Slowing-down of Charged Particles in a Plasma

Size: px
Start display at page:

Download "Slowing-down of Charged Particles in a Plasma"

Transcription

1 P/2532 France Slowing-down of Charged Particles in a Plasma By G. Boulègue, P. Chanson, R. Combe, M. Félix and P. Strasman We shall investigate the case in which the slowingdown of an incident particle Р г can be explained by binary collisions with the constituents of the plasma, collective oscillations entering only in a weak or negligible manner. 1 EPRESSION FOR THE SLOWING-DOWN (NON-RELATIVISTIC CALCULATION) The particle Р г (mass m v velocity v v kinetic energy E 1} charge Z x e) moves in an ensemble of elements P 2 (mass, charge Z 2 e) undergoing Maxwellian agitation (temperature T 2 ). At a given moment the velocity of one of these particles P 2 is F 2. We designate by u the velocity of Р г relative to P 2 : u = Vj - V 2 When a collision occurs the trajectory of Р г is deviated by an angle a in the coordinate system of P 2. One supposes that the function oc(u, a) is known, a being the " collision parameter " (distance of P 2 to the initial relative velocity vector). After the collision the relative velocity becomes u/, with Let us call dn 2 ) the density of particles P 2 whose speed is between F 2 and F 2 + dv 2 in absolute value and whose velocity vector is in the small element of solid angle ( 1, SI + dsi). At the zero instant all the elements P 2 having velocity V 2 which should interact with P x during the unit time following, with collision parameter a and azimuth ф (with respect to any origin, e.g., the plane Vx, u) are contained in a volume иааааф. Their average number is then The momentum conservation in the collision (l)-(2) allows one to write, after a simple calculation: = F-, 2 F/ 2 = - ( F-, 2 Fo 2 -\ ^ u 2 1 (1 cos OL) sin в sin a cos, being the angle between the vectors "V^ and u. ш ф is obtained by multiplying by dn 2 ) Letting I (1 cos oc)ada = a{u), jo R is the interaction radius of the particles, a is a function of u, obtained from the laws of scattering. Evaluating the integrals over ф and a we obtain: v. = dn 2 ) - uo(u) by collisions with particles (2) having velocities V 2. Let us express dn 2 ) as follows: n 2 ) - «^(F 2 ) represents the Maxwellian distribution. Here n 2 is the total number of particles P 2 per unit volume, в being the angle (V^, V 2 ). One obtains after integration over the azimuth : n = 71-2 т л 4- n. ( ( l J JO<e<n This represents, but for a numerical coefficient, the mean loss of energy of a particle (1) per unit of time The mean loss of energy per unit time is Original language: French. * Laboratoire Central de l'armement. 242

2 SLOWING-DOWN OF CHARGED PARTICLES 243 and per unit length : It is convenient to use as variables и and F 2. finds: One Taking into account ^(F 2 ) = ^4 2 F 2 2 exp ( y 2 2 V 2 2 ) A 2 = (2/л)ЦкТ 2 /т 2 )-''' and y 2 2 = /2kT 2, one arriyes dz - " nn 2 f " *(u)i(u)du, Uo The calculation of dejdz is therefore reduced to the evaluation of a single integral, the function a(u) being assumed known. I(u) can be transformed in various ways of which the most useful for us is the following: \du Г si I _ 2y 2 2 -± sinh (2y 2 W^ u 2 a(u)i(u)du = exp ( y 2 2 u 2 ) sinh 2y 2 exp (- 1 exp ( - sinh (2y 2 2 F 1^) -^ 1 ^(T(w) exp ( Vfv-x COULOMB INTERACTION LIMITED BY THE DEBYE LENGTH (the subscript e refers to the electrons), one deduces, setting Let m be the reduced mass defined by one knows that Шл Wo = ^ i + that x 9 9 = У2 dz from which, exp l Е -т«^ w x kt G = I exp ( m^x^m^) sinh /3: Letting D be the Debye length (1) d Чп (1 +«т 1

3 244 SESSION A-5 P/2532 G. BOULÈGUE et al. and 2kT 2 D If E 1 /kt 2 is large, a good approximation is obtained by calculating the term in d/dx for a mean value : nu»/* kt 2 m x TIME OF EQUIPARTITION We now assume that an ensemble Р г is slowed down by an ensemble P 2. The particles (1) have a Maxwellian distribution of velocities at temperature T v Per cm 3 and per second, the particles (1) lose energy due to interaction with particles (2) that is given by: Ф being the error function Ф{х) = 2; - x 2 )dx. From this G^VTrexp -_ V m-, Ф A x is the coefficient of the Maxwellian distribution for the particles (1). dejdz is expressed by an integral over u, W lt 2 by a double integral. These calculations give 7\ T 2 (m, and u 5 a(u) exp du. te E-L If we assume a Coulomb interaction limited to the Debye length, it is necessary to calculate This is practically the formula arrived at by the elementary considerations. On the other hand, for arbitrary values of E x \kt 2y the evaluation of G may be rather long; a series expansion of sinh fix is convenient. Some results for a particular case will be given later. It is interesting to investigate the case of slowingdown of a heavy particle by electrons (fn 2 f small), when EJkT 2 is small. One then finds (putting y = x 2 ) : + x 2 y 2 )dy 2k exp ^-+^ du. This integral is derived from that in the previous equation by replacing a(u) by its value given in (1). Letting and g = 2^( ^ 4 \ /i/i/i /мл т л J(x) =2 cos х~ г Í xr 1 cos xdx Jy sin xr 1 I ^ л;" 1 sin x dx one has w b2 = In and finally = 2[cos ÍHH- Ci^- 1 + sin ^-1 Sbr- 1 ], (2) and further: ^ / m \i/2 dz ~ 2 x^/ \3 /гт 2 the function /(g) having been defined in Eq. (2). When g is large (> 10) one has One notes that EJife is zero for E x = Ъ Г 2 \2 } С = In у is the Euler constant: С = which is completely reasonable, " С rapidly becomes negligible in comparison to In g.

