BAND SPECTRUM. A spectrum consists of lines showing certain definite regularities of arrangement. The so-called "line spectra" (q.v.) are attributable to atoms, and band spectra are due to molecules. Many of the features characteristic of band spectra may be seen in the Plate. All comprise a very large number of lines, which in places may be so closely packed as to be separately indistinguishable. In cases where analysis is pos sible the lines are found to close up in a perfectly regular manner, giving an appearance resembling that of a fluted column (see No. 5) . Such a series appears to terminate abruptly at the point where the separation of the lines is least ; this is called the head of the band, and is a prominent feature of most band spectra (see Nos. 4, 5, 6, 7, IL), i I) . Such heads frequently occur in groups (see 7 and io), and since a long series of lines is associated with each head there is apt to be considerable overlapping of series and consequent difficulty in analysis (see 5 and 13). Fur ther, there is commonly a number of associated groups of heads forming what is known as a band system (see
and finally the totality of systems originating from a particular compound is termed the band spectrum of that compound. Thus, for ex ample, cyanogen gives rise to one system (see 5) stretching from the blue into the ultraviolet, and another (see 6), very different in many respects, lying in the red, yellow and green.
Still further complexity, due to the double or triple character of the component lines, is quite common (see 4, 3, 8, 9) . Again, a series may fade out before the head is reached (see 3), in which case the band structure is not always evident. It may be ex tremely difficult to recognize if there are numerous overlapping bands of this headless variety, each comprising few lines with relatively large spacing. The secondary hydrogen spectrum (see 1) is of this nature; the helium band spectrum (see 2) is of a character intermediate between this and the more usual type of band spectrum.
The spacing of the lines is roughly inversely proportional to the moment of inertia of the molecule. An extreme instance is seen when comparing the case of helium (see 2), for which the moment of inertia is about 3.6XIo
and the spacing about I oA.U., with that of iodine (see 13), of moment of inertia
I o 4° and spacing
Classification and Origin of Band Spectra.—Band spectra may be classified in various ways; e.g., according to the process which gives rise to them (emission, absorption, fluorescence), to the nature of the energy change involved (electronic, vibrational, rotational), or to the type of molecule (diatomic, polyatomic, polar, symmetrical). On account of the complexity of the subject a general survey is given in the first part of this article, the second part being devoted to a somewhat more detailed treatment of the class which has been most extensively studied. As in the case of line spectra, the emission or absorption of a band line is a con sequence of a transition from one to another of a set of stationary states and the frequency of the line is proportional to the differ ence of energy in the two states. A particular state is only found to combine in this way with a limited number of other states, but the number of possible states of a molecule is so great that in spite of this limitation the spectrum is usually exceedingly rich in lines.
The energy of molecules may differ in any or all of three re
The electron configuration, which appears to re semble that characteristic of atoms, in that it can be classified in groups or sequences having a particular quantum number in com mon. (b) The vibration of the atomic nuclei. In the case of a diatomic molecule this consists of a to and fro motion along the line joining them. Its precise nature is determined by the force called into play when the nuclei are displaced from their equi librium positions, but the energy, and therefore also the amplitude, of the vibration can only assume certain definite values. (c) The rotation of the molecule as a whole, which again is quantized, so that only certain definite speeds of rotation occur.
The contribution of (a) to the total energy is usually larger than that of (b), which again exceeds that of (c). Since also the change in the rotational quantum number is limited to one unit the bands which are due solely to rotation changes are of very low frequency, lying far out in the infra-red towards I
(I ,000 p.= I mm.) , where ex perimental difficulties are very great. Nevertheless, sufficient data have been secured (e.g., for HC1, H2O,
in absorption) to confirm the general validity of the quantum predictions. In the case of diatomic molecules the spectrum consists of a series of almost equally spaced lines, as predicted by theory; but for polyatomic molecules the structure is more complex.
Bands due to a change of vibrational energy, with which a rota tion change may also be associated, are of higher frequency and lie in the nearer infra-red, usually between i,u and 'oil. Even here the experimental difficulties are still considerable, and adequate resolving power has only been obtainable in a very few cases (for diatomic molecules such as HC1). The kind of structure which is typical of this class of band is shown in fig. I. The band as a whole originates in the absorption by non-vibrating molecules of sufficient energy to set up a one-quantum vibration. The structure is due to the fact that the molecules, originally in various rotation states, pass over to others simultaneously with the vibration change. The right-hand branch (known as the R branch) corresponds to an increase of unity in the rotational quantum number and the left-hand (P) branch to a decrease of one unit in the rotational quantum. The number of lines in each branch thus indicates the number of different rotational states of the molecules, and the intensity distribution of the lines cor responds to the distribution of rotational energy among the mole cules. Both characteristics vary with the temperature in accord ance with the requirements of the kinetic theory of gases.
