Adv Micro Electronics
Adv Micro Electronics PHYS 497
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Abundances Hoyle39s Cycle smrformat39ian 43 395 wagiuir39agl in this 5911153 a radiation density constant 755 X 1016 J m393 K394 c velocity of light 300 X 108 m s1 G gravitational constant 667 X 103911 N m2 kg2 h Planck s constant 662 X 103934 J s k Boltzmann s constant 138 X 103923 J K 1 me maSs of electron 911 X 103931 kg mH mass of hydrogen atom 167 X 103927 kg NA Avogadro s number 602 X 1023 mol1 o Stefan Boltzmann constant 567 X 10398 W m392 K394 G ac4 R gas constant kmH 826 X 103 J K391 kg1 e charge of electron 160 X 103919 C LD luminosity of Sun 386 X 1026 W MltD mass ofSun 199 X 1030 kg 7quot9fo effective temperature of sun 5780 K RD radius ofSun 696 X108m Parsec unit of distance 309 X 1016 m 1 Abundance of a nu ve abundanCes of nuclei of nut alsovdensity changes for example due to compression p a ri si on ofthe material 39 quotample that is made up by nucleus of species i p mass density 39gcms mass of nucleus of Species i cos only and ltuI m120212i1NA as atomic mass unit AMU note Abundance has no units only valid in C68 39o39ar quot y 0 moquot en sity but changes only if the ne39ears eciesget destroyed or produced Changes in density are factored out n V ans and as with nuclei electron density Me p39 NAYQ i t i39Zin prop to number of protons can also wrlte39 A prop to number of nucleons 139 I I So Ye is ratio of protons to nucleons in sample counting all protons including the ones contained in nuclei notjust free protons as described by the proton abundance UI Some secial case 2 nuclei Ye05 53 The Fermi Gas Model This model quite simple in its stnucture is based on the fact that the nucleons move almost freely inside the nucleus due to the Pauli principle Since two of them cannot occupy the same energy state they do not scatter as all possible nal states that could scattered to are already occupied by other nucleons But when a nucleon approaches he surface and tries to y offthe nucleus it suffers an attractive force by the nucleons that are left behind forcing it to return toward the interior Inside the nucleus it feels the attraction forces of all the nucleons that are around it resulting in a net force approximately equal to zero We can imagine the nucleus as a balloon inside of which the nucleons more freely but occupying states of different energy gure 55 The nucleons in the Fermi gas model obey the Schrodinger equation fora free particle hZ v2 11 Emir 514 where m is the nucleon mass and t its energy to Simplify let us assume that instead of a sphere the region to which the n ncleons are limited to is the interior ofa cube The nal results ofour calculation will be independent of this hypothesis In this way Kl will have to satisfy the boundary conditions Wlx yz0 515 for x0y0zl and xza In2a where a is the side ofthe cube The solution of 5 4 and 515 is given by 441xle A sin bx sintkyy sin kg 516 since lira 24327 kra nyrr and kza nsz 51 where in my and n are positive integers and A is a normalization constant I I Is WI I ml f DIE mI k kdk Figure 56 Allowed states in the part ofthe momentum space contained in the lax Elf plane Each state is represented by 1 point in the lattice For each group on my til we have an energy l1sz at rst smirking a og k kg r12 513 llama2 where n1 nf ME Equations 51 and 118 represent the quantization of a particle in a box where k E ix It i2 is the rnomentmn divided by it of the particle in the bore Due to the Pauli principle a given momentum can only be occupied byquot at most four nucleons two protons wi th opp nsi 9 spin 9 anrl two men 5 Fons wi Ih npposi I39e spin 9 Con sirler the spa re of u ertnrs k byquot Virtue of 518 for each cube of side length 175 a in this space only one point exists that represents a possible solution of the relationship 516 The possible number of solutions see gure 56 ntk with magnitude k between it and i ll is given by the ratio between the volume of the spherical slice displayed in the gure antl the volume mm For each allowetl solution in the k space 1 2 mm8mt a am where 43rl 2ril is the volume of a spherical box in the kspace with radius between it and l dls Only of the shell is considered since onlyr positive values of kyle and ls are necessary for counting all the states with eigen mctious de ned by 5161 With the aid of 518 we can make the energy appear explicitly in 519 If 32 a m a E omvhw Ws mm dnlEl 0 mm m Figure 57 Fermi distribution for T 0 The total number of possible slates of the nucleus is obtained by integrating 320 from O to the minimum Value needed to include all the nucleons This value EF is called the Feimi magu Thus we obtain Ii 32 3 A maqijggitykz nan wl39iei39e the last equality is due to the mentioned fact that a given state can be occupied by four nucleons lnverting 52 we obtain F 22 m hhp tun hz where p 2 Ala We assume that the maximum energy is Ilie same for both nucleons wl39tich means equal values for protons and neutrons If that is not true the Fermi energy for protons and neutrons will be different If pp Z jul and p N for are the respective proton and neutron densities we will have til Hill9 linz pll 523 and Erin Tllzrlpp 3313 524 for the corresponding Ferrni energies The number orquot nucleons with energy between 39E and E d5 given by 520 is plotted in gure 5 as a function of E This distribution of particles is referred to as the Fermi distribution For T 0 which is characterized by the absence of any particle with E r E p This corresponds to the ground state of the nucleus An excited state I gt 0 cart be obtained by the passage ofa nucleon to a state above the Fermi level leaving a vacancy hole in the energyr state it previously occupied If we use p 12 it 1038 nucleons icrn3 012 nttcleonsffrn3 which is the approxi tnate density of all nuclei with A312 we obtain k 136 rm 525 bl Cal Figule 58 a Potential well and states ofa Fermi