4 SLOWING-DOWN OF CHARGED PARTICLES 245 Figure 1. Loss of energy of a tritium nucleus in a plasma of 6 Li-D at a temperature of 5 kev, as a fraction of triton energy Figure 2. Loss of energy of a tritium nucleus in a plasma of 6 Li-D at a temperature of 2 kev, as a function of triton energy (feev)

5 246 SESSION A-5 P/2532 G. BOULÈGUE et ai. Let us assume, in particular, that the temperatures T lf T 2 and T e are of the same order of magnitude T. From the equations Txr 3,dT 1 3 jdt 2 W = n k ^ = n k one easily finds dt-, 3 kn^f and from an analogous equation for dtjdt one has t eq is a time for equipartition : Numerically, 3 kn 1 n К dt As-IZJ 2& LJ-]j f bt ht \ /Í1 j /vi 2 \ ' 7~ + ) J{g) Ш г Ш 2 J{g) ' Let us compare these results with those of Spitzer : 2 4леМ= 2ЬЛШ ' For large values of g and Л, we have /(g) ^ 2 In g = 2 In In Л. Our value for the equipartition time approaches that of Spitzer if g is large, but it diners considerably for average or small values of g. We feel, therefore, that our formula is more general than that of Spitzer, which it approaches for large values of Л. APPLICATIONS We shall apply the formulae of a previous section to the problems of the slowing-down of tritium nuclei by a plasma composed of 6 Li nuclei, deuterons and electrons at various temperatures, the densities being n u = xlo 23 /cm 3 Njy = n e = Slowing-down by the Electrons It is especially for this case that our formulae are applicable since the collisions are very frequent and each has only a small effect ; the slowing-down is thus quasi-continuous and our expressions can give a good approximation. When the energy Е г of the triton approaches the thermal energy, the slowing-down factor loses much of its importance and approaches zero for E x = ( ) kt. Slowing-down by the Ions It is precisely in this zone that the slowing-down by the ions becomes predominant. It is certain that in this case our formulae no longer have the same validity, due to the marked discontinuity of the slowing down. Nevertheless, it seems reasonable that they can represent the average slowing-down. Figures 1 and 2 show the results of calculations for kt 5 kev and kt = 2 kev. Probability of the Reaction D (t, n) 4 He The probability that such a reaction takes place during the slowing-down of the triton, calculated for the energy range 1 Mev to AkT, is expressed by : (- dejdz) J а(е г ) is the reaction cross section. /гг(кеу) p x x x x 10-2 One finds: Without being completely negligible, this probability stays small. We conclude that the majority of the tritons will be thermalized before producing a thermonuclear reaction. REFERENCES 1. D. Pines and D. Bohm, Phys. Rev., 85, 338 (1952). 2. L. Spitzer, Physics of Fully Ionized Gases, p. 72, Eqs. (5-14), p. 80, Eqs. (5-31), Interscience Publishers, Inc., New York (1956).

Fusion Chain Reaction

Fusion Chain Reaction P/1941 Poland Fusion Chain Reaction By M. Gryziñski CHAIN REACTION WITH CHARGED PARTICLES With the discovery of the fission chain reaction with neutrons, the possibility of obtaining a chain reaction with

More information

L 1 ~ ехр гы. * = Am) Thermodynamics of Deuterium-Tritium Mixtures. By G. Boulegue, P. Chanson, R. Combe, M, Feix and P.

L 1 ~ ехр гы. * = Am) Thermodynamics of Deuterium-Tritium Mixtures. By G. Boulegue, P. Chanson, R. Combe, M, Feix and P. P/2495 France Thermodynamics of Deuterium-Tritium Mixtures By G. Boulegue, P. Chanson, R. Combe, M, Feix and P. Strasman ;î We propose to study deuterium-tritium mixtures which, when heated to an elevated

More information

Individual Particle Motion and the Effect of Scattering in an Axially Symmetric Magnetic Field

Individual Particle Motion and the Effect of Scattering in an Axially Symmetric Magnetic Field P/383 USA Individual Particle Motion and the Effect of Scattering in an Axially Symmetric Magnetic Field By A. Garren,* R. J. Riddell,* L. Smith,* G. Bing,t L. R. Henrich,* T. G. Northrop f and J. E. Roberts

More information

Interaction of Particles and Matter

Interaction of Particles and Matter MORE CHAPTER 11, #7 Interaction of Particles and Matter In this More section we will discuss briefly the main interactions of charged particles, neutrons, and photons with matter. Understanding these interactions

