1. Orbital Magnetic Dipole Moment ; Bohr Magneton ‘We continue our study of one-electron atom, in which the orbital quantum number [ determines the magnitude of the electron’s angular momentum. Now, an electron revolving in an orbit about the nucleus of an atom is a minute current-loop and produces magnetic field. It thus behaves like a magnetic dipole. ‘We compute its magnetic moment. Let us consider an electron of mass* m and charge —e moving with velocity of magnitude 7 ina rid cireular Bohr orbit of radius r, as shown in Fig. 1. It constitutes a current of magnitude. where T is the orbital period of the electron. Now. T=2mr/v, and so 2D. 2ur From electromagnetic theory, the magnitude of the =i orbital magnetic dipole moment |1; fora current i in He a loop of area A is Fig. 1) H=iA and its direction is perpendicular to the plane of the orbit, as shown. Substituting the value of i from above and taking A = 7, we have ev 2 eur att x np a. wi) We oR? 2 . ‘ F : =, Because the electron has a negative charge, its magnetic dipole moment p,; is opposite in > siae ge at direction to its orbital angular momentum L, whose magnitude is given by L=mvr. veel) Dividing eq. (i) by €q. (ii), we get ra wii) ‘Thus, the ratio of the magnitude [ly of the orbital magnetic dipole moment to the magnitude L of the orbital angular momentum for the clectron is a constant, independent of the details of the orbit, This constant is called the 'gyromagnetic ratio’ for the electron. We can write eq. (iii) asa vector equation :Pa ate -(35, 8 ny ? ‘The minus sign means that jiy is in the opposite directica to L, me ‘The unit of electron magnetic dipole moment is ampere-mel aT), ‘ It is usual to write the last relation as 5 e (iv) fl--5; in |E 1 ) or joule/tes|a The gy 2 is called the ‘orbital g fas BW krbied have tm poesseve symmetry with further equations involving g factors which are different from 1. Bohr Magneton ; From quantum mechanics, the permitted scalar values of orbital angular momentum L’ are given by Lewy) fe ‘where Lis the ‘orbital quantum number’. Therefore, the magnitude of the orbital magnetic ‘moment of the electron is w= +b ih. The quantity ee forms a natural unit forthe measurement of atomic magnetic dipole m ‘Moments, and is called the ‘Bohr ‘Magneton’, denoted by jig. Its value is +1 i by = CL. 060 x 107? &) 663 x 10 J) S927 x 10™ Amt 4mm 4314 x OAL XI kg) Thus, 4 = TTF py. Finally, equation (iv) can be written as Hs ~ ai 2. Behaviour of Magnetic Dipole in External Magnetic Field : Larmor Precession ‘An electron moving around the n Hence, when the atom is placed in about the field direction as axis, frequency of this precession is cal (i)where —e is the charge on the electron of mass am ‘The minus sign signifies that ji? is directed opposite toD Asa result of its interaction ni with: Magnetic field B. the dipole experiences a torque z _given by Pe iv x BR ii) According to eq. (i) and (ii), the torque f*acting on the dipole is always perpendicular to the angular momentum L ‘We know that a torque causes the angular momentum to change according to a form of Newton’s law : pat dr’ and the change takes place in the direction of the torque. The torque ? fon.the electron, therefore, produces a change dL” in L? in a time dt. The change dT’ is perpendicular to L(because the change is in the direction of torque, and the torque is perpendicular to L”). Hence Feo the angular momentum remains constant in aa magnitude, but its direction changes. As time goes on, the vector [ traces a cone around BP such that the angle between L’ and B° remains constant. This is the precession of L’, and hence of the electron orbit , around B If @ be the angular velocity of precession, then L’ precesses through an angle « dt in time dt. From Fig. 2, we see that dt = Hb angle = Lsin® radius popilt TS g I - = dt Lsin® ~ Lsind But from eq. (ii), t = Hy B sin @. sow! .on be Thus, the angular velocity of Larmor precession is equal to the product of the magnitude of the magnetic field, and the ratio of the magnitude of the magnetic moment to the magnitude of the angular momentum Again, from eq. (i), fe =: ‘ lal Tie The Larmor frequency (frequency of