gas for T 3911 b When one takes into account the Coulomb force The potentials for protons and neutrons are different and we can inlagine a well for each nucleon type In two of the levels we show the spins associated to each nucleon which con39resp01391ds to E 37 MeV 526 We know ihalt the separailou energy of a nucleon is of the order of 8 MeV Thus the nucleons are not inside a well with in nite walls as WE supposed but in a well with depth V0 2 3 3 MeV 45 MEV gure 3821 Fraknol Voyages Through the Universe 2e Figure 246 also Figure 165 Volume 2 is k Perseus Elemental I w Halo and Isotoplc A C mms o composmon orior r i 3quotquot of Galaxy at f quot location of solar system at the time of it s formation E I l vuvvu Ion13v ulge 0 I C O 0 Thin layer 7 LY of dust 40 quot 6 D 4 2000 LY I 7 Globular clusters 4quot Harcourt Inc ilems and derived items copyrlghl 2000 by Harcourt Inc De rer39mined from solar spectra and me reor39i res See for example JEESLEQZJJQJM abundances b2 LfZ l ZI JIllleCJ 2 Earth material Problem chemical fractionation modified the local composition strongly compared to pre solar nebula and overall solar system for example Quarz is 13 Si and 23 Oxygen and not much else This is not the composition ofthe solar system But Isotopic compositions mostly unaffected as chemistry is determined by number of electrons protons not the number of neutrons gt main source for isotopic composition of elements 2 Solar spectra Sun formed directly from presolar nebula largely unmodified outer layers create spectral features 3 Unfractionated meteorites Certain classes of meteorites formed from material that never experienced high pressure ortemperatures and therefore was never fractionated These meteorites directly sample the presolar nebula l391931 riUI a f f QJl l Cylw O Any object which is heated glows This is the principle under which amp lihtbulbs work amp As temperature increases the low becomes brihter and chanes COM Red HOt W hlte Hot color from reddish to white Wedgewood noticed 1792 that his kilns all change color at the same temperature regardless of size and makeup Kg 5 39 3 Cold Points to a universal law connectin temperature to color of low What is Temperature Hot The atoms within an object are constantly in motion even if the object itself does not move Temperature is a measure of the enery tied up in these random motions Thermal properties are often independent of microscopic properties like the species of atom involved etc precisely because the motion is so random The Kelvin temperature scale is defined so that 0 K called absolute zero corresponds to no atomic motion Motion Stops 273 C 0 K Water freezes 0 C 273 K Water boils 100 C 373 K 14 7 direction from which the light comes which gives the direction to the object since light travels in straight lines 7 The spectrum of the light which gives the temperature and chemical composition of the emitting object What is Light Visible light Xrays radio signals yray s ultraviolet light e 9 0 etc all consists of moving particles called photons 9 e6 7 The different types and the different colours of visible light 9 7 all just correspond to photons having different energy Phomns Regardless of energy all photons move at the same speed The spectrum of a light beam is a counting of the number of photons arriving per unit time per unit area as a function of their energy or color l Elhezftm may My 33 Visible Radiation Energy Wavelength Frequency AM Radio 12 neV 100 m 3 MHz FM Radio 1 prev 1 m 300 MHz Microwave 01 meV 1 cm 30 GHz Infrared 04 eV 3 pm 900 THz Red 2 eV 700 nm 4 PHz Blue 3 eV 400 nm 7 PHz Ultraviolet 12 eV 100 nm 28 PHz soft Xray 1 keV 1 nm 3 EHz yRay gt 1 MeV lt 1 pm gt 3000 EHz A continuous spectrum F Eh i E i l lll Mn 10 nrn 500 nm CID nm T130 mm mm uJ r39awbl39e f Visibfe fair 1 m fared n Let us assume that we have a closed system containing a large number Nof distinguishable molecules in thermal equilibrium with its surroundings at a temperature Tand under constant volume conditions We also suppose that only the two lowest nondegenerate energy levels 60 and E are occupied and contain 110 and 111 molecules respectively Traditionally the ground7state energy E0 is set to zero for simplicity because only the relative energies and not their absolute values are relevant here The number of ways in which this configuration can be realized ie the number of microstates W is given by 17 l 1 O l with N 110 111 The connection between the statistical mechanical description of the system embodied in the concept of the number of microstates and the bulk thermo7 dynamic description given by the entropy is expressed in Boltzmann39s famous equation lr 1n VI g of course Boltzmann39s constant This equation can be shown to be reasonable since students should already be familiar with the fact that the approach to equilibrium corresponds to maximizing entropy and it can be easily shown that this also corresponds to maximizing W or in thereby implying a link between these two concepts Substitution of eq l into eq 2 gives S anVD 7111110 7 lnnll beino 3 Now suppose that a small amount of energy E corresponding to the separation between the energy levels 0 and El is added to the system and promotes one molecule from the lower to upper energy level ie 170 a 170 7 l 111 gt 121 l Hence 5quot 1rlnNi 7 lnl 170 7 ll 7111171 l 4 The change in entropy A5 is thus A5 5397 5 11n170111 1 111170111 171gtgtgt l 5 From the laws of thermodynamics we obtain a connection between the entropy and total energy of the system IS dUT 6 since under constant volume conditions dU dr lf We choose a reversible path dLwmmble dU and by definition dq wmble TdS so that 1U TdS Since Uand Sare state functions the latter equation is always true regardless of whether we choose a reversible path or not Consequently AS IdS IdUT iTIdU 7 assuming that temperature remains effectively constant and with the integral over Staken from Sto