Measurement of rotational temperature, vibrational temperature, and molecular concentration, in nonradiating flows of low density nitrogen

RADIATING FLOWS OF LOW DENSITY NITROGEN

by

E. P. MUNTZ

APRIL, 1961 UTIA REPORT NO. 71

TEMPERATURE, AND MOLECULE CONCENTRATION, IN NON-RADIATING FLOWS OF LOW DENSITY NITROGEN.

by

E. P. MUNTZ

APRIL, 1961 _{UTIA REPORT NO. 71}

,

ACKNOWLEDGEMENTS

The author wishes to thank Dr. G. N. Patterson for pro-viding the opportunity to undertake this investigation. The continued interest of Dr. 1. 1. Glass is gratefully acknowledged.

The appreciation of the author is extended to Dr. J. H. deLeeuw for many interesting discussions and valuable suggestions .

Thanks are expressed to Mr. D. J. Mars den for.his assistance, to Dr. R. Senate for supplying the electron guns used in the experiments and to the author' s wife for proof reading the text.

The financial assistance of the Defence Research Board of Canada and the United States Air Force Office of Scientific Research made this work possible.

A method has been developed for measuring rotational temperature, vibrational temperature, and molecule concentration, in non-radiating flows of low density nitrogen, The method involves the passing of a narrow beam of high energy electrons through the flow. The beam electrons excite a relatively strong emission from the first negative system of the nitrogen molecular ion N2

+

It has been shown thatthe rotational and vibrational temperatures, as well as number density, of the ground state nitrogen molecules can be inferred from observations of the beam excited first negative system.

Particular emphasis has been placed on the measurernent of rotational temperature. A theoretical prediction has been made of the relative intensities of the rotational lines in the first negative system. The prediction of the relative intensities, as a function of the rotational temperature of the ground state molecules, has been checked experimentally and found to apply to within the accuracy of the experi-ments (± 2% for rotational temperature measurements).

I. Il. lIl. J SYM,aOLS iv .. INTRODUCTION 1 THEORETICAL,.. 3 2~ 1

Excitation-Emission Path

... ..

3

2.. 1.

1 Cascade Populations of

~

B

Z.

3a

.2..1..2

Double

.

.Excitation to

N;

elI,

'

3a

2

.

. 1.3 Ex.citation to N:8"-E

by

Secondary Electrons

5

2. •. 1. 3. 1 Secondarie.

s with.

Energies

in Excess of

18.7 ev 6

2.

1..

a.

2.

Secondaries with En.ergies

in Exèess of

3 .

.. 1.

.ev . 6

2

.

• 1..3.3 Effect of Flow Velocity on Secondary

._

Excited

.

.Emission

7

2. 1.3.4 Conclusions Concer.ning.

.

.s.

econdar.y

Excited Emission

7

2.

1.

4 Self-Absorption

8

2.

1.

5 Conclusions Concerning Excitation Process

8

2.

1.

6 Excitation - Em.

ission P

.

rocess

8

2. 2 Vibrational Excitation and Vibrational Ternperature

9

2. 3 R.

otational Excitation and Rotational Temperature

11

2. 3. 1 Case of Low Vibrational Temperature

16

2.3. 2 Case of

.

High Vibrational Temperature

18 2.4

Theoretical Background ior Measurement of

Nitrog,en M.olecule Concentration

20

2.4 ..

1

.

Light Output a .

..

a Function of Number..

Dsnaity

20

2.4.2

.

Effect of Thermal Diffus10n of the Exc1ted

Particles on Density Measurements

22 2.4 .. 3

Effect of Flow Velocity on Bearn. Pos1t1on

23

2.4.4 Secondary Electrons and The1r Effect on

Dens1ty Measurements

23

2.4.5

Conclus10ns, Section

2 .... 4 23

EXPERIMENTAL

3. 1 Equ1pm ent

3. 1. 1 Intermed1ate Plasma Tunnel

3.