precession) is therefore Oty» f- Qn 4nm Iv is independent of the orientation angle @ between orbit normal (L} and field direction B.portance considerable importance in atomic structure as jt my + This theorem is of bd te enables an easy calculation of energy levels in the presence of an external magnetic field 3. Space Quantisation ‘When an atom is ploced in an external magnetic field’, the electrpa orbit precessey about the field direction as axis (Larmor precession). The electron orbital angular momentum vector L’ traces a cone around B” such that the angle @ between E° and BP remains constant (Fig. 3) ; If the magnetic field HF is along the c-axis, the \ component of 1’ parallel tothe field is L, = Loos or cos = Quantum mechanically, the magnitude of the angular momentum [and its z-component L, are quantised according to the relations 9.3) h Levey ft -e ELECTRON ‘Ons)T and heme where ! and m; are orbital and magnetic quantum numbers respectively. Hence the angle @ between C’and the z- axis is determined by the quantum numbers | and m, according as my cute Es WD Since, for a given J, there are (2/ + 1) possible values of my (= 0, £ By #2, sun 1), the angle @ can assume (2/ + 1) discrete values. In other words, the angular momentum vector L’can have (21 + 1) discrete orientations with respect to the magnetic field. This quantisation of the orientation of atoms in space is known as ‘space quantisation’, The space quantisation of the orbital angular momentum vector E corresponding to 1=2 {or Lev Aren} is shown in Fig. 4, For /=2, we ve m = 2,1,0, hos Fi i sothat 2,220 Ao 9 oh jh ba ae Oe, aA c Alternatively, the orientations @ of i respect to the field B (c-axis) are given bycos 0 = H(i + 1) 2 1 7 "16 V6 8 Tg Y 6° 6 = 08165, 04082, 0, 0.4082, - 0.8165 o = 35°, 66°, 90", 114", 145° z Fi Po. We note that L’can never be aligned exactly parallel or antiparallel to B since | m1 i always smaller than Vi( + 1) 4, Electron Spin ‘The Bohr-Sommerfeld quantum theory of elliptic orbits with relativity correction wa in fair numerical agreement with the observed fine structure of hydrogen spectral lines. It however, suffered from two major drawbacks : (i) Firstly, Sommerfeld’s relativistir explanation could not be applied for the spectral lines of atoms other than hydrogen. Fo example, the spectral lines of alkali atoms are doublets. having two close fine-structure components. In alkali atoms the (single) optical electron moves in a Bohr-like orbit 0 large radius at low vel '. Therefore, the relativity effect would be too small to accoun for the large (compared with hydrogen) fine-structure splitting observed in alkali lines, (i Secondly, the simple quantum thecry failed to explain anomalous Zeeman effect, that is the splitting of atomic spectral Lines into four, six or more comonents when the light source was placed in an external magnetic field. In an effort to remove these two drawbacks of the theory. Goudsmit and Uhlenbeck Proposed in 1925 that an electron must be tooked upon as a charged sphere spinning about its own axis, having an intrinsic (built-in) angular momentum and consequently ‘au intrinsic magnetic dipole moment. These are called ‘spin angular momentum’ 5” and ‘spin magnetic dipole moment’ [2* respectively. (These are in addition to the orbital angular momentum L’ and orbital magnetic dipole moment ji). Let us write the magnitude S of the spin angular momentum of the electron in terms of a ‘spin quantum number’ s , as we do for the orbital angular momentum L in terms of orbital quantum number f. Thus which conforms to the observed fine-structure doubling*. Thus ht _ Wh 2x” 22n The component of 5” along a magnetic field parallel to the z-direction is** s=¥! ++1) i)oc be takes (25 +1) =2 values which where m, isthe ‘spin magnetic quantum num! +s and —5, thats Thus S28 ic ratio for ch Itnas been concluded fom experimental dat that the gyromagnetic ratio for electrons (. & ¢lectron orbital motion. Thus, twice the corresponding ratio" | = 2 py for the spin magnetic moment {17 of electron is elated tothe spin angular momentum ay ‘The minus sign indicates that ji? is opposite in direction 10 F because electron