y and the integral over Ufrom Uto U Since the change in energy U 7 is simply the energy E added to the original system then it fol7 lows from the previous equation that AS 6 T 8 Hence from eqs 5 and 8 11n10111 e T 9 and HlIID exp ekj 10 This of course is the Boltzmann distribution law which may be generalized for any two arbitrary energy levels O 2 Waiting ces the distribution of ng light in space e a e ll leiIa wave Iher th eory of thijs behaviour is called Quantum Mechanics The waveleng rh of a pho ron is r ela red ro i rs energy The eecfmn iof 2V is a commonly used ener gy uni r for pho rons because The binding energy of an a rom is a few ei w s behave as if their positions c 4 Qt be determined With precision grie ater than their wavelength Planck39s 39i hammi Sysc39rwm The d1str1bution of photons em1tted by an incandescent object is universal and depends only on the object s temperature This distribution is called a Thermal Spectrum 1 00 Max Planck gives in 1900 the first theoretical understanding of the experimentallymeasured thermal spectrum High Tempera rur e Low Tempera rur e 10 39 7 He predicts the intensity of emitted light as function of the light s energy Photon Energy E L emitquot iiadiiiii on To calculate the number of modes of oscillation of electromagnetic radiation possible in a cavity consider a onetlimensional box of side L In equilibrium only standing waves are possible and these will have nodes at the ends X 0 L Ln X 11X 1 2 and since Av c speed of propagation for all wave motion v nxL There will be two modes for each trio of integers one for each dimension nx n 112 because there are two independent polarizations possible To find the number of modes with frequency between v and v iv look at an array of points Vz Each cube has side L 2L There is one point per cube of volume 2 3 13 and only positive integers 11X ny HZ are acceptable Thus the number of triplets of positive integers is equivalent to the volume of one octant of the space divided by the volume 2 x i x 41w2 dv no modes of oscillation 7 8 a 87 V Vzdv betweenv and v dv T L 3 C3 2L The factor 47cv2 iv is the volume of a thin spherical shell L3 has been replaced by V the volume of the cavity It is convenient to express this density of states in terms of other variables gv dv Mvzdv g0 dA Md ge d5 SJquotquot3 l 1222ci2 gp dp 8ngpzdp the h s IN pc The expressions involving frequency v energy e and wavelength A are classical physics If we assume that each mode of oscillation represents a harmonic oscillator with Lid each potential and kinetic energy on the average in accordance with the equipartition theorem we get the Rayleigh Jeans law Energy Volume 2 uvdv a T kTVZdV or we Mm Bartram c3 Volume A4 The divergence of this relation at high frequency or low wavelength was known as the ultraviolet catastrophe Planck s new idea was to assume that the possible energies of the oscillators were quantized ie that oscillators of frequency v could only have energy 2 nhv n012 where h was a new constant he introduced Now known as Planck39s constant it was determined by fitting the theoretical curve to the experimental data The average energy per oscillator was calculated from the Maxwell Boltzmann distribution 2 En HEART E n z eEH39IKT H The denominator is called the partition function and is often represented by Z It is easily evaluated by summing the geometric series U U Z T e EnfkT T 9 112 1 h 1 EX where X h V RT n0 110 The numerator can then be found from the denominator T nhvem In L M 110 TX 1 6quotsz and the average energy per oscillator is seen to be 11V g 9X l ehvfkT 1 Thus the energy per unit volume of the radiation in the cavity is ampL Bethe 1 1 dv C3 Elwin 1 dquot or mm d is enamel OD The total energy per unit volume energy density is the integral over all frequencies or wavelengths 4 UT 8 h x vs dv 90 X3 IX C3 0 ehm39lkT l 0 9X l The integral is obviously a pure number It happens to be 314115 Thus the energy density in a black body is 5 4 u T w E 15hc3 This may be thought of as one form of the Stefan Boltzmann law Josef Stefan in 1879 showed experimentally that the ux from a cavity in thermal equilibrium is proportional to the fourth power of the absolute temperature and Ludwig Boltzmann in 1884 derived this fourth power relation from thernmdymamic theory Until Planck s work there was no theoretical method of determining the constants of pmportionalityj The ux radiated from the surface of a black body is related to the energy density Fv u 2AA or F 11 LE3 4 quot 62 81mm 1 i 4 quot k5 Emmet 1 energy 7 n where Pquot dv flux with frequency between v and v Iv area time The total flux obtained by integrating over all frequencies is F U T4EUT4 4 4 This is the usual form of the Stefan Boltzmann law The constant 0 5670 gtlt 13908i 567 105 Stefan Boltzmann constant In2 K39 cm2 s K It is of interest to look at the limits of the Planck distribution At low frequency or large wavelength uVTfI gt and mum gt S T Rayleigh Jeans law c A Note that Planck s constant drops out This is one example of the correspondence principle as 11 becomes negligible compared to other quantities in the quantum mechanical law the result approaches the classical law At high frequency or small wavelength avgquot LyeIR39JERT and min Lime bests c The frequency or wavelength of maximum ux can be found by setting the derivative with respect to v or ii equal to zero 2 v ehwquotkT1 0 d1 2 2R 1 3 IV V CB email which simpli es to 3 with X h quot 91 l kT The transcendental equation may be solved graphically graph y X and y 31 ex and find the nonzero point of intersection or numerically The result is X 2821 meaning that flux as a function of frequency is a maximum at vmax 2821 kTh Maximization of PA is similar 0 AF 2 5 with y h C which yields y 4965 R e l ART quot Flux as a function of wavelength is a 111aximum at he 290111111K 4965kT T 39 At room temperature T 2 290 K and thermal radiation is a maximum