The non-lte Rate Equations

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1 Th o-lte Rat Equatos Statsta quatos Popuato umbrs LTE: popuato umbrs foow from Saha-Botzma quatos,.. pury oa probm T, No-LTE: popuato umbrs aso p o raato f. Ths, tur, s pg o th popuato umbrs a pths,.. o-oa probm. T,, J Th Saha-Botzma quatos ar rpa by a ta osrato of atom prosss whh ar rsposb for th popuato a -popuato of atom rgy s: Extato a -xtato Iozato a rombato by raato or osos

2 Statsta Equbrum Chag of popuato umbr of a wth tm: Sum of a popuato prosss to ths - Sum of a -popuato prosss out from ths P P t O suh quato for ah Th trasto rat P omprss raat rats R a oso rats C I star atmosphrs w oft ha th statoary as: h P P for a s t Ths quatos trm th popuato umbrs. t P P Raat rats: bou-bou trastos Two atrat formuatos: a Est offts B B A b L absorpto offts σ ν aatag a: usfu for aayta xprssos wth smpf mo atoms aatag b: smar xprssos as of bou-fr trastos: goo for fft programmg Numbr of trastos u by tsty I ν frquy tra ν u so ag ω Bϕ I ω/ 4 π absorb Ergy / Itgrato or frqus a ags ys Or atraty R wth σ B ϕ / 4π R B ϕ J σ 4π J 4

3 Raat rats: bou-bou trastos I aaogy, umbr of stmuat mssos: R B g σ R 4π J g Numbr of spotaous mssos: R Tota owwars rat: ϕ J B g R 4π g A ϕ B σ g g ϕ J ϕ g σ R R R 4π J g σ T R R [ 4π ] J 5 Raat rats: bou-fr trastos Aso possb: ozato to xt stats of part o Examp C III: Grou stat s S Photoosato prous C IV grou stat s S C III frst xt stat sp P o Two possbts: Iozato of p tro C IV grou stat s S Iozato of s tro C IV frst xt stat p P C III two xt tros,.g. p P Photoozato oy to xt C IV o p P 6

4 4 7 Raat rats: bou-fr trastos Numbr of photoozatos absorb rgy ν, by photo rgy, tgrat or frqus a so ag Numbr of spotaous rombatos: 4 J R I p σ π ω T T G m p h F R m h R m R h ω π π σ π 8 Raat rats: bou-fr trastos Numbr of u rombatos Tota rombato rat J R m h J h m p R m h J G R I G T T σ π π π ω 4 J R T σ π

5 Raat rats Upwar rats: Dowwar rats: R R R wth σ R 4π J 4π wth σ J T Rmar: TE w ha J B R R 9 Cosoa rats Star atmosphr: Pasma, wth atoms, os, tros Part osos u xtato a ozato Coo stars: mattr mosty utra frqut osos wth utra hyrog atoms Hot stars: mattr mosty oz osos wth os bom mportat; but muh mor mportat bom tro osos / / tro o mass 4 o tro mass m A H m Thrfor, th foowg, w oy osr osos of atoms a os wth tros. A 5

6 Etro osoa rats Trasto : bou or fr, σ tro oso ross-sto, tro sp Tota umbr of trastos : C σ f Ω T mmum oty ssary for xtato thrsho f oty strbuto Maxw I TE w ha thrfor C C Tota umbr of trastos : C C Etro osoa rats W oo for: osoa ross-stos σ xprmts quatum mhaa auatos Usuay: Bohr raus πa as ut for ross-sto σ σ πa Q usuay tabuat as futo of rgy of og tro Q m m T Ω T σ f wth m E a f 4π π T u 8 C T Q ut u u wth u: E T a C πa m u π u x T C T T wth T Q E x u T T Ω Γ Γ xt x, x: E -E 6

7 Etro osoa rats Ω u T C T Γ T Aatag of ths ho of otato: Ma tmpratur p s srb by Γ T oy way aryg futo of T H, smp poyoma ft possb Importat for umra appato C ΩT E T T Now: xamps how to omput th C Computato of osoa rats: Extato Va Rgmortr 96: Vry usfu approxmat formua for aow po trastos C E f H E H u C T4.5 f u E hyrog ozato rgy Γ u osator strgth of raat trasto E u E / T Γ u max u [ g,.76 E u ].7 for trastos btw s wth qua prpa quatum umbr g. s Thr xst may formua, ma for partuar os a trastos,.g., optay forb trastos btw s H I Mhaas & Sto 968 C C wth og Γ T u Γ T ogt ogt offts tabuat for ah trasto 4 7