negatively charged), Teis usual (o write the lst elation as | me- #(35) 5 i, where nen. The quantity g, is called the ‘spin g factor’. We can express spin magnetic dipole moment in terms of Bohr magnetor Ha (= € 4/4 xm), Equation (iv) then becomes > ate rm w=" Gam > (H) ‘The possible component of [1 along z-axis are given by sing Be Bee Am =- a mn, * Iby equation (i) == Bem, Now,g, = 2 and m, = + } (by eq, ili) Hee = + He. The spinning electron proved to be successful in explaining not only fine structure andthe anomalous Zeeman effect but other atomic effects also, although electron spin was introduced as a postulate, but in 1928 Dirac proved on the basis of relativistic quantum mechanics that an electron must have an intrinsic angular momentum and an. intrinsic ‘magnetic moment which were just the same as attributed to it by Goudsmit and Uhlenbeck ‘Thus, electron spin was put on a firm theoretical foundation, The practical evidence of is «existence came from Stern-Gerlach experiment 5, Vector Model of Atom : Coupling of Orbital and Spin Angular Momenta ‘ = {otal angular momentum of an atom results from the combination of the ori! Ui angular momenta of its electrons. Since angular momentum is a veel ‘Wuantity, we can represent the total angular momentum by means of a vector, obtained 6Ythe addition of orbital and spin angular momentum vectors. This leads to the vector model of the atom. given by L=Nii+a * and its z-component is h am *, where | is orbital quantum number and m, i is the corresponding magnetic quantum number, with values MMT WD le hel, Similarly, the magnitude of the spin angular momentum is given by pos Ss vse +1) oR and its z-component is S, =m, In where sis the spin quantum number (which has the sole value + ~ and m, is magnetic spin quantum number (m, = +4 = +5), ‘The {otal angular momentum of the one-clectron atom, J” is the vector sum of Poa Pas Dand 3?thacis PaP+?. ‘The magnitude and the z-component of J*are specified by two quantum numbers jand 'm,. According to the usual quantisation conditions 7 A J=VG+) In h and moe J is called the ‘inner quantum number’ and my is the corresponding magnetic quantum ‘umber. The possible values of mj range from +j 10 ~j in integral steps : Myo O sae ft Ley Let us obtain the relationship among the various angular momentum quantum numbers, Since J, ,L, and S, are scalar quantities, we may write J= Lk, tS. This gives mj = my £ my. mySince my isaminlegerand m, is 4. so mj must be hall-integra. The maximum values are j. land s respectively. Therefore, from the last expression, we have Like m,, j is always half-integral, Since J, 1 and S*are all quantised, they can have only certain specific relative orientations. In case of a one-electron atom, there are only two relative orientations possible, comesponding to pelts, sothat Job and gel-a, so that dems “The two ways in which i and S’can combine to form J%when 1= 1, s=4) are shown in Fig. 5 of my. m, and mt, = “? (F135) “The angular momenta of the atomic electron, L’ and 5? interact magnetically; which is known as ‘spin-orbit interaction’, They exert torques on each other. ‘These internal torques do not change the magnitudes of the vectors L” and 5? but cause them to precess uniformly around their resultant J (Fig. 6). If the atom is in free space so that no external torques act on it then the total angular momentum J’is conserved in magnitude and direction. Obviously, the angle between D’ and §? remains invariant. From the cosine law, the have Pals $+ 2L5e0s(0,5)). . £-v-s 2608 (1,8) = STG . W=1d + = sG+1 IG+ Noe) This is the vector model of one-electron atom. It can be extended to many-electron atoms. The vector model enables us to explain the phenomena which could not be understood from Bohr-sommerfeldtheory such as fine structure of spectral lines, anomalous Zeeman effect anu hyperfine structure, ‘The conception of spinning electron, which is a salient feature of the vector model, is responsible for the fine-structure doubling of the alkali spectral lines. The energy of an electron in a given quantum state will be higher or lower (depending upon the orientation of its spin vector 5°) than its energy in