at k E 001 mm 10 pm in the infrared Fortunately our eyes are not sensitive to this wavelength The maximum intensity of the sun s radiation is at k 39239 500 nm implying that the sun39s sur ace temperature is T E 5800 K The variation of intensity with wavelength of the sun and other stars is not Amax exactly that of a black body but it is rather close The universal 111icrowave background radiation peaked at A quot 1 mm fits the Planck curve for a black body of T 2728 K to great precision The deviation of order 6 parts in 106 is of course of great interest an awiiexplajns the change of color of a hot body occurs because the energy of the photons which are most strongly emitted grows with temperature 7 This precise relation betWeen temperature and color had been experimentally discovered by Wien in 1893 Wien rs Law states that the temperature T and the wavelength lmax or energy Emax of the most abandantlyprodaced photons is Am in m 0002897 7 5624T in Kelvins or Emax in eV 00004279 T in Kelvins F or example 7 Human Body T 37 C 310 K so 1 max 94 pm infrared light 7 Sun s Surface 1 max 500 nm yellow light so T 5770 K 53 2me 39m qu Law Planck s Law also explains the connection between an object s temperature and overall Luminosity L 7 The precise relation was experimentally discovered by Stefan 1879 and partially understood theoretically by Boltzmann 1884 The StefanBoltzmann Law states that the luminosity of a hot body is proportional to 1 its surface area A or equivalently its radius squared R2 2 its temperature to the 4th power T4 according to L 0A T 4 The constant s is the same for all materials 6 56705119 X 10 8 Wm2 K4 If L A and T are measured in solar units then 039 I Sim 11m yellow light and so 27 107 of energy at the solar 0K then a T4 567 X 10398 gtlt57704 Wm2 k1 mm 19911 339 11 9 we The distribution function fE is the probability that a particle is in energy state E The distribution function is a generalization of the ideas of discrete probability to the casewhere energy can be treated as a continuous variable Three distinctly different distribution functions are found in nature The term A in the denominator of39e ach distribution is a normalization term Which may change With temperature MaxwellBoltzmann classical f AeiEkT Identical but distinguishable particles BoseEinstein quantum Identical indistinguishable particles With integer spin bosons FermiDirac quantum AeEkTH Identical indistinguishable particles With halfinteger spin fermions Emth fusion releasas energy E 4 103912 J I P 4 H become 1 He 2 pOSiTrons 2 neuTrinos plus energy Fusion RaTe 41026 Wf410 12 J 1038 fusionsec Sun burns 4 1038 H nuclei 67 1011 kg of H per sec Solar LifeTime Since The Sun is 75 H by mass iT would Take 075 199 103 kg67 1011 kgsec 23 1018 sec 71 1010 y To burn all H in The Sun Only nuclear reacTions can provide so much energy for billions of years Why Dozsn T The Sun Exalogla Fusion requires very high TemperaTures Hydrogen nuclei musT geT pasT enormous elecTrical repulsions in order To geT close enough To one anoTher To allow nuclear reacTions To sTarT GraviTy causes high TemperaTures in The Solar core Cenfra femperafure 15 million K Cem ra dens1y 150 gcm3 The Sun is a balance beTween graviTy39s conTracTion and The escape of energy from The core 391 Made The Sun massive bu r no39i very dense Average Density 14 gcm3 Like most of The Universe The Sun is almos r co mple rely made of hydrogen wi rh a bi r of helium Observed Solar Elemen ral Abundances as a frac rion of a rom s pr esen r are 912 H 87 He 008 0 004 C How care These abundances known soon 7 sunsquot 5915 Abunrvkng Element Symbol Abundance Abundance by number by mass Hydrogen H 9120 710 Helium He 87 271 Oxygen O 0078 097 7 Carbon C 0043 040 r Nitrogen N 00088 0096 Silicon Si 00045 0099 Magnesium Mg 00038 0076 Neon Ne 00035 0058 Iron Fe 00030 014 r Sulphur S 00015 0040 Cut Away View of the Sun ii z The Sun s ah atmosphere The Sun s lower atmasphere Convection zone 39Sunspnts Phutas here I The Sun39s 1wisilnule urface T quot3 EDDD K aun k Earth her I h Wu H I comparison 5wng SifrlLJCMPa Photosphere surface that emits radiation what we see w5 00 km Chromosphere sun lower atmosphere 1500 km Transition zone 1500 1000 km upper part of the atmosphere where T rises dramatically Corona 10000km tenuous atmosphere gt extend to solar wind Convection zone 200000 km below the photosphere where solar energy is transported by radiation Radiation zone 300000 km where solar energy is transported to Surface by radiation Core 200000 km radius Site of nuclear reactions EIIJIIE JQ IEBXFI QMJ stellar yachts 1591 ZliiilIIlEJJZ rhz 51111 corona hot thin gas u p to 2 Who K S s gt emission lines ch romo sphere hot thin gas 10000 km gt emission lines up to 10000 K s PhOtOSPhere a photons escape still dense enough for 500 km freey photons to excite atoms 6000 K when frequency matches I gt absorption lines convective zone radiation continuous spectrum transport short photon mean free path Emission lines from atomic deexcitations Wavelength gt Atomic Species Absorption lines from atomic excitations Intensity gt Abundance a K Chromos here 411 39lUJBiQJQJLEJU3 lQP p 7 w Photosphere Base 3 on Computer Madeling Convection i A iuolear39Fus ion occurs Radiation Rad Hie Zane Energy direc ll carried by Core rqdm rmn Pho rons Take up To 1 y ro diffuse ou r Convec ve Zane Pho rons absorbed so energy carried ro surface by convec rion o Phafasphere Pho39rons escape Chromasphere Cooler lower a rmosphere Corona Ho r upper a rmosphere nucleus Radiative Zone eleCtTOH N a Pho lons carry energy rowards surface from The core Pho rons can Take millions of years To diffuse ou r since They frequen rly collide wi rh H nuclei and elec rrons Pho rons are no r absorbed because Tempera rures are so high Tha r a romic nuclei and elec rrons are knocked apar r ela roms are ionzed Atoms absOrb photons more efficiently and so upcoming photons are