-range behavior Phys. Rev A 54, 2824(1996) 26. March CHIN 09 10

11 Interatomic potential 26. March CHIN 09 11

12 Interatomic potential σ=3.67ǻ Minimum: ε=-6.49 K r m =4.14 Ǻ σ=2.556ǻ in 4 He 26. March CHIN 09 12

13 Spin-polarized H, equation of state olid points: equation of tate of H RB 75, ) iangles HS gas RA 60,5129) ne: E 128 1/2 = 4π x 1 x N + 15 π 3 = ρa mean-field result Lee-Huang correction E/ N in units 2 h 2ma 26. March CHIN a=0.697ǻ

14 pin-polarized H condensate fraction -solid points H -triangles HS gas (PRA 60,5129) -line: Bogoliubov approximation n 0 8 = 1 x 3 π 1/2 26. March CHIN 09 14

15 Liquid T - equation of state 26. March CHIN 09 15

16 Liquid T - equation of state the analytical form fits DMC data from the spinodal point to the highest density calculated, high in the overpressurized regime e = E/N ρ 0 = (7) Ǻ -3 is the equilibrium density (0.369σ -3 ), e 0 =-3.656(4)K, e1=6.86(7)k, e 2 =4.70(5) K (in 4 He ρ 0 = 0.365σ -3, e=-7.267(13)k) 2 3 ρ ρ e( ρ) = e0 + e1 1 + e2 1 ρ0 ρ0 26. March CHIN 09 16

17 Liquid T - equation of state ρ(ǻ -3 ) E/N T/N P (bar) c(m/s) (1) 5.578(15) -1.43(2) 71(3) (2) 7.120(12) -0.73(1) 158(3) (4) 8.85(2) 1.25(2) 237(3) (4) 13.06(3) 11.6(1) 402(4) (13) 27.05(7) 117.3(1.0) 884(7) 26. March CHIN 09 17

18 Liquid T - pressure and speed of sound P( ρ) = 2 ρ e ρ c ( ρ) = 2 1 m P ρ March CHIN 09 18

19 Liquid T - pressure - spinodal denisty in T : ρ=0.0056ǻ -3 =0.277σ -3, P=-1.48(2) bar - spinodal density in 4 He ρ=0.264 σ -3, P=-9.30(15)bar 26. March CHIN 09 19

20 gr () = Liquid T two-body radial distribution function N( N 1) 2 ρ N d N d 1 Ψ( r, r,.. r ) r... r Ψ( r, r,.. r ) r... r N N obtained using unbiased (pure) estimators Casulleras, Boronat (1995) 26. March CHIN 09 20

21 Liquid T - static structure function ( q) = ρρ q q / N q N i i = e qr i= 1 m Sq ( ) = hq/ 2mc March CHIN 09 21

22 Liquid T - condensate fraction n0 = lim ρ( r) r f ( x) = Aexp( bx) 26. March CHIN 09 22

23 Solid T - equation of state f ( x) = a1x + a2x a3x 26. March CHIN 09 23

24 -pressure and speed of sound Solid T 26. March CHIN 09 24

25 Solid T two-body radial distribution function 26. March CHIN 09 25

26 Solid T static structure function 26. March CHIN 09 26

27 Liquid solid transition in T -Maxwell double tangent construction 26. March CHIN 09 27

28 Liquid solid transition in T BCC FCC HCP ρ f Ǻ Ǻ Ǻ -3 =0.477 σ -3 ρ m Ǻ Ǻ Ǻ -3 =0.528 σ -3 P 9.9 (1.0) bar 9.5 (1.0) bar 8.9 (1.0) bar Comparison: -H P=173(15) bar ρ f = Ǻ -3, ρ m = Ǻ -3 -He P=25 atm ρ f = σ -3, ρ m = σ March CHIN 09 28

29 Liquid solid transition in T Lindeman at the transition is 0.26 γ = ( r r ) / 2 I a L For solid 4 He γ=0.26, H γ=0.25 Discontinuity of the kinetic energy ~ 3.5 K At the freezing density condensate fraction n 0 =0.03 (in H 0.04) 26. March CHIN 09 29

30 Liquid solid transition in T Two-body radial distribution function 26. March CHIN 09 30

31 Liquid solid transition in T Static structure factor 26. March CHIN 09 31

32 Pure T clusters N = 20,..,320 EN ( N E E E x = E E E 0 s c 2 )/ = 0 + sx+ cx, N 1/3 = 3.66(3) K = 10.2(2) K = 6.1( 4) K surface tension t: t = E /(4 π r ), 4π r 3 s ρ = 1 t = 0.08 KǺ March CHIN 09 32

33 Pure T clusters -red crosses Blume et al. PRL 89, March CHIN 09 33

34 Pure T clusters -surface tickness (difference between the radii where the density has decreased from central values by 10% and 90%) N= Ǻ N= Ǻ N= Ǻ N= Ǻ N= Ǻ -equilibrium density of the liquid 0= (7) Ǻ March CHIN 09 34

35 Pure T clusters 26. March CHIN 09 35

36 Mixed T clusters µ = E(T H ) E(T ) N N µ = E(T D ) E(T ) N N 26. March CHIN 09 36

37 Mixed T clusters (T ) 60 D and (T ) 60 H 26. March CHIN 09 37

38 Small mixed T clusters Italic- clusters which appear to be unstable 26. March CHIN 09 38

39 Small mixed T clusters D 2 D March CHIN 09 39

40 Summary we have determined the ground state properties of bulk H, bulk T, pure and small mixed T clusters at low denisties equation of stated of gas H is well expressed in terms of the gas parameter ρa 3 transition pressure in H is around 170 bar, in T around 9 bar mixed T -H clusters are unbound, binding of T -D depends on the number of atoms and nuclear spin states 26. March CHIN 09 40

41 Thank you for your attention! 26. March CHIN 09 41