the absence of spin-orbit interaction, This means that each quantum state (except state for which is zero) is splited into two separated sub-states. This results in the spliting of every spectral line into two component lines, The rmumerical value of the splitting agrees with that experimentally observed. Similarly. spin-orbit interaction together with relativity correction explains the hydrogen fine-structure. Inthe vector model, L” and 5° precess around J When the atom is placed in an external magnetic field By then J” precesses about the direction of B, while { T’and 5° continue precessing about J*(Fig. 7). The 7 diserete orientations of J” relative to B, which involve slightly different energies, give rise 1 anomalous Zeeman effect in agreement with experiment. Like clectrons, the atomic nuclei also have smaller intrinsic (spin) angular momenta an* magnetic moments. When these vectors are added to the atomic model, the experimentally observed “hyperfine structure’ of spectral lines is explained. 6. Spectroscopic Terms and their Notations The quantised energy states (simply called as quantum states) of an atomic electron are described in terms of the quantum numbers n, 4s and j. The electrons having orbital Quantum number |= 0,1,2,3,4 oon are named as 5*,p, dif. g ou. electrons, The atomic slates in the atom are specified by writing the corresponding principal quantum ‘number alongwith these letters. Thus, an electron for which n = 2 and 1 = 0 isina 25 ‘omic state, and one for which n = 3 and [= 1 isina 3p atomic state, and so on The energy levels of electrons of an atom are called ‘terms’ of the atom. ‘The Corresponding energies, expressed in wave numbers, are called ‘term values’, For a ‘one-electron atom, the energy levels corresponding to 1=0,1,2,3,4, cu, ane called SPD FG cue terms respectively, By spin-orbit interaction, each energy level al a given | is splitted into two sub-levels corresponding to jsltss and jul-sel1 1 at : with the exception of $ term (J = 0) for which j = > only. Gy which determines the total angular momentum of the electron, cannot be negative). The number of different possible orientations of and 5” and hence the number of different possible values of j is known as the ‘multiplicity’ of the term. It is equal to (25+ 1). Thus, the multiplicity of the terms of a one-electron atom is (25 + 1) = 2. It is added as a left superscript to the term symbol, thus : 75,P,2D, and so forth. These are called “doublet terms"*. In addition, the j value is added as a right subscript. Thus i Full Notation 172 Win ME Psa. Pia 5/2, 3/2 2D Dis Sometimes the principal quantum number is also added to the term symbol, such as Py Hee n= d= 1s = 5, j= V2. In an atom containing several electrons, the orbital angular momenta G. i=1,2,3 of the individual electrons couple among themselves to give a Fesultant electronic orbital angular momentum L’ for the atom whose magnitude is NVL(L + 1) h/2m, where L is the orbital quantum number for the atom. Similarly, ist = 1,2, 3)... add up to give a resultant spin angular momentum 5’ of magnitude Weep * where S is spin quantum number for the atom. Finally, T and 3° couple to give the total electronic angular momentum J” for the atom, with magnitude WF) £ J is the inner quantum number for the atom, The terms of the atom may now be singlets, doublets, triplets and so forth. They are denoted in the same way as the terms of a one-electron atom. We shall return to this point in a subsequent chapter.4, Spin-Orbit Interaction ‘The spin-orbit interaction is an interaction between an electron’s spin magnetic dy moment and the internal magnete field of an atom which arises from the orbital man the electron through the nuclear electric field, Since the internal magnetic field is related) the electron’s orbital angular momentum, this is called the spin-orbit interaction. Its relatively weak interaction but 1s responsible, partly, for the ‘fine-structure of the excita} states of one-electron atoms* TLet us write the electric field E’, in which the electron is moving, as 2 gradient of potential function V(r), where ris electron-nucleus distance. EB’ = grad V(r) + -t av, [ gad =] ro