An important characteristic of such bands is the absence of a line at the centre, concerning which there has been a certain amount of controversy. This phenomenon is also found in the rotation structure of electronic bands, and is well illustrated in the Plate, No. 12, where the location of the missing ("null") line is indicated. It now appears that such a line, if it existed, would belong to another branch altogether and therefore could not occur without the other members. The branch in question, designated Q, corresponds to zero change of rotational quantum number, and is frequently absent, not only in vibration bands but also in the electronic type. The exact location of a vibration band, and also its intensity, is dependent upon the particular vibration transition concerned e.g., o-->i,
etc. The first is much the strongest under ordinary conditions, and it is very significant that the relation between the frequencies is not exactly integral, as it would be if the classical theory applied. The actual expression for the frequency of the nth harmonic is approximately of the form new (I —nx), where x is a small con stant depending upon the law of force between the nuclei. It is to be noted that up to the present vibration and rotation bands have been observed only in absorption spectra.
Turning now to electronic bands, which lie mainly in the visible and ultra-violet regions, we find a much greater complexity of structure, but on the other hand the ex perimental technique available is much more highly developed, so that a great wealth of data has been accumulated, the inter pretation of which has reached a fairly advanced stage. We have here to do with a simultaneous change in electronic, vibrational and rotational energy, and each makes its own individual and characteristic contribution to the structure of the spectrum. The electronic contribution is the largest and determines, broadly speaking, the location of the band system. It may be represented as the difference of two terms to which the designations S, P, D, etc., may be attached, and which may have multiple values, just as for atomic spectral terms. The chief difference between the two cases appears to be that transitions such as S--'S and P—P, which are "forbidden" for atoms, may occur in molecules. But the total number of electron transitions which actually occur is for tunately much less than in atoms.
With a given electron transition is associated a variety of vibration and rotation changes, which are reflected in the spectrum as a coarse and a fine structure respectively. Postponing for the moment any detailed description of these, we may note that for diatomic molecules the structure is capable in both cases of being represented by an expression of the type
.. . the higher terms being usually negligible. In the case of rotation the coefficient B gives the moment of inertia (B = h/4iI) and C is determined by the change in the moment of inertia due to the transition. The moment of inertia can thus be evaluated if the values of is are correctly assigned, and if the constitution of the molecule is known the separation of the nuclei then follows. In the case of vibration the coefficient B gives the vibration fre quency for small amplitudes, and C the variation of frequency with amplitude. If the vibrations were simple harmonic, B would give the force per unit displacement and C would be zero. This is never the case however, nor, indeed, is it to be expected on any obvious molecular model, but from the values of B and C it is possible to determine the amplitude of the vibrations and the law of force, that is to say, the manner in which the force varies with the nuclear separation, in the neighbourhood of the equi librium position. A typical example of vibration structure is represented in fig. 2, which shows on a scale of wave-numbers the arrangement of the chief band heads of the violet cyanogen system. It will be seen that they group themselves in sequences for each of which the change An of vibrational quantum number is the same. As each band comprises a large number of lines (several hundred in this case) there is a great deal of overlapping in one sequence. The larger the moment of inertia the more numerous and closely packed are the lines of one band, so that it eventually becomes impossible to analyse the structure. Never theless, the analysis has been carried out more or less completely for more than a hundred band systems already, and various general conclusions of interest have emerged.
One of the most striking of these conclusions is the small range of variation in the nuclear separation for the 6o or so diatomic molecules concerned. For practically all of them it lies between I and 2A.U. (I o
. For molecules of similar structure, e.g., the hydrides, it exhibits a regular pro gression, decreasing as the atomic number increases, and increas ing suddenly at the beginning of each new period, when the formation of a new electron shell is begun. The vibration fre quency shows larger variations (of the order of ioo to 4,000
[in practice the wave-number in
i.e., the reciprocal of the wave-length, is used instead of the actual frequency, this may be obtained from the wave-number by multiplying by 3X'°'°] ) which are generally in the inverse sense. By considering both of these features in relation to an assumed force function some indication of the nature of the binding can be deduced. This may be classified as :—(I) polar, in which two oppositely charged ions are bound by electrostatic attraction varying nearly inversely as the square of their distance apart (e.g., HC1, OH, CuH, ZnH, etc.) ; (2) due to electron sharing, in which some electrons from each atom form a common shell (e.g., CO, NO, CN, etc.) ; (3) due (probably) to mutual polarization of the atoms, each retaining its own electron system (e.g.,
and perhaps some other elemen tary molecules) .
