The Band Spectrum of Water Vapour. III

The Band Spectrum of Water Vapour. IIIAuthor(s): David JackSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 120, No. 784 (Aug. 1, 1928), pp. 222-234Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95039 .

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222

The Band Spectrum of Water Vapour.-III. By DAVID JACK, M.A., B.Sc., Assistant and Carnegie Teaching Fellow. The

University, St. Andrews.

(Communicated by 0. W. Richardson, F.R.S.-Received June 6, 1928.)

1. Introduction.

In previous papers,*t details have been given of bands in the spectrum of water vapour, including the arrangement of the bands 2608 and 3428 into P, Q, and R branches. Measurements have also been given of a series of singlet lines* lying close to the head of the band 2811 and similar in structure to the

singlet series observed by Watson+ near the head of the band 3064. The advances which have recently been made in the theory of band spectra throw a considerable amount of light on their structure, and in the present com- munication it is proposed to apply the new methods to the analysis of the

water-vapour bands. The conception which has been emphasised by several writers in the last

few years is the close analogy which exists between molecular and atomic

spectra. The electronic state of a molecule is characterised by the same features as are found in atoms, and in particular is associated with an electronic quantum number corresponding to Sommerfeld's inner quantum number for an atom. The outer electrons in a molecule are responsible for energy changes which are comparable in their nature and amount with those associated with the outer electrons of "

corresponding " atoms. Mulliken, in a series of papers? in the '

Physical Review,' has discussed at length this analogy between atomic and molecular spectra, and in the sixth paper of the series has proposed a suitable notation which will be employed here.

The various electronic states are classified as S, P, D, ..., on analogy with the S, P, D, ... states in line spectra. These states are subject to multiplicity which is classified into two types. The first type is analogous to the multi-

plicity observed in line spectra, and the various sub-levels are distinguished by numerical subscripts, 1, 2, ..., added to the term symbol, F. The second

type has no parallel in line spectra and in this case literal subscripts, A, B, * 'Roy. Soc. Proc.,' A, vol. 115, p. 373 (1927). t 'Roy. Soc. Proc.,' A, vol. 118, p. 647 (1928).

' Astrophys. J.,' vol. 3, p. 145 (1924). ? ' Phys. Rev.,' vol. 28, pp. 481 and 1202 (1926); vol. 29, pp. 391 and 637 (1927); vol. 30,

pp. 138 and 785 (1927).



Band Spectrum of Wcter Vapour. 223

are employed: This type, frequently referred to as a-type multiplicity, is attributed to the effect of rotation on the component, a, of the electronic

angular momentum parallel to the axis of figure. The angular momentum of the rotating nuclei is denoted m (in units h/27),

and the component of the electronic orbital angular momentum parallel to the axis of figure, ak. The resultant of m and ak is denoted jk, while the resultant of j, and s, the angular momentum of electron spin, is denoted j. Cansider the case in which both the initial and the final levels are doublet electronic levels (1, 2), and suppose also that both are further subject to doubling of the rotational type (A, B). Then corresponding to each value of j, in both initial and final states, there are four sub-levels. If transitions between any initial and any final level be permitted, subject to the selection principle for j by which Aj = i 1, 0, then there are 48 possible transitions. Transitions for which Aj =-- 1, 0, and + 1 are denoted respectively P, Q, R, where

Aj= - j". The actual sub-levels involved are indicated by subscripts, for example, a transition from a 1A initial to a 2B final level when Aj - 1, is denoted PIA2B (j) j being the final value, j".

Further selection rules are, however, put forward which limit the number of

transitions, and these will now be considered. In Hund's case b which is of

special interest here, s is parallel or antiparallel to jk so that j = j ? s, and if j, and j$ correspond to the substates 1 and 2, then j = jk + s, and

j = jk - s. This in fact defines the meaning of the subscripts, 1 and 2. If now s is given the usual value, 2, it is evident that in a 1 to 2 transition where j, and j2 have the same value, Aj, = - 1. Such a transition would

normally be denoted Q, but now in view of the transition Aj = - 1 of j, it is called a P-form Q and is written PQ. Similarly, for a 2- to I transition where j2 and jl have the same value, Aj = - 1, and the result is an R-form Q, 'Q.