More information

Diffusion of Arc Plasmas across a Magnetic Field

Diffusion of Arc Plasmas across a Magnetic Field P366 USA Diffusion of Arc Plasmas across a Magnetic Field By Albert Simon* The effect of a magnetic field В is to reduce the coefficients of diffusion, ) p, across the magnetic field to the values equations

More information

Chapter V: Interactions of neutrons with matter

Chapter V: Interactions of neutrons with matter Chapter V: Interactions of neutrons with matter 1 Content of the chapter Introduction Interaction processes Interaction cross sections Moderation and neutrons path For more details see «Physique des Réacteurs

More information

Slowing down the neutrons

Slowing down the neutrons Slowing down the neutrons Clearly, an obvious way to make a reactor work, and to make use of this characteristic of the 3 U(n,f) cross-section, is to slow down the fast, fission neutrons. This can be accomplished,

More information

Outline. Chapter 6 The Basic Interactions between Photons and Charged Particles with Matter. Photon interactions. Photoelectric effect

Outline. Chapter 6 The Basic Interactions between Photons and Charged Particles with Matter. Photon interactions. Photoelectric effect Chapter 6 The Basic Interactions between Photons and Charged Particles with Matter Radiation Dosimetry I Text: H.E Johns and J.R. Cunningham, The physics of radiology, 4 th ed. http://www.utoledo.edu/med/depts/radther

More information

By T. Hesselberg Jensen, O. Kofoed-Hansen and С F. Wandel*

By T. Hesselberg Jensen, O. Kofoed-Hansen and С F. Wandel* P/2506 Denmark Energy Balance in a Thermonuclear Reacting Plasma containing Deuterium, Tritium and Reaction Products under Isothermal Pulsed or Steady-State Conditions By T. Hesselberg Jensen, O. Kofoed-Hansen

More information

The Equipartition Theorem

The Equipartition Theorem Chapter 8 The Equipartition Theorem Topics Equipartition and kinetic energy. The one-dimensional harmonic oscillator. Degrees of freedom and the equipartition theorem. Rotating particles in thermal equilibrium.

More information

Lecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization

Lecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization 22.101 Applied Nuclear Physics (Fall 2006) Lecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization References: R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York,

More information

Waves in plasma. Denis Gialis

Waves in plasma. Denis Gialis Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous.

More information

Stability and Heating in the Pinch Effect

Stability and Heating in the Pinch Effect P/347 USA Stability and Heating in the Pinch Effect By M. N. Rosenbluth One of the most promising types of thermonuclear device is the stabilized pinch. 1 ' This consists of a pinched cylindrical plasma

More information

Electrodynamics of Radiation Processes

Electrodynamics of Radiation Processes Electrodynamics of Radiation Processes 7. Emission from relativistic particles (contd) & Bremsstrahlung http://www.astro.rug.nl/~etolstoy/radproc/ Chapter 4: Rybicki&Lightman Sections 4.8, 4.9 Chapter

More information

A Method of Knock-on Tail Observation Accounting Temperature Fluctuation Using 6 Li+T/D+T Reaction in Deuterium Plasma

A Method of Knock-on Tail Observation Accounting Temperature Fluctuation Using 6 Li+T/D+T Reaction in Deuterium Plasma A Method of Knock-on Tail Observation Accounting Temperature Fluctuation Using 6 Li+T/D+T Reaction in Deuterium Plasma Yasuko KAWAMOTO and Hideaki MATSUURA Department of Applied Quantum Physics and Nuclear

More information

SECTION C: NUCLEAR RADIATION AND NUCLEAR ENERGY LOSS PROCESSES. " N & = '!t and so N = N 0. implying ln! N $

SECTION C: NUCLEAR RADIATION AND NUCLEAR ENERGY LOSS PROCESSES.  N & = '!t and so N = N 0. implying ln! N $ SECTO C: UCLEAR RADATO AD UCLEAR EERGY LOSS PROCESSES n this section we discuss decay and transmutation processes in nuclei (including α, β, and γ decay, as well as fission and fusion processes), using

More information

V.K. Gryaznov and I.L. Iosilevskiy Moscow Institute of Physics and Technology

V.K. Gryaznov and I.L. Iosilevskiy Moscow Institute of Physics and Technology Construction of Effective Interpolating Equation of State for One-and Two-Component Classical Plasma V.K. Gryaznov and I.L. Iosilevskiy Moscow Institute of Physics and Technology In equilibrium plasma

More information

Heating of a Confined Plasma by Oscillating Electromagnetic Fields

Heating of a Confined Plasma by Oscillating Electromagnetic Fields P/357 USA Heating of a Confined Plasma by Oscillating Electromagnetic Fields By J, M. Berger,* W. A. Newcomb, f R. M. Kulsrud * and A. Lenard J. M. Dawson, * E. A. Fneman,* We wish to consider the heating

More information

Storm Open Library 3.0

Storm Open Library 3.0 S 50% off! 3 O L Storm Open Library 3.0 Amor Sans, Amor Serif, Andulka, Baskerville, John Sans, Metron, Ozdoby,, Regent, Sebastian, Serapion, Splendid Quartett, Vida & Walbaum. d 50% f summer j sale n