8 Computato of osoa rats: Iozato Th Sato formua s aaogy to th a-rgmortr formua as of xtato. Hr, th photo absorpto ross-sto for ozato s utz: u C.55 σ g T u σ thrsho photo ross - sto for ozato. for os wth harg Z g. for os wth harg Z. for os wth harg Z > Atrat: sm-mpra formua by Lotz 968: E H C C T.5a u E u u a,b, [ E u b u E u / u ] mpra quatts, aust to ua atoms For H u H spf ft formua ar us, mosty from Mhaas a Mhaas & Sto 968 Autoozato a tro rombato gat post Ergy o I,.g. H I ozato rgy b o II,.g. H II b bou stat, ouby xt stat, autoozato grou stat of xt Io : Autoozato. ays to grou stat of xt ozato stag pus fr tro b: Dtro rombato. Rombato a a ouby xt stat of xt owr ozato stag. autoozs aga wth hgh probabty: A auto... 4 /s! But somtms a stabzg trasto b ours, by whh th xt ays raaty. 6 8

9 Computato of rats Numbr of tro rombatos from to b: R b As As probabty for spotaous stabzg trasto I th mt of wa raato fs th rrs pross a b gt. Th w obta Bats 96: A a A / A a a A s wth / trasto probabty for autoozato C T E Io / T Φ T So, th umbr of tro rombatos from to b s: R Φ T A A / A A b s a a s 7 Computato of rats Thr ar two ffrt rgms: a hgh tmpratur tro rombato HTDR b ow tmpratur tro rombato LTDR for th ass that th autoozg s ar os to th ozato mt b or far abo t a a Most mportat rombato pross H II H I th soar oroa T~ 6 K b Vry mportat for spf os photosphrs T< 5 K.g. N III λ464-4å msso ompx Of stars Raso: uppr s orpopuat, baus a stabzg trasto s gog to t. Baus as b A >> A a s 8 9

10 LTDR Th raato f photosphrs s ot wa,.., th rrs pross b s u Rombato rat: R b Φ T As J J ma tsty stabzg trasto,.., g by otuum au ry broa, baus short ftm Rrs pross: g b Rb bbb J b As J gb Ths rats ar formay a to th usua ozato a rombato rats a o ot show up xpty th rat quatos. 9 Compt rat quatos For ah atom of ah o, of ah hma mt w ha: P P I ta: R C > < > < R C R C R C xtato a ozato rats out of -xtato a rombato -xtato a rombato rats to xtato a ozato

11 Cosur quato O quato for ah hma mt s ruat,.g., th quato for th hghst of th hghst ozato stag; to s ths, a up a quatos xpt for th fa o: ths rat quatos oy y popuato ratos. W thrfor a osur quato for ah hma sps: Abua fto quato of sps, wrtt for xamp as umbr abua y rat to hyrog: y popuato umbrs of sps popuato umbrs of hyrog Abua fto quato Notato: Popuato umbr of ozato stag :, LTE s o ot appar xpty th rat quatos; popuatos p o grou of xt ozato stag:,,φ, T NLTE s E

12 Abua fto quato Notato: NION umbr of ozato stags of hma mt NL umbr of NLTE s of o LTE umbr of LTE s of o Aso, o of th abua fto quatos s ruat, s abuas ar g rat to hyrog othr ftos o t hp harg osrato Φ Φ,,,,, H LTE protos H NL NION LTE NL protos H LTE H NL NION LTE NL T y T y 4 Charg osrato quato Notato: Popuato umbr of, o, mt : NELEM umbr of hma mts q harg of o NELEM NION LTE NL NELEM NION LTE NL T q q Φ,,,,,

13 Compt rat quatos: Matrx otato Vtor of popuato umbrs,, L, NLALL NLALL tota umbr of NLTE s A b rat quato matrx otato O suh systm of quatos pr pth pot Examp: hma mts Emt : NLTE-s: o: 6, o: 4, o: Emt : NLTE-s: o:, o: 5, o: Emt : NLTE-s: o: 5, o:, hyrog Numbr of s: NLALL6,.. 6 x 6 matrx 5 x x x o Iozato to xt stats x x x x x x o x x x abua f. z z y y z x x x o x LTE otrbutos x x x x o x abua f. z H harg osrt. q q z q q z q q z q q z z x x x abuas x z y y x x x x x x z 5 p 6