The second selection rule which is usually imposed is that Ajk = + 1, 0. This makes impossible both P transitions from a 1 state to a 2 state and R transitions from a 2 state to a 1 state. The former change makes Ajk = - 2

giving a " double P-form," the latter makes Ajk = - 2 giving a " double R-form " transition. This cuts out 8 transitions, reducing the total number to 40.

At present no definite rules are laid down as to how the subscripts, A and B, are to correspond to the physical properties of the substates concerned, but if the normal (intense) P and R branches be taken to correspond to transitions, A to B or B to A, then the following additional selection rule may be stated.



D. Jack.

In A to B or B to A transitions Ajk = 1, and in A to A or B to B transitions Ajk = 0. This reduces the total number of transitions from 40 to 20.

In the special case where one or other of the states is a rotationally single S

state, half the above transitions automatically drop out and the total number of transitions permitted by the selection rules is 10.

2. Simplification of Notation.

The notation defined above is somewhat cumbersome but it can generally be simplified considerably without loss of definiteness. The Ajk index can

always be omitted when it is the same as the letter denoting the j transition,

e.g., PP may be written simply P. In special cases, such as that of the water-

vapour bands considered here, further simplification may be introduced. The water-vapour bands arise from transitions between 28 (doublet S) initial

and 2P final states.* Following Mulliken,t the rotationally single initial states will be denoted A. Since, therefore, the B subscript does not appear in the initial states, combinations like PlB1A, Q2B2B, do not occur, and the symbols,

PA1B, RIA1B, QOIAA may, without ambiguity, be contracted to P1, R1, Q. For convenience a table showing the 10 transitions "permitted by the above selection rules is given below.

Table I.

Main branches.

Satellites.

Here. Mulliken. Here. Mulliken.

P1 PPA1B Q1 Q1A1A PQ1A2B

P2 P2A2B Q2 Q2A2A Q2A11

R, RR1AlB P2A1A

R2 RA2B R12A

The table includes both the complete symbols according to Mulliken and the contractions used here. The satellites will be given in full. The classification into main branches and satellites is the result of intensity considerations.

' Phys. Rev.,' vol. 30, p. 395 (1927).

t Phys. Rev.,' vol. 30, p. 792 (1927).

224



Band Spectruwm of Water Vapour. 225

3. Old and New Notation.

According to the new meaning attached to subscripts I and 2, transitions between two levels with subscripts 1 give rise to the higher frequency com-

ponents of the doublets, and transitions between levels with subscript 2 to the lower frequency components. As a result, the subscripts 1 and 2 formerly applied to the water-vapour bands have to be interchanged, e.g., a line formerly denoted P2 now becomes P1. The next point which arises is that of the connection between the old empirical "m" of Heurlinger, and j. This

question, as that of the interchange of the numerical subscripts, has already been referred to* in connection with Kemble's work on the water-vapour bands. It appears, then, that the two components of a doublet have different values of j, these values differing by unity. They have, however, the same value of jlc so that if, for the higher-frequency component of a doublet, j1 = 2, then for the lower-frequency component 2 = 1. Accordingly Heurlinger's P1 (m), min 2, becomes P2 (j), j 1, and Heurlinger's P2 (m), m -2, becomes P1 (j), j = 2. A complete list of old and new symbols will be given later.

4. OH Bands. Main Branches. From consideration of intensities, the most probable transitions are those

corresponding to the symbols classed under main branches in Table I. For the P and R branches the transitions are between 1A and iB, or 2A and 2B

Atates, while for the Q branches the transitions are 1A to 1A, and 2A to 2A. The a-type doubling (A, B), and crossing over are capable of accounting for the failure of the combination principle,

R (j- )-Q (j)= Q (j -1) - P (j), (1) when applied to the main branches. Kemblet has given an explanation of the variation of the doublet separation in the water-vapour bands with j, but his

theory does not take account of the c-type doubling. The main branches are defined, in terms of Mulliken's notation, by the

relations, P (j) =F'A (j 1)- F"B (j) 1L Q, (j)='A) -= F" A (j) (2) R '

(j) =F'A (j +1) - F"lB (j) J

and similarly for P2, Q2, R2. Here the quantity v0 is supposed for convenience to be included in F'. Making use of the above expressions the F1 terms were evaluated for the band 3064, apart from an additive constant. Similarly an

independent set of values was obtained for the F2 terms. *' Roy. Soc. Proc.,' A, vol. 118, p. 652 (1928).

