1 Adiabatic expansion of an electron gas.1Kazunori Takahashi, 2Rod Boswell and 2Christine Charles 1 Tohoku University, Sendai, Japan. 2 Australian National University, Canberra, Australia.
2 Thermodynamic systemsLTE Non-LTE Example Fluids Low pressure plasmas Collisional Highly Weakly or Collisionless Knudsen Number Small (particles behaving locally) Large (particles behaving nonlocally) Parametric Representation Zero-dimensional Multiple-dimensional Thermodynamic Interpretation Yes ? LTE: Local Thermodynamic Equilibrium
3 Helicon Double Layer ThrusterRF power: 250 W, MHz Argon pressure: 3×10-4 Torr
4 Electron Energy Probability Function with Yunchau Zhang and Igor KaganovichDruyvesteyn theory for probe Additive inverse of biased voltage reflects absolute energy state of electrons Nonlocal EEPFs Electron kinetic energy is replaced by total mechanical energy Adiabatic process across potential drop Low energy electrons are depleted, and high energy ones overcome the barrier
5 Plasma Parameters Definitions of effective parametersPressure Density Temperature Correlation between parameters Thermodynamic relation ?
6 A polytropic process is a thermodynamic process that obeys the relation:pvn = C where p is the pressure, v is specific volume, n is the polytropic index (any real number), and C is a constant. n=0 (isobaric), n=1 (isothermal), n=5/3 (adiabatic), n=γ (isentropic), n= (isochoric). For our experiment, the correlation between the effective electron temperature and density obtained along the divergent magnetic field can be fitted by a polytropic relation Te = Te0neg-1 where ne is the effective electron density normalized by the density at start of the divergent magnetic field.
7 Polytropic Relation Enthalpy & Potential ConservationFor LTE means heat absorption from surroundings, contradicting the real adiabatic process governed by collisionless electrons Revisiting non-LTE thermodynamics Enthalpy & Potential Conservation General relation for nonlocal EEPFs ?
8 And now for some thermodynamicsEnthalpy is a thermodynamic state function, designated by the letter "H", that consists of the internal energy of the system (U) plus the product of pressure (p) and volume (V) of the system: H = U + pV The enthalpy is the preferred expression of system energy changes at constant pressure and is the energy transferred from the environment through heating or work other than expansion work. It is a thermodynamic potential, so all that can be measured is the change in enthalpy, ΔH.
9 And now for some more thermodynamicsdU = dQ – dW (1st law of thermodynamics) where dQ is a small amount of heat added to the system and dW is a small amount of work done by the system. For a homogeneous system only reversible processes can occur so dQ = TdS (2nd law of thermodynamics) if work is only done by the pressure then dW = pdV resulting in: dU = TdS – pdV dH(S,p) = TdS + Vdp
10 Space Plasmas Solar wind A typical representation for space plasmasDirectly measureable by satellites Limitation of satellite measurements Inherent circular orbits Limited range to the sun Analysis from temporal signals Affected by satellite condition Low pressure space plasma Compared to laboratory plasma?
11 Small magnetic flux tubes and photospheric granulationWhite: Flux tubes Scale 100 km
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13 Magnetic network loops and funnelsStructure of transition region Magnetic field of coronal funnel Hackenberg, Marsch and Mann, Space Sci. Rev., 87, 207, 1999 Dowdy et al., Solar Phys., 105, 35, 1986
14 Magnetic funnel in the solar coronaIn the funnel the magnetic field magnitude typically changes from 50 gauss in the chromosphere at 1~Mm to 5 gauss at 20~Mm (A) Emphasis on open field lines and correlation with the Ne7+ outflow speed (dark blue shading at 20,000 km means speeds larger than 8~km/s); (B) Illustration of the funnel boundary and magnetic uni-polar flux constriction by many surrounding small bi-polar loops
15 Optical measurements of the corona taken from Hinode satellite.Oscillations in velocity and intensity of Fe XII 195Å spectral line interpreted as a slow mode MHD wave. The data analyzed here is taken on the 8th of 2007 February near the West limb of the solar disk starting at 13:06 UT. The Extreme-ultraviolet Imaging Spectrometer slit crosses a region of increased brightness, which is interpreted to be the footprint of active region loops.
16 Derivation of polytropic index on the (active) sun’s surfaceThe plasma density was derived from from the line ratio of the two Fe XIII spectral lines (assuming a constant temperature) and the temperature from the line ratio of the Fe XIII 202Å and Fe XII 195Å spectral lines (while taking into account the density variations). The Astrophysical Journal Letters, 727:L32, 2011
17 Similar Electrons in lab and spaceElectrons in the solar wind and laboratory plasmas share dominant similarities: Nearly collision-less due to the long free path Having a thermal velocity larger than the convective velocity of the plasma flow due to their negligibly small mass compared to ions Closely associated with a potential drop structure along a divergent magnetic field Moving and bounding nonlocally across the potential structure Polytropic Index Solar wind Laboratory HDLT about 1.15 (depending on specific locations) 1.17 A possible new scenario: adiabatic solar electrons with a polytropic index determined by nonlocal EEPFs
18 Interregnum The electric fields generated by the Boltzmann expansion of the plasma serves to accelerate the ions and to reflect the electrons in order to maintain equal fluxes. The ions make one pass through the system but low energy electrons are reflected, trapped by the axial electric fields, and eventually thermalised, resulting in a polytropic index near unity. New experiment at Tohoku to test what would happen if all the electric fields in the expanding plasma were removed and there were no trapped electrons
19 Using empirical calorimetry by a number of people, principally Joule, that the change in the internal energy of a system (ΔU) is equal to the nett energy added as heat (Q) minus the nett work (W) done by the system: ΔU = Q +W The meaning of Q has received considerable attention, and sometimes generated heated debate, over the whole of the 20th century. A working hypothesis that the first law of thermodynamics provides a definition of heat via the law of conservation of energy and the definition of work in terms of changes in the external parameters of a system. These changes are commonly effected using a calorimeter, which can be calibrated by adiabatically doing external work on it: the most accurate method is by passing an electric current from outside through a resistance inside the calorimeter. Our experiment can be seen as a calorimeter where the amount of work being done by the plasma is measured by the change in the electron temperature.