dr ~ The term © is aunit vector inthe radial direction which gives. Bits proper direction. ‘The magnetic field in a reference frame fised with the electron, arising frp orbital motion of the electron with velocity ¥" inthe electric field B’ (due to the nucle] is given by ‘Now, the orbital angular momentum of the electron is Dem expression may be written as mor dr ‘The electron and its spin magnetic moment {F2 can assume different orientation” intra magni eld B of the atom, and its potential energy is different fo" ait ions. The expression for the magnetic potential energy of orientation S__4a= - 0B € ig 2 Bul fy =~ a 2m \e where g, = 2. Therefore, the last expression can be written in terms of the electron’s spin angular momentum Sas af, = £5. ot 2 m Substituting for B from above, we get ec idvne p MEL = 5 sc mar dr ‘This is the energy in a frame of reference in which the electron is at rest. On relativistic transformation to the normal frame of reference in which the nucleus i at rest, the energy is reduced by a factor of 2. This is known as “Thomas precession’. Taking this into account, the spin-orbit interaction energy is Be Pe a date dr Toexpess this in terms of quantum numbers 4,5 and j. we write Pa ly 8 Taking the self dot product of this equality, we have Peps (Ps FP). (24 PF) =D.74 8-94 29.2 PT. Pee. 7 a ee So P.Ce5[PP-o.0- #54 1 2 -j[7-v-s] 1 a tigen-te+y-see 0] glide nt seen] __e# of. 1 avn Thus, bE, . = wawalld #1 te t= siee ty] te In general, + 2412 nor constant during the ekcton motion. Heace we must ake its average value over the unperturbed motion. So eh oy. _ Ve AE, = aaralio+ y-s ysis] et G) This is the general expression for the spin- iteration energy ofan atom. The average value L va is calculated by using the potential function V(r) forthe given atom and r dr the radial probability density for the state of interest For ahydrogen-like atom in which the electron moves ina Coulombian field. we have l_ ze VO Tre - tod 90 aig _l_ Ze dr 4 Rey Therefore, equation (i) can be written as——_— T [io+ p-ms+ st v5 aye -—— SE = Tye (lemme) s evaluated by is Ra using a generating function representation for ‘The resultis provided | > 0 ‘The average value of vr is the hydrogen radial cigenfunctio a — 1 abatifts 3} 1) is the radius of the smallest B johr orbit of the hydrogen where a| = 4% &— anime om, Introducing this value in the last expressions we get a j AB eh Zz jge- tds DG 4D} _ zee Cae + 7) +1 + worked out to the Following form : ? —Rathe [igen teense + n]. Tanita} ‘ where (= ae] is Rydberg constant for an infinitely heavy nucleus, and wie ‘This equation i AE.» : £__ | ig fine-structure constant. 2eghe “The term shift due to spin-orbit interaction is ats =~ Fee Rao Z = [fg+y-tg+ nse ey). ani(isy +d Now, for a hydrogn-like atom (single electron), we have 1 gen 2 and $0 jaltsets}. Making this substitution, we get iG+ Mel) -s@ele! for jeted | a-(+l for j=! ‘The term shifts corres 1 pondingto j= 14} and j= 1-4 ae Rat af, / = - ; i aei(is sueykot 1 ai = an't(t+ )a41) Gh a+b gin-orbitinteracton causes cach term (energy evel) ofa giver | to split into ‘of different j’s, one displaced upwards and the other dos wards. The ce between them is AT = aT," - ah! oon antl fers +) “Pits p= 1097 x 10 me! and a = 1/137, weget Zz 1 AT = $84——= miti+ 1) z AT = $84 5" __ em"! fe wile). | ps tespin-rbit interaction spiting increases with increasing atomic number (Z) aad | ponalle for higher and higher [. Its zero for Sterms (J = 0) for which j = 4 |. “2. Quantum Mechanical Relativity Correction Besides the energy shifts duc to spin orbit interaction, the relativistic effect is equally |inportan as it produces energy shifts in the hydroven atom comparable to those produced iy spin-orbit interaction. In order to calculate the relativistic shift, we consider the relativistic Hamiltonian fancion HY ofthe electron. We know that erry, where X is relativistic kinetic energy and W is potential energy. Now, Ke Gie + mci! — myc?, where p is linear momentum and nigis rest mass of ‘decron, Thus webreailles *y Img &myc’ Th he change in Harntonian due to relativity isWe ent a perturbation term. The first-order enersy shift can be