Quantum Mechanics and Stellar Spectroscopy

Transcription

1 Quantum Mechanics and Stella Spectoscopy Recall the electic foce. Like gavity it is a 1/ 2 foce/ That is: e = esu e 2 = dyne cm 2 F elec = Z 1 Z 2 e2 2 whee Z 1 and Z 2 ae the (intege) numbes of electonic chages. Similaly, the electic potential enegy is The Electical Foce Z 1 m Z 2 M e 2 G E elec = Z 1Z 2 e 2 Ze Potons in nucleus. Electons obit like planets. The neuton was not discoveed until 1932 (Chadwick) e Ruthefod Atom (1911) F elec = F cent Ze 2 2 = m e v2 v = Ze2 m e = Ze2 m e v 2 fo a single electon Z = 1,2,3, H, He, Li, etc classically, any value of v o is allowed. Much like planets. Ze Potons in nucleus. Electons obit like planets. The neuton was not discoveed until 1932 (Chadwick) e Ruthefod Atom (1911) F elec = F cent Ze 2 2 = m e v2 v = Ze2 m e Total enegy: = Ze2 m e v 2 E tot = KE + PE= m e v2 2 Ze2 Z = 1,2,3, classically, any value of v o is allowed. = Ze2 2 Ze2 = Ze2 2 i.e., 2KE = -PE (if PE is negative) Viial theoem still woks fo the electic foce.

2 Ruthefod Atom (1911) The solution lies in the wave-like popety of the electon and of all matte BUT, Ze e E tot = Ze2 2 As the electon moves in its classical obit it is acceleated, and theefoe emits adiation. Because enegy is being adiated, the total enegy of the system must decease become moe negative. This means must get smalle and v must incease. But smalle and lage v also imply geate acceleation and adiation. In appoximately 10-6 s the electon spials into the nucleus. Goodbye univese Fo wavelike phenomena e.g., light, intefeence is expected spacing = D λ d Thomas Young ealy 1800 s fo light. Young s expeiment Same basic esult obtained using electons! 8e - 270e - λ = h p h = eg sec = gm cm 2 whee p is the momentum of the electon, m e v which has units gm cm/sec sec Hitachi labs (1989) 2,000e. - 60,000e -

3 In 1924, Louis-Victo de Boglie fomulated the DeBoglie hypothesis, claiming that all matte, not just light, has a wavelike natue. He elated the wavelength (denoted as ) and the momentum (denoted as p) λ = h p A popety of ou univese This is a little like the elation Planck had fo photons p = hν c = h λ λ = h p paticle_duality Light and paticles like the electon (and neuton and poton) all have wavelengths, and the shote the wavelength, the highe the momentum p This is also known as the Heisenbeg Uncetainty Pinciple. The moe accuately you locate a paticle ( ), the moe unbounded is its momentum HEISENBERG UNCERTAINTY RELATION Conside one electon in a contacting box The condition that a paticle cannot be localized to a egion x smalle than its wavelength = h/p implies λ < Δx p Δx > h p> h Δx m # v # m One cannot confine a paticle to a egion x without making its momentum incease p= h Δx is the "degeneate" limit As you squeeze on the box, the paticle in the box has to move faste. This is in addition to any themal motion the paticle may have h h λ = = p mv λ v The squeezing povides the enegy to incease v

4 A little thought will show how this is going to solve ou poblem with the stability of matte (and also, late, the existence of white dwafs) As the electon is foced into a smalle and smalle volume, it must move faste. Ultimately this kinetic enegy can suppot it against the electical attaction of the nucleus. Enegy KE 1 2 At lage distance electical attaction dominates. At shot distances the quantum mechanical kinetic enegy is lage. Since p = h λ KE = 1 2 m e v2 = p2 2m e 1 λ 2 ~ 1 2 but PE = Ze2 1 The kinetic enegy inceases quadatically with 1/, the electical potential, only linealy. Thee comes a minimum adius whee the electon cannot adiate because the sum of its potential and kinetic enegies has eached a minimum. Radius PE 1 Gound state of the hydogen atom Neils Boh (1913) (lowest possible enegy state) Must fit the wavelength of the electon inside a cicle of adius, the aveage distance between the electon and the poton. e - + p λ = 2π = h p h h p = = λ 2π mv m v p h KE = = = = m 2 m 2 m(4 π ) 2 e PE = as befoe The 2 hee is athe abitay but gives the ight answe and omits deepe discussion of wave functions Assuming Z 1 = Z 2 = 1 Enegy At o 1 KE 2 o KE = 1 2 PE h 2 2m e (4π ) = Ze2 2 o h 2 2m e Ze = 4π o h 2 o = 4π 2 Z e 2 m e 2 PE 1 i.e., p = Fo a single electon bound to a single poton, i.e., hydogen. h 2π and KE = 1 2 m ev 2 = (m ev) 2 2m e h 2 = p2 = 2m e 8π 2 2 m e Enegy would have to be povided to the electon to make it move any close to the poton (because it would have to move faste), moe enegy than e 2 / can give. Note that PE goes as 1/ and KE goes as 1/ 2 o Fo Z=1 (hydogen) o = A = x10 9 cm

