Journal of Energy and Power Engineering 15 (2021) 20-38 doi: 10.17265/1934-8975/2021.01.004 D DAVID PUBLISHING
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy
Bahman Zohuri1, 2, Farahnaz Behgounia1 and Masoud J. Moghaddam2 1. Golden Gate University, Ageno School of Business, San Francisco, California 94105, USA 2. Galaxy Advanced Engineering, Albuquerque, New Mexico 87111, USA
Abstract: In the 1970s, scientists began experimenting with powerful laser beams to compress and heat the hydrogen isotopes to the point of fusion, a technique called ICF (Inertial Confinement Fusion). In the “direct drive” approach to ICF, powerful beams of laser light are focused on a small spherical pellet containing micrograms of deuterium and tritium. The rapid heating caused by the laser “driver” makes the outer layer of the target explode. In keeping with Isaac Newton’s Third Law “For every action, there is an equal and opposite reaction”, the remaining portion of the target is driven inwards in a rocket-like implosion, causing compression of the fuel inside the capsule and the formation of a shock wave, which further heats the fuel in the very center and results in a self-sustaining burn. The fusion burn propagates outward through the cooler, outer regions of the capsule much more rapidly than the capsule can expand. Instead of magnetic fields, the plasma is confined by the inertia of its own mass—hence the term inertial confinement fusion. A similar process can be observed on an astrophysical scale in stars and the terrestrial uber world, that have exhausted their nuclear fuel, hence inertially or gravitationally collapsing and generating a supernova explosion, where the results can easily be converted to induction of energy in control forms for a peaceful purpose (i.e., inertial fusion reaction) by means of thermal physics and statistical mechanics behavior of an ideal Fermi gas, utilizing Fermi-Degeneracy and Thomas-Fermi theory. The fundamental understanding of thermal physics and statistical mechanics enables us to have a better understanding of Fermi-Degeneracy as well as Thomas-Fermi theory of ideal gas, which results in laser compressing matter to a super high density for purpose of producing thermonuclear energy in way of controlled form for peaceful shape and form i.e. CTR (Controlled Thermonuclear Reaction). In this short review, we have concentrated on Fundamental of State Equations by driving them as it was evaluated in book Statistical Mechanics written by Mayer, J. and Mayer, M. in this article.
Key words: renewable, nonrenewable source of energy, fusion reactors, super high density matter, laser driven fusion energy, Fermi-Degeneracy, Thomas-Fermi theory, return on investment, total cost of ownership.
1. Introduction gain in the NIF target chamber will be a significant step toward making fusion energy viable in commercial NIF (National Ignition Facility), located at LLNL power plants. LLNL scientists also are exploring other (Lawrence Livermore National Laboratory), will be the approaches to developing ICF as a commercially viable first laser in which the energy released from the fusion energy source, a process that is considered as FI (Fast fuel will equal or exceed the laser energy used to Ignition) [1]. produce the fusion reaction—a condition known as FI is the approach, which is being taken by the NIF ignition. Unlocking the stored energy of atomic nuclei to achieve thermonuclear ignition, and burn is called will produce ten to 100 times the amount of energy the “central hot spot” scenario. This technique relies on required to initiate the self-sustaining fusion burn. simultaneous compression and ignition of a spherical Creating ICF (Inertial Confinement Fusion) and energy fuel capsule in an implosion, roughly like in a diesel engine. Although the hot-spot approach has a high Corresponding author: Bahman Zohuri, Ph.D., adjunct professor, artificial intelligence and machine learning. probability of success, there also is considerable
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 21 Controlled Thermonuclear Energy interest in a modified approach called FI, in which compression is separated from the ignition phase. FI uses the same hardware as the hot-spot approach but adds a high-intensity, ultra-short-pulse laser to provide the “spark” that initiates ignition. A deuterium-tritium target is first compressed to high density by lasers, and then the short-pulse laser beam delivers energy to ignite the compressed core—analogous to a sparkplug in an internal combustion engine [1]. Moreover, to accommodate the symmetry conditions needed, the absorption of laser energy must be carefully determined starting from the early stages [7, 8]. The absorption data for dense plasmas are also required for fast ignition by ultra-intense lasers due to creation of plasmas by the nanosecond pre-pulse [9].
Least understood are laser-plasma interactions that Fig. 1 Filamentation of counter-streaming electron beams involve strongly coupled Γ>1 and partially (red/blue density isosurfaces) leads to filamentation of the degenerate electrons. Such conditions also occur in background ions (green isosurfaces). The colder (blue) electron filaments attract the ions [1]. warm dense matter experiments [10, 11] and laser cluster interactions [1]. Because modern thermonuclear weapons use the To explain the apparent fluid-like behavior of the fusion reaction to generate their immense energy, current filamentation instability in FI scenarios, many scientists will use NIF ignition experiments to examine experts in the field of ICF have developed analytical the conditions associated with the inner workings of theory for the coupling of electromagnetic instabilities nuclear weapons. to electrostatic modes. This theory shows that as cold Ignitions experiments also can be used to help electrons tend to filament at a faster rate than hot ones scientists better understand the hot, dense interiors of they pull with them the ions. Because hot electrons are large planets, stars and other astrophysical phenomena. usually in the minority, the filaments of the bulk of the Here in this short review, we consider an elementary electron population overlap with those of the ions, account of thermal physics. The subject is simple, the which is exactly what one would have expected from a methods are powerful, and the results have broad fluid instability (such as Rayleigh-Taylor). Nevertheless, applications. Probably no other physical theory is used this is a purely kinetic electromagnetic phenomenon. more widely throughout science and engineering. In The predicted growth rate was confirmed using order to study of plasma physics and its behavior for a Particle-In-Cell simulations and the physics is source of driving fusion for a controlled thermonuclear illustrated in Fig. 1. reaction for purpose of generating energy, in particular The analytical theory was only available for beam-like using high power laser or particle beam source, distributions, such as drifting Gaussian and water bag requires an understanding of the fundamental distribution. However, in order to overcome this problem knowledge of thermal physics and statistical mechanics and to resolve this, one can calculated the stability theory as part of essential education. As part of this properties (growth/damping rate) of electromagnetic education, we need to have a better concept of the EOS modes for arbitrary electron distribution functions. (Equation of State) for ideal gases, which proves to be
