Return

The pressure effect on the vibrational-rotational levels will, basically considered, consist of the following three effects:(1) shift, (2) broadening of widths, and (3) splitting of the degenerate levels. A simple shift of (1) obviously exerts no influence on the mean absorption. Moreover, the splitting phenomenon of (3) may not also appear at low pressures from the two-body impact theory especially in the microwave region, so that only from the standpoint of the pressure broadening of (2), were the investigations and mathematical calculations carried out. At pressures near atmospheric, however, a deviation between experiment and calculation was seen to occur; namely, the introduced correction factor

There are now to be initially considered as the important factors which cause degenerate rotational levels that split the l-doubling[4] produced by a Coriolis interaction[8] between vibrations and rotation of a molecule, the L-doubling[4, 9] produced by an interaction between the electronic and rotational motions in a molecule, Stark effects[10], Zeeman effects and so on. The former two, however, do not depend on the gas pressure surrounding the molecule due to their dependence merely on the intra-molecular interactions. The latter two, which are due to macroscopic electromagnetic fields, may not take place when using electrically neutral gases.

During the course of deriving at high pressures the approximate theoretical formula if the validity of the effect of pressure splitting of (3) is to be recognized which the pressure causes in degenerate vibrational-rotational levels and which, in general, may not occur in the microwave region but may occur in the infrared, the previously derived formula shall be revised again by introducing a novel notion of equivalent effective pressure for the effect of pressure splitting and thereby a better agreement with the experiments will be determined.

(It should also be noted that as an important pressure effect other than the above three, there is also that of asymmetry, which is, however, considered to be of little consequence at pressures around atmospheric, and therefore is not taken into consideration, only that due to the splitting effect itself.)

For a degenerate state, the first approximation

where

The selection rule for the parallel vibration n

where

Accordingly, the splitting phenomenon will produce in a transition between two degenerate vibrational-rotational levels, therefore in a degenerate absorption line, the following width of deviation (or shift)

When more than two overlapping degenerate lines of Lorentzian shape slightly shift in a mutual fashion, in general, the entire outline is not Lorentzian, but will be asymmetric as if distorted. However, by assuming that it may approximately keep a symmetric Lorentzian form because of a slight shift, the entire profile of the split degenerate lines, with the absolute intensity constant, that is, the integrated value of the line profile of overlapping absorption coefficients being kept invariable, will be considered to lower the height of its own peak and broaden. In fact, this is nearly the same phenomenon as that of the foreign gas broadening in which the unchanged integrated absolute intensity increases owing to the broadening of its own width the integrated absorption intensity, or in other words the mean absorption intensity according to Lambert-Beer's law.

This splitting phenomenon due to collisions may be a comparatively trivial one at low pressures even in the infrared region, but it will be worth noticing that it gives rise to a significant effect on the mean absorption intensity at high pressures around atmospheric, where simultaneously the asymmetric effect[11] also due to the statistical types of theories[4] might happen to lead to an increase in the half width, but in the case of low absorbing gas pressures such an effect on the width may be regarded as quite slight.

where

On the other hand, the peak height

where d

where

Secondly, it should be emphasized that the more flattened the absorption coefficient

Taking into consideration its effect, the half width d

where d

where s'

Because it is probably related to those higher terms of "prohibitive" difficulty in Anderson's theory[7], the functional dependence of

By the way the following approximation may hold

where s

By using Eq.(11), Eq.(9) may be rewritten in the form

where - means the average over

and

which was obtained[2] based solely on the pressure broadening theory, where

As a final formula

where

and

where

Furthermore, Eq.(15) may be rewritten as

where

where

Eq.(18) is then

where

and assuming that e~1 and

which gives

the expression

By leaving only the linear term in Eq.(14) in which s'

Here, the relation s

In particular, in the case of the self-broadening where

Eqs.(26) and (27) are apparently the same expressions as those without the splitting effect, which can be assumed to be negligible at low pressures.

where

and for low pressures, by the way,

which is approximately the same expression as that without the splitting effect.

