Electron Spin. I = q T = e 2πr. (12.1)

Transcription

1 ectron Spin I Introduction Our oution of the TIS in three dienion for one-eectron ato reuted in quantu tate that are uniquey pecified by the vaue of the three quantu nuber n,, Thi picture wa very uccefu in expaining any characteritic of the hydrogen ato, even ore o than the ohr ode However, experient howed that thi picture wa incopete Thi gap wa fied by the introduction of the concept of intrinic anguar oentu or pin anguar oentu of the eectron Much ike an orbiting panet, an eectron in an ato ha two independent anguar oenta: (i) the orbita anguar oentu (due to orbita otion) (ii) an intrinic or pin anguar oentu, which i anaogou to a panet pinning about it axi To undert the conequence of pin, we ut firt reviit the idea of agnetic dipoe oent To treat the topic of atoic agnetic dipoe oent, we ha ue caica ei-caica idea becaue they are uch iper to work with The fina reut are, however, in agreeent with the fu quantu echanica treatent Orbita Magnetic Dipoe Moent Conider an eectron of a, charge e veocity agnitude v executing a circuar orbit of radiu r in a ohr ato The circuar otion of the eectron contitute a current oop The current i given by r I = q T = e ( πr v) = ev πr (11) e Reca that the conventiona current i directed oppoite to the v direction of the eectron otion Thi current oop contitute a agnetic dipoe, with a agnetic dipoe oent (ao caed agnetic oent) given by = IAnˆ, where A i the area of the oop ˆn i a unit vector whoe direction i perpendicuar to the pane of the oop i given by the right h rue (finger in the direction of the current, outtretched thub point in direction of ˆn ) The agnitude of i given by (1) ev evr = IA = ( π r ) = π r (13) The agnitude L of the orbita anguar oentu i given by L = vr Thu, el = (14) Since L i quantied in unit of, we can write 1

2 e 1 = L, (15) where e ha unit of agnetic oent We define the ohr agneton e (16) Note that = = J/T ev/t Rewriting q (15) give L = (17) In vector for, we obtain L =, (18) where the inu ign appear becaue the eectron ha a negative charge, o it veocity i oppoite in direction to the conventiona current The ubcript indicate that the agnetic oent i aociated with the orbita anguar oentu q (18) i the ae reut obtained uing quantu echanic Since L L are quantied according to L = ( + 1) (19) L =, (110) repectivey, we have = ( + 1) (111) = (11) Hence, i quantied, the vector, ike L, can ony point in certain direction Now if a agnetic dipoe i iered in a agnetic fied, it wi experience a torque given by τ =, (113)

3 which tend to aign the dipoe with the fied (jut ike a agnet upended in the arth fied) There i an orientationa potentia energy aociated with thi torque (externa work i needed to twit the dipoe oent out of aignent with the fied, which increae the potentia energy) The potentia energy i given by U = (114) Thu, the potentia energy i owet when i parae to highet when i antiparae to Stern-Gerach xperient In a unifor fied, no net force i exerted on a agnetic dipoe However, in a non-unifor agnetic fied, a force i exerted on a agnetic dipoe [Show diagra of apparatu] Conider an atoic dipoe in a quantu tate whoe agnetic oent depend ony on the orbita anguar oentu If the fied varie aong the -axi, then one can how that F = (115) Note that i quantied according to q (11): =, with =, + 1,, 1, Thu, if a bea of ato i in a tate with orbita anguar oentu quantu nuber, then the force exerted on an ato due to the non-unifor fied wi depend on the vaue of If the force i tranvere to the otion of the ato, it foow that the defected bea wi be pit into +1 dicrete egent Note that +1 wi away be an odd nuber In 19, Stern Gerach perfored uch an experient with a bea of iver ato It wa found that the bea wa pit into two egent Though thi confired the idea of patia quantiation, it wa inconitent with the expected odd nuber of coponent if = 0, 1,, 3, etc Phipp Tayor did a iiar experient with hydrogen ato in 197 The H ato were prepared o that they woud be in their ground tate, ie, = 0 = 0 In thi cae, the ato bea houd be unaffected by the fied gradient However, the bea wa pit into two egent One i forced to concude that the theory a deveoped up unti thi point i incopete Fro the pitting of the bea into two coponent, one infer that there i an additiona dipoe oent, apart fro that aociated with the orbita anguar oentu Thi additiona dipoe oent i due to the intrinic anguar oentu, or pin anguar oentu, of the eectron The pin anguar oentu i quantied in an identica fahion to it orbita counterpart: S = S = ( + 1) (116) S =, (117) where i the pin anguar oentu quantu nuber i the pin agnetic quantu nuber Ao, 3

