s of the two electrons are strongly coupled together to give a l couple together to give a resultant APPENDIX I

Transcription

1 APPENDIX I Cupling Schemes and Ntatin An extensive treatment f cupling schemes and ntatin is given by White r Kuhn. A brief review is given here t allw ne t read this manual with sme insight. The mtins f the electrns in an atm are gverned by: (a) the electrstatic frces f attractin between the nucleus and electrns and f repulsin between pairs f individual electrns; (b) the magnetic frces due t rbital mtins and the spins f the electrns. These interactins are cmplex but fr particular cnfiguratins ne may make simplifying assumptins in rder t perfrm calculatins. The assumptins specify a mdel fr the system and different mdels will require different ntatins which specify the varius angular mmenta. Atmic states can be described by a set f numbers and letters which specify the angular mmenta. Often a real situatin can be described adequately by different mdels and the same set f states will be described in different bks by different ntatins. In fact, fr the same mdel there may be different ntatins used because a ntatin which desn't specify as many quantities may be sufficient t describe a particular example and in fact may be easier t use. In the examples that fllw ne electrn systems, such as the alkali metals, culd be used but all examples will be tw r mre electrn systems taken frm He and Ne. Russell-Saunders r L, S Cupling Russell-Saunders cupling arises frm the predminance f electrstatic ver magnetic s f the tw electrns are strngly cupled tgether t give a interactins. In this mdel, the spins resultant S = s + s2 while their interactin with the rbital angular mmenta are much weaker. The rigin f the strng interactin which cuples the spins is the electrstatic repulsin between the electrns due t the Pauli exclusin principle. In this mdel the rbital mtins f the tw electrns cuple tgether als because f electrstatic interactin. The rbital mtins are described by the rbital angular mmenta s we may say that the rbital angular mmenta rbital angular mmentum L = l + l2. Finally, bth L and S cuple t frm the ttal angular mmentum J = L + S. Nte that J has magnitude J which is the ttal angular mmentum f the atm. (Frm quantum mechanics, J is really J(J +) where J is a nn negative integer. Russell-Saunders ntatin is given by n 2S + LJ l cuple tgether t give a resultant Where n is the principle quantum number fr the rbit which specifies the shell (larger n fr larger rbits), where numbers are substituted fr S and J, and where the letters S, P, D, F are substituted fr L = 0,, 2, 3, respectively. Mst ften the designatin f n is nt included in the ntatin. As an example, cnsider the grund state f He. D nt cnfuse the designatin f the cnfiguratin f the electrns with the designatin f the state. There are tw s electrns in the n = state and thus the cnfiguratin f the grund state is s 2. These tw electrns have ppsite spin s s + s2 = 0 and since bth electrns have l = 0 then L = 0 and f curse L + S = J = 0. Therefre in L, S ntatin the grund state f He is S. If ne f the electrns is prmted t the n = 2 state we can have the cnfiguratin s2s. This can als be a S state if the electrn spins are ppsite but since the excited electrn is in the n = 2 state this is designated as a 2 S r t emphasize that it is an excited state we write as in Appendix II, He * 2 S. The grund state f Ne is als a S state and can be described using Russell-Saunders ntatin.

2 Hwever, fr excited states f Nen, calculatins using this apprximatin d nt give the crrect energy levels and a different mdel using j, l cupling is emplyed. j,j and j,l Cupling In rder t understand j, l cupling and t distinguish it frm L, S r Russell-Saunders cupling ne shuld first understand the mechanisms f interactin between the electrns and which mechanism is dminant is a particular atmic system. As mentined abve, in L, S cupling we have the extreme where the electrstatic interactin dminates the magnetic interactin and the cupling fr tw systems can be symblically written {(ll2)(ss2)} = {LS} = J At the ther extreme, the primary interactin is the magnetic interactin between the spin and rbital mtins f the individual electrns. The spin electrns are strngly cupled t frm The electrstatic frces between the electrns nw cause the different resultant cupling and may be symblically written as s and rbital l angular mmenta f the individual l + s = J the ttal angular mmentum fr an individual electrn. j t be cupled and t frm the J = j + j 2. This extreme, which is the secnd mst cmmn type f cupling is called j,j {(ls)(l2s2)} = {j j 2 } = J A third intermediate type f cupling which is valid fr Ne is called j,l cupling and it ccurs when sme f the magnetic interactins are small and thers large cmpared with the electrstatic effects. In Ne, the cnfiguratin f the grund state is s 2, 2s 2, 2p 6 and when ne f the electrns is excited, the system is s 2, 2s 2, 2p 5 + "excited electrn" r fr shrt, 2p 5 + "excited electrn", r "parent in" + "excited electrn". The parent in has ne p electrn missing frm a clsed shell and s the term diagram f the in is the same as if it had nly ne p electrn except that terms will be inverted. The parent in cre thus has levels 2 P/2 and 2 P3/2 (with the 2 P3/2 level lying deeper) and the energy levels in Ne apprach these tw limits as is shwn in Figure 2. Russell-Saunders ntatin is used fr the parent in because the spin rbit interactin in the in is large. When the excited electrn is added t the system, the rbital and spin vectrs in the in cre are nt unlinked. This means that the electrstatic interactin between the external electrn and the cre is weak cmpared t the spin rbit interactin in the parent in. Hwever, the electrstatic interactin between the excited electrn and the cre is strng cmpared t the cupling f the spin f the electrn t the cre. This means that the j f the cre and the l f the excited electrn cmbine t frm a resultant K = j + l and hence the name j, l cupling arises. The spin f the external electrn then interacts with K t frm a resultant J = K + s (i.e. J = K ½) and thus j,l cupling may be symblically written as {[(ls)l2]s2} = {[ j l2]s2} = {Ks2} = J It shuld be nted that j, l cupling and j, j cupling are identical when the excited electrn ccupies an s rbit. In this situatin, since l2 = 0 then K = j and j2 = s2 and hence { j j } = { K 2}. 2 s 2