There is one special case of band emission which calls for notice, both on account of the interesting nature of the phenomenon and of the simplicity of the theoretical explanation. This is the fluorescence of certain elements in the state of vapour (e.g., S, Se, Te, I) as excited by monochromatic radiation. If iodine vapour, for example, is ex cited by a wave-length corresponding to one of the lines in its very complex absorption spectrum (see Plate), it is found that the fluorescent light emitted consists of a regular series of doublets, the origin of which may be indicated most conveniently by an actual example. The mercury line X 5461 happens to co incide with an iodine absorption line due to the transition (o, 34) to (26, 35), where the first figure in the bracket refers to the vibration and the second to the rotation state. There is an electron transition as well, but this need not be considered for our present purpose. The excited molecule (26, 35) then emits radiation, but does not necessarily re-emit in one process the whole of the energy previously absorbed. If it should do so it will of course emit the original line, but it may alternatively return to the state (o, 36), giving out a line of slightly different wave-length. There are many other final states possible, such as (I, 34) or (I, 36), (2, 34) or (2, 36), and so on. Each pair gives rise to a doublet, and the doublets themselves form a series of approxi mately constant wave-number interval. Further, the comparison of accurate wave-length data with the theoretical expressions per mits an evaluation of the molecular constants, thus evading the difficulty, hitherto insurmountable, of completely analysing the ab sorption spectrum. (See FLUORESCENCE and PHOSPHORESCENCE.) Rotation Structure of Bands.—We proceed now to the more detailed consideration of electronic bands due to emission by dia tomic molecules. A single band of this type arises from a variety of rotation changes associated with a particular electron and vibration transition. The contribution of these to the radiated frequency may thus be regarded as constant,
say. By applying the usual quantum restriction to the rotational motion and making use of Bohr's Correspondence Principle (see LINE SPECTRA), we find that a molecule in the m'th rotational state is capable of emit ting one of three different frequencies, namely: These are designated R(m— I), Q(m) and P(m+ r) in the nota tion which is most commonly used at present. The appearance of such a band is represented diagrammatically in fig. 3, but it should be noted that m has been given half-integral values since this is most frequently the case in practice. The Q branch is often absent.
Either the P or the R branch may run to a head, according to whether C is positive or negative. The former is the case when the moment of inertia increases as a result of emission, and vice versa. But the position, or even the existence of a head, is of no immediate physical significance, for it is simply determined by the relative values of B and C. The band may fade out before the head is reached, as in
or the head may be so close to
that it is not recognizable by the crowding together of the lines, so that the band appears to be headless, as in
The rotation terms, from which the molecular constants are calculated, are evaluated as follows. For brevity we may write: P(m) =
I) —F"(m) where F' refers to the state Q(m) =
—F"(m) of higher electronic excita R(m) =
—F"(m) tion and F" to the lower, whence Q(m)— P(m) = R(m— i) —Q(m— i) =F'(m)
i) and Q(m) — P(m+ r) = R(m) —Q(m+ I) = F"(m+ I) —F"(m) . These relations serve at once to fix the relative numeration of the branches and to isolate the initial and final term differences, from which the actual term values can readily be obtained. A similar procedure can be followed even if the Q branch is absent.
The intensity distribution in all three branches is similar, show ing an increase with m up to a maximum and then a more gradual decrease to zero. The position of the maximum shifts to higher m values with increasing temperature in agreement with the Max well-Boltzmann law of distribution of rotational energy amongst the molecules. In practice it rarely happens that the branches are single ; doublets are quite common, and higher multiplicities, up to fivefold, have been recorded. Nor are the simple expressions given above usually adequate for the accurate representation of band branches. It is necessary to modify them on account of several disturbing factors, e.g., the distortion of the molecule due to rota tion and vibration. Further terms, in
and
must be intro duced in the rotation term values, and the value of B adjusted according to the particular state of vibration. With such a cor rected formula a very accurate representation can usually be achieved, and it is then possible to calculate the vibration fre quency solely from observations of the rotation structure.