5 This is the (aveage) adius of the gound state of the hydogen atom, A. It is pemanently stable. Thee is no state with lowe enegy to make a tansition to. Fo atoms with a single electon H, He +, etc. n=3 Solve as befoe: Boh s Fist Postulate n 2 h 2 = = 0.53 n2 4π 2 Ze 2 m e Z Angstoms The only possible states of the electon ae those fo which mv = nh 2π Howeve, thee also exist excited states of atoms that have a tansitoy existence. E tot = Ze2 2 = 2π 2 Z 2 e 4 m e n 2 h 2 E tot = 13.6 ev Z 2 n 2 Fo atoms with only a single electon. Fo hydogen Z = 1 1eV eg n = 1 is the "gound state" n = 1 n=2 n=3 * Boh s Second Postulate In the full quantum mechanical solution the electon is descibed by a wave function that gives its pobability fo being found at any paticula distance fom the nucleus. In the simplest case these distibutions ae spheical. The adius in the Boh model is the aveage adius but the enegy is pecise. Only the gound state, n = 1, is pemanently stable

6 Boh s Second Postulate Radiation in the fom of a single quantum (photon) is Emitted (o absobed) as the electon makes a tansition Fom one state to anothe. The enegy in the photon is the Diffeence between the enegies of the two states. emission absoption E m E n + hν E n +hν E m m > n hν = hc λ = E m E n 1 λ = E E m n = 2π 2 Z 2 e 4 m e 1 hc h 3 c n 1 2 m 2 1 = Z 2 1 λ mn n 1 2 m 2 λ mn = A 1 Z 2 n m 2 (fo atoms with only one electon) cm 1 E.g., m = 2, n = 1, Z = 1 m = 3, n = 1, Z = o λ = A = o = = 1216A 3 m = 3,n = 2, Z = λ = = o = = 1026 A 8 λ = = = = o 5 =6563A 1 λ mn = A 1 Z 2 n m 2 Lines that stat o end on n=1 ae called the Lyman seies. All ae between and 1216 A. Lines that stat o end on n=2 ae called the Balme seies. All ae between 3646 and 6564 A. BALMER SERIES H α,β,γ, H # H # Hydogen emission line spectum Balme seies H # Adjusting the enegy of each state in hydogen by adding 13.6 ev (so that the gound state becomes zeo), one gets a diagam whee the enegies of the tansitions can be ead off easily. Ly α,β,γ,...

7 Peak numb e Wavelength of peak (nm) mecuy mecuy Species poducing peak tebium fom Tb tebium fom Tb mecuy possibly mecuy mecuy o euopium in Eu +3 :Y 2 O 3 o tebium likely Tb possibly tebium fom Tb likely euopium in Eu +3 :Y 2 O 3 Fluoescent Light Fixtue ed geen violet likely euopium in Eu +3 :Y 2 O likely euopium in Eu +3 :Y 2 O euopium in Eu +3 :Y 2 O likely tebium fom Tb likely euopium in Eu +3 :Y 2 O likely euopium in Eu +3 :Y 2 O likely euopium in Eu +3 :Y 2 O 3 How ae excited states populated? Absob a photon of the ight enegy Collisions Ionization - ecombination

8 Emission Emission H-alpha Absoption Absoption Ly-alpha Absoption Line Spectum (not the sun) Stas show absoption line specta Hydogen Flux Wavelength 7000