14 Emts of rat matrx For ah o wth NL NLTE s o obtas a submatrx wth th foowg mts: R C < owr ft A R > uppr rght C m Rm Cm Rm Cm agoa m< m> KNL KNL K hghst part o, to whh o a oz; os ot ha to b NL! 7 Emts orrspog to abua fto q. Ar oat fa row of th rspt mt: NION NL NION NL xpt of grou stat of xt os A Φ, m grou stat of xt os m LTE y [ NLALL - NL H ] [ NLALL ] y Φ m NLALL m LTE H 8 4

15 Emts orrspog to harg osrato q. Ar oat th ry fa row of rat matrx,.., NLALL q NLALL, xpt of A grou stat of xt os q q Φ, m s m LTE Not: th homogty tor b rght-ha s of statsta quatos otas zros xpt for th ry ast mt NLALL: tro sty from harg osrato quato 9 Souto by arzato Th quato systm A b s a ar systm for a a b so f,, T, J ar ow. But: ths quatts ar gra uow. Usuay, oy approxmat soutos wth a trat pross ar ow. Lt a ths arabs hag by δ, δt, δj.g. orr to fuf rgy osrato or hyrostat qubrum. Rspos of popuatos δ o suh hags: Lt A b wth atua quatts A A δ A δ b δb Ngtg orr trms, w ha: A b δ δ A δb wth w quatts, T, J 5

16 Larzato of rat quatos N: xprssos for: δ A, δb O possbty: A A NF A δ A δt δ δj T J If ato to th arabs, T, J ar trou as uows, th w ha th Mtho of Compt Larzato Jν srtz NF frquy pots J Othr possbty: mats from th quato systm by xprssg J through th othr arabs, : T J f, T, As a approxmato o uss J ~ S, T, a trats for xat souto Larzato of rat quatos A A NF A δ A δt δ δs T S S S S δs δt δ δ NLALL T Mtho of approxmat Λ-oprators Arat Lamba Itrato aaogous, δ b : NF b b b δ b δt δ δ S T S, L,, δ 6

17 Larzato of rat quatos NF A A S A b δa δt T S T A NF A S δ S NLALL NF A S δ S Larz quato for rspos δ as aswr o hags δ, δt, δj NF A S Exprssos S show th ompx oupg of a arabs. A hag th raato f a, h, th sour futo at ay frquy auss a hag of popuatos of a s, f a partuar aot absorb or mt a photo at that ry frquy! Larzato of rat quatos I orr to so th arz rat quatos w to omput ths rats:,,, wth rspt to A,b,S T S A rats a b omput aaytay! Irass auray a stabty of umra souto. Mor tas atr. 4 7

18 LTE or NLTE? Wh o parturs from LTE bom mportat? LTE s a goo approxmato, f: Cosoa rats omat for a trastos R << C P R C C C baus C souto of rat quatos LTE J B s a goo approxmato at a frqus R R souto of rat quatos LTE 5 LTE or NLTE? Wh o parturs from LTE bom mportat? LTE s a ba approxmato, f: Cosoa rats ar sma C ~ / T, T C α Raat rats ar arg R ~ T, α> T R Ma fr path of photos s argr tha that of tros Examp: pur hyrog pasma z ~ / H sty of utra H T Saha: H ~ pt z ~ p, T z / / E / T E / T Dparturs from LTE our, f tmpraturs ar hgh a sts ar ow 6 8

19 LTE or NLTE? 7 LTE or NLTE? 8 9

20 LTE or NLTE? DA wht warf, T ff 6K, og g LTE or NLTE? DAO wth og g 6.5 DO wth og g 7.5 4

21 Summary: o-lte Rat Equatos 4 Compt rat quatos For ah atom of ah o, of ah hma mt w ha: P P I ta: R C > < > < R C R C R C xtato a ozato rats out of -xtato a rombato -xtato a rombato rats to xtato a ozato 4

and one unit cell contains 8 silicon atoms. The atomic density of silicon is

and one unit cell contains 8 silicon atoms. The atomic density of silicon is Chaptr Vsualzato o th Slo Crystal (a) Plas rr to Fgur - Th 8 orr atoms ar shar by 8 ut lls a thror otrbut atom Smlarly, th 6 a atoms ar ah shar by ut lls a otrbut atoms A, 4 atoms ar loat s th ut ll H,