absorbed by upper layers of solar material Since the upper material is heated from below it convects Energy physically carried to surface by movement of solar material Convection Mobile material when heated from below convects Heated material becomes less dense and rises Rising material displaces material above which cools and sinks Sinking material moves to replace material which has risen Occurs in planetarysolar interiors and atmosphere 3 3 N0 Absorpf39on The passage of ighT Thr ough or iTs r eflecTion from a maTer iaI causes phoTons having specific energies To be removed The paTTer n of phoTon energies which are removed in This way Tur n ouT To be char acTer isTic of The aToms which are pr esenT in The maTer iaI Thr ough which The ighT passes or from which iT r eflecTs NKE WiTh Absor pTion PhoTon Ener gy E or bifing ia 39 mo ve n orbiTs having gI ose preic39ise value is Th pecjes of aTom gies 39in aToms are in The range r me R 100000fm Excitation N a r AbsorpTion of lighT occurs when a phoTon hiTs an aTom Ionization and iTs energy is absorbed by changing The ener y of one or more of The e ecTrons in The aTom K Since only a limiTed number of energies are available To The elecTrons in an aTom only phoTons having The righT energy To Take an elecTron beTween orbiTs having allowed energies can be absorbed UI 39 Emission The opposiTe process can also occur whereby an eecTron in an exciTed orbiT can drop To a lowerenergy orbiT by emiTTing a phoTon Again only phoTons having energies specific To The allowed eecTron orbiTs in The aTom can be emiTTed Spech39umfor lithium which has only three electrons Speclmmfor Mon which has ten electrons Excitation from Collision The phoTons emiTTed by an exciTed gas of aToms has a specTrum ThaT is characTerisTic of The kind of aToms doing The emiTTing Since emission and absorpTion involve The same processes The phoTon energies which can be absorbed by an aTomic species are idenTicaI To Those which can be emiTTed Comparison of an observed emission or absorpTion specTra wiTh specTra measured on EarTh allows The idenTificaTion of a celesTiaI objecT39s aTomic composiTion The elemenT Helium was firsT discovered in This way in solar specTra Emission and Absongion i tlbsw o tiwi Soas i r39a provide majority of data because by far the largest number of elements can be observed least fractionation as right at end of convection zone still well mixed well understood good models available Cross section 039 adf reactions O39 of incoming projectiles persecond and target nucleus per second and cm2 or in S mb ols with as particle number current density y rate 0 Of course n v with particle number density n Units for cross section 1 barn 1O3924 cm2 100 fm2 or about halfthe size cross sectional area of a uranium nucleus 38 Continuum i Intensityof Emission o AI Residuai Emission 9 A i 1 effective line width total absorbed intensity aiffimpie model consideration for absorption in a slab of thickness Ax I I0 observed and initial intensity 6 absorption cross section 11 number density of absorbing atom So if one knows cs one can determine n and get the abundances There are 2 complications ii i r iplleg tgn 1 Dahliahriiina g Thevcross section is a measure of how likely a photon gets absorbed when an atom is bombarded with a flux of photons It depends on Oscillator strength a quantum mechanical property of the atomic transition Needs to be measured in the laboratory not done with sufficient accuracy for a number of elements Line width the widerthe line in wavelength the more likely a photon is absorbed as in a classical oscillator E JAE w ET Atom excited state has an energy width AE This leads to a range of photon energies that can be absorbed and to a line width Heisenbergs uncertainty principle relates that to the lifetime 1 of the excited state photon energy range need lifetime of final state 4 0 of an atomic level in the stellar environment depends on The natural lifetime natural width lifetime that level would have if atom is left undisturbed Frequencv of Interactions of atom with other atoms or electrons Collisions with other atoms or electrons lead to deexcitation and therefore to a shortening of the lifetime and a broadening of the line Varying electric fields from neighboring ions vary level energies through Stark Effect gt depends on pressure gt need local gravity or massradius of star Doppler broadening through variations in atom velocity thermal motion gt depends on temperature micro turbulence Need detailed and accurate model of stellar atmosphere Atom ictrans itio ns depend on the state of ionization l The number density n determined through absorption lines is therefore the number density of ions in the ionization state that corresponds to the respective transition to determine the total abundance of an atomic species one needs the fraction of atoms in the specific state of ionization Notation l neutral atom H one electron removed llltvvo electrons removed Example a Call line originates from singly ionized Calcium Notation for following slides ni is the density of atoms in the ith state of ionization that is with 139 electrons removed gi is the degeneracy of states for the iions Si is the energy required to remove 139 electrons from a neutral atom creating an i level ion ne is the electron density The Saha Euation In Local Thermodynamical Equilibrium the distribution of atoms with ionization state i met the various energyr states In is proportional to exp A39T where mm is the excitation energy of state m relative to the ground state Then Mm grim m 39 ex 41 quot t1 gm p kT j and further after summing over all m Ntm Q im im quot 2 quot 12 N ulT 9quot AT where Xer litT Zen EXPL 43 is called the level partition function We can extend this treatment to the continuity of states with positive energy by using differen tials and state densities We thus derive for the ratio of singly ionized atoms to neutral both in