,

1. 2 Plasma Generator

3. 1. 3 Known Te

,

mperature Gas Supply

3. 1. 4 Spectroiraph

3.1.5

Electron Beam Generator

24

24 25 25

26

26

b,

3.2 Photographic Procedure and Film Reduction 3.2. 1 Film Calibration 3.2.2 Film Development 3.2.3 Film Reduction 28 28 29 29 3.3 Experimental Establishment of the Electron Beam Method

As A Quantitative Probe for Measurements in Lovi"

Density Flows of Nitrogen 29

3.3.1 Rotational Temperature 29

3.3.1.1 Experimental Apparatus 3. 3. 1. 2 Experimental Procedure 3.3.1.3 R'eduction of Films

3.3. 1. 4 Results

3. 3. 1. 5 Discussion of Results, Conclusions ",

3.3. 2 Density Measurements - Establishment of the Calibration Curve

3. 3', 2. 1 Experimental Apparatus 3.3.2.2 Experimental Procedure

3.3.2.3 Discussion of Results, Conclusions

31 31 32 32 33 34 34 35 36

3. 3, 3 Beam Attenuation as a Function of Density 37 3.3.3.1 E::l.{perimental Procedure

3.3.3.2 Discussion of Results, Conclusions 3.3.4 Effect of Beam Voltage Variations Over a

Limited Range of Voltages

3, 3. 5 Other Properties of the Beam That Were Investiga ted

3.4 Conclusions, Section III

38 38

38 39 39 IV., PROBING OF SEVERAL LOW DENSITY FLOWS WITH AN

ELECTRON BEAM 40

4. 1 E?,perimental Apparatus 40

4.2 Experimental Procedure 40

4.3 Experiments 41

4. 3. 1 Flow From a Low Density Nozzle at Normal

Temperature 41

4.3. 2 Density and Static Temperature Profiles

of the Flow From a SecoI)d Nozzle 42 4.3.3 Rotational Temperature Profile Through the

Stagnation Region in Front of an Impact Tube

in Supersonic Flow 44

4.3.4 Temperature and Degree of Diss.ociation for

a High Tem perature Flow of Nitrogen 44 4.3.5 Rotational and Vibrational Temperature in a

'y

V.

4.3.6 Some Remarks Concerning the Performance of the Plasma Head at Low Arc Chamber

Pressure CONCL USIONS

5. 1 The Rotational Temperature of Ground State

47 47

Nitrogen Molecules 47

5. 2 The Electron Beam as a Probe for Low Density

Flows of Nitrogen 48

5.3 Limits on the Measurement of Nitrogen Molecule

Number Density 48

5.4 Electron Beam Voltage and Current Requirements 50 5.5 Limitation of the Beam as a Probe for Low Density

Flows of Nitrogen

5.5.1 Flow Velocity 50

5.5.2 Flow with Self-Emission 50

5.5.3 Accuracy of Rotational Temperature

Measure-ments with Increasing Temperature 50

5.5.4 Spatial Resolution 50

5.6 Suggested Improvement and Extension of the Probing Technique REFERENCES TABLES I to VI FIGURES 50 52

a(v) c F

[G]

h I J K k L N n SYMBOLS

Einstein's spontaneous transition probability for emission between states n and m.

rate of photon emission of wave number V

rotational constant corresponding to the v vibrational level

mean random speed of a gas particle

mean speed of electrons (not necessarily random) speed of light

excitation function of the electrons in the electron beam. function of rotational quantum number and TR - see Sec. 2.3

term values for vibrational levels (see Ref. 10, p. 92) Planck's constant

intensity of radiation

intensity in various vibrational bands

an unknown constant associated with the absolute intensity of radiation from the source used in the film calibration.

rotational quantum number - includes electron spin rotational quantum number - excludes electron spin Boltzmann constant

particle m ean free path number of particles

number of NlX 12 molecules neutral particle number density

n_I nes ne NiSll:

Nix

·

l

_1.

NIX

I

!