4 g S = (118) = g, (119) where g i the eectron pin g factor, =, +1,, Now, ince the hydrogen ato bea i pit into two coponent, one find that for a inge eectron (uch a in a hydrogen ato) = ; =, + (10) (Thi give the required nuber of coponent +1 = ) y eauring the defection of the H ato, it wa found that g = (A ore precie vaue i 003) Thee concuion were confired by any other experient, incuding the Zeean effect, which wi be tudied next 1 Since S =, it foow that the coponent of S can have ony two vaue: ±, iiary for the coponent of the pin agnetic oent = g = (11) 1 The two tate correponding to = ± are uuay caed pin up pin down The copete pecification of the quantu tate of an eectron in the H ato therefore require four quantu nuber: n,,, In the abence of a agnetic fied, the pin up pin down tate are degenerate Thu, the degeneracy of each ψ quantu tate ut be doubed to account for pin Finay, we note that both the orbita pin anguar oenta contribute to the tota anguar oentu: J = L + S (1) Likewie, the tota agnetic oent i given by n = L + S ( ) (13) It i worthwhie to note that q (13) indicate that the tota agnetic oent vector i not antiparae to the tota anguar oentu vector J becaue there i no ipe proportionaity reationhip between the two vector [Do PhT Stern-Gerach iuation] 4

5 xape: Obtain an expreion for the tranvere defection of a bea of hydrogen ato in the ground tate in a Stern-Gerach-type experient The ength of the path i D the peed of the ato i v Soution: Let the tranvere defection be aong the x-direction Then F = = g = The tranvere defection ditance veocity i non-reativitic) Thu, d 1 1 F D = at = v (We aue that any tranvere d D = v (For typica vaue, d 1 ) Zeean ffect We aw that the potentia energy of a agnetic dipoe in a agnetic fied i given by = U (14) If we take the direction of the fied to be the direction, then q (14) becoe U = (15) Conider an ato in a tate in which the tota pin anguar oentu i ero Thi occur in ato in which the eectron are paired, ie, one eectron i in a pin-up tate the other i in a pin-down tate Thi produce an effective ero-pin tate In uch cae, the agnetic oent i due to orbita anguar oentu aone, the orientationa P becoe U = = (16) Hence, the energy of a tate with orbita anguar oentu quantu nuber wi hift in a agnetic fied The energy hift Δ = U depend on the vaue of If the energy of the degenerate quantu tate before the fied i witched on i n, then after the fied i witched on, the energy becoe + Δ = + (17) n n Since there are +1vaue of, the degenerate energy eve wi be pit into +1 eve of different energy, ie, the degeneracy i reoved by a agnetic fied One of the reut of thi pitting i the Zeean effect 5

6 Conider a tranition fro a tate in which = 1 to one in which = 0 In the abence of a agnetic fied, the upper eve i degenerate a inge oton of energy i eitted, where i the energy difference between the two quantu tate If the fied i turned on, the upper eve pit into 3 eve, now there are three poibe tranition fro each of the pit eve to the non-degenerate ower tate The energie of the eitted oton are,, + Thu, a inge pectra ine i pit into three coey paced ine (ince i a copared to the energy difference between the origina two eve) Thi i an exape of the Zeean effect [Show picture of pitting tranition] For hitorica reaon, the pitting of a inge pectra ine into three ine i caed the nora Zeean effect The nora Zeean effect wa atifactoriy expained by Lorent uing a caica ode However, in ore genera cae, the pitting of a pectra ine produce ore than three coponent Thi wa caed the anoaou Zeean effect it coud not be expained caicay The reaon i that in the ore genera anoaou cae, the pin anguar oentu i not ero therefore contribute to the tota anguar oentu the tota agnetic oent The anayi of the genera cae ha to be done in ter of the tota anguar oentu J (except at very high fied) i rather copicated, o we ha not treat it here It houd be noted that not a poibe tranition occur between a pair of Zeean-pit tate Aowed tranition are governed by eection rue, eg, Δ = ± 1, Δ = 0, ± 1 Waveength Shift The hift in waveength of a pectra ine due to Zeean pitting can be eaiy cacuated Since 5 the energy hift ( 10 ev) i very a copared to the tranition energy, we ue differentia So, λ = dλ = d Taking the differentia quantitie to be equa to finite change, we have In ter of aboute vaue, we have Δ λ = Δ (18) Δ λ = Δ (19) To obtain the hift in the frequency of the oton, we have 6

7 f Δ =, o Δ f = (130) h h xape: Copute the change in waveength of the eitted oton in the p 1 tranition for an iaginary hydrogen ato without pin paced in a 00-tea agnetic fied 136 ev = 136 ev = 10 ev Soution: With the agnetic fied off, ( ) For the p tate, = 1, o = +, where = 1, 0, +1, p i the energy of the upper p degenerate eve with the fied off The ower tate i an tate, ie, = 0 o there i no pitting: ower = 1 Hence, upper ower = ( p 1 ) + = + Thu, the hift in oton energy Δ = When = 0 for the upper tate, there i no hift When = ± 1 for the upper tate, Δ = ± Hence, Thu, Δλ = Δ ( ) Δ = ± = ± ev/t (00 T) ev 140 ev n = ( ± ev) = n (10 ev) For the tranition invoving = 1 tate, the waveength decreae (correponding to the negative 3 hift) For = 1, the waveength increae The ie of the hift i, of coure, n Fine Structure A coe exaination of the eiion ine of the hydrogen pectru how that any of the are not inge ine but doubet, ie, two coey paced ine Thi i an exape of fine tructure Fine tructure i due to pitting of eve due to the interna agnetic fied of an ato In the ret frae of an eectron, the nuceu orbit it Thi contitute a current oop, which produce a agnetic fied int at the poition of the eectron The pitting i due to the pin agnetic oent interacting with int : Δ = int 7