22 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy the central to the development of early molecular and Confinement Fusion) for purpose of CTR (Controlled atomic physics. In our case, it will lay down the ground Thermonuclear Reaction), for laser or high-energy particle beam compression of D + T → 4He + n + 17.6 MeV (2) matter to super-high densities as [1, 2]: The Lawson number, also known as Lawson 1. Thermonuclear applications, an event that in criterion, is defined as illustrated in Eq. (3). nature, takes place at extraterrestrial in stars and nτ ≥104 s/cm3 (3) 2. Surface of the sun and terrestrially in a nuclear explosion. where in this case, n is the plasma density in units of Nuckolls, et al. [2] in their published paper in 1972 particles per cm3 and τ is the time in seconds for under the title of “Laser Compression of Matter to which the plasma τ of density n is going to be Super-high Densities: Thermonuclear (CTR) confined. Applications” established the ground based on their However, in case of ICF, the same Lawson criterion initial research at LLNL, that hydrogen may be of Eq. (3) shapes in different form as ρr ≥1 2 compressed to more than 10,000 times liquid density gram/cm , where ρ and r are the compressed fuel by an implosion system energized by a high energy density and radius pellet, respectively. In order, for the laser [2]. confinement criteria also known as Lawson criterion to This scheme makes possible efficient thermonuclear be satisfied, it needs to take place, before the burn of small pellets of heavy hydrogen isotopes and occurrence of Rayleigh-Taylor hydrodynamics feasible fusion power reactors using practical lasers. instability would happen for uniform illumination of The implosion process to a focal point first was the target’s surface, namely pellet of Deuterium and solved numerically by Guderley [3] utilizing the Tritium. See Section 3.4 of Ref. [1] written by Zohuri application of self-similarity of the second kind for for further information. deducing a closed solution in one-dimensional form Note that: in conventional CTR approaches, the [3-6]. density is limited by material properties, and the For a thermonuclear burning to take place in objective is to achieve sufficiently long confinement extra-terrestrially in stars as well as in a nuclear times by the use of electromagnetic fields [8]. explosion, we need to look at a specific thermonuclear In the case of laser driven pellet fuel of D-T reaction burn rate that is proportional to density and is given by to induce a controlled fusion energy (i.e. ICF), the main Eq. (1). objective is to achieve a breakeven by satisfying ∂ρ Lawson criterion (i.e. at the minimum amount of laser ~ ρσυ (1) ∂t energy in needs to be equal to the minimum fusion of where ∂∂ρ t is the fractional burnup, ρ is the energy out). For this matter to be achieved, a sufficient density, and σ v is the Maxwell velocity-average high fuel density needs to take place, while the reaction cross-section. With this relationship being confinement time is determined by the inertia of satisfied, except at high fuel of let say D-T (e.g. matter. Deuterium (D) and Tritium (T), the two isotopes of “Spherical compression of 104 fold via the laser Hydrogen (H) element) depletions, the thermonuclear implosion scheme described here reduces the laser energy production at a fixed ion temperature is energy required for CTR by more than one proportional to the Lawson number [7] ()n×τ , a thousand-fold, from more than 108-109 Jules, which is product of density n and confinement timeτ . so large as to be currently impractical to 105-106 Jules, In case of D-T reaction of Eq. (2) in MCF (Magnetic assuming laser and thermal/electric efficiencies of 10%
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 23 Controlled Thermonuclear Energy and 40% respectively” [2]. This is the way that we can reproduce the thermal “Note that one kJ of laser energy may be sufficient to energy on earth similar to what we observe at the sun generate an equal thermonuclear energy” [2]. surface as well as other terrestrial stars. Bear in mind that electron degeneracy pressure [13] 2. Implosion Driven by Laser Compression is a particular manifestation of the more general of Fuel Pellet phenomenon of quantum degeneracy [14] pressure. In In nature, hydrogen in the center of the sun is quantum mechanics, an energy level is degenerate if it believed to exist at more than one thousand times corresponds to two or more different measurable state liquid density, and at pressures greater than 1011 of quantum system as illustrated in Fig. 2. atmospheres (temperature ~ 1-2 keV) [9]. Conversely, two or more different states of a These pressures are maintained gravitationally by quantum mechanical system are said to be degenerate if the overlying enormous solar mass, ~ 1033 grams. they give the same value of energy upon measurement. Matter in the cores of white dwarf stars is believed to The number of different states corresponding to a exist at more than 105 gram/cm3, and at pressures particular energy level is known as the degree of greater than 1015 atmospheres [10]. degeneracy of the level. It is represented Moreover, we would be able to have a gripe of the mathematically by the Hamiltonian for the system situation by having a better understanding of the having more than one linearly independent eigenstate behavior of electron in white cores that are obeying with the same energy eigenvalue. Fermi-Degenerate, so the pressure is a minimum Note that degeneracy plays a fundamental role in determined by the quantum mechanical uncertainty quantum statistical mechanics. principles [11] and for hydrogen with Also, from the knowledge of our quantum mechanics Fermi-Degenerate electrons is given by the Eq. (4) and quantum energy state, we know that, per Pauli [12]. exclusion principle, we are disallowed by having two 24identical half-integer spin particles (electrons and all 23ππ24kT 3 kT Pn=+−eFε + other fermions) from simultaneously occupying the 3 5 4εεFF 80 same quantum state. The result is an emergent pressure (4) against compression of matter into smaller volumes In this short review here, our task would be to derive of space. The principle and fundamental aspect of this equation for an ideal Fermi gas, first by defining it, secondly via thermal physics utilizing statistical mechanics come to the conclusion of ICF generating CRT that is driven by Fermi energy of Eq. (5) via pressure given in Eq. (4).
2/3 h2 3 ε Fe= n (5) 8m π where ne is the electron density, kT is the thermal energy, h = 6.62607004 × 10-34 m2 kg/s, and m is 4 the electron mass. At 10 times liquid density ( ne = 5 x 26 10 ), the minimum hydrogen pressure occurs, when 12 kT ε F , and is approximately 10 atmospheres. Fig. 2 Degenerate states in a quantum system [14].