'Phys. Rev.,' vol. 30, p. 395 (1927). VOL. CXX.-A. Q



226 D. Jack.

5. Satellites. P and R.

Heurlinger* observed that some of the lines of the bands 3064 and 3122 are

accompanied by faint lines lying very close to the main lines. These faint lines he called satellites and gave data on 18 satellites of the band 3064.

Fortratf gives much more extensive lists of satellites in a paper in which he discusses the Zeeman effect in these bands. More recently Watson+ measured two series of Q satellites not mentioned by Heurlinger or Fortrat.

Consider first the satellites of the P and R branches. The most probable P- and R-form satellites, according to Table I, are PQ1A2B and RQ2A1B. These

are defined by Q1A2 (j) F FlA (i) - F"2B (j)

and , (3)

Now, Q1A2B (0) - P2 ) F'1B (j)-F B (j- ) F

and . (4) R (j -- 1)- Q2AB (j - 1) F'IB (j) - F2B (j 1) J

In Table II are given the differences on the left-hand side of equation (4). The results should be the same both for the P and for the R differences. Since the data on the satellites are subject to considerable irregularities, the values from both Heurlinger's and Fortrat's tables are given. The results in columns 2 and 3 are in good general agreement with those in columns 4 and 5, as they ought to be if the above interpretation is correct.

Table II.

. .__ _ . ,

PQ1A2B (J) P2 (j)- Ri(- 1)- Q2A1B(j 1),

Heurlinger. Fortrat. Heurlinger. Fortrat.

3 - 0-74 - - 4 0-77 0-76 0*73 0.73 5 0-92 0-96 1-02 0.99 6 - - 1.06 1.23 7 1-41 1.38 1-47 1-45 8 1-45 1-30 - 0.93* 9 -- 180 1-84 1.60

10 - 2.10 2-19 0-79* 11 - 2.64 - 0.91* 12 - 2-50 --

* These values appear to be irregular, probably due to inaccuracy in the measurement of the faint lines.

* "Untersuchungen iiber die Struktur der Bandenspektra," Lund (1918). ' J. de Physique,' vol. 5, p. 20 (1924).

j 'Nature,' vol. 117, p. 157 (1926).



Band Spectrcum of Water Vapour. 227

6. Term Values.

Table II gives the values of F'1B (j) - F'2B (j- 1), and these may be used to obtain the connection between the F1 and F2 levels. The differences of the

values of F'iB (j) and F'2B (j- 1) already determined (with different arbitrary zeros) were compared with the values in Table II, and the correction which must be applied to the F1 values to correlate them with the F2 values, ascer- tained.

Fig. 1 is a diagram showing a few of the energy levels in the band 3064. The F1 levels have all been drawn to a uniform scale. The spacing of the A and B levels has been magnified five times, and the spacing of the 1 and 2 initial levels 10 times. Transitions between the various levels are indicated in the usual way by arrows. The transition marked S will be referred to in ?8.

F, 12 F' II F )2 F."

F, c- r2 KV

. In.

112

o02

2r rw ,

F, F,

2 SA . I

pI Q A28 QIA,B

J QZOAI QZA2B A2E

A

r nA

1I A- 102

91 0 . . I

FiT. 1.

q 2

r2 F,

F;' F," F:,

IVA 9

II

n.

v

k2A

?R.



D. Jack.

7. Q Satellites.

The Q satellites measured by Watson may next be considered. Watson states that these satellites obey the combination relation (1) if Q is taken to refer to Watson's satellite lines. It follows directly from this relation, the numerical test of which is contained in Table III, that

Q*1 (j)= F'lA(j)- F"lB (j).