More information

Chapter 18 Thermal Properties of Matter

Chapter 18 Thermal Properties of Matter Chapter 18 Thermal Properties of Matter In this section we define the thermodynamic state variables and their relationship to each other, called the equation of state. The system of interest (most of the

More information

12. MHD Approximation.

12. MHD Approximation. Phys780: Plasma Physics Lecture 12. MHD approximation. 1 12. MHD Approximation. ([3], p. 169-183) The kinetic equation for the distribution function f( v, r, t) provides the most complete and universal

More information

Plasmas as fluids. S.M.Lea. January 2007

Plasmas as fluids. S.M.Lea. January 2007 Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider

More information

Space Charge in Linear Machines

Space Charge in Linear Machines Space Charge in Linear Machines Massimo.Ferrario@LNF.INFN.IT Egham September 6 th 017 Relativistic equation of motion dp dt = F p = γm o v γm o dv dt + m ov dγ dt = F β = v c dγ dt = d dt " a v % m o γ

More information

Exam 2: Equation Summary

Exam 2: Equation Summary MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Physics Fall Term 2012 Exam 2: Equation Summary Newton s Second Law: Force, Mass, Acceleration: Newton s Third Law: Center of Mass: Velocity

More information

PHYS 352. Charged Particle Interactions with Matter. Intro: Cross Section. dn s. = F dω

PHYS 352. Charged Particle Interactions with Matter. Intro: Cross Section. dn s. = F dω PHYS 352 Charged Particle Interactions with Matter Intro: Cross Section cross section σ describes the probability for an interaction as an area flux F number of particles per unit area per unit time dσ

More information

Generation and Thermalization of Plasma Waves

Generation and Thermalization of Plasma Waves P/361 USA Generation and Thermalization of Plasma Waves By T. H. Stix A fully ionized gas immersed in an axial magnetic field may exhibit low-frequency oscillations the electric field and macroscopic velocity

More information

PHYS 5012 Radiation Physics and Dosimetry

PHYS 5012 Radiation Physics and Dosimetry Radiative PHYS 5012 Radiation Physics and Dosimetry Mean Tuesday 24 March 2009 Radiative Mean Radiative Mean Collisions between two particles involve a projectile and a target. Types of targets: whole

More information

About One way of Encoding Alphanumeric and Symbolic Information

About One way of Encoding Alphanumeric and Symbolic Information Int. J. Open Problems Compt. Math., Vol. 3, No. 4, December 2010 ISSN 1998-6262; Copyright ICSRS Publication, 2010 www.i-csrs.org About One way of Encoding Alphanumeric and Symbolic Information Mohammed

More information

Space Charge Mi-ga-on

Space Charge Mi-ga-on Space Charge Mi-ga-on Massimo.Ferrario@LNF.INFN.IT Hamburg June nd 016 OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons

More information

Nuclear Fusion and Radiation

Nuclear Fusion and Radiation Nuclear Fusion and Radiation Lecture 3 (Meetings 5 & 6) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Nuclear Fusion and Radiation p. 1/39 Fusion Reactions

More information

Solid State Physics FREE ELECTRON MODEL. Lecture 17. A.H. Harker. Physics and Astronomy UCL

Solid State Physics FREE ELECTRON MODEL. Lecture 17. A.H. Harker. Physics and Astronomy UCL Solid State Physics FREE ELECTRON MODEL Lecture 17 A.H. Harker Physics and Astronomy UCL Magnetic Effects 6.7 Plasma Oscillations The picture of a free electron gas and a positive charge background offers

More information

Chapter II: Interactions of ions with matter

Chapter II: Interactions of ions with matter Chapter II: Interactions of ions with matter 1 Trajectories of α particles of 5.5 MeV Source: SRIM www.srim.org 2 Incident proton on Al: Bohr model v=v 0 E p =0.025 MeV relativistic effect E p =938 MeV

More information

Single Particle Motion

Single Particle Motion Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic

More information

Interactions of particles and radiation with matter

Interactions of particles and radiation with matter 1 Interactions of particles and radiation with matter When the intervals, passages, connections, weights, impulses, collisions, movement, order, and position of the atoms interchange, so also must the

More information

APPENDIX Z. USEFUL FORMULAS 1. Appendix Z. Useful Formulas. DRAFT 13:41 June 30, 2006 c J.D Callen, Fundamentals of Plasma Physics

APPENDIX Z. USEFUL FORMULAS 1. Appendix Z. Useful Formulas. DRAFT 13:41 June 30, 2006 c J.D Callen, Fundamentals of Plasma Physics APPENDIX Z. USEFUL FORMULAS 1 Appendix Z Useful Formulas APPENDIX Z. USEFUL FORMULAS 2 Key Vector Relations A B = B A, A B = B A, A A = 0, A B C) = A B) C A B C) = B A C) C A B), bac-cab rule A B) C D)

More information

Scattering of an α Particle by a Radioactive Nucleus

Scattering of an α Particle by a Radioactive Nucleus EJTP 3, No. 1 (6) 93 33 Electronic Journal of Theoretical Physics Scattering of an α Particle by a Radioactive Nucleus E. Majorana Written 198 published 6 Abstract: In the following we reproduce, translated

More information

Energy Dependence of Neutron Flux

Energy Dependence of Neutron Flux Energy Dependence of Neutron Flux B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2015 Sept.-Dec. 2015 September 1 Contents We start the discussion of the energy