ground state le 1 gl 1 X0 d3 dgp V 2 39 emu 7 44 439 39o1 901 MP ha where the factor 2 takes care of the two independent spin positions of a free electron Integrating over momentum we derive de 9m 3 Au IQ1 r 2 2 AT 2 r 4f NM 9M mu exp LT 73 J The electron shares the available volume with all electrons and thus 321 u Hence we E nally derive the Soho equation fir1 1 I V n 2 3 than 911 h 3927quot RT 32 5 L m 3quot eitH z D ate that using the level partition sums can be written as an equation for the ionization equilibrium 7v is ilT 31quot 4 1 1 in 53971 V n 2 exp l f Ni I of 3913 H 39J 439 if The density of electrons is determined by the density of ionized atoms of course in the statistical sense the Boltzmann exponential factor prei eres the bound state Whereas the large phase space volume available to a freed electron favors the ionized state Let us see what we obtain for the ionization fraction of hydrogen in the solar photosphere where 136 eV Tim 2 2 16000 K TEES 239 6000 K 11H 392 10 0111 3 48 So the actual gas temperature is much smaller than the temperature equivalent of the ionization energy Naiveij one would expect a very low ionization fraction of hydrogen in the solar photospherem Hydrogen is the dominatingquot element hence the electron density should be related to the hydrogen density by Nt1 N1 NW NH us 49 53 because the rst excited level of atomic The ratio of level partition sums is approximater 0 hydrogen is high compared with the thermal energy inserting the numbers Saha s equation then gives the ionization fraction l39 2 3 Hi 5 i it at 1 a 2 enrT 523104 41011 A slightly hotter star with photospherie temperature T 12000 Ii of the same photospherit density would have an ionization fraction 1 12000 K 03 411 much higher than what the ionisation energy would suggest Zed atom through lines tal nce 11m 9 Pacific state of ionization euitral atoms 39ami c equilibrium LTE which means mbination reactions are in thermal equilibrium Alt gtAe39 ne electron number density me electron mass B electron binding energy 9 statistical factors 2J1 need pressure and strong temperature temperature dependence with higher and highertemperature more ionized nuclei of course eventually a second third ionization will happen I again one needs a detailed and accurate stellar atmosphere model glimmer1533 Practically one sets up a stellar atmosphere model based on star type effective temperature etc Then the parameters including all abundances ofthe model are fitted to best reproduce all spectral features incl all absorption lines can be 10039s or more a c r 10 Example for a rprocess star Sneden et al ApJ 572 2002 861 08 06 x Io4 ED17 3245 E H J HD 122563 g IIIAAIHIIIIIAI1AIAIII varieerlI quot1 39 quotquot 39 39 39 39 39 39 39 39l 39 abundance H 39 quot 12008 quot bl Gel 06 2139 I 04 saunas llllillllllllllllllllll 3038 3037 3033 3039 3040 Wavelength A 39 g ix g iwl 5935mm Disadvantages less understood more complicated solar regions it is still not clear how exactly these layers are heated some fractionationmigration effects for example FlP species with low first ionization potential are enhanced with respect to photosphere possibly because of fractionation between ions and neutral atoms Therefore abundances less accurate But there are elements that cannot be observed in the photosphere for example helium is only seen in emission lines this is how Helium was discovered by Sir Joseph Lockyer of England in 20 October 1868 Solar Chromosphere red from Hot emission lines Meteorites can provide accurate information on elemental abundances In the tiresola r nebula More precise than solar spectra if data are available Butsame gases escape and cannot be determined this way for example hydrogen or noble gases Not all meteorites are suitable most ofthem are fractionated and do not provide representative solar abundance information One needs primitive meteorites that underwent little modification after forming Classification of meteorites Group Subgroup Frequency Stones Chondrites 86 Achondrites 7 Stony Irons 15 Irons 55 Wise ailiond i39as m 9f falls Q trwl lrjvia gt Have Chondrules small 1mm size shperical inclusions in matrix believed to have formed very early in the presolar nebula accreted together and remained largely unchanged since then Carbonaceous Chondrites have lots of organic compounds that indicate very little heating some were never heated above 50 degrees Chondrule Elmndrit s Eafe 39stny meteorites that have not been modi ed due to melting or g 39 ion of the parent body They formed when various types of dust and small grains that were present in the early solar system accreted to form primitive asteroids Prominent among the components present in chondrites are the enigmatic chondrules millimetersized objects that originated as freely oating molten or partially molten droplets in space most chondrules are rich in the silicate minerals olivine and pyroxene Chondrites also contain refractory inclusions including CaAl Inclusions which are among the oldest objects to fOrm in the solar system particles rich in metallic FeNi and sul des and isolated grains of silicate minerals The remainder of chondrites consists of negrained micrometersized or smaller dust which may either be present as the matriX of the rock or may form rims or mantles around individual chondrules and refractory inclusions Embedded in this dust are presolar grains which predate the formation of our solar system and originated elsewhere in the galaxy Most meteorites that are recovered on Earth are chondrites 86 of witnessed falls are chondrites as is the overwhelming majority of meteorites that are found There are currently over 27000 chondrites in the world39s collections The largest individual stone ever recovered weighing 1770 kg was part of the Jilin meteorite shower of 1976 Chondrite falls range from single stones to extraordinary showers consisting of thousands of individual