24 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy
which defines the degenerate Fermi gas and this
equation β =1 kTB is the inverse temperature, and
T is the temperature and kB is the Boltzmann constant. Note that β is called thermodynamic beta in the field of statistical thermodynamics subject, which is also known as coldness, is the reciprocal of the thermodynamic temperature of a system as defined above. In Eq. (6), the parameter µ is representing the total chemical potential (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero
temperature Fermi energy ε F of Eq. (5) by a Sommerfeld expansion, where we are assuming Fig. 3 Image of white dwarfs Sirius A and B [18]. kTBF ε :
24 compressing pellet fueled by D-T to a FI steps driving 24 ππkTBB kT µ()T =+−−+ εε0 F 1 ICF. εε 12FF 80 For bulk matter with no net electric charge, the attraction between electrons and nuclei exceeds (at any (7) Hence, the internal chemical potential µε− is scale) the mutual repulsion of electrons plus the mutual 0 repulsion of nuclei; so absent electron degeneracy approximately equal to the Fermi energy at pressure, the matter would collapse into a single temperatures that are much lower than the nucleus. characteristic Fermi temperature TF. This characteristic 5 In 1967, Freeman Dyson showed that solid matter is temperature is on the order of 10 K for a metal, hence stabilized by quantum degeneracy pressure rather than at room temperature (300 K), the Fermi energy and electrostatic repulsion [15-17]. internal chemical potential are essentially equivalent. Because of this, electron degeneracy creates a barrier In the limit, defined by Eq. (6), the quantum to the gravitational collapse of dying stars and is mechanical nature of the system becomes especially responsible for the formation of white dwarfs as important, and the system has little to do with the illustrated in Fig. 3, an image of Sirius A and B taken classical ideal gas. by the Hubble Space Telescope [18] Furthermore, electrons are part of a family of particles known as fermions. Fermions, like the proton Sirius B, which is a white dwarf, can be seen as a or the neutron, follow Pauli’s principle and faint point of light to the lower left of the much brighter Fermi-Dirac statistics. In general, for an ensemble of Sirius A [18]. non-interacting fermions, also known as Fermi gas, 3. Ideal Fermi Gas each particle can be treated independently with a single-fermion energy given by the purely kinetic term To describe and define an ideal Fermi gas, we may ε , presented in Eq. (8) as: state that it is strongly determined by the Pauli principle by considering the limit as presented in Eq. (6) p2 ε = (8) here, which defines the degenerate Fermi gas as: 2m
µ kTB ⇔ βµ 1 (6) where p is the momentum of one particle and m is
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 25 Controlled Thermonuclear Energy
4/3 degeneracy pressure is proportional to ρe . Note that again, in white dwarfs, electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the Chandrasekhar limit (1.44 solar masses) and presented by Eq. (11) [20]:
3/2 ωπ0 3 c 1 = 3 M Limit 2 Eq. (11) 2Gm ()µeH where: h Fig. 4 Pressure vs. temperature. = is the reduced Plank constant; π its mass. Every possible momentum state of an electron c is the speed of light; within this volume up to the Fermi momentum pF is G is the gravitational constant; being occupied. µ is the average molecular weight per electron, Fig. 4 is the pressure vs temperature curves of e classical and quantum ideal gases (Fermi gas, Bose gas) which depends upon the chemical composition of the in three dimensions. Pauli repulsion in fermions (such star; as electrons) gives them an additional pressure over an mH is the mass of the hydrogen atom; 0 equivalent classical gas, most significantly at low ω3 ≈ 2.018236 is a constant connected with temperature. the solution to the Lane-Emden equation; Moreover, the degeneracy pressure at zero c m = is the Plank mass, the limit is the temperature can be computed as [19]: P G 2 32 22E p order of MmPHlimit =Total = F . P 23 (9) 3Vm 3 10π The limiting mass can be obtained formally from the Chandrasekhar’s white dwarf equation by taking the where V is the total volume of the system and ETotal is the total energy of the ensemble. Specifically for the limit of large central density. electron degeneracy pressure, m is substituted by the A more accurate value of the limit than that given by this simple model requires adjusting for various factors, electron mass me and the Fermi momentum is obtained from the Fermi energy, so the electron including electrostatic interactions between the degeneracy pressure is given by Eq. (10) as: electrons and nuclei and effects caused by nonzero temperature [21]. π 2 2/3 2 (3 ) 5/3 Lieb and Yau [22] have given a rigorous derivation Pee= ρ (10) 5me of the limit from a relativistic many-particle where ρe is the free electron density (the number of Schrödinger equation. free electrons per unit volume). For the case of a metal, This is the pressure that prevents a white dwarf star one can prove that this equation remains approximately from collapsing. A star exceeding this limit and true for temperatures lower than the Fermi temperature, without significant thermally generated pressure will about 106 Kelvin. continue to collapse to form either a neutron star or When particle energies reach relativistic levels, a black hole, because the degeneracy pressure provided modified formula is required. The relativistic by the electrons is weaker than the inward pull of gravity.
26 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy
Note that in the non-relativistic case, electron For a fully relativistic treatment, the EOS used degeneracy pressure gives rise to an EOS of the form 5/3 interpolates between the equations PK= 1ρ for 5/3 ρ 4/3 PK= 1ρ , where P is the pressure, is the small ρ and PK= 2 ρ for large ρ. When this is mass density, and K1 is a constant. done, the model radius still decreases with mass, but Solving the hydrostatic equation leads to a model becomes zero at M limit . This is the Chandrasekhar white dwarf that is a polytropic of index 3/2 – and limit [23]. therefore has radius inversely proportional to the cube The curves of radius against mass for the root of its mass, and volume inversely proportional to non-relativistic and relativistic models are shown in its mass [23]. Fig. 5. They are colored blue and green, respectively. As the mass of a model white dwarf increases, the µe has been set equal to 2. Radius is measured in typical energies to which degeneracy pressure forces standard solar radii or kilometers, and mass in standard the electrons are no longer negligible relative to their solar masses [24]. rest masses. The velocities of the electrons approach Fig. 4 shows radius-mass relations for a model white the speed of light, and special relativity must be taken dwarf. into account. In the strongly relativistic limit, the EOS The green curve uses the general pressure law for an 4/3 takes the form PK= 2 ρ . This yields a polytropic ideal Fermi gas, while the blue curve is for a of index 3, which has a total mass, M limit say, non-relativistic ideal Fermi gas. The black line marks depending only on K2 [23]. the ultra-relativistic limit.
Fig. 5 Radius-mass relations for a model white dwarf. (Source: www.wikipedia.org)
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 27 Controlled Thermonuclear Energy
thermodynamic quantities, such as pressure, energy, density, entropy, specific heat, etc., and is related to both fundamental physics and the applied sciences. Important branches of physics were developed or originated from the equations of state, while in return more complex formulations of the EOS were due to the development of modern physics. For super-fast moving objects, Einstein extended classical mechanics to what we know as the theory of relativity by revolutionizing a new era for quantum
Fig. 6 Phase diagram of a second order quantum phase mechanics and Planck-Bohr-Heisenberg-Schrodinger transition. for very small objects, thermodynamics and statistics (Source: www.wikipedia.org) were developed to describe EOS of matter in extreme As a lump sum of this section, we may state that an density and temperature domains. ideal Fermi gas is a state of matter which is an Later on, the EOS of high pressure was studied ensemble of many non-interacting fermions. Fermions experimentally in a laboratory by using static and are particles that obey Fermi-Dirac statistics, like dynamic techniques driven by a sample being squeezed electrons, protons, and neutrons, and, in general, between pistons or anvils and the pressure and particles with half-integer spin. These statistics temperature are limited by the strength of the determine the energy distribution of fermions in a construction materials for a few megabars and a few Fermi gas in thermal equilibrium, and are characterized hundred degrees of Celsius. This is the approach that by their number density, temperature, and the set of the scientists and physicists working in the field of ICF available energy states, as illustrated in Fig. 6. are taking, by pressing the ablation surface of the fuel The model is named after the Italian physicist Enrico pellet of D-T adiabatically, so the corona of the pellet Fermi [25]. containing two hydrogen isotopes of Deuterium (D) This physical model can be accurately applied to and Tritium (T) reaches a FI level for fusion purpose. many systems with many fermions. Some key Bear in mind that in the dynamic experiments shock examples are the behavior of charge carriers in a metal, waves can be created by means of the Hugoniot nucleons in an atomic nucleus, neutrons in a neutron relationship, driven by measuring the shock wave star, and electrons in a white dwarf. parameters, such as shock wave speed and particle flow velocity. The EOS are studying the fluid equations of 4. Fundamentals of EOS motions, where the state of a moving fluid or gas can be The EOS gives an impressive demonstration of how defined in terms of its velocity, density, and pressure as great a fraction of the inorganic world is explained by functions of a position, and time. These functions are the revolutionary deviation from classical mechanics to obtained by integral equations or differential equations quantum mechanics. At the same time, the EOS which are derived from the conservation laws of mass, presents the tools by which our experimental momentum and energy [26]. knowledge can be extended into regions of extreme In conclusion, since the passage time of the shock is concentrations of energy and matter. short in comparison to the disassembly time of the In a very holistic definition way, the EOS describes a shocked medium, one can study EOS for any pressure physical system by the relation between its that can be supplied by the source of driver.