More information

The Fluxes and the Equations of Change

The Fluxes and the Equations of Change Appendix 15 The Fluxes nd the Equtions of Chnge B.l B. B.3 B.4 B.5 B.6 B.7 B.8 B.9 Newton's lw of viscosity Fourier's lw of het conduction Fick's (first) lw of binry diffusion The eqution of continuity

More information

Radiation Physics PHYS /251. Prof. Gocha Khelashvili

Radiation Physics PHYS /251. Prof. Gocha Khelashvili Radiation Physics PHYS 571-051/251 Prof. Gocha Khelashvili Interaction of Radiation with Matter: Heavy Charged Particles Directly and Indirectly Ionizing Radiation Classification of Indirectly Ionizing

More information

1 v. L18.pdf Spring 2010, P627, YK February 22, 2012

1 v. L18.pdf Spring 2010, P627, YK February 22, 2012 L18.pdf Spring 2010, P627, YK February 22, 2012 18 T2 Nuclear Information Service at LANL: http://t2.lanl.gov/data/ ENDF/B VI Neutron Data : http://t2.lanl.gov/cgi bin/nuclides/endind Thermal neutron x

More information

Department of Chemical Engineering, Slovak Technical Bratislava. Received 8 October 1974

Department of Chemical Engineering, Slovak Technical Bratislava. Received 8 October 1974 Calculation of the activity coefficients and vapour composition from equations of the Tao method modified for inconstant integration step Ax. П. Ternary systems J. DOJČANSKÝ and J. SUROVÝ Department of

More information

Passage of particles through matter

Passage of particles through matter Passage of particles through matter Alexander Khanov PHYS6260: Experimental Methods is HEP Oklahoma State University September 11, 2017 Delta rays During ionization, the energy is transferred to electrons

More information

Hydromagnetic Shock Waves in High-temperature Plasmas

Hydromagnetic Shock Waves in High-temperature Plasmas P/374 USA Hydromagnetic Shock Waves in High-temperature Plasmas By C. S. Gardner, H. Goertzel, H. Grad, С S. Morawetz, M. H. Rose and H. Rubin INTRODUCTION Shock waves are one of the most important tools

More information

Lecture 22 Highlights Phys 402

Lecture 22 Highlights Phys 402 Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a

More information

University of Illinois at Chicago Department of Physics SOLUTIONS. Thermodynamics and Statistical Mechanics Qualifying Examination

University of Illinois at Chicago Department of Physics SOLUTIONS. Thermodynamics and Statistical Mechanics Qualifying Examination University of Illinois at Chicago Department of Physics SOLUTIONS Thermodynamics and Statistical Mechanics Qualifying Eamination January 7, 2 9: AM to 2: Noon Full credit can be achieved from completely

More information

Hydromagnetic Instability in a Stellarator

Hydromagnetic Instability in a Stellarator P/364 USA Hydromagnetic Instability in a Stellarator By M. D. Kruskal,* J. L. Johnson,! M. B. Gottlieb* and L. M. Goldman;; Kruskal and Tuck 1 (in a paper hereafter referred to as KT) have examined the

More information

504 BOOK REVIEWS [July

504 BOOK REVIEWS [July 504 BOOK REVIEWS [July The book leaves much to be done but this fact only enhances its interest. It should be productive of many extensions along the lines of economic interpretation as well as of mathematical

More information

1 Stellar Energy Generation Physics background

1 Stellar Energy Generation Physics background 1 Stellar Energy Generation Physics background 1.1 Relevant relativity synopsis We start with a review of some basic relations from special relativity. The mechanical energy E of a particle of rest mass

More information

Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation 19. Deriving the Vlasov Equation From the Klimontovich Equation

Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation 19. Deriving the Vlasov Equation From the Klimontovich Equation Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation 19 Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation Topics or concepts to learn in Chapter 2: 1. The microscopic

More information

Physical models for plasmas II

Physical models for plasmas II Physical models for plasmas II Dr. L. Conde Dr. José M. Donoso Departamento de Física Aplicada. E.T.S. Ingenieros Aeronáuticos Universidad Politécnica de Madrid Physical models,... Plasma Kinetic Theory

More information

APEX CARE INSTITUTE FOR PG - TRB, SLET AND NET IN PHYSICS

APEX CARE INSTITUTE FOR PG - TRB, SLET AND NET IN PHYSICS Page 1 1. Within the nucleus, the charge distribution A) Is constant, but falls to zero sharply at the nuclear radius B) Increases linearly from the centre, but falls off exponentially at the surface C)

More information

Nuclear Physics and Astrophysics

Nuclear Physics and Astrophysics Nuclear Physics and Astrophysics PHY-30 Dr. E. Rizvi Lecture 4 - Detectors Binding Energy Nuclear mass MN less than sum of nucleon masses Shows nucleus is a bound (lower energy) state for this configuration

More information

PROBLEM SET. Heliophysics Summer School. July, 2013

PROBLEM SET. Heliophysics Summer School. July, 2013 PROBLEM SET Heliophysics Summer School July, 2013 Problem Set for Shocks and Particle Acceleration There is probably only time to attempt one or two of these questions. In the tutorial session discussion