These satellites are therefore to be represented, in the present notation, by QIA1B (j) and Q2A2B (j)

Now QlAlB (J)- Q () Ft (j) - FIB

(i) and . (5) Q2A2B () Q2 (j) - F"2 (j) - F B ) J

Columns 2 and 4 of Table IV show the values of the differences on the left-

hand side of (5), while columns 3 and 5 give the corresponding values of the differences on the right-hand side, obtained from the calculated values of F.

The figures in column 2 should agree with those in column 3, and similarly for columns 4 and 5. When allowance is made for the obvious irregularity of the experimental data on QIA1B and Q2A2B, it is seen that the results are

cqnsistent with this interpretation of the satellites.

Table III.

j. R1(j-l)-Q*i (j). Q*I (J-l)-P1 (j). R2 (j-l1)-Q2 (j). Q*2 (j - 1) -P2 (j)-

8 - - 28901 - 9 292-27 - 325-30 326-36

10 330-48 331-62 361-15 361-38 11 363417 363-00 396-50 395-82 12 399-21 399-57 431-28 429-52 13 433-41 431.77 465-42 463*00 14 466-58 465-27 496-79 495-60 15 498-63 498-06 529-26 529-81 16 532-47 531-15 561-93 561-93 17 - 561-19 > 591-63 592-26 18 595-20 - 626-51 624-25 19 624-18 622.91 650-78 649-53 20 65374 653-61 683-35 684- 06 21 683-64 681-98 709-66 708-55 22 709-27 709 67 737-73 737-91 23 739-70 739-41 - 763-52

Q* represents Watson's Q satellites.

* The star is added to indicate that the lines in question are Q satellites and not main.

branch lines.

228



Band Spectrum of Water Vapour. 229

Table IV.

. QlAlBUJ) - Q(j)) F 1A(j)

- FIB(J). Q2A2B(j)-Q 2(j)' F"2A(j) - F"2B ()'

8 -- 183 1-17 9 . 5-20 2-65 2-00 1-60

10 2-80 4-21 2-07 2-24 11 5-70 4-46 2-17 2-68 12 4-89 5-77 2-59 3-84 13 5-89 6-65 2-82 3-99 14 6-66 7-21 5-35 5-37 15 8-54 8-56 6*50 5-93 16 8*01 9-31 6-34 6*91 17 - 9-76 8-71 7-51 18 10'23 11-37 5-29 8-75 19 12-62 12-75 11-98 9-77 20 13-28 14-28 9-35 10-85 21 14-39 16-05 12-51 / 12-12 22 17-86 16-80 12-51 12-72 23 15-85 18-20 - 24 10-92 - - 25 18-06 -

In these Q satellites the transitions violate the a-selection rule in that they arise from transitions from A to B states while at the same time Ajk = 0. That some of the OH satellites probably violate the a-selection rule was

suggested by Mulliken.* There still remain several series of Q satellites to be accounted for. There

appear to be at least five different series of Q satellites in Fortrat's lists, which do not include those just discussed. There only remain two unused symbols in Table I, and to those may be added the two, QP2A1B and QRlA2B, which violate the a-selection rule. These four are probably capable of accounting for four of the above satellites, but the precise interpretation of the satellites is a little uncertain on account of errors of measurement, which, judging from the figures in Fortrat's tables, are likely to be considerable. To account for the fifth series it would seem necessary to assume further multiplicity of the

energy levels. It is just possible that very narrow rotational doubling may actually be present in the initial levels. If this were so Watson's satellites

might be represented by B to B transitions. The evidence for the existence of rotational doubling in the initial levels is very slender and will not, therefore, be further stressed here.

8. The Singlet Series.

The question now arises as to the possibility of explaining the singlet series

by transitions between the levels represented in fig. 1. The singlet series

*' Phys. Rev.,' vol. 30, p. 793 (1927).



230 D. Jack.

measured by Watson will here be referred to as singlet series I, and that measured by the author as singlet series II. Owing to the proximity of singlet series I to the band 3064 it is natural to expect some close relation between the two, and similarly for singlet series II and the band 2811.