28 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy 5. Degenerate Gases
By definition in a simple form, we can state that a degenerate gas is the type of gas in which, because of high density, the particle concentration is so high that the Maxwell-Boltzmann distribution does not apply, and the behavior of the gas is governed by quantum statistics. Examples of degenerate gases are the conduction of electrons in a metal, the electrons in a white dwarf, and the neutrons in a neutron star. Fig. 7 White dwarf stars. With high density concentration within the nature of (Source: www.wikipedia.org) degenerate gas, comes degeneracy pressure, and briefly, which usually reached at high densities of a gas we may state that the pressure in a degenerate gas of composed of subatomic particles with half-integral fermions caused by the Pauli exclusion principle and intrinsic angular momentum (i.e. spin). the Heisenberg uncertainty principle. Because of the Such particles are called fermions, because their exclusion principle, fermions at a high density, with microscopic behavior is regulated by a set of quantum small interparticle spacing, must have different mechanical rules—Fermi-Dirac statistics. momenta; from the uncertainty principle, the momentum difference must be inversely proportional These rules state, in particular, that there can be only to the spacing. Consequently, in a high-density gas one fermion occupying each quantum-mechanical state (small spacing) the particles have high relative of a system. As particle density is increased, the momenta, which leads to a degeneracy pressure much additional fermions are forced to occupy states of greater than the thermal pressure. higher and higher energy, because the lower-energy Furthermore, stars are supported by the degeneracy states have all been progressively filled. This process pressure of the electron gas in their interior, which is of gradually filling in the higher-energy states obeying Eq. (1) formulation. Degeneracy pressure is increases the pressure of the fermion gas, termed the increased resistance exerted by electrons degeneracy pressure. A fermion gas in which all the composing the gas, as a result of contraction of stellar, energy states below a critical value (designated Fermi as we expressed at the begging of this article. The energy) are filled is called a fully degenerate, or application of the so-called Fermi-Dirac statistic and of zero-temperature, fermion gas. Such particles as special relativity to the study of the equilibrium electrons, protons, neutrons and neutrinos are all structure of white dwarf stars (see Fig. 7) leads to the fermions and obey Fermi-Dirac statistics. The electron existence of a mass-radius relationship through which a gas in ordinary metals and in the interior of white dwarf unique radius is assigned to a white dwarf of a given stars constitutes two examples of a degenerate electron mass; the larger the mass, the smaller the radius. gas. White dwarfs and neutron stars are supported against Note that: Fermion, any member of a group of collapse under their own gravitational fields by the subatomic particles having odd half-integral angular degeneracy pressure of electrons and neutrons, momentum (spin 12, 32), named for the Fermi-Dirac respectively. statistics that describe its behavior. Fermions include To put what just stated in general perspective, the particles in the class of leptons (e.g., electrons, muons), degenerate gas, in physics is a particular configuration, baryons (e.g., neutrons, protons, lambda particles), and
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 29 Controlled Thermonuclear Energy nuclei of odd mass number (e.g. tritium, helium-3, function of Eq. (12) is always modified by omitting the uranium-233). unity in the denominator. The difference between the Fermions obey the Pauli exclusion principle, which two kinds of system disappears and classical or forbids more than one particle of this type from Boltzmann statistics is obtained. This approximation α occupying a single quantum state as illustrated in Fig. 2. can certainly be made if e is large, or NCjj1 This condition underlies, for example, the buildup of for all regions j . The parameter α can be electrons within an atom in successive orbitals around determined for monatomic gases by setting the nucleus and thereby prevents matter from Σ NNj = , with neglect of the unity in the collapsing to an extremely dense state. Fermions are j produced and undergo annihilation in denominator. The result is presented in Eq. (14) [12] particle-antiparticle pairs. here as: 32 From Quantum Mechanical Distribution Function 2π mkT V M32 T 52 α = = point of view, for the number of molecules per eg2 0.026g NPAtm quantum cell, one can derive Eq. (12) as [12]: (14) N 1 j = (12) This is found to justify, a posterior, the neglect that αε+ j kT C j e 1 was introduced in Ref. [12]. In the above relation parameter of N , it denotes the The parameters in Eq. (14) are defined and given as: number of particles, which at equilibrium, are found in M = the atomic weight; g = the multiplicity of the ground level in one a region j consisting of C j cells, or quantum states atom; of one particle, the energies of which lie between ε j = molar volume of the gas. and εεjj+∆ . The parameter α is determined by V the condition that the total number N of particles in More details that are beyond scope of this short review article can be found in the book by Mayer and the system is fixed and that is Σ N j , summed over all j Mayer [12]. Bear in mind that, the Fermi-Dirac gas at zero rejoins j , must be equal to N number of particles or the relation as presented in Eq. (13) [12]. temperature in case of metals especially their electrical conductivity, may be rather satisfactorily explained by Σ=NNj Eq. (13) j assuming that metals contain a perfect gas of electron. In Eq. (12), the parameter k is considered as If each metal atom contributes one or as many Boltzmann constant, while T is temperature in electrons as its valency, to the electron gas, the density Kelvin. of particles in the gas is very high. The molar volume Note that Eq. (12) is applicable to all systems V of the electron gas is the atomic volume of the composed of mechanically independent particles. The metal divided by a small integer. Note that the atomic minus (-) sign in Eq. (12) is to be used if the particles volumes of metals are of the order of 10 cc. have symmetric eigenfunctions, in which case they are Under this condition, the electrons obey Fermi-Dirac statistics and therefore, the distribution function of Eq. said to obey Bose-Einstein statistics. While, the plus (+) (12) is given in Eq. (15): sign must be used if the particles have antisymmetric N 1 eigenfunctions, in which case they are said to obey j = (15) ()εµj − kT Fermi-Dirac statistics. [1, 12] C j e +1 In case of treating a perfect gas, the distribution The classical distribution function is that in which