More information

UNIVERSAL HYBRID QUANTUM PROCESSORS

UNIVERSAL HYBRID QUANTUM PROCESSORS XJ0300183 Письма в ЭЧАЯ. 2003. 1[116] Particles and Nuclei, Letters. 2003. No. 1[116] UNIVERSAL HYBRID QUANTUM PROCESSORS A. Yu. Vlasov 1 FRC7IRH, St. Petersburg, Russia A quantum processor (the programmable

More information

Chapter NP-4. Nuclear Physics. Particle Behavior/ Gamma Interactions TABLE OF CONTENTS INTRODUCTION OBJECTIVES 1.0 IONIZATION

Chapter NP-4. Nuclear Physics. Particle Behavior/ Gamma Interactions TABLE OF CONTENTS INTRODUCTION OBJECTIVES 1.0 IONIZATION Chapter NP-4 Nuclear Physics Particle Behavior/ Gamma Interactions TABLE OF CONTENTS INTRODUCTION OBJECTIVES 1.0 IONIZATION 2.0 ALPHA PARTICLE INTERACTIONS 3.0 BETA INTERACTIONS 4.0 GAMMA INTERACTIONS

More information

Exercise 1 Atomic line spectra 1/9

Exercise 1 Atomic line spectra 1/9 Exercise 1 Atomic line spectra 1/9 The energy-level scheme for the hypothetical one-electron element Juliettium is shown in the figure on the left. The potential energy is taken to be zero for an electron

More information

The Scattering of Electromagnetic Waves. By Noboru HOKKYO Department of Physics, Osaka City University (Read. May. 15, 1956; Received June 23, 1956)

The Scattering of Electromagnetic Waves. By Noboru HOKKYO Department of Physics, Osaka City University (Read. May. 15, 1956; Received June 23, 1956) The Scattering of Electromagnetic Waves by Plasma Oscillations By Noboru HOKKYO Department of Physics, Osaka City University (Read. May. 15, 1956; Received June 23, 1956) Abstract Theory of plasma oscillation

More information

Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma

Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma Z. Bonaventura, D. Trunec Department of Physical Electronics Faculty of Science Masaryk University Kotlářská 2, Brno, CZ-61137,

More information

Chapter 3. Coulomb collisions

Chapter 3. Coulomb collisions Chapter 3 Coulomb collisions Coulomb collisions are long-range scattering events between charged particles due to the mutual exchange of the Coulomb force. Where do they occur, and why they are of interest?

More information

PHYS 3313 Section 001 Lecture # 22

PHYS 3313 Section 001 Lecture # 22 PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for

More information

Neutrino and Dark Matter Detections via Atomic Ionizations at sub-kev Sensitivities

Neutrino and Dark Matter Detections via Atomic Ionizations at sub-kev Sensitivities Neutrino and Dark Matter Detections via Atomic Ionizations at sub-kev Sensitivities Chih-Pan Wu Dept. of Physics, National Taiwan University Collaborators: Jiunn-Wei Chen, Chih-Liang Wu (NTU) Chen-Pang

More information

PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT. Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford

PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT. Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford 1 PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford 2 Overview Why do we need plasmas? For fusion, among other things

More information

Sawtooth mixing of alphas, knock on D, T ions and its influence on NPA spectra in ITER plasma

Sawtooth mixing of alphas, knock on D, T ions and its influence on NPA spectra in ITER plasma Sawtooth mixing of alphas, knock on D, T ions and its influence on NPA spectra in ITER plasma F.S. Zaitsev 1, 4, N.N. Gorelenkov 2, M.P. Petrov 3, V.I. Afanasyev 3, M.I. Mironov 3 1 Scientific Research

More information

Nuclear Reactions. Fission Fusion

Nuclear Reactions. Fission Fusion Nuclear Reactions Fission Fusion Nuclear Reactions and the Transmutation of Elements A nuclear reaction takes place when a nucleus is struck by another nucleus or particle. Compare with chemical reactions!

More information

3 /,,,:;. c E THE LEVEL DENSITY OF NUCLEI IN THE REGION 230 ~ A < 254. A.L.Komov, L.A.Malov, V.G.Soloviev, V.V.Voronov.

3 /,,,:;. c E THE LEVEL DENSITY OF NUCLEI IN THE REGION 230 ~ A < 254. A.L.Komov, L.A.Malov, V.G.Soloviev, V.V.Voronov. ~ - t-1 'I I 3 /,,,:;. c E4-9236 A.L.Komov, L.A.Malov, V.G.Soloviev, V.V.Voronov THE LEVEL DENSITY OF NUCLEI IN THE REGION 230 ~ A < 254 1975 E4-9236 AX.Komov, L.A.Malov, V.G.SoIoviev, V.V.Voronov THE

More information

Physics 100 PIXE F06

Physics 100 PIXE F06 Introduction: Ion Target Interaction Elastic Atomic Collisions Very low energies, typically below a few kev Surface composition and structure Ion Scattering spectrometry (ISS) Inelastic Atomic Collisions

More information

Basic equations of motion in fluid mechanics

Basic equations of motion in fluid mechanics 1 Annex 1 Basic equations of motion in fluid mechanics 1.1 Introduction It is assumed that the reader of this book is familiar with the basic laws of fluid mechanics. Nevertheless some of these laws will