found by “This may be regarded 35 cratuating itsaverae value OVE the unperturbed wave function. ‘The equivalent mechanical Hamiltonian operator w een differential operator of p #8 ~ = 2. Therefore, the perturbed par could be ‘of the quantum se 1 Bmgc 16K ave fanetion of the hydrogen atom, the fistorder cncrey shift IF Yo is the unperturbed w ‘due to relativity is given by ‘ie -fet (a jeu Con evaluating this integral. we obtain mer nowe he . ‘i iat and a is the fine-structure constant, The relativistic where R. is the Rydberg constant term shift is AE, Meg 3. Hydrogen Fine-structure uc to different effects combine linearly corrections to spin-orbit ‘The first-order perturbation istic effect in Therefore, the net term-shift due nydrogen-like ato AT = At, + AT, interaction and the relativ’ jge n-ne y-sar dl in a = + -td+ = sGtN | ai(ee ger» } 1 where 3 = 3 Cn substring j = 1+ and j =f — 3 inturn, we cba 1 for j 2443" (+ D-C+ D-see del a-('+ 1)This equation is identical with Sommerfeld’s relativistic equation for the energy levels of hydroger-like atoms; namely ar SPE! 4 xceptthat £ has been replaced by j + }~-This cquation was independently obtained by Dirae from a completely relativistic quantum mechanical treatment of the hydrogen-like atoms, and hence known as Dirac equation. Let us now compare the energy levels of the hydrogen atom obtained from Sommerfeld’s theory with those obtained from —Dirac's theory. Taking R= 1097 x 10'm', a = 1/137 and Z= 1 (for hydrogen), we have = 841 3) = S841 3) et oe OE &)'= (i al Sommerfeld Formula waar = 4/1. lr Prd” “fe Dirac Formula. an “lat ist ST is the term-shifi from the Bohr level. We consider Bohr levels corresponding to nw 23 {Bohr Level | Sommerfeld Levels Cra ale a & | attem) 1 jf= ly) | atten) ' 1 146 0 1 1.46 2 2 0091 1 21 0-091, 0456 ; 22 1 0456 0 1 0456 2 O08 2 53 | o018.00s8 } 2°2 3 2 0084 1 ak 0-054, 0162 az 0162 0168296 aspredicted by Bohs, Sommerfeld, and Dirac for ly. We see that the energy levels hydrogen atom, ; Fig. | (a), (b) and (c) nespestivel BOHR LEVEL _m=2 ‘Tne cnerg) levels of 1,2.) are shown in cevel_me a ee) S0MMERFELD. urac ww tba BOHR LEVEL n=3 a - gangiaal Be3 _Ipoipenr! EEE? 3°55 fuz | 9 bat, J292 BaP 0054 en? 4 A 3 Daye 3 Py, kel 19-162 em™ 1 i 1 ‘4 i SOMMERFELD! «od Fig.) ‘due to Sommerfeld and those due to Dirac are essentially the same. ‘This coincideas ‘oceurs for the hydrogen atom because the errors made by Sommerfeld in 4gnorint spin-orbit interaction and sn using classical mechanics to evaluate the enerEY shift selativity happen to cancel forthe case of hydrogen alom. ‘The only difference between the results of the wo theories is that Dis teat ot Sommerfeld, predicted « double-degeneracy of mos levels, because the ene"8) deren is the cuenta nanos 8 and j, bui noton the quantum number [. Generally ther two values of | correspondit j eve! my dosh ponding to same j. Hence Dirae theory predicts that most![Let us now consider the fine-structure of H,-line (h = 3m = 2) on the basis of Dirac theory. The fine-siructure energy levels, as deduced above, with proper spacings in terms fem” | and designations are drawn in Fig. 2 ab n. Ta Db $108 em"? == 0:329Cn'—= —v (Theoretical Structure of Has predicted by spin-reiativity quantum-mechanical theory) Fig.2) ‘The selection rules for dipole transitions are aed! Aj= 0,4) but j= 0+ f=, ‘These rules allow five transitions, resulting in five components spaced as shown in Fig. 2 ‘The relative intensities of these componcnis, as shown, have been calculated on the s3sumption that all the quantum siates corresponding ton = ¥ are equally excited. Comparison with Experiment : The theoretically deduced structure of the Hg line has a general agreement with Hansan’s observauion. Still « is far from perfect agreement,= extensively studied the fine-structure of the Hg line, and also of the oe ee ineeeh spectrograph and a microphotometer. He found two main De ier "between theory and experiment set) Th-component La was observed tobe weaker than the component Il b, which i one evan theory. Similay, the component IIIb wid observed’ ke te Thuch = than as predicted theoretically. These deviations could, however, ha