30 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy
3/2 the unity in the denominator of this equation is omitted µµ 002m εε= π ε1/2 ε [12]. Cd() 4 V2 d ∫∫00h (19) Note that, the lowest energy of classical or a 3/2 8π 2mµ Bose-Einstein gas at T = 0 and ε 0 by taking the Fig. = 0 V 2 1 into consideration, at the zero temperature all 3 h particles crowed into the lowest state and lose all Since this number must be equal to a number of kinetic energy. particles or electrons N in the system, one can obtain For a Fermi-Dirac gas, this is not a possible case and the result as Eq. (20) as: the particles, in this case, are subject to the Pauli 3/2 8π 2mµ exclusion principle. This principle indicates that no = 0 NV2 (20) more than one particle may be in one quantum state, or 3 h cell. Considering, these circumstances, the lowest And in term of µ0 explicitly, we can obtain Eq. (21) energy of the gas of N particles, is therefore, as: obtained if the N cells of lowest energy are filled with 2 2/3 ε hN3 one particle in each. The energy 0 of the gas at µ0 = (21) 8mVπ T = 0 is therefore different from zero and one can easily calculate the quantity of this energy. Note that: the quantity µ0 , which is the uppermost The number of quantum states, C()εε∆ of one energy of the cells, frequently is called the Fermi particle, the energy of which lies between ε and energy. εε+∆ , is given by Eqs. (16) and (17) consequently as Substituting Eqs. (20) and (21) into Eq. (18), we [12]: obtain another result for C()ε as Eq. (22): mV C(εε )∆= 4 π gm (2 ε )12∆ ε (16) 3 ε 1/2 h3 CN()ε = (22) 2 µ 3/2 and 0
3/2 Looking at Eq. (22), it easily can be noted that we 2m 12 C()εε∆= 2 π gV ε∆ ε (17) define a cell by both the translational quantum numbers, h2 kkkxyz,, and the internal quantum numbers of where m denotes the mass of the particle and V is particle, which in this case consist of the two spin the total volume, as well as g representing the directions. One sometimes defines a cell by the degeneracy of the internal ground level of the particles. translational quantum numbers only and says that two Thus, for electrons with g = 2 , owing to the two electrons of opposite spin may occupy this cell, which possible orientations of the spin, Eq. (17) reduces to the is in alignment with Pauli excursion principle. The Eq. (18): difference, obviously, is one of nomenclature only.
3/2 Going forward, we can now express the total energy 2m 12 CV()επ= 4 ε (18) of the N particles in this distribution, namely, the h2 energy ε 0 of the Fermi gas at T0 = 0 , is given by Eq. With above result, we now can determine the (23) as: µ number of cells with energy less or equal to µ0 (i.e. 0 ε0 = εεεCd() (23) εµ= 0 , the chemical potential at ground zero) is given ∫0 by Eq. (19) as: Now utilizing Eqs. (21) and (22) and then integrating the result, leads to Eq. (24) as:
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 31 Controlled Thermonuclear Energy
2/3 distribution function driven by Eq. (15) is zero if 3 33hN2 εµ> (0) and it would be equal to 1 if εµ< (0) . εµ00=NN = (24) j j π 5 40mV Moreover, the distribution function of Eq. (15) under The average energy per electron in the Fermi gas at above conditions represents the state that all cells with
T0 = 0 is 3/5 of that of the energetically highest energy lower than µ(0) are filled and all the cells particle or 3/5 of the Fermi energy µ0 . with higher energy would be empty.
Furthermore, the energy µ0 depends inversely on What remains is the case, when εj = εµ = (0) . the mass of particles by inserting for m the mass of With this condition in hand, we encounter the situation the electron. that the distribution function has a discontinuity, as it For molar volume V ≈10 for most metals, it is suddenly drops from unity to zero. In this case, the seen that the Fermi energy of an electron gas is Fermi energy µ of the filled level of highest energy extremely high and in the next sections, we would 0 show that thermodynamic properties of the gas above is equal to the value µ that is taking place in the
T = 0 can be obtained as a power series in kT µ0 distribution function provided by Eq. (15) at zero as it is shown in Eq. (7). A series of that type must be temperature or T = 0 . expected to converge very rapidly, so that the behavior But in Chapter 6 of Ref. [12], it is shown that in of the electron gas at room temperature is not greatly general the quantity µ in Eq. (15) is the chemical different from that at T = 0 . potential which is 1 N times the free energy F , thus The Eq. (21) shows that both the small mass and the the free energy of the electron gas at T = 0 is high density of the electron gas favor this high value of FN00= µ and easily can be verified by direct µ0 . calculation of the various thermodynamic functions at Obviously, atoms or molecules have masses more T = 0 and at a temperature above zero, the than two thousand times that of an electron, so that the distribution temperature of Eq. (15) must be used for value of µ0 for a chemical Fermi-Dirac gas, even at the evolution of thermodynamic functions as well [12]. the same density, is very much smaller. With respect to all these above debates, a development with respect to 6. Fermi-Dirac Gas Integrals Equations kT µ0 for a chemical gas obeying the Pauli principle In order to derive such a relationship, we can take the would lead to a series which converges at very low number of electrons N()εε∆ into account for the temperatures only, and at room temperature the energy ranging between ε and εε+∆ by driving it thermodynamic functions are radically different from as a function of the temperature and volume by taking those at T = 0 which is discussed in Mayer and Eq. (18) and substituting C()ε into Eq. (15), then we Mayer [12] chapters 5 through 8 and we encourage the obtain the number of electrons per cell as Eq. (25) here: reader to refer to this reference for more details that is 3/2 2m ε 1/2 beyond the scope of this short review. επ= NV() 4 2 (εµ− )/kT (25) However, note that the same results could be he +1 obtained for the electron gas at T = 0 using the Now using Eq. (22), we also can find another form distribution function as it is given in Eq. (15). In Eq. of Eq. (25), which is presented here as Eq. (26): (15), the quantity µ()T is a function of temperature 3N ε 1/2 T and would be determined by the condition that the ε = N() 3/2 (εµ− ) kT (26) total number of particles should be fixed. 21µ0 e + Furthermore, at zero temperature ( T = 0 ) in which µ0 , defined by Eq. (21), is the chemical
