The Super Fragment Separator of FAIR. Future Research Work and Education in Theoretical and Experimental Physics

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1 The Super Fragment Separator of FAIR. Future Research Work and Education in Theoretical and Experimental Physics C.-V. Meister Technische Universität Darmstadt The Fragment Separator at GSI (Photo: A. Zschau/GSI) C.-V. Meister 1

2 Plasma classification Parameters of nonideal plasmas (courtesy V. Fortov, 2013) C.-V. Meister 2

3 Plasma parameters of FAIR experiments Phase diagram of plasmas (courtesy D.H.H. Hoffmann 2009) C.-V. Meister 3

4 Material destruction analysis related to future Super-Fragment-Separator experiments In the Super-FRS target and in the beam catchers at GSI stress waves are generated by intense, fast-extracted ion beams which deposit a high amount of energy within a very short time into target material. measurements of thermal parameters contactless electrical resistivity measurements registration of thermal parameters (specific heats, heat capacities, thermal conductivity) C.-V. Meister 4

5 Inelastic thermal spike model for heat transport electrons and lattice are two coupled subsystems kinetic energie of the projectiles deposited into electrons electron thermalization within about s transfer of electron energy to the ions by electron-phonon coupling in lattice thermal equilibrium within s The heat distribution in the electron and lattice subsystems described by classical heat transport equations C.-V. Meister 5

6 Inelastic thermal spike model for heat transport Heat transport equations for cylindrically symmetric penetration of projectiles into the target: ] C e (T e ) T e t = 1 r C a (T a ) T a t [ rk e (T e ) T e r r = 1 r [ rk a (T a ) T a r r g(t e T a ) +A(r,t), ] +g(t e T a ). electron-ion energy exchange described by g(t e T a ), g - coupling parameter T j, C j, K j are temperature, specific heat coefficient, thermal conductivity of the electrons (j = e) and the lattice (j = a) A(r, t) - spacio-temporal energy deposition rate. Most of the electrons deposit their energy close to the ion path within s dt d r A(r,t) - total energy loss by the electrons C.-V. Meister 6

7 Inelastic thermal spike model for heat transport Recently used parameters: K e = C e D e (D e - electron diffusivity) hot electrons in the conduction band of insulators: C e 1.5k B n e 1 J cm 3 K 1 (n e 1 - number of excited electrons per atoms) D e = v F d 2cm 2 s 1, v F - Fermi velocity, d - inter-atomic distance λ 2 ea = C ed e /g, λ ea - electron-phonon mean free path parameters of the lattice C a, K a, solid and liquid mass density, melting and vaporisation temperature (latent heat, sublimation energy) taken from experiments C.-V. Meister 7

8 Calculation of the equation of state and specific heats Virial expansion in density order 5/2 known of F(V,n e,n i,t ) From there, equation of state (Meister et al., Astron. Nachr., 1999), inner energy ( ) F U(T,V,N) = F(T,V,N) +TS(T,V,N), S =. T V,N=const and specific heat capacities as well as adiabatic coefficient found ( ) ( ) 2 ( ) U p p c v =, c p c V = T /, T V,N=const T V,N=const V T,N=const γ t = c p, γ t = ρ ( ) p, ρ mass density. c v p ρ T =const C.-V. Meister 8

9 Calculation of heat conductivity Electrical currents j E and heat currents j Q are related to the corresponding generalized forces, the gradient of the temperature in the material T and the gradient of the electrochemical potential ζ = (ϕ + µ/e), where ϕ(r) is the external potential and µ(r) the chemical potential, by the linear relations (in isotropic systems) j E = e 2 L 11 ζ el 12 T T, j Q = el 21 ζ el 22 T T. Electrical conductivity σ, thermopower α, and thermal conductivity K defined by Onsager coefficients L ik = L ki (Kraeft et al., Quantum statistics of charged particles, Akademie-Verlag 1986; Reinholz et al., Phys. Rev. E52, , 1995 σ = e 2 L 11, α = 1 L 12, et L 11 K = 1 T ( L 22 L 12L 21 L 11 ) C.-V. Meister 9

10 Calculation of heat conductivity By non-equilibrium statistical mechanics the transport coefficients are determined by the relevant scattering processes of the particles in the system. An interpolation between Spitzer and Ziman-Faber formula is found. σ S = (4πε o) 2 e 2 3/2 k B T 1 m e ln(3/γ) 3 σ Z = 4 (4πε o ) 2 (k B T ) 3/2 2π e 2 m e Φ Φ = 2V ( ) ( ) 2k F me k B T df (E) de(k) dq q 3 V ei (q) 2 ε o /e 2. N e 2π de C.-V. Meister 10

