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: F elec = Z 1Z 2 e 2 2 whee Z 1 and Z 2 ae the (intege) numbes of electonic chages. Similaly, the electic potential enegy is 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 = 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. E elec = Z 1Z 2 e 2

2 Ze Potons in nucleus. Electons obit like planets. The neuton was not discoveed until 1932 (Chadwick) KE = 1 2 m ev 2 = Ze2 2 v= Ze2 m e e Ruthefod Atom (1911) F elec = F cent Ze 2 2 = m e v2 Total enegy: = Ze2 m e v 2 E tot = KE + PE= m e v2 2 Ze2 = Ze2 2 Ze2 = Ze2 2 i.e., 2KE = -PE (if PE is negative) Viial theoem still woks fo the electic foce. Z = 1,2,3, classically, any value of v o is allowed. BUT, Ze e Ruthefod Atom (1911) v = Ze2 m e F = m e a = m ev 2 a= v2 = Ze2 m e 2 E tot = Ze2 2 As the electon moves in its 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 The solution lies in the wave-like popety of the electon and of all matte Fo wavelike phenomena e.g., light, intefeence is expected L L = D/d! Young s expeiment Thomas Young ealy 1800 s fo light.

3 Same basic esult obtained using electons λ = h p 8e - 270e - 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 2,000e - 60,000e - Hitachi labs (1989). A popety of ou univese This is a little like the elation we had fo photons E = hν = hc/λ but if E = pc The condition that a paticle cannot be localized to a egion x smalle than its wavelength = h/p also implies λ < Δx p Δx > h p> h Δx 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. Electons always have some motion egadless of thei tempeatue because thei wavelength cannot be zeo (aka Heisenbeg Uncetainty Pinciple) One cannot confine a paticle to a egion x without making its momentum incease p= h is the "degeneate" limit Δx

4 Conside one electon in a contacting box m # v As you squeeze on the box, the paticle in the box has to move faste. h h λ = = p mv # m λ v The squeezing povides the enegy to incease v 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. Since p = h λ 1 2 m e v2 = p2 2m e 1 λ 2 ~ 1 2 but 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. Enegy KE 1 2 At lage distance electical epulsion dominates. At shot distances the quantum mechanical kinetic enegy is lage. 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 2m 2 m(4 π ) 2 e PE = as befoe Note that PE goes as 1/ and KE goes as 1/ 2 The 2 hee is athe abitay but gives the ight answe and omits deepe discussion of wave functions new QM enegy

5 Enegy 1 KE 2 PE 1 Fo a single electon bound to a single poton, i.e., hydogen. At o 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 i.e., p = h 2π and KE = 1 2 m ev 2 = (m e v)2 2m e = p2 2m 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. 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. Howeve, thee also exist excited states of atoms that have a tansitoy existence. o Fo Z=1 (hydogen) o = A = x10 9 cm Fo atoms with a single electon H, He +, etc. Solve as befoe: n 2 h 2 = = 0.53 n2 4π 2 Ze 2 m e Z Angstoms E tot = Ze2 2 = 2π 2 Z 2 e 4 m e n 2 h 2 E tot n=3 = 13.6 ev Z 2 n 2 Fo atoms with only a single electon. Fo hydogen Z = 1 Boh s Fist Postulate The only possible states of the electon ae those fo which mv = nh 2π 1eV eg n = 1 is the "gound state" n = 1 n=2 n=3 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. *

6 * Boh s Second Postulate All obitals fom n = 1 though 4 Numbe electons pe shell is 2n 2, but don t always completely fill one shell befoe stating on the next. 2, 8, 18 2, 10, 18, 36 He, Ne, A, K Only the gound state, n = 1, is pemanently stable 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 (o E n +hν E m ) m > n hν = hc λ = E m E n 1 λ = E m E 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 = = 1216 A 3 m = 3,n = 2, Z = λ = = o = = 1026A 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.

7 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,... 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

8 How ae excited states populated? Absob a photon of the ight enegy Collisions Ionization - ecombination Emission H-alpha Absoption Ly-alpha

9 Stas show absoption line specta Absoption Line Spectum (not the sun) Hydogen Flux Wavelength 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 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 Detemined fom spectal analysis but the most abundant elements (H) do not always have the stongest lines as we shall see

10 The sola spectum C = Balme alpha F = Balme beta f = Balme gamma B = oxygen D = sodium H, K = singly ionized calcium othes = Fe, Mg, Na, etc. 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. Wollaton (1802) discoveed dak lines in the sola spectum. Faunhaufe ediscoveed them (1817) and studied the systematics 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.

11 Some of the stonge lines in stas 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 Main sequence stas would look like this to the human eye Ou sun s spectal class is G2-V

12 Spectal Sequence H β H α Cannon futhe efined the spectal classification system by dividing the classes into numbeed subclasses: Ca II Fo example, A was divided into A0 A1 A2 A3... A9 Between 1911 and 1924, she classified about 220,000 stas, published as the Heny Dape Catalog.

13 Summay of spectoscopic types He II stong, He I inceasing fom O4 to O9 H pominent 10 4 in He II has same wavelength as 5 2 in H I Balme Seies Tansition 3 -> 2 4 -> 2 5 -> 2 6 -> 2 7-> 2 Name H H H H H Wavelength He I lines dominate H inceasing in stength Colo Red Bluegeen Violet Violet Ultaviolet

14 H lines each maximum stength. Ca II gowing. Fe II, Si II, Mg II each Ca II lines stongest, H lines weak, neutal metal lines stong. G-band of CH is stong. H γ = 4341A H δ = 4102 A H lines stat to decease in stength. Ca II stong. Fe I gowing in stength. Mg II deceasing. H lines weak. Lines of neutal metals pesent but weakening. Majo chaacteistic is bands fom molecules like TiO and MgH (Pat of) the sola spectum DISTINGUISHING MAIN SEQUENCE STARS 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