The main branches of the water-vapour bands do not follow exactly a para- bolic law. In the higher-frequency component branches (new subscript 1) the second differences of the wave-numbers always show a marked numerical increase with increasing j, while in the lower-frequency component branches

(new subscript 2) there is a corresponding numerical decrease. In this respect the singlet series show a strong similarity to the higher-frequency branches. It seems natural, therefore, to expect combination relations between the

singlet series and P1, Q1, R1, rather than P2, Q2, R2. Let the lines of the singlet series be represented by S (j), the lowest value of

j being taken as 2 on analogy with P1, Q1, R1. If the series, S, be then represented on the same Fortrat diagram as the P1, Q1, Ribranches, fig. 2, it appears, so far as the accuracy of the diagram goes, that

S (j)- RI (j+ 1) = Q (j)- PI (j + 1). (6)

Ist rff /Q P

33,000 32.500 32.0500 -V 31,00X

FIG. 2.

This relation is subjected to a numerical test in Table V. The first part of the table relates to combinations between the band 3064 and singlet series I. If the relation held accurately S (j) should correspond to the transition, F'1A (j + 2) to F"1A (j). This follows immediately on replacing P, Q, R in

(6) by the appropriate terms. It is observed, however, that the values in column 2 are less than those in column 3, which fact would indicate that the initial S (j) level is F'2A (j + 1) and not F'IA (j + 2). Column 4 gives the

difference, column 3-column 2, which, if the initial S (j) sub-level is really F'2A (j + 1), should be equal to F'A (j + 2) - F'A (j + 1). The values of



Band Spectrum of Water Vapour. 231

Table V.

(1) (2) (3) (4) (5)

. S(j)-R (j+ 1). Ql(j)-Pi (j+ 1). Column3-Column2. F'tA(j+2)-F'2A(j+l).

Band 3064 and Singlet Series I.

2 82-82 83-61 0 79 076 3 116-76 117-77 1.01 0.96 4 151-37 153-91 2-54 1-15 5 186-04 187-73 1-69 1-50 6 220-84 222-70 1-96 1-73 7 255-53 257-37 1*84 1-60 8 289-92 292-15 2-23 2-27 9 324-12 226-42 2-30 2-03

10 357-78 360-20 2-42 2-60 11 390-88 393-87 2-99 2-67 12 424-00 426-88 2-88 3-03 13 456-04 459.38 3-34 3.17 14 487-57 491-40 3.93 3-44 15 518-92 522-61 3-69 3-67 16 549-36 553-18 3-82 3-88

Band 2811 and Singlet Series II.

2 82-76 83-52 0-76 0'85 3 117-23 117-90 0-67 0-74 4 151-48 152-60 1-12 0-87 5 186-70 187-72 1.02 1.11 6 220-59 222-74 2-15 1-56 7 255-50 257-76 2-76 1.11 8 289-96 292-23 2-27 1 86 9 324-40 326-58 2-18 2-09

10 358-12 360-37 2-25 2-22 11 389-00* 394-01 5-01* 2-78 12 423-50* 427-22 3-72* 2-61 13 456-98* 549-20 2-22* 2-77 14 487-65 491-65 4-00 3-07 15 519-41 522-90 3-59 3-69 16 549-47 553-33 3-86 3-91 17 579-41 583-16 3-75 4-19 18 608-79 612-31 3-52 3-87

* Turning point of S series. Lines not completely resolved.

the latter difference, previously obtained, are given in column 5. The agreement between columns 4 and 5 is very good indeed, so that the interpretation of the singlet series,

S (j) F2A (j + 1)- FrA (j), (7) receives very strong support from the experimental data.

The second part of the table shows the corresponding results for the band 2811 and singlet series II. The figures in the last column are obtained by



232 D. Jack.

making use of the fact that the final levels in both bands, 3064 and 2811, are the same.

The actual combination relation should therefore be

S (j)- Q2AIB (j + 1) - (j) - Pi (j + 1), (8)

instead of (6). The experimental data by which this relation may be tested

directly are very unsatisfactory, since they involve the rather irregular values for the R satellites, of which only a few are available in any case. The less direct method used in Table V is, therefore, much to be preferred as it is likely to be very much more reliable. It should be observed that, although the S series of the bands 3064 and 2811 are very weak compared with the main branches of these bands, yet they are capable of being measured with the same

accuracy as the main branches since they lie in regions free from other lines.