32 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy potential at T = 0 and µ the potential at the in metals this assumption of µ kT 1 is justifiable temperature in question. up to temperatures above those values at which the The integration of Eq. (25) over all values of ε metals start melting. Furthermore, in order to integrate serves to derive the value of µ as a function of the Eq. (28), it will be necessary to play a unique temperature by equating the result the left-hand side of mathematical technique as follows. −−µ kT 40 ∞ Since it can be found that e ≈10 , the value an integrated form of Eq. (26), mainly Nd()εε to ∫0 of g()ε in Eq. (29) reduces to 1 at ε = 0 and the total number of particles N . decreases monotonously to 0 at ε = ∞ as they are Analyzing Eq. (26), one can realize that for N()ε , presented here as Eq. (31): µ will necessarily result as a function of µ0 and 10ε = −−µ kT 40 kT alone. Furthermore, the energy may be ge()ε = ⇒≈10 (31) 0 ε = ∞ determined from Eq. (27) as: ∞ The derivation of g()ε as it is given by Eq. (29) E= ε Nd() εε (27) ∫0 designated by g'(εε ) = dg d , is always a negative In order to be able to integrate any form of the equation such as Eq. (27), we need to deal with the result, but has one single sharp minimum at εµ= , as problem of making integration of the type: long as µ is a positive value. ∞ I= f()()ε gd εε (28) For µ kT 1, this maximum of −g '(ε ) is very ∫0 where the function f ()ε is some simple continuous sharp and the function −g '(ε ) is negligibly small for function of ε such as ε 1/2 or ε 3/2 and can be all values of ε differing greatly from εµ= . defined as: Utilizing technique of partial or part-by-part integration 1 g()ε = (29) mainly (i.e. u dυ= u υυ − du over Eq. (28), may e()εµ− kT +1 ∫∫ We have seen already that at T = 0 this function be applied and transformed into an integral over g()ε is a step function and is defined as: −Fg()εε '()(i.e. the F()ε being defined as ε 1 εµ< 0 F(ε )= fd ( εε ') ' ) as it can be seen here, and g()ε = (30) ∫0 0 εµ> 0 because of the form of −g '(ε ) , only the values in the According to Mayer and Mayer [12], also for most neighborhood of εµ= contribute to the integral. µµ= The limits of integration are actually are set from metal at T = 0 , the value of 0 is extremely ε = 0 to ε = ∞ , but since −g '(ε ) is practically high, and µ kT will be of the order of magnitude of 4 5 zero for ε ≤ 0 , no greater error is introduced by 5 × 10 to 5 × 10 K. Now our task is to show the analysis of evaluating changing the limits of integration which can be the integral type as it is presented in Eq. (28) given the performed by developing the function F()ε as a conditions provided by Ref. [12], which is named Taylor’s series in powers of ()εµ− about the place under the assumption that µ kT 1 and their of maximum −g '(ε ) [12]. values can be obtained in form of power series for For the purpose of part-by-part integration, we are in small quantity of kT µ0 . need of the first and second derivatives of the function The result will show, a posteriori, that for electrons g()ε that is presented in Eq. (29) and they are
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 33 Controlled Thermonuclear Energy presented here as Eqs. (32) and (33): From this, it can be seen that the relative abruptness dg()ε e()εµ− kT of the descent of g()ε from almost unity to almost ε = = − g '( ) 2 (32) zero increases with the value of µ kT . dε kT e()εµ− kT +1 Providing all the above information and equations and that we derived, we can generate a set of plots for the Fermis distribution function as it was a derivative for dg2()ε e()εµ− kT g "(ε ) = = − 2 2 2 various temperatures as illustrated in Fig. 8. However, dε kT e()εµ− kT +1 ( ) note that in Fig. 8, µ should be read in place of µ0 2e()εµ− kT everywhere. + 2 εµ− 3 Note that the function g()ε and g '(ε ) are (kT) e()kT +1 plotted in Fig. 8 for various values of kT . ()εµ−−kT () εµkT ee−1 By partial integration of Eq. (28) of the integral I , = 2 ()εµ− kT 3 we can easily find that: (kT) e +1 ∞ I= f()()ε gd εε (33) ∫0 (38) ∞ The first derivative is always negative. The second =F ( ∞ )( g ∞− ) Fg ()ε '() εε d derivative is zero when the following condition is ∫0 satisfied as given in Eq. (34): where ε e()εµ− kT −=10 F(ε )= fd ( εε ') ' (39) ∫0 for Analyzing the results of these integral operations, εµ= (34) we can easily see that, if f ()ε is not infinity at ε = At εµ= the function −g '(ε ) has a maximum, 0 , then F(0) and the product Fg(0) (0) are , zero. If f ()ε does not go exponentially to which is sharper at the lower temperature. The negative infinity with ε , the product Fg()()∞∞ will be zero of the slope of the original function at this point is since g()ε approaches zero as e−ε kT with greatest. increasing ε Using εµ= in Eqs. (29) and (32), we can easily One may consequently write the result as Eq. (40): find that the value of the function g()ε at this point is ∞∞ I= f()()ε gd εε= − Fg () ε '() εε d (40) given by: ∫∫00 1 We now introduce a new variable that we can g()ε = (35) 2 transfer to as Eq. (41): and its derivative namely Eq. (32) is given by: εµ− x = (41) with kT 11 and develop the function Fx() as a power series of g '(µ ) = − (36) 4 kT x as demonstrated in Eq. (42) as: The Napierian logarithmic decrease in g()ε with ∞ xn ln(ε ) is given by Eq. (37) as: Fx() = ∑ F()n ( x= 0) (42) n=0 n! dgln (εµ ) 1 −= (37) where, in the old variable form, we establish Eq. (43): d ln(ε ) 2 kT εµ=
34 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy
Fig. 8 Fermi distribution function for gε() and g'ε() illustration.
µ x Fx(= 0) = f (εε ) d (43) 1 e ∫0 gx'( ) = − kT( ex − 1) 2 Thus, we have a new form as Eq. (44) here: (47) 11 = − − dFn ()ε kT( exx++ 1)( e 1) ()nn= = F( x 0) ( kT ) n dε εµ= The function gx'( ) in Eq. (48) is completely dFn−1 ()ε symmetrical in x , and can be written as = n ()kT n−1 (44) gx'( )= g '( − x ) . The function approaches zero dε εµ= exponentially as x approaches −∞ . If µ kT is nn(− 1) = (kT ) f ()µ larger, the value of the function is already negligible at the lower limit, x= −µ kT , of the integral in Eq. By substituting Eq. (44) into Eq. (40), we may be (45). No error is introduced and consequently by able to write Eq. (45) as: changing the limits of integration of the terms in the ∞ I= − F(0) gd '(εε ) sum of Eq. (45) to x = −∞ and x = +∞ . ∫0 We must now evaluate the integral of the form as Eq. ∞ ()kT n ∞ − f (nn− 1) (µ )x g '( x ) dx (48) here: ∑ ∫x=−µ kT n=1 n! +∞ ⌠ xn (45) xx− dx (48) (ee++ 1)( 1) The integral of the first term is given as Eq. (46) as: ⌡−∞
∞ From the symmetry of the denominator, it is seen −∫ gdg'(εε ) = (0) −∞ g ( ) that the integrand is anti-symmetrical in x if n is 0 (46) −µ odd, that is, it changes sign if x is replaced by −x =+≅ (1e kT ) 1 and the integral is therefore zero for odd n . For even The function gx'( ) is obtained by using the values of n , we may integrate from zero to infinity expression of Eq. (35) for x in Eq. (32) and write the and multiply by 2 and since then the integrand is new equation as Eq. (47) here: symmetrical in x .