stones as occurred in the Holbrook fall of 1912 where an estimated 14000 stones rained down on northern Arizona From Wikz gedia the ee encyclopedia achOndr39ite is a stony meteorite that is made of material similar to terrestrial basalts or plutonic rocks Compared to the chondrites they have all been differentiated and reprocessed to a lesser or greater degree due to melting and recrystallization on or within meteorite parent bodies As a result achondrites have distinct textures and mineralogies indicative of igneous processes Achondrites account for about 8 of meteorites overall and the majority about two thirds of them are HED meteorites originating from the crust of asteroid 4 Vesta Other types include Martian Lunar and several types thought to originate from other asyet unidenti ed asteroids These groups have been determined on the basis of e g the FeMn chemical ratio and the 170180 oxygen isotope ratios thought to be characteristic quot ngerprintsquot for each parent body From VVikipedia the free encyclopedia more on meteorites httpwwwsaharametcom httpwwwmeteoritefr Element Abundmiccs in the Solar pholosphcrc and in McIcorilcs El Photosphere39 Meteorites PhltMet El lelosphere Meteorites thMcl 01 H 1200 7 42 N10 192 i035 197 002 17115 02 He 1093 10004 44 Ru 184 i007 183 i004 00 03 Li 110 31110 331 004 221 45 Rh 112 i012 111 i004 002 04 Be 1310 009 142 004 002 46 Pd 169 31004 170 1004 7001 05 B 255 030 279 i005 024 47 Ag 094 1025 124 i004 030 06 C 852 006 7 48 Cd 177 011 176 004 001 7 N 7 9 0 06 40 In 1 6 015 082 004 084 08 0 883 i006 50 Sn 20 j03 214 1004 4114 09 F 456 103 448 i006 008 51 Sb 10 i03 103 1007 003 10 N 808 006 52 Te 224 004 11 Na 633 i003 632 i002 001 531 151 008 12 Mg 758 31105 758 100 000 54 Xe 217 3008 13 Al 647 i007 649 212001 002 55 C5 113 21002 14 Si 755 i005 756 i001 4101 56 Ba 213 005 222 i002 009 15 P 545 11004 556 006 7011 57 La 117 1007 122 1002 7 005 16 S 733 2101 720 006 013 58 Ce 158 21009 163 002 005 17 C1 55 033 528 21006 022 59 Pr 071 008 080 002 7009 units given is A lognnH 12 log of number of atoms per 1012 H atoms often also used number of atoms per 106 Si atoms log of photosphere abundnnieei i byji kmda 1o 39 39 r g 05 U 3 0 E I 00 Us I S 0 e fl 05 10 L L VI 1 gig Haggai kid I I i i f H l I 1 1 M leg llLJ I l 1 1 I I l O 20 4O 60 Z atomic number generally good agreement numberflaction mg en mass fraction X 071 umiim asis fraction Y 028 Metallicity mass fraction of everything else 2 0019 Heavy Elements beyond Nickel mass fraction 4E6 ocnuclei 139ZC1 O1 Ne24Mg Ca 111 l J sprocess peaks nuclear shell closures 11 I width l l l l All P Fe 50 100 150 200 mass number 5 9 general trend less heavy elements BBeLl I l I 1 I I rprocess peaks nuclear shell closures 1 UTh llllllll lllllHWllllllll llllllll HHHI HHHH wmu HHHH HHHH HHHH HHHI wmu mm 1 0 I l r O N 01 0 UI ou sig la We 59sz neighborhood iA39b undances outside the solar system can be determined through 39 Stellar absorption spectra of other stars than the sun is Interstellar absorption spectra Emission lines from Nebulae Supernova remnants Planetary nebulae y ray detection from the decay of radioactive nuclei Cosmic Rays What do we expect NGC 6543 The Cat39s Eye Nebula gIg l ik 395 oklgc ulg prmassi Star Formation 39 Nucleos nthesis continuous enrichment increasin metallicity Nucleosynthesis BH Black Hole NS Neutron Star WD White Dwarf Star ISM Interstellar Medium UI Therefore 39I39he comp39osi39l39ion of 139he u quot DISK OLD METAL39 POOR quotPRESSUREquot SUPPORT SCARCE BULGE OLD YOUNG BWI mganayuS I 39I environment OLD METAL39POOR TO SUPER METAL RICH SOLAR CYLINDER MODERATELY METAL RICH ROTATIONAL SUPPORT UI Also metallicify gradient in GGIQZC r e T u 2 S 6 Oler quot F h I I I I I I l I 0 4 8 12 16 Rstkpc 8IIIIIIIIIIIIIJFIIIIIOIII 39 39 39 4 3 12 16 kpc E 76 I39D II L g 72 quot Age lt 10 Gyr 0 Age 9 10 Gyr N I v 68 Welghted T Q unweighted j x Observation I I I I I I I I l 4 8 12 16 kpc Hou et a Chin J Astron Astrophys 2 2002 17 FeH Fen nsolar FerH log 51515375le 97 ci U03 1 Pop I metal rich like sun E quot Pop ll metal poor FeHlt 2 E 395 Poplll first stars not seen but today situation is much more complicated many mixed case model calculation Argast et al AampA 356 2000 873 finally found I I 1000 2000 5000 4000 5000 Age Hm metallicity age relation old stars are metal poor BUT large scatter idi reotly at or near nth esis products from the interior t3 i39n39 sjtars To has no stable isotope and Miyears Merrill 1952 STARS SHOWING RESULTS OF saPROCESS J was WWMMWWWWWWWWMNWIIWW Cl b mullIl WWUWWWWII39IMIWNIIIHmII quotIquot I PrICeII SmII PrJI SmIl Ball Unid 4429 4467 450 4538 4554 4563 I C dquot quot39l l l I ll L ZrO Ball ZrI TiO SrI ZrO TIO ZrO 4534 4554 4576 4584 4607 4620 4626 464 x H erllblllllll C A Call Toll FIT IT 4227 4238 4262 429 proof for ongoing nucleosynthesis in stars 62 Hydrogen orange Nitrogenred Sulfurpink Oxygengreen Pb ThU massive stars hydrostatic buring LJ 13C 17O Novae rp process sprocess Unknown SN type NS merger rprocess 9294 M09698 Ru Unknown Xray bursts rp process Possible type II SN vprocess contribution to Note yellowred all related to massive stars gt8 12 solar masses The Origin of the Elements 1 D Mass known 39 f I Half life known I D nothing known p process 9 Supernovae 7 7 l Calculate dEdN the energy required to add a particle for a Fenni Dirac distribution Usling our distribution Jnctions Boltzmann FermiDirac or BoseEintein calculate the number of degrees of freedom relevant to the very early hightemperature universe assuming the relativistic particles are the muons electrons photons and three neutrino species We define the photon weight in this sum to be 2 3 Use the Saha equation to calculate the temperature of recombination which you can de ne as the time when there is an equal amount of neutral hydrogen and free protonselectrons It is fine to simplify hydrogen by considering it as a system with a single l36 