More information

CHAPTER 12 The Atomic Nucleus

CHAPTER 12 The Atomic Nucleus CHAPTER 12 The Atomic Nucleus 12.1 Discovery of the Neutron 12.2 Nuclear Properties 12.3 The Deuteron 12.4 Nuclear Forces 12.5 Nuclear Stability 12.6 Radioactive Decay 12.7 Alpha, Beta, and Gamma Decay

More information

One dimensional hybrid Maxwell-Boltzmann model of shearth evolution

One dimensional hybrid Maxwell-Boltzmann model of shearth evolution Technical collection One dimensional hybrid Maxwell-Boltzmann model of shearth evolution 27 - Conferences publications P. Sarrailh L. Garrigues G. J. M. Hagelaar J. P. Boeuf G. Sandolache S. Rowe B. Jusselin

More information

Relativistic hydrodynamics for heavy-ion physics

Relativistic hydrodynamics for heavy-ion physics heavy-ion physics Universität Heidelberg June 27, 2014 1 / 26 Collision time line 2 / 26 3 / 26 4 / 26 Space-time diagram proper time: τ = t 2 z 2 space-time rapidity η s : t = τ cosh(η s ) z = τ sinh(η

More information

energy loss Ionization + excitation of atomic energy levels Mean energy loss rate de /dx proportional to (electric charge) 2 of incident particle

energy loss Ionization + excitation of atomic energy levels Mean energy loss rate de /dx proportional to (electric charge) 2 of incident particle Lecture 4 Particle physics processes - particles are small, light, energetic à processes described by quantum mechanics and relativity à processes are probabilistic, i.e., we cannot know the outcome of

More information

Mechanization of Non-linear Calculations in Fusion Reactor Theory

Mechanization of Non-linear Calculations in Fusion Reactor Theory P/1183 France Mechanization of Non-linear Calculations in Fusion Reactor Theory By P. Braffort and M. Chaigne < From Kurchatov's report 1 and recent British publications, 2 the physical foundations of

More information

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 2015 07. Kinetic Theory I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons

More information

Interactions of Particulate Radiation with Matter. Purpose. Importance of particulate interactions

Interactions of Particulate Radiation with Matter. Purpose. Importance of particulate interactions Interactions of Particulate Radiation with Matter George Starkschall, Ph.D. Department of Radiation Physics U.T. M.D. Anderson Cancer Center Purpose To describe the various mechanisms by which particulate

More information

CANONICAL EQUATIONS. Application to the study of the equilibrium of flexible filaments and brachistochrone curves. By A.

CANONICAL EQUATIONS. Application to the study of the equilibrium of flexible filaments and brachistochrone curves. By A. Équations canoniques. Application a la recherche de l équilibre des fils flexibles et des courbes brachystochrones, Mem. Acad. Sci de Toulouse (8) 7 (885), 545-570. CANONICAL EQUATIONS Application to the

More information

Stopping, blooming, and straggling of directed energetic electrons in hydrogenic and arbitrary-z plasmas

Stopping, blooming, and straggling of directed energetic electrons in hydrogenic and arbitrary-z plasmas Stopping, blooming, and straggling of directed energetic electrons in hydrogenic and arbitrary-z plasmas This model Monte Carlo 1 MeV e 1 MeV e C. K. Li and R. D. Petrasso MIT 47th Annual Meeting of the

More information

PHYSICS CET-2014 MODEL QUESTIONS AND ANSWERS NUCLEAR PHYSICS

PHYSICS CET-2014 MODEL QUESTIONS AND ANSWERS NUCLEAR PHYSICS PHYSICS CET-2014 MODEL QUESTIONS AND ANSWERS NUCLEAR PHYSICS IMPORTANT FORMULE TO BE REMEMBERED IMPORTANT FORMULE TO BE REMEMBERED 1. Identify the correct statement with regards to nuclear density a) It

More information

We begin our discussion of special relativity with a power point presentation, available on the website.

We begin our discussion of special relativity with a power point presentation, available on the website. Special Relativity We begin our discussion of special relativity with a power point presentation, available on the website.. Spacetime From the power point presentation, you know that spacetime is a four

More information

4. Complex Oscillations

4. Complex Oscillations 4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic

More information

Lecture Note 1. 99% of the matter in the universe is in the plasma state. Solid -> liquid -> Gas -> Plasma (The fourth state of matter)

Lecture Note 1. 99% of the matter in the universe is in the plasma state. Solid -> liquid -> Gas -> Plasma (The fourth state of matter) Lecture Note 1 1.1 Plasma 99% of the matter in the universe is in the plasma state. Solid -> liquid -> Gas -> Plasma (The fourth state of matter) Recall: Concept of Temperature A gas in thermal equilibrium

More information

Lecture 6.1 Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular,

Lecture 6.1 Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular, Lecture 6. Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular, Newton's second law. However, this is not always the most

More information

PARTICLE DYNAMICS IN THE LINEAR ACCELERATOR

PARTICLE DYNAMICS IN THE LINEAR ACCELERATOR DOCUUEMT ROOM 36-41T PARTICLE DYNAMICS IN THE LINEAR ACCELERATOR J. R. TERRALL J. C. SLATER -- wdl TECHNICAL REPORT NO. 204 MAY 31, 1951 RESEARCH LABORATORY OF ELECTRONICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY

More information

CHARGED PARTICLE INTERACTIONS

CHARGED PARTICLE INTERACTIONS CHARGED PARTICLE INTERACTIONS Background Charged Particles Heavy charged particles Charged particles with Mass > m e α, proton, deuteron, heavy ion (e.g., C +, Fe + ), fission fragment, muon, etc. α is

More information

Detecting high energy photons. Interactions of photons with matter Properties of detectors (with examples)

Detecting high energy photons. Interactions of photons with matter Properties of detectors (with examples) Detecting high energy photons Interactions of photons with matter Properties of detectors (with examples) Interactions of high energy photons with matter Cross section/attenution length/optical depth Photoelectric

More information

Chapter IX: Nuclear fusion

Chapter IX: Nuclear fusion Chapter IX: Nuclear fusion 1 Summary 1. General remarks 2. Basic processes 3. Characteristics of fusion 4. Solar fusion 5. Controlled fusion 2 General remarks (1) Maximum of binding energy per nucleon

More information

2. Passage of Radiation Through Matter

2. Passage of Radiation Through Matter 2. Passage of Radiation Through Matter Passage of Radiation Through Matter: Contents Energy Loss of Heavy Charged Particles by Atomic Collision (addendum) Cherenkov Radiation Energy loss of Electrons and

More information

Fokker-Planck collision operator

Fokker-Planck collision operator DRAFT 1 Fokker-Planck collision operator Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK (This version is of 16 April 18) 1. Introduction In these

More information

Notes on fusion reactions and power balance of a thermonuclear plasma!

Notes on fusion reactions and power balance of a thermonuclear plasma! SA, 3/2017 Chapter 5 Notes on fusion reactions and power balance of a thermonuclear plasma! Stefano Atzeni See S. Atzeni and J. Meyer-ter-Vehn, The Physics of Inertial Fusion, Oxford University Press (2004,

More information

1.1 A Scattering Experiment

1.1 A Scattering Experiment 1 Transfer Matrix In this chapter we introduce and discuss a mathematical method for the analysis of the wave propagation in one-dimensional systems. The method uses the transfer matrix and is commonly

More information

Particle Interactions in Detectors

Particle Interactions in Detectors Particle Interactions in Detectors Dr Peter R Hobson C.Phys M.Inst.P. Department of Electronic and Computer Engineering Brunel University, Uxbridge Peter.Hobson@brunel.ac.uk http://www.brunel.ac.uk/~eestprh/

More information

On materials destruction criteria. L.S.Kremnev.

On materials destruction criteria. L.S.Kremnev. On materials destruction criteria. L.S.Kremnev. Moscow State Technological University "STANKIN", Moscow, Russia, Kremnevls@yandex.ru. Abstract. In terms of nonlinear material fracture mechanics, the real

More information

Synchrotron Power Cosmic rays are astrophysical particles (electrons, protons, and heavier nuclei) with extremely high energies. Cosmic-ray electrons in the galactic magnetic field emit the synchrotron

More information

Kinetic theory of the ideal gas

Kinetic theory of the ideal gas Appendix H Kinetic theory of the ideal gas This Appendix contains sketchy notes, summarizing the main results of elementary kinetic theory. The students who are not familiar with these topics should refer

More information

Integrated Modeling of Fast Ignition Experiments

Integrated Modeling of Fast Ignition Experiments Integrated Modeling of Fast Ignition Experiments Presented to: 9th International Fast Ignition Workshop Cambridge, MA November 3-5, 2006 R. P. J. Town AX-Division Lawrence Livermore National Laboratory

More information

xkcd.com It IS about physics. It ALL is.

xkcd.com It IS about physics. It ALL is. xkcd.com It IS about physics. It ALL is. Introduction to Space Plasmas! The Plasma State What is a plasma? Basic plasma properties: Qualitative & Quantitative Examples of plasmas! Single particle motion

More information

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.

More information

The Vlasov equation for cold dark matter and gravity

The Vlasov equation for cold dark matter and gravity The Vlasov equation for cold dark matter and gravity Alaric Erschfeld Ruprecht-Karls-Universität Heidelberg Master seminar, 25.11.2016 1 / 50 Table of Contents 1 Introduction 2 The Vlasov equation 3 Virial

More information

On the Distribution o f Vertex-Degrees in a Strata o f a Random Recursive Tree. Marian D O N D A J E W S K I and Jerzy S Z Y M A N S K I

On the Distribution o f Vertex-Degrees in a Strata o f a Random Recursive Tree. Marian D O N D A J E W S K I and Jerzy S Z Y M A N S K I BULLETIN DE L ACADÉMIE POLONAISE DES SCIENCES Série des sciences mathématiques Vol. XXX, No. 5-6, 982 COMBINATORICS On the Distribution o f Vertex-Degrees in a Strata o f a Rom Recursive Tree by Marian

More information

Nuclear Binding Energy

Nuclear Binding Energy Nuclear Energy Nuclei contain Z number of protons and (A - Z) number of neutrons, with A the number of nucleons (mass number) Isotopes have a common Z and different A The masses of the nucleons and the

More information

Charged particle motion in external fields

Charged particle motion in external fields Chapter 2 Charged particle motion in external fields A (fully ionized) plasma contains a very large number of particles. In general, their motion can only be studied statistically, taking appropriate averages.

More information