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 35 Controlled Thermonuclear Energy
By developing the following relationship as: One finds that f ()εε= 1/2 in this problem. The 1 e−x integral F()µ is becoming as: = xx−− x2 µµ2 (ee++ 1)( 1) (1 + e ) fd()εε= ε1/2 d ε = µ3/2 (54) ∫∫00 −− − 3 =−+−eexxx2323 e (49) The derivatives are as Eq. (55): ∞ m− mx =−−( 1)me ( n even) ∑ df 1 −1/2 m=1 = µ Using Eqs. (46) and (47) with Eqs. (43) and (49), we dε εµ= 2 come to the conclusion as Eq. (50): and (55) 3 df 3 − = µ 5/2 ∞∞ I= f()()ε gd εε= − Fg () ε '() εε d dε εµ= 8 ∫∫00 ∞ ∞∞m − (− 1) Using Eqs. (54) and (55) with Eq. (52) in Eq. (53), =f ()εε d − 2 ( kT ) f (2n 1) (µ ) ∫0 ∑∑2n nm=11= m one finds Eq. (56) here as: 3/2 21n− 2424 (2n− 1) df()ε µπ π f µ = kT7 kT () 21n− 11+ + += dε µ µµ εµ= 0 8 640 (50) (56) The sums occurring have the numerical values as which determines µ as a function of µ0 and T . demonstrated in Eq. (51) here: ∞ m 2 µ (− 1) π In order to make the equation explicit in , we use −= ∑ 2 the development as Eq. (57) here: m=1 m 12 and (51) 1 25xx2 =−+ − (57) 2/3 1 ∞ (− 1)m 7π 4 (1+ x ) 3 9 −= ∑ 4 m=1 m 720 which is used to obtain the following for µ as Eq. so that (58): ∞ 2424 ⌠ f ()ε ππkT kT Id= εµ− ε µµ=−++0 1 (58) ()kT µµ ⌡0 e +1 12 720 ∞ 2 ⌠ π 2 df =f()εε d + ( kT ) (52) −−22 2 2 and here µµ=0016 + ( π)(kT µ) is ⌡0 6 dε εµ= 7π 43df substituted in the quadratic term. In the quartic term, ++ ()kT 3 which is the last correction, µ is simply substituted 360 dε 0 εµ= for µ . One obtains Eq. (59) for chemical potential µ Eq. (52) will now be applied to calculate µ . Using as: Eq. (26), we obtain Eq. (53) as: 24 ππ24kT kT 1/2 µµ=−−+1 ∞∞3Ndεε 0 12µµ 80 N= Nd()εε = 00 ∫∫0021µ 3/2e (εµ− ) kT + 0 (59) (53) as an equation for µ , it is the chemical potential as we
36 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy stated above, in terms of kT and µ0 . temperature T and volume V , can be obtained as: µ0 is the chemical potential at T = 0 which is given in turn as function of the volume V as it is 24 35ππ24kT kT indicated in Eq. (21). EN=+−+µ0 1 5 12µµ 16 The energy E may be calculated by using the 00 following relationship as Eq. (60) driven by Eq. (26): 24 33ππ24kT kT ∞ =+−+Nµ E= ε Nd() εε 0 5 4µµ 80 ∫0 00 3/2 (60) 3Nd∞ εε = (64) 3/2∫0 (e−µ )/ kT 21µ0 e + With help from Eq. (21), we can then write the following set of Eq. (65) as: In the above integral f ()εε= 3/2 and then we can 2 2/3 hN3 dµµ002 write: µ0 = = − (65) 83mπ V dV V ε 2 5/2 fd()εε= µ which would be sufficient to permit the calculation of ∫0 5 all other thermodynamic functions at entropy of df ()ε 3 1/2 = µ (61) S = 0 at the temperature T = 0 . [12] dε 2 The heat capacity at constant volume CV is found df3 ()ε 3 1 = − by direct differentiation of results of Eq. (64). Then we 3 3/2 dεµ8 can write: so that 2 ππ22kT3 kT 3/2 24C= Nk 1 −+ (66) µπ24 π V 3 57kT kT 2µµ00 10 EN=µ 1 +−+ 5 µ0 8 µµ 384 The entropy S may be obtained by integration of (62) T ()C T dT , thus we can write: is obtained. ∫0 V By using Eq. (59) to replace µ with µ , one can 2 0 T 22 ⌠ CV ππkT kT find out Eq. (63) for the energy E as: S== dT Nk 1 −+ ⌡0 T 2µµ 10 24 00 35ππ24kT kT EN=+−+µ 1 24 0 24 5 12µµ 16 Nµ0 ππkT kT 00 = −+ T 2µµ 20 (63) 00 (67) 7. The Thermodynamic Functions of a The work function of A= E − TS is from Eqs. (64) Degenerate Fermi-Dirac Gas and (67) in the following form of Eq. (68): The chemical potential µ as illustrated in Eq. (59) 24 3 ππ24kT kT and the energy E from Eq. (63) for the generate AN=−+−µ 0 5 4µµ 80 Fermi-Dirac gas have been derived as a power series of 00 24 the temperature T in terms of µ as well volume 24 0 35ππkT kT V as Eq. (21). =−+−Nµ0 1 5 12µµ00 48 Eq. (63), giving energy E as a function of
Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a 37 Controlled Thermonuclear Energy
(68) All the above equation for degenerate Fermi-Dirac gas was derived based on kT ε , which is an The pressure P is given by −∂( AV ∂ ) and F T indication of strongly degenerate gas case. with help from Eq. (65) being substituted in Eq. (68), It is to be noted that in all the equations for the one can get the result that is shown in Eq. (4), and that thermodynamic properties of the gas the is the result we are looking for, thus we may establish temperature-dependent part occurs as kT µ0 . Since the Eq. (69) as: 5 µ0 k is about 10 K for the molar volumes of an 24 -3 -2 Nµ 2 ππ24kT kT electron in metals, kT µ0 is about 10 K to 10 K P =+−+0 V 5 6µµ 40 for ordinary temperatures. The thermodynamic 00 functions of degenerate gas, at the concentrations 24 24 35N 2 ππkT kT considered, do not depend greatly on temperature T . =µ0 +−+ 5 3V 5 6µµ 40 In particular, the heat capacity is almost negligible 00 24compared to that due to the vibrations of the ions up to 24 23N ππkT 3 kT considerable temperature. =µ0 +− + 3V 5 4εε 80 FF In the case of kT ε F , which is considered to a 2 E non-degenerate gas, the considerations will be very = 3 V similar to the corresponding case discussed in Eliezer, et. al. [26] for a Bose-Einstein gas except that we (69) would have to use the following expansion as The equation PV= 23 E V is found at T = 0 illustrated in Eq. (73) [26]: Kelvin, and Eq. (70) is seen to be independent of ∞ 1 ⌠ xn−1 dx temperature T [12]. gy()≡ n Γ+ x ∂∂EE003µ 22N 0 ()n⌡0 ( 1 ye) 1 PN00=−=−=µ = (70) ∂∂V5 VV 53 V 1 ∞ −1 =∫ ye−x(1 + ye −− xn) x1 dx We can also derive the heat content with help of Γ 0 H= E + PV and it accordingly yields Eq. (71) as: 1 ∞ =ye−x1 −+ ye −− x y22 e xn + x −1 dx ∫0 5 Γ HE= yy23 3 =−+− y nn 24 23 5ππ24kT kT =+−+Nµ 1 (73) 0 12µµ 16 00 The final results for the various thermodynamic functions have already been given in the book Eliezer (71) et. al. [26], chapter (5.64) and (5.68); the lower signs = − Finally, forming F H TS would provide us correspond to Fermi-Dirac statistics and for an electron with Eq. (72) as: gas G=2. 