In order to account for the vibration structure it is necessary to express the vibrational energy in terms of the quantum number n; the expression which is usually employed is an—big, where a is the vibration frequency for neg ligibly small amplitudes and b expresses the departure from simple harmonic motion. The vibration structure is then determined by the range of possible values for n', n" and n'—n" (n' and n" refer to the upper and lower electronic levels respectively) . This varies greatly in different molecules; in some (e.g., CaH,
vibration is almost, if not entirely, precluded owing to instability of the molecule, whilst in others a fairly long series of vibration states is developed (e.g.,
with n' up to S9). The observed vibration transitions, n'—n", show a correspondence with the change in the moment of inertia, I'—I", that is to say, relatively high values of the two tend to occur together. Franck has pointed out that this is to be expected, since a sudden change of size of the molecule must disturb the equilibrium of the nuclei and set up a vibration which is more intense the greater the change. We find a particularly clear example of this in the absorption spectra of
Br, and Cl, in the visible region. In all three cases I'—I" is exceptionally large compared with other molecules. In the case of the non-vibrating molecule of
the electron transition sets up a vibration of at least 18 units, and the earlier portion of the n"=o progression, for which n'<18, is therefore entirely absent.
shows a similar peculiarity, whilst in the case of Cl, the change is apparently so great that dissociation occurs, giving continuous instead of banded absorption, and the first observed absorption band therefore corresponds to n"= r instead of n" =o as in the other two cases. Incidentally, this explains why the banded absorption is relatively so weak in
for at ordinary temperatures the majority of the molecules are in the vibration less state and so give rise only to continuous absorption. The ap pearance of the spectrum is also much influenced by the relative values of a' and a". If they do not differ much the bands forming a sequence, for which (n'--n") =constant, lie fairly close to gether, as in CN (see fig. 2 also Plate, Nos. s and io), but if a'—a" is large, as in the halogens, the sequences are widely spaced and frequently are intermingled with one another, so that the vibra tional structure is much less apparent. (See Plate, No. r3.) Since the molecular vibration frequency is directly dependent upon the masses of the nuclei we should expect the vibration structure to furnish evidence as to the existence of isotopes (q.v.), and this is, in fact, the case. The matter has been thoroughly investigated, both theoretically and experimentally, by Mulliken, with completely satisfactory results. It is found that the bands due to the different isotopic combinations are separated by an amount proportional to their distance from the origin of the system, and accurate measurements of the effect have proved of service in resolving uncertainties as to the identity of the emitting molecule. It is quite possible that intensity measurements might be utilized to determine the relative abundance of the isotopes.
There remains to be considered the electronic energy change involved in the emis sion or absorption of a band spectrum. As already mentioned, the electronic terms are generally similar to those of atoms, but only in
and
has the existence of well-developed Rydberg sequences been established. Some striking similarities, as regards multiplicities and term values, have, however, been recognized between certain molecular electronic levels and those of "corres ponding" atoms. For example, the electron levels of the molecules BeF, BO, COP,
and CN, which all have nine outer electrons if it is assumed that the K shells of the constituent atoms remain intact, exhibit a certain similarity to those of Na, which also has nine electrons outside the K ring. This suggests that, as in Na, there is one electron less firmly bound than the other eight, and the analogy is supported by the resemblance which is traceable between CO,
and Mg (ten outer electrons) and between
NO and Al (II outer electrons). The matter must still be regarded as sub judice, however, as Mecke has brought forward evidence which appears to conflict with the above view. It seems clear, in any case, that in these molecules there must be some sort of com mon shell, which is concerned with the binding of the nuclei but not directly with the emission. There are also molecules (Cu halides,
in which the radiating electron appears to be linked with one of the constituent atoms instead of with the molecule as a whole. (See fig. 3.) Absorption Band Spectra.—Although emission band spectra have been much more extensively studied, band spectra which can be obtained in absorption are of particular interest and im portance, for several reasons. The identity of the absorbing mole cule is definitely known, whereas in emission there is frequently much uncertainty, which can thus be removed if any of the same levels can be recognized in both spectra. Further, absorption spec tra are in general much less complex than emission, since the only initial levels are those normally present in the unexcited vapour, whereas in emission the initial levers refer to excited states and are therefore much more numerous. These absorption spectra, however, lie mostly, apart from the halogens, in the far ultraviolet (Schumann) region, and have therefore only recently been inves tigated. Unfortunately it is not possible to obtain an absorption spectrum corresponding to every emission spectrum, since the substance concerned may not be obtainable in the vapour state under conditions suitable for the examination of its absorption spectrum. Indeed, many of the band spectra to which reference has here been made are due to very unstable molecules (OH,
the metallic hydrides) which have only a transitory existence un der the special conditions obtaining in a discharge tube. The spectra of even the common chemical molecules consisting of more than two atoms are still practically unknown.
See E. C. C. Baly, Spectroscopy (3rd ed., vol. iii., 1927) ; "Molecular Spectra in Gases," Bulletin No. 57 of the National Research Council, 1927).