11 Zubarev formalism for electrical conductivity It is assumed to use the linear response theory to calculate the thermal conductivity. Therefore the Zubarev formalism will be used, as it takes also non-mechanical perturbations into account. Transport coefficients are expressed by force-force correlation functions, which are found in T-matrix approximation for the particle interactions. Numerical results are obtained for hydrogen and the alkali plasmas. L = e2 k 2 B T K σ C.-V. Meister 11

12 Lorenz numbers L non-degenerate plasma: L = 4 for electron-ion-interaction non-degenerate plasma: L = adding electron-electron interaction (Reinholz et al. 1995) degenerate plasmas with free electrons: L = π 2 /3 (L = WΩ/K 2 ). metal κ [W/m K] 10 8 L [WΩ/T 2 ] metal κ [W/m K] 10 8 L [WΩ/T 2 ] Al Na Ag Pb Au Pt Cd Sn Cu Nb Fe In W Mo Zn Experimental values of thermal conductivityκ and Lorenz numberl = κ/(σ T) at T = 272 K (Kaye et al., 1966) C.-V. Meister 12

13 Structure factor for chemical components in mean spherical approximation Structure factor for caesium and aluminum (C.-V. Meister, 2014) C.-V. Meister 13

14 Electrical conductivity σ as function ofγ. Comparison of theoretical results for Cs at T = 10 4 K with experimental values for Cs and inert gases. Theory at T = 10 4 K: (1) Spitzer thory, (2) statically screenedt-matrix approximation, (3) staticallyscreened Born approximation. Experiments: Black square - Ar at K T K [Günther et al., 1981]; with error bares at K [Ivanov et al., 1976], black point - Ar, white point - Xe, black-whitepoint - Ne; black point - Ar bei K T K [Bakeev and Rovinskii, 1970]; white point - Xe bei 9000 K T K [Bakeev and Rovinskii, 1970];x with error bars - Cs at 4000 K T K [Seshenov et al., 1975]; + - H at K T K [Radtke et al., 1981]; white square - air at K T K [Andreev and Gavrilova, 1975];x - C 2 H 2 Cl at K T K [Ogurzova et al., 1974]. (C.-V. Meister and G. Röpke, 1982) C.-V. Meister 14

15 Electrical conductivity σ as function ofγ. Comparison of theoretical results for Cs at T = 10 4 K with experimental values for Cs and inert gases. Theory at T = 10 4 K: (1) Spitzer thory, (2) statically screenedt-matrix approximation, (3) staticallyscreened Born approximation. Experiments: Black square - Ar at K T K [Günther et al., 1981]; with error bares at K [Ivanov et al., 1976], black point - Ar, white point - Xe, black-whitepoint - Ne; black point - Ar bei K T K [Bakeev and Rovinskii, 1970]; white point - Xe bei 9000 K T K [Bakeev and Rovinskii, 1970];x with error bars - Cs at 4000 K T K [Seshenov et al., 1975]; + - H at K T K [Radtke et al., 1981]; white square - air at K T K [Andreev and Gavrilova, 1975];x - C 2 H 2 Cl at K T K [Ogurzova et al., 1974]. (C.-V. Meister and G. Röpke, 1982) C.-V. Meister 15

16 Electrical conductivity Influence of the structure factor S(q) on the normalized electrical conductivityσ. Dotted line: Coulomb potential for (1) S(q) = 1 and (2) S(q) in Debye approximation. Full line: Hellmann potential for (3,4) S(q) = 1 and (5,6) S(q) in mean sphericalapproximation (MSA). x - experimentalvalue for cesium at the melting point. The Hellmann potential is recently used to describethe transportcoefficientsof aluminum C.-V. Meister 16

17 Conclusions A model for the electrical conductivity of warm dense matter is given. The thermal conductivities are yet estimated multiplying with the relevant Lorenz numers. The presented models for thermal and transport parameters have to be further developed for non-fully ionized plasmas of high density and lower temperatures. New models for the coupling coefficients between electrons and ion grids in solids have to be proposed C.-V. Meister 17

18 Conclusions The construction of the Super Fragment Separator of FAIR demands already now for a careful destruction analysis of the used materials. Performing the analysis, the tasks get more and more specialized, so that they cannot be managed by one scientist. But working in an interdisciplinary group means also to understand the basics of the co-workers. For instance, as observational data of the materials are only available in a plasma parameter region γ < 4, theoretical models have to be developed and applied for γ 4. Thus engeneers should know the theoretical models and their regions of application in general - even if they will not perform the analysis themselves C.-V. Meister 18

19 THANK YOU VERY MUCH FOR YOUR ATTENTION The Fragment Separator at GSI (Photo: C.-V. Meister/TUD) C.-V. Meister 19

Half Yearly Exam 2015

Half Yearly Exam 2015 GOZO COLLEGE Secondary School KULLEĠĠ TA GĦAWDEX Skola Sekondarja Half Yearly Exam 015 Year 9 Track 3 CHEMISTRY Time: 1½ hours Name: Class: Useful Data: Atomic numbers and relative atomic masses are given