20 The first law is constructed on the concept of a wall with certain properties: those enclosing arbitrary systems that allow them to remain in their own states of internal thermodynamic equilibrium are defined as adiabatic. Easy to state, difficult to construct in the real world since a good 40% of the internal energy is lost as heat, somewhere, somehow. The primary culprit is the walls, and where do we draw the boundaries of our little universe? The expansion of a gas has been extensively studied and is well understood using traditional thermodynamics for both atomic and molecular gases. So well, in fact, that the experiment actually works in undergraduate laboratories and returns values of the ratio of specific heats of the gas with accuracies of the order of 1%!
21 For an ideal gas with no heat transfer to the surroundings the system is adiabatic and the polytropic index corresponds to the specific heat ratio ga i.e., g = ga = cp=cv = (Nd + 2)=Nd, where Nd is the number of degrees of freedom of the gas, 3 for monatomic (ga = 5/3), 5 for diatomic and so on and cp and cv are the specific heats at constant pressure and constant volume, respectively. The polytropic relation can be rewritten using the normalized temperature T=T0 and density n=n0 as T/T0 = (n/n0)g-1
22 (a) Electron beam source on the left; spiral filament heated by a floating power supply (not shown) creates thermal electrons accelerated into quartz tube by the power supply VD producing a high density plasma at the maximum of the magnetic field. The plasma then flows into the diffusion chamber along the diverging magnetic field to the grounded end plate. A voltage VA can be applied to the floating anode of the electron beam source; (b) axial profile of the magnitude of the magnetic field.
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24 EEPFs for zero current in the diffusion chamberlow-energy thermal electrons seem to be non-local. Clear depleted tail at the zero voltage (chamber wall); hence the cooling occurs by the wall. The electron beam component decays along the axis although the mean-free path with the neutrals is much larger than the system scale.
25 From the series of the eepfs for the chamber current free mode, the low energy electrons seem to be non-local. A depleted tail at the chamber potential is clearly seen. The beam decay is NOT due to e-n collisions. (possible scenario is two-stream instability?)
26 Remnant Electron Beam along the axisScale length of the beam decay is ~ 10 cm, much smaller than the collisional mean free path. This length is unchanged by the neutral pressure indicating the beam electrons suffer a collisionless scattering process.
27 Cartoons of work done by electron gas on magnetic wall.(a) Simple magnetic field aligned cylindrical plasma having a radial gradient in the electron density produces a gradient in the electron pressure. Insert: cartoon of how the gyro motion of the electrons produces an azimuthal diamagnetic current j (green arrow). (b) The electron pressure exerts a force normal to the magnetic field and does work on the magnetic wall (blue arrows); a force equal in magnitude and opposite in direction is exerted on the electron gas (red arrows). (c) This electron pressure-gradient-driven azimuthal current (j) exerts a force component given by j ~Br and (d) the other orthogonal force component j ~ Bz exerts a force on the magnetic field lines.
28 Structures of the plasma potential and electron density.a) Axial profile of the local plasma potential Vp for VA = 60 V where the electron current flows to the anode on the left and 0 V with the electron current flowing to the earthed wall on the right. A triangle shows the grounded wall potential at z = 51 cm. The arrow shows electrons that have energy less than the local plasma potential and are trapped by the electric field for VA = 60 V, whereas all the electrons escape from the system for VA = 0 V b) Axial profile of the electron density obtained by integrating the measured EEPFs for the same values of VA c) Radial profile of the plasma potential taken at z = 20 cm for VA = 0 V.
29 Changing the plasma potential changes the number of trapped and free electrons.a) The local plasma potential measured at z = 25 cm, as a function of VA b) EEPFs also measured at z = 25 cm. For the higher potential case (typically for the VA = 60 V case), most of the electrons are trapped in the system, whereas no electron reflection occurs at the grounded end wall for the VA = 0 V case.
30 Expansions of the free and trapped electrons are adiabatic and isothermal, respectively.Measured EEPFs over the range of 0-80 eV as a function of z position for (a) VA = 0 V (b) VA = 60 V; (c) Polytropic relation together with the calculated curves ranging from isothermal to adiabatic expansion. For the trapped electrons at VA = 60 V, the polytropic index is ∼ The polytropic index approaches 5/3 when the electric field is removed from the system.
31 Effect of removing possible errors in the data due to noise, remnant beam electrons and the influence of the magnetic field EEPFs (less than 25 eV and z greater than 15 cm) as a function of z for (a) VA = 0 V (b) VA = 60 V (c) Effective electron temperature obtained by integrating the EEPFs in Figs. 6(a) and 6(b) for VA = 0 V (open circles) and VA = 60V (filled squares); (d) Polytropic relation from the data in Figs. 6(a) and 6(b), together with the calculated curves from Eq.(3). When the electric field is eliminating by setting VA = 0V, ∼ 1:4 is close to the adiabatic value of = 5=3 whereas with the electric fields it is very close to unity.
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