3 Racah ntatin is used t describe this cupling and is given by ( 2S+ LJ)nl[K]J where the ( 2S+ LJ) refers t the parent in and fr Ne can be either 2 P/2 r 2 P3/2. The n and l refer t the excited electrn and fr Ne in this experiment we will nly cnsider s and p electrns where l = 0 and l = respectively. This ntatin can be shrtened by the use f primed and unprimed symbls as in the Racah ntatin f Table I (Appendix II). Fr an excited s electrn in Ne, there are fr states, tw frm the parent in in the 2 P3/2 state and tw frm the parent in in the 2 P/2 state. Fr the parent in in the 2 P3/2 state, j = 3/2 and l = 0 s K = j + = 3/2. Fr the parent in in the 2 P/2 state, j = /2 and l = 0 s K = j + l = /2. Since J = K ½ then J = 2,, and 0. The fur states are n s [/2] n s [/2] 0 ns[3/2] 2 ns[3/2] 3

4 Fr an excited p electrn in Ne there are 0 states, six fr the parent in in the 2 P3/2 state and fur fr the parent in the 2 P/2 state. Fr the parent in in the 2 P3/2 state, j = 3/2, l = and s K = j + l -, j - = 5/2, 3/2, /2 which gives rise t states with J = 3, 2, 2,, and 0 and term designatins np[5/2, 3/2, /2]3, 2, 2,,, 0. Fr the parent in in the 2 P/2 state the fur excited states are n p [3/2,/2] 2,,,0. rules The subscript J is used t determine which transitins are allwed accrding t the selectins where J = 0, is frbidden. J = 0 J = 0 An lder and still cmmnly used ntatin fr the excited states f Ne is the Paschen ntin which is given in Table I. Nte that the 3s levels in the Paschen ntatin are the 5s and 5s levels f the Racah ntatin. Paschen ntatin was an attempt t fit the Ne spectrum t a hydrgen-like thery. Althugh the apprach did nt wrk, the ntatin remains and the numbers and letters can be cnsidered t be simply the names f the states and nthing mre. 4

5 APPENDIX II Excitatin Transfer in He-Ne Mixtures Figure 3: Energy Level Diagrams fr Helium and Nen A cllisin invlving interchange f ptential energy between excited atms such as is called a "cllisin f the secnd kind". The difference in energy E cmes frm r ges int kinetic energy f the clliding species. Fr resnant cllisins, ( E 0), crss sectins tw rders f magnitude larger than gas kinetic crss sectins are realized. When E 800 cm - (0. ev), the crss sectin fr excitatin transfer is negligible. Ttal spin cnservatin ( S = 0) is favured in a cllisin invlving excitatin transfer. Thus fr the fllwing tw prcesses, the first is mre likely. Len J. Radziemski, Spectrscpic ntatin fr the energy levels f helium and nen, Optics News, January 989, p5-6. 5