plained as due to unequal excitation of the m= 3 levels. (B) The separation between the two min components .a and Hb was observed io he 0319 cm! against the theoretical separation of 0:329 em”. Sinitaly, the separation between II b and IIT & was observed to be 0:134 em’! in stead of 0:108 env Lamb Shift: Pasternack, in 1938, pointed out thatthe discrepancy inthe separation between the main components of the H line could be explined by assuming the ware 25,2 to be about 0-03 cm” ' higher than 2*P,,, in contradiction to Ditac’s theory Which haa shown them to have exactly the same energy. Lamb and Retherford, in 1947, provey the correctness ofthis assuruption. They performed a microwave experiment on hy drogen toms and showed tha for hydrogenlike atoms the sates ofa particular” value hong terms with the same j value but different 1 values, such as 27P,/: and 275,,s, are nop degenerate, but are separated. Such a separation is known as ‘Lamb shift’, Tye plan of the Lamb-Retherford experiment is shown in Fig. 3. Molecular hydrogen (Hi, ) entering an oven O- was dissociated into atomic hydrogen which left the oven a. Passed through slits 5, S. This beam of hydrogen atoms passed through a Vacuum did: in ‘which electrons were being emitted from heated cathode K and accelerated toward annie A. Some of the mormal atoms (1 *5,/2) passing through this region collided with the electrons ard were excited into 2°5,..27P,q and 27Pyq siates. These excited atoms proceeded toward a tungsten plate P and collided with it. During this journey the atoms in the 277i and 27P\ states retumed to their ground state 175, /4, but those in the (metastable) sate 2 "5, * could not do s0 because of the selection rule Al @ 0. These metastable atoms returned to their ground state by collision with the plate P- from whichtherefore, electrons were emitted. The stream of electrons so produced was collected and passed on to a galvanometer whose reading was a measure of the metastable atomic bean intensity. Any mechanism which causes the metastable 2 *5),. aloms to undergo a transition to the 27Piq state will result in a fall in the galvanometer reading, which is sensitive only 1 metastable atoms. Such transitions were induced by passing the atoms through a waveguide WW in which microwaves of variable Irequency were being generated ‘was found that at a certain frequency the metastable atomic beam intensity suddenly reduced. It was concluded that the microwaves of this frequency were absorhred by the 235,q atoms which were excited to the 27P,,2 state from which they decayed atonce to- the ground state. Thus, the atoms reaching the tungsten plate were in their ground state and could not eject electrons from it. Hence this frequency was a measure of the term difference between 275yq aud 2*Pyq states. Such measurements showed that this difference was not 0365 cm’ ' (as predicted by Dirac) but 0-0353 em’ ' less. This meant thatthe state 275,72 was higher than 27P, 2 by 00353 om”! (Fig. 4) 2 2 Pin 0-365 em"? METASTABLE: ‘STATE ra 2Sye 2 Pp GROUND 2 STATE "Si, Fg.) In practice, instead of adjusting the frequency of the microwaves for maximum reduction in the metastable atomic beam intensity, the energy levels themselves are ‘adjusted by means of a magnet NS, under Zeeman effect. ‘Soon after the discovery of this shift, Bethe showed that aevised theory of interaction hetween matter and radistion causes all 5 terms to be raised by an amount which agreed well with the experimental value for hydrogen. (Hence Lamb shift 1s also called as Fadiation shift)$e igh of Laz shift is drawn in Fg. S “Tae erm diagrams revised in te 2-036.on 3g 2 2 sags Be nes US 1 ¥ Py (Fine structure of Hy with Lamb shift) (Fig. §) The line 37Pyq + 245 jq thus mo longer coincides exactly wih 3'Dyq427Pia. If the 3*Pyq and 37Dyq levels were equally excited, it can t calculated that the intensity ratio of these lines would be 2-08 : 5-00, so that the two would actually be observed as a single line shifted by 208 - 28 Pls 1 5.09 ~ 00353 cm = 00ldcem, “1 which is im good agreement with William's observation of a separation of 0-319 « ‘between the two main components I a and II 6 (Fig. 2).