Exposures may be increased sufficiently to give really intense photographs of the singlet series without any difficulty.

9. Selection Rules.

'The combination principle provides very strong evidence for the interpretation of the singlet series as stated in equation (7). The most striking feature of the singlet series is, then, that it violates the Ajk selection rule, Ajk = i 41, 0. The transition, F'2A to F"A, together with the transition, Aj + 1, leads to the conclusion that Aj. -= + 2.

Remarkable as this result* may appear, it is, however, to be expected from the position of the S curve in the Fortrat diagram, fig. 2. The P branch and all its satellites lie close together and correspond to Ajk =- 1, irrespective of Aj. The Q branch and all its satellites correspond to Ajk - 0, the R branch and its satellities to Ajk = -4-1. Now, each group (main branch and its

satellites) is defined by a particular value of Ajk, irrespective of Aj, and

occupies a definite narrow region in the diagram. The regions corresponding to each value of Aj. are well separated, and consequently, from the position of S on the high-frequency side of R, it seems reasonable to attribute the S branch to a transition, Ajk -- 2. The S branch is therefore to be considered as a double R-form R branch, say SR.

* Cf. 'Nature,' vol. 121, p. 793 (1928). Dieke, Takamine, and Suga, referring to helium, make the following statement: " The band 2p - 3y has three branches which have the appearance of a Q-branch, a P-branch and a branch in which the effective rotational quantum number decreases by two units."



Band Spectrum of Water Vapour.

This interpretation suggests the question as to the possible occurrence of a double P-form branch, where Aji= - -2. Unfortunately, the experimental data available cannot be expected to provide any evidence on this point, since if such a branch does occur, its low intensity together with its position in a dense region of intense lines will make its observation a matter of considerable

difficulty. In most band spectra there is considerable overlapping of the

bands, but in this respect the water-vapour spectrum is particularly suitable for the study of double R-form branches.

From the foregoing discussion it seems that the only seection rule which holds rigidly is the rule, Aj = i 1, 0. That is to say, the resultant of all the

angular momenta can change only by i I or 0 in any one transition, whereas the component momenta may change by an amount greater than unity under certain conditions. At the same time it may be said that the most probable transition of any momentum component is ? 1 or 0, although this rule cannot be taken as final.

10. Summary. The water-vapour bands can be interpreted as arising from transitions

between 2S and 2p levels. The 2p levels are subject to o-type doubling (A, B), while the 28 levels are probably rotationally single. The P and R main branches are due to transitions, 1A to 1B, and 2A to 2B, while the Q branches show "

crossing over," IA to 1A, and 2A to 2A. Other combinations give rise to satellites, in some of which the a-selection rule does not hold.

The most striking feature exhibited by these bands is the presence of a singlet branch for which the combination principle definitely indicates a transition of two units in jk. It is therefore concluded that, although the j selection

rule, Aj += 1, 0, probably holds rigidly, the other selection rules (jk and a) are only an expression of the most probable transitions and cannot be taken as final.

The connection between the old and new not abon is given in Table .It <

(p. 234), which also serves to summarise the interpretation of the various branches.

233



Band Spectrum of Water Vapour.

Table VI.

Main branches. Satellites.

Old. New. Old. New.

P1 (2) P2 (1) { iP1(2) QIA2B (1) H, F

P2 (2) P (2) aRg2 (2) Q2AlB (2)

Q1(2) Q2 (1) cQ01l(2) Q2A2B(1)

Q2(2) Q1(2) LrQ2 (2) Q1AB (2)

R1 (2) R2 (1) P2A1A(2) Fortrat's

R2 (2) R1 (2) a (2) R1A(1)

oQ~ (2) J

i P2A1 (2)

oQ2 (2)J QR R1lA2B (1)

Sin.le , , .Double B-form Singlet series Double Rform

. S2A1A (2)

F., Fortrat. H., Heurlinger. W., Watson. * Probable interpretation.

The writer is deeply indebted to Prof. H. Stanley Allen for the keen interest he has taken in the work of this paper and for much helpful discussion.

234



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The Band Spectrum of Water Vapour. III