24 ππ24kT kT With this, we come to the conclusion of the short FN=−−+µ 1 0 12µµ 80 review. 00 (72) 8. Conclusion which is, of course, can be seen to be as Nµ by In conclusion, we are hoping that we have managed comparison with Eq. (59). to bring the functions of a degenerate Fermi-Dirac gas
38 Thermal Physics and Statistical Mechanics Driven Inertial Confinement Fusion (ICF) Inducing a Controlled Thermonuclear Energy together as a short review via statistical mechanics January 22, 2021. https://en.wikipedia.org/wiki/Degenerate_energy_levels. approach and readers are no longer need to search for [15] Dyson, F. J., and Lenard, A. 1967. “Stability of Matter I.” them and their related derivation by looking all over the J. Math. Phys. 8 (3): 423-34. textbooks as well as internet these days, such as Bibcode:1967JMP.....8..423D. doi:10.1063/1.1705209. Wikipedia.org or any other resource. [16] Lenard, A., and Dyson, F. J. 1968. “Stability of Matter II.” J. Math. Phys. 9 (5): 698-711. References Bibcode:1968JMP.....9..698L. doi:10.1063/1.1664631. [17] Dyson, F. J. 1967. “Ground‐State Energy of a Finite [1] Zohuri, B. 2017. Inertial Confinement Fusion Driven System of Charged Particles.” J. Math. Phys. 8 (8): Thermonuclear Energy. Springer International Publishing. 1538-45. Bibcode:1967JMP.....8.1538D. [2] Nuckolls, J., Wood, L., Thiessen, A., and Zimmerman, G. doi:10.1063/1.1705389. 1972. “Laser Compression of Matter to Super-high [18] Wikipedia. “White Dwarf.” Accessed January 22, 2021. Densities: Thermonuclear (CTR) Applications.” Nature https://en.wikipedia.org/wiki/White_dwarf. 239 (5368): 139-42. [19] Griffiths, D. J. 2005. Introduction to Quantum Mechanics. [3] Guderley K G. 1942. “Starke kugelige und zylindrische Second Edition. London, UK: Prentice Hall. verdichtungsstosse in der nahe des kugelmitterpunktes [20] Mazzali, P. A., Röpke, F. K., Benetti, S., and Hillebrandt, bnw. der zylinderachse.” Luftfahrtforschung 19: 302. W. 2007. “A Common Explosion Mechanism for Type Ia [4] Behgounia, F., Nezam, Z. Z., and Zohuri, B. 2020. Supernovae.” Science 315 (5813): 825-8. “Dimensional Analysis and Similarity Method Driving arXiv:astro-ph/0702351. Bibcode:2007Sci...315..825M. Self-similar Solutions of the First and Second Kind doi:10.1126/science.1136259. PMID1728999. Inducing Energy.” Journal of Energy and Power [21] Timmes, F. X., Woosley, S. E., and Weaver, T. A. 1996. Engineering 14: 233-50. “The Neutron Star and Black Hole Initial Mass Function.” doi:10.17265/1934-8975/2020.07.003 Astrophysical Journal 457: 834-43. [5] Zohuri, B. 2015. Dimensional Analysis and Self-similarity arXiv:astro-ph/9510136. Bibcode:1996ApJ...457..834T. Methods for Engineers and Scientists. Springer Publishing doi:10.1086/176778. Company. [22] Lieb, E. H., and Yau, H. T. 1987. “A Rigorous [6] Zohuri, B. 2017. Dimensional Analysis Beyond the Pi Examination of the Chandrasekhar Theory of Stellar Theorem. Springer International Publishing. Collapse.” Astrophysical Journal 323: 140-4. [7] Lawson, J. D. 1957. “Some Criteria for a Power Bibcode:1987ApJ...323..140L. doi:10.1086/165813. Producing Thermonuclear Reactor.” Proc. Phys. Soc. [23] Chandrasekhar, S. 1931. “The Density of White Dwarf (Lond.) B70 (1): 6. Stars.” The London, Edinburgh, and Dublin Philosophical [8] Gough, W. C., and Eastlund, B. J. 1971. “The Prospects Magazine and Journal of Science 11 (70): 592-6. of Fusion Power.” Scientific American 224 (2): 50-67. doi:10.1080/14786443109461710. [9] Schwartzschild, M. 1965. Structure and Evolution of the [24] Chandrasekhar, S. 1935. “The Highly Collapsed Stars. New York: Dover, p206. Configurations of a Stellar Mass (Second Paper).” [10] Chandrasekhar, S. 1938. Stellar Structure. Chicago: Monthly Notices of the Royal Astronomical Society 95 (3): University of Chicago Press, p427. 207-25. Bibcode:1935MNRAS..95..207C. [11] Landau, L., and Lifshitz, E. 1969. Statistical Physics, ch. 5. doi:10.1093/mnras/95.3.207. Addison/Wesley, Reading. [25] Ochsenbein, F., and Centre de Donn´ees astronomiques de [12] Mayer, J., and Mayer, M. 1940. Statistical Mechanics. Strasbourg (CDS). 2000. “Standards for Astronomical Wiley, p385. Catalogues, Version 2.0, Section 3.2.2.” Accessed [13] Wikipedia. “Electron Degeneracy Pressure.” Accessed January 12, 2020. January 22, 2021. http://vizier.u-strasbg.fr/vizier/doc/catstd.pdf. https://en.wikipedia.org/wiki/Electron_degeneracy_press [26] Eliezer, S., Ghatak, A., and Hora. H. 1986. Fundamentals ure. of Equations of State. New Jersey: World Scientific [14] Wikipedia. “Degenerate Energy Levels.” Accessed Publishing Company.