6 Figures 3 and 4 illustrate the energy level cincidence fr the excitatin transfer reactin frm He 2 S metastables t the Ne 3s states. The level 5s in figure 3 is the 5s (3s2 and 3s3) and 5s (3s4 and 3s5) f figure 4. Als the 3p[K] levels f figure 2 and table I which are in Racah ntatin, are the 2p levels in the Paschen ntatin f Table. Althugh the grund states f He and Ne can be described in terms f L, S cupling, the excited states f Ne cannt as was pinted ut in appendix I. If the excited s states f Ne culd be described by L, S cupling, then, since there is ne missing p electrn (r hle) with (l = and s = /2) and ne s electrn (l = 0 and s = /2), there wuld be fur excited states. Fr S = 0 there wuld be a singlet P and fr S = there wuld be three triplet states 3 P2, 3 P and 3 P0 with the singlet state having higher energy than the triplet states. Why? The magnetic interactin prduces an energy difference between the three triplet states and since the p electrn is a result f the in cre being shrt ne p electrn frm a clsed shell, the three states are inverted in energy frm what ne wuld nrmally expect frm a single p electrn cmbining with an s electrn. This means that in the triplet, the 3 P0 state is the highest in energy and the 3 P2 is the lwest. If L, S cupling were valid, ne wuld expect the three triplet states t be clse tgether cmpared t their separatin frm the singlet state. In figure 4, the fur excited states arising frm the ps electrnic cnfiguratin are labeled with bth Russell-Saunders and Paschen ntatins. It shuld be bvius that the predictins f L, S cupling are nt really valid but the Russell- Saunders ntatin is included t indicate the state which wuld be a singlet ( ) if it were valid. As was pinted ut in appendix I, the cupling is really j, l cupling which fr the ps electrnic cnfiguratin is equivalent t j, j cupling which may be easier t visualize (see White p 97). As Russell-Saunders 6

7 cupling breaks dwn and prceeds tward j, j cupling (j, l cupling is intermediate between the tw), the fur states predicted by Russell-Saunders cupling change energy and mve t states with the same J values in j, l cupling. Using Figure 4 and Table I ne can see that this is true. With this infrmatin in mind ne can see why in a mixture f He and Ne the preferential selective excitatin f the Ne 3s2 level rather than the 3s3 level ccurs in the endthermic reactin in which ttal spin is cnserved, and a E f 386 cm - (abut 2 kt) has t be prvided frm kinetic energy supplied by the discharge. TABLE Wavelength (nm) Paschen Ntatin Racah Ntatin s2-2p s2-2p s2-2p s2-2p s2-2p s2-2p5 [632.8] 3s2-2p s2-2p s2-2p s2-2p [/ - 3 [/ 2] [/ - 3 [/ 2] 3 [/ - 3 [/ 2] 2 [/ - 3 [/ 2] [/ - 3 [/ 2] 2 [/ - 3 [/ 2] [/ - 3 [/ 2] 2 [/ - 3 [/ 2] 0 [/ - 3 [/ 2] [/ - 3 [/ 2] s3-2p s3-2p s3-2p s3-2p s3-2p6 [/ 0-3 [/ 2] [/ 0-3 [/ 2] 3 [/ 0-3 [/ 2] 2 [/ 0-3 [/ 2] [/ 0-3 [/ 2] 2 7

8 63.4 3s3-2p s3-2p s3-2p s3-2p s3-2p [/ 0-3 [/ 2] [/ 0-3 [/ 2] 2 [/ 0-3 [/ 2] 0 [/ 0-3 [/ 2] [/ 0-3 [/ 2] 0 Wavelengths are fr transitins between 3s and 2p levels in Ne. Levels with dd parity have a "" as a right superscript. Levels with even parity have n right superscript. TABLE I (cntinued) Wavelength (nm) Paschen Ntatin Racah Ntatin s4-2p s4-2p s4-2p s4-2p s4-2p s4-2p s4-2p s4-2p s4-2p s3-2p - 3 [/ 2] - 3 [5/ 2] 3-3 [5/ 2] 2-3 [3/ 2] - 3 [3/ 2] 2-3 [3/ 2] - 3 [3/ 2] 2-3 [/ 2] 0-3 [/ 2] - 3 [/ 2] s5-2p s5-2p s5-2p s5-2p7 2-3 [/ 2] 2-3 [5/ 2] [5/ 2] [3/ 2] 8

9 s5-2p s5-2p s5-2p s5-2p s5-2p s5-2p 2-3 [3/ 2] [3/ 2] 2-3 [3/ 2] [/ 2] [/ 2] 2-3 [/ 2] 0 When using this table ne shuld be aware that nt all lines listed actually appear because sme transitins are frbidden. Als, sme f the lines, especially thse with lwer p values in the Paschen ntatin may be very faint. 9