9 When we examine the specta of stas, with a few exceptions to be discussed late, we see blackbody specta with a supeposition of absoption lines. The sun though a low esolution spectogaph The identity and intensity of the spectal lines that ae pesent eflect the tempeatue, density and composition of the stella photosphee. Blackbody Emission line spectum Absoption line spectum The sola spectum C = Balme alpha F = Balme beta f = Balme gamma B = oxygen D = sodium E = ion H, K = singly ionized calcium othes = Fe, Mg, Na, etc. Wollaton (1802) discoveed dak lines in the sola spectum. Faunhaufe ediscoveed them (1817) and studied the systematics (Pat of) the high esolution sola spectum

10 Almost evey element has been obseved spectoscopically in the sun and has an accuate abundance detemination. The est, except fo noble gases and volatile elements, have an accuate detemination fom pimitive meteoites (cabonaceous chondites) O F Cl W Asplund et al (2009; ARAA) Ag Pb

11 Ionization As the tempeatue in a gas is aised, electons will be emoved by collisions and inteactions with light. The gas comes ionized. The degee of ionization depends on the atom consideed and the tempeatue. Notation: Ionization stages H I neutal hydogen 1 p 1 e H II ionized hydogen 1 p 0 e He I neutal helium 2 p 2 e He II singly ionized helium 2 p 1 e He III doubly ionized helium 2 p 0 e C I neutal cabon 6 p 6 e C II C + 6 p 5 e C III C ++ 6 p 4 e etc. The ionization enegy is the enegy equied to emove a single electon fom a given ion. The excitation enegy is the enegy equied to excite an electon fom the gound state to the fist excited state. ae Ion Excitation enegy (ev) Ionization enegy (ev) H I He I He II Li I Ne I Na I Mg I Ca I Li is He plus one poton, Na is Ne plus 1 poton, Ca is A plus 2 potons. The noble gases have closed electon shells and ae vey stable.

12 Some of the stonge lines in stas Spectal Sequence H β H α Ca II

13 Cannon futhe efined the spectal classification system by dividing the classes into numbeed subclasses: Fo example, A was divided into A0 A1 A2 A3... A9 A0 is hotte than A9 B9 comes befoe A0 OBAFGKM F1 comes afte A9 Faction MS stas sola neighbohood O > 25,000 K Delta Oionis 1/3,000,000 B 11,000 25,000 Pleiades bightest 1/800 A ,000 Siius 1/160 F Canopus 1/133 G Sun 1/13 K Actuus 1/8 M < 3500 Poxima Centaui 3/4 He II pesent, He I inceasing fom O4 to O9 H pominent 10 4 in He II has same wavelength as 5 2 in H I Main sequence stas would look like this to the human eye He I lines dominate H inceasing in stength

14 Balme Seies Tansition 3 -> 2 4 -> 2 5 -> 2 6 -> 2 7-> 2 H lines each maximum stength. Ca II gowing. Fe II, Si II, Mg II each Name H H H H H H γ = 4341A H δ = 4102 A Wavelength Colo Red Bluegeen Violet Violet Ultaviolet H lines stat to decease in stength. Ca II stong. Fe I gowing in stength. Mg II deceasing. Ca II lines stongest, H lines weak, neutal metal lines stong. G-band of CH is stong. H lines weak. Lines of neutal metals pesent but weakening. Majo chaacteistic is bands fom molecules like TiO and MgH DISTINGUISHING MAIN SEQUENCE STARS FROM RED GIANTS OF THE SAME COLOR The suface gavity g = GM R 2 of a sta is clealy lage fo a smalle adius (if M is constant) To suppot itself against this highe gavity, a the stella photosphee must have a lage pessue. As we shall see late fo an ideal gas P = n k T whee n is the numbe density and T is the tempeatue. If two stas have the same tempeatue, T, the one with the highe pessue (smalle adius) will have the lage n, i.e., its atoms will be moe closely cowded togethe. This has two effects: 1) At a geate density (and the same T) a gas is less ionized 2) If the density is high, the electons in one atom feel the pesence of othe neaby nuclei. This makes thei binding enegy less cetain. This speading of the enegy level is called Stak boadening

15 Note: Suface gavity on the main sequence is highe fo lowe mass stas R M 0.65 GM R 2 deceases with inceasing M All 3 stas have the same tempeatue but, The supegiants have the naowest absoption lines Small Main-Sequence stas have the boadest lines Giants ae intemediate in line width and adius Luminosity Classes In 1943, Mogan & Keenan added the Luminosity Class as a second classification paamete: Ia = Bight Supegiants Ib = Supegiants II = Bight Giants III = Giants IV = Subgiants V = Main sequence And so the sun is a G2-V sta