From the solid line in Figure 1[3]

therefore,

is obtained. From Eqs.(30) and (20) in the case of the foreign gas broadening for CO+N

in which the quadratic and higher terms except the first may be neglected, because they make almost no contribution. Table 1 shows the calculated values of

Figure 1. Pure CO. Slope of the line = 6.20×10^{-3}cm^{-1/2}. cmHg^{-1}. Cell length of the condenser microphone detector: *l*'=5cm. Cell length: 5cm, 10cm, and 30cm

Figure 2. CO+N_{2} at total pressure *P*=70cmHg. *l*'=5cm. Cell length: 5cm, 10cm, and 30cm

w (cm.cmHg) | D (w, 70) | C (w, 70) (cmHg) |
---|---|---|

1 | 0.084 | 147 |

2 | 0.086 | 137 |

5 | 0.090 | 120 |

10 | 0.101 | 80.7 |

15 | 0.107 | 64.4 |

20 | 0.115 | 46.5 |

30 | 0.124 | 30.5 |

and at 3700cm

where <

Moreover, the following average values from Eqs.(33) and (34) are given below,

where

From Figure 3[3]

therefore

In the case of the foreign gas broadening for CO

where the cubic and higher terms have been dropped for the same reason as in the case of CO.

Table 2 shows the values of

Figure 3. Pure CO_{2}. Slope of the line = 1.60×10^{-2}cm^{-1/2}. cmHg^{-1}. *l*' = 5cm. Cell length: 5cm, 10cm, 20cm× and 30cm

Figure 4. CO_{2}+N_{2} at total pressure *P*=70cmHg. *l*'=5cm. Cell length:5cm, 10cm, and 30cm

w (cm.cmHg) | D (w, 70) | C (w, 70) (cmHg) |
---|---|---|

0.5 | 0.345 | 72.8 |

1 | 0.383 | 48.0 |

2 | 0.423 | 30.0 |

3 | 0.433 | 26.2 |

5 | 0.460 | 17.1 |

8 | 0.495 | 7.42 |

10 | 0.510 | 3.87 |

Figure 5. CO+N_{2}. Dependence of *D* on *w* at total pressure *P*=70cmHg

Figure 6. CO_{2}+N_{2}. Dependence of *D* on *w* at total pressure *P*=70cmHg

The two effects of splitting *S _{m}*(

Table 3. Comparison of E.E.P. and *S _{m}*(

w (cm.cmHg) | Ratio of E.E.P. (CO/CO_{2}) | Ratio of S (CO/CO_{m}_{2}) |
---|---|---|

1 | 3.06 | 1.13 |

2 | 4.57 | 1.69 |

5 | 6.96 | 2.58 |

10 | 21.6 | 8.00 |

In the following section, the influence of the modified Bessel function of order zero

The empirical values slightly deviated from the experimental curve of CO

It cannot be explicitly determined as one of the problems of how much influence on the derived formulas is the approximation by the rectangular shapes with a width of

In the case of CO

As for N

Finally, as a matter of course, the above investigations will also apply to atmospheric pressure or a little higher, which may be considered, however, to bring on a somewhat greater effect of splitting than that at 70cmHg so that the cubic term of (

and as with the cubic term in Eq.(38a), though with a somewhat poorer accuracy,

is obtained with a negative sign.

In the case of CO, the <

and

For higher pressures than those around atmospheric, the spectral lines become more flattened and more overlapped with the lines of the neighboring levels (and then e may no longer be regarded as const ). The basic assumption of the two-body impact theory cannot hold good due to the appearance of many-body impacts, say, the frequency of three-body impacts, which is related directly to the half width, is no longer proportional to the pressure but to its square. At the same time, the Lorentz line shape obtained from the pressure broadening theory may no longer be assumed to be correct because of the appearance of the asymmetry of lines due to the statistical types of theories , so that other different considerations will be needed.

[ 2] T. Yonezawa and E. Niki,

[ 3] T. Yonezawa and E. Niki,

[ 4] C. H. Townes and A. L. Schawlow,

[ 5] M. Baranger,

[ 6] J. M. Ziman,

[ 7] P. W. Anderson,

[ 8] Jack D. Graybeal,

[ 9] L. D. Landau and E. M. Lifshitz,

[10] P. Debye,

[11] G. Herzberg,

[12] E. Bright Wilson, Jr., J. C. Decius and Paul C. Cross,

[13] G. Herzberg,

Return