10 APPENDIX III BEAM DIVERGENCE The fllwing frmulae and facts are useful in determining the bserved beam divergence.. Equatin (56) f Blm (page 74) gives the spt size at the mirrr surface inside the cavity. Nte that is the /e radius f the field strength r the /e 2 pint f the intensity. The radii f curvature f the mirrrs, b and b2 are identical and are equal t 3 m. The separatin f the mirrrs is d. 2. The exit mirrr is really a thin plan-cncave lens s that inside the laser is very nearly just utside the laser. 3. The curvature f the wavefrnt inside the laser is just the radius f curvature f the mirrr which is 3 m. 4. Since the exit mirrr is a plan-cncave lens, the radius f curvature R just utside the laser is the radius f curvature just inside the laser divided by n. Yu may take n, the index f refractin, t be abut.5. Yu may derive this change in curvature using single surface frmulae (see Jenkins and White pp ). The curvature f the wavefrnt inside the mirrr (which is nw acting as a lens) is the curvature f the mirrr. The curvature changes as it exits thrugh the plane surface int air. 5. Facts () and (4) allw yu t calculate the beam radius and the beam curvature R f the gaussian beam just utside the exit mirrr f the laser. 6. Equatin (69) f Blm (page 03) allws yu t find the effective distance z f the beam waist r fcus frm the exit mirrr. (This distance is different frm what yu wuld calculate if yu were inside the laser.) 7. Equatin (70) allws yu t calculate, the beam waist r beam radius at the fcal pint. 8. Equatin (7) then allws yu t find the beam radius at any psitin z utside the laser. Here z is z plus the distance frm the mirrr t the pint f measurement. 9. Fr very large distances z, the half angle f beam divergence in radians is just /z as z. This value f the half angle f the beam divergence is /( ). 0. Calculatins using (7) shuld be cmpared with measurements f spt sizes at least tw different distances. 0

11 APPENDIX IV Chapter f Principles f Lasers by O. Svelt See r

12 APPENDIX V Lngitudinal Mdes in a Laser Cnsider the situatin f tw plane mirrrs set parallel t ne anther. T a first apprximatin the mdes f this plane parallel resnatr can be thught f as the superpsitin f tw plane e.m. waves prpagating in ppsite directins alng the cavity axis. One shuld nte the analgy with standing waves n a string. The resnant frequencies are btained by impsing the cnditin that the cavity length L must be an integral number f half-wavelengths s that the electric field f the e.m. standing waves is zer n the tw mirrrs. S L = n l 2 () and since c = v (2) the resnant frequencies are given by æ v = n e ö ç (3) è 2Lø This treatment is nt sufficient t accunt fr spatial mdes because it assumes that the waves are travelling exactly parallel t the axis when in fact the plane waves may be prpagating at very small angles t the z axis. The mdes f a rectangular cavity are well knwn and are given by v = c é æ nö ê ç 2ë è Lø 2 æ + m ö ç è 2aø l 2 æ ö ù ç ú è 2aø û where a is the lateral dimensin f the cavity and n, m and l are integers giving the number f halfwavelengths in the z, x and y directins. T a gd apprximatin the mdes f the plane parallel resnatr are described by the mdes f the rectangular cavity when m and l are much less than n. Frm equatin (), ne can see that this apprximatin is valid fr any values f m and l that might be investigated. Frm equatin (4), the frequency difference between tw mdes having the same values f m and l and whse n values differ by is These tw mdes differ nly in their field distributin alng the z axis (i.e. lngitudinal). Fr this reasn vn is ften referred t as the frequency difference between tw cnsecutive lngitudinal mdes. Althugh the terms "lngitudinal mde" and "transverse mde" are used, it is incrrect t think that there are tw distinct types f mdes. In fact, any mde is specified by three numbers n, m and l. The electric and magnetic fields f the mdes are nearly perpendicular t the resnatr axis. The variatin f these fields in a transverse directin is specified by l and m while the field variatin in a lngitudinal (i.e. axial) directin is specified by n, When ne refers, rather lsely, t a (given) transverse mde, it means that ne is cnsidering a mde with given values fr the transverse indexes (l, m), regardless f the value f n. Accrdingly a single transverse mde means a mde with a single value f the transverse indexes (l, m). A similar interpretatin can be applied t the "lngitudinal mdes". Thus tw / 2 (4) (5)