eV bound state 4 Generalize the Saha equation to consider the temperature for a process in the early universe where two protons and two neutrons combine directly to fOI H l 4He That is find the temperature where half of the neutrons are bound in 4He Later we will compare this result with the BBN predictions 5 An atom with a single electron is in a heat bath at a temperature of T6 l The atom is high Z so the electron is bound at this temperature and only three states have appreciable occupations The ground state has spin 52 The first excited state at 210 eV has spin 32 The second excited state at 380 eV has spin 32 What are the occupation probabilities for these three states APPENDIX More facts about the Sun Th3 5939s i ir959ilar 3 Altitude Temperature km Corona K Transition 2600 Zone 1000000 I Helloselsmlc sound waves travel throughout the Sun making it vibrate 2300 25 000 like an enormous bell Chromos here 2 7 Sound is stimulated by convection in p solar interior 7 The sound waves have a characteristic 5 00 4400 frequency determined by the temperature and density as a function 7 5770 of depth 0 7 7 i 6 500 7 Comparison of predictions with the SOlar Interlor observed oscillations on the surface tests the solar model The chromosphere is the lower atmosphere which extends 72000 km above the photosphere 7 It is usually invisible in the photosphere s glare 7 It can become visible during solar eclipses depending on the apparent size of the Moon s disk The intensity of the chromosphere s spectral lines dominantly a red Hydrogen line indicates it is 10000 times less dense but hotter than is the photosphere 7 How can temperature rise as one moves away from the core which is the source of heat The 511 in XaRrys If both the photosphere and chromosphere are covered during an eclipse the hotter but dimmer Camma becomes Visible The corona mostly radiates Xrays The corona is dim despite being hot because the density of gas there is comparatively small The intensity of spectral lines and the presence of highlyionized atoms indicate that the corona is very hot Manama K 1 reveal an active corona conditions coronal particles mostly protons and electrons n or are expelled into space and replenished from below t near e ares V ggpelled into space the corona becomes less dense forming coronal uns 993 s 39 The solar surface is marked by dark spots called Sunspots often occurmg in pairs or groups 7 Sunspots are dark because they are 1000 K cooler 7 They are typically 10 000 km across like the Earth 39 Sunspots form and fade lasting I for up to 100 days quot39A r 45 7 Sunspots move W1th the Sun s quot quot I 3 A a I Cc r 3er rotatlon L 4 associated with The sunspoTs mans magneTiicfields are found on The surface of The Sun and are Magnetic fields are measured on The Sun using The Zeeman effecT Wimi are Iblilflfl3 l l FiaJszz Ele cTrically neuTral wires carrying elecTric currenTs aTTracT one anoTher Moving elecTric charges exerT a velociTy dependenT magne c force on oTher charges disTincT from elecTrosTaTic forces The moTion of elecTrons in some aToms also produces magneTic forces MagneTs are objecTs for which The force due To each aTom is noT randomly aligned and so does noT cancel when summed over a macroscopic sample 9 Magnetic Q Non Magnetic I Magnetic No Magnetic F161 d Field 1b fe 39 as tiny magnets and so W ang their energy in a magnetic eld r This change ofelectron energies 1 changes theenergiesof photons which Photon Photon an torn can emit or absorb i Observing this change in emitted and absorbed photon energies permits I I I I I magnetic elds to be detected in space Observed Absorptlon Lines SUM59918 Eli lii J IMCUJZ HC Flak Magnetic eld 39 Magnetic fields are found in quotquot95 sunspots 7 Magnetic elds cool the regions where they penetrate the solar surface thereby making the area appear darker Sunspmpalr 7 Each sunspot of a pair has opposite polarity one is magnetic N and one is magnetic S 2006 2008 The Number of SunspoTs The number of sunspoTs is observed To change over In The Sun39s norThern hemisphere he Year39s W39 393 roughly one polariTy leads as The Sun 11 yecfrr39 qllcle Th mOST roTaTes buT The opposiTe polariTy 236150 Gr max39mum was leads in The souThern hemisphere Annual sunspot number I I 1920 1930 1940 1950 1960 I 1 970 1 990 1900 1910 1950 Year 40 on TheiSun yar39y WI I39h a g 30o 3 20 510 T numbers cycle wu rh a period 5 on olar y fars bu r magnetic field 5 10 QIhriTies reverse in successive cycles 8 a Few Sunspofs during a solar39 minimum 8 40 A H E Minimum Maxumum Many sunspo rs during a solar39 maxumum a Last one 1991 3150 g 100 7 5o 3 C E lt S equator Minimum 1935193719391941 19431945194quot Year 7 6 mng 313 CW3 Sunspm Dair Solar cycle is believed to be due to the wrapping of magnetic fields due to the Sun s differential rotation After sufficient wrapping the lines break and untangle and the cycle starts once more Sunsuo rs 5mg Cllrrigi ia Are sunspot cycles linked to Earth39s weather L ffe Ice Age coincides with the Maunder Minmum in the sunspot cycle in late 160039s Meager evidence for a correlation between sunspots and climate There is no known mechanism l 1 1600 1700 1808 1900 2000 for causing climate change ear N O O 01 O Maunder 0 minimum f H 01 0 Annual sunspot number 0 Pwn uamas Solar Prominences are jets of particles which erupt from the photo sphere and loop through the inner corona following magnetic eld lines 7 Quiescent Pruminences can persist for days or weeks 7 Active Prominences are more erratic changing on scales as short as hours r v H b 5 on rJaras l cgq er Soar Fares are even more 39 violenT explosions of parTicles from The solar surface Flares release as much energy as a large prominence buT over hours or mnufes raTher Than days or weeks Released parTicles Typically escape from The Sun as soar cosmc rays reaching The EarTh in 30 minuTes S w 4quot 7
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