13 cnsecutive lngitudinal mdes means tw mdes with cnsecutive values f the lngitudinal index n [i.e. n and (n+) r (n-)[. In practice ne uses spherical mirrrs instead f plane mirrrs because the fcussing prperty f spherical mirrrs tends t cncentrate the electric and magnetic fields alng the resnatr axis and thus light lsses are avided. Fr a spherical mirrr the fcal length is f = R/2. Fr this reasn, the resnant cavity cnsisting f tw identical spherical mirrrs separated by their radius f curvature is said t be a cnfcal arrangement. Fr this arrangement light lsses are minimal. The resnant frequencies fr the cnfcal arrangement turn ut t be é 2n + (+ m + l) ù v = cê ë 4L ú (6) û The frequency spectrum is given in Figure 5. Nte that mdes having the same value f 2n + m + have the same resnance frequency althugh they have different spatial cnfiguratins. These mdes are said t be frequency degenerate. Nte als that, unlike the plane mirrr case where nly plane waves were cnsidered, the frequency spacing is nw c/4l. The frequency spacing between tw mdes with the same (l, m) values (e./g. TEM00) and with n differing by (i.e. the frequency spacing between tw adjacent lngitudinal mdes is, hwever, c/2l as fr the plane case). As a numerical example, if R = L = 3 m, then in the apparatus fr this experiment there will be beat frequencies at 25 and 50 MHz. If the mirrrs are nt identical r if the arrangement is nt cnfcal, ne generalizes t the case f a resnatr cnsisting r tw spherical mirrrs with radii f curvature R and R2 separated by a distance L. Fr cnvenience g and g2 are defined by g =- L R (7) and g 2 =- L R 2 (8) The resnance frequencies fr the general spherical resnatr are v = c é 2L n + (l + m +) cs- (g g 2 ) / 2ù ê ú (9) ë p û Equatin (6) fr a cnfcal resnatr is a special case f equatin (9) fr if R = R2 = L, then frm 3

14 equatins (7) and (8), g = g2 = 0 and equatin (9) reduces t equatin (6). Nte that the frequency degeneracy which ccurs fr a cnfcal resnatr which is shwn in figure 5 is lifted in the case f a general spherical resnatr. This is illustrated in Figure 6. As a numerical example, ne can shw that if F = 3m and L = 2.8m then in the apparatus fr this experiment there will be beat frequencies fr 25.7, 27.9, 5.3, 53.6 and 55.8 MHz. The value f 53.6 MHz is the value that wuld be btained if ne simply assumed a cnfcal arrangement with L = 2.8m. 4

15 APPENDIX VI Emissin Spectrum f Nen 5

16 APPENDIX VII Gaussian Beam Optics Melles Grit prvides a discussin f Gaussian Beam Optics in its catalgue at Gaussian beams are cnfusing nly because there are many different chices f the imprtant parameters that varius authrs use t describe them. It might help t keep in mind besides the wavelength there are nly tw independent parameters required. One pssible chice fr the pair is the minimum beam waist w0 and its lcatin in space. A cmmn practice is t place the beam waist at the rigin f a cylindrical crdinate system, with r giving the radial crdinate and z the displacement alng the beam directin. The catalgue gives the radius f curvature and the beam radius at a psitin z, in terms f z and w0. Other pssible chices invlve replacing w0 by the far field divergence r the Rayleigh range/distance zr. The parameter zr is ften called z0, the cnfcal parameter. A better discussin, than that given by the Melles Grit catalgue, f the prpagatin f Gaussian beams thrugh ptical cmpnents is in terms f the ABCD law. See Yariv, Quantum Electrnics, secnd editin equatin 6.6-4a fr a definitin f the cmplex beam parameter and equatin fr the ABCD rule. [Nte w(z) f Melles Grit = (z) f Yariv.] Althugh nt at all clear frm Yariv, the values f A, B, C and D are the cmpnents f the 2 2 matrices é ABù ê ë CDû úgiven n page 0 f Yariv. T relate the beam radius and curvature at ne plane t the radius and curvature at anther plane frm the prduct f the 2 2 matrices fr all ptical elements between the tw planes. This will give yu the net A, B, C and D t use in equatin Thus yu can find the new R and w. Once yu have these, then the equatins in the sectin n Real Beam Prpagatin in the Melles Grit catalgue permit yu t find the lcatin and size f the beam waist. A discussin f laser r Fabry-Pert cavities can be given in terms f Gaussian beam parameters. The cnnectin is simple. Mirrrs are surfaces f cnstant phase. Thus apply subscripts t R and z fr mirrr and 2 and demand that z + z2 = L, the length f the laser. Yu can then slve fr z, z2 and w0. The simplest case is the symmetric cavity. Yu knw that z = z2 = L/2. Thus w0 can be fund immediately. A cnfcal interfermeter has R = R2 = L. It has the advantage that all mdes, nt just the (0,0) r Gaussian mde have a frequency that is a multiple f c/(4l). 6