N

+

2 •

_N"

Kil 1

N"

_K'+l

N

'1 K'-l N K, p(v'. v")

Pp

PR

QN QI ~ QIS q(v'. v") (QRt'ra» " v 1

ion number density

secondary electron number density primary electron number density

upper electronic state of nitrogen ion N 2

+

ground state of nitrogen ion N

2

-

:r

ground state of nitrogen molecule nitrogen molecular ion

number of molecules in a rotational level of N~X'~ number of molecules in (K'+1) numbered rotational level of N2, X I!.

number of molecules in (K' -1) numbered rotational level of

N2,

X I

t

... 2.(" number of molecules in a rotational level K' of N_z

B

G

relative vibrational transition probability

relative rotational transition probability for the P branch

relative rotational transition probability for the R branch

cross-section of neutral molecule for excitation to

N:

Bl.L

by an electron

cross-section of ground state ion for excitation to ... "-c:'

N2,. B G by an electron

cross-section for excitation of ground state ion to

N;

B't2 by a "slow" electron

Franck-Condon factor for transition between

vibrational level v' and vtl of two electronic states. rotational state sum for vï vibrational level

v tfJ_N

~rs

Subsc,ripts 1 2 A a

vibrational state sum for Na. X IJ: state gas constant for a specific gas

rotational temperature vibrational temperature

vibrationallevel or vibrational quantum number degree of dissociation

electron-ion recombination coefficient

electron excitation coefficient for the transition

N:X~ to NtB~

. . f d N~tB~~

excltatlon rate rom groun state ions to .. " excitation rate from ground state molecules to

NI

Fir.

N:I32.(" excitation rate of ground state ion to ... .:;, "slow" electrons

excitation rate to

K'

Qf

~ v' density

collision diameter; cross-section =

wave number

wave number of rotational line (Kt, K2)

=

(3, 2) in the (0, 0) band of the first negative system constants defined in text

designates the state N2,X \~ designates the state

Nix

'l.g

refers to atoms

due to

conditions for nitrogen molecule number density calibration

c conditions for measurements in nitrogen jet M refers to molecules

v"

₁ refers to vibrational levels of the N°i X

Ir.

state

VI _{refers to vibrationallevels of the} N:B2..~ _state

(

v"

₂ refers to vibrational levels of the

Ni

x

2

Z

state

Superscripts

1 designates a signlet quantum state 2 designates a doublet quantum state

indicates upper state in a transition

I. INTRODUCTION

This research was undertaken as part of the development program of a low density (1'\

<

1. 5 x 10 16/cm3 ), high stagnation tempera-ture wind tunnel (Ref. 1). One of the first steps in producing a useful facility of this type is the development of reliable methods for measuring the physical quantities which describe gas flows at low density and high temperature.

For the densities to be used in the tunnel, the partic1e collision frequencies will be relatively low. As aresult, for the flows occuring in the tunnel! it is expected that the various degrees of freedom, or energy modes of the gas, will not always be in thermodynamic equili -brium. In addition, because of the low densities and high temperatures involved, the flows are expected to be seriously influenced by viscous effects.

When viscous non-equilibrium flows exist in a test facility, several experimental difficulties may be encountered. These difficulties arise because the properties of the gas from one point to another in the flow are not nearly as simply related as they are in the case of the inviscid flow of a perfect or equilibrium gas. In the viscous non-equilibrium flow, for a detailed and accurate interpretation of experimental results, it is necessary that the state of the gas can be determined at any point in the flow, independent of upstream conditions.

lf a gas flow is strongly self-emitting and ionized, then some of the desired properties of the flow may be obtained from an analysis of the emission spectra (e. g. , Refs.2,-3 and 4), and by the use of Langmuir probes (Refs. 5 and 6). The use of the stream self-emission fails, if the flow is non-radiating or only weakly radiating. One example of this type of flow would be the expansion of partially dissociated nitrogen through a supersonic nozzle in a low density wind tunnel. There is a wide range of such flows which are of interest in both applied and basic research.

There were, prior'to the work reported here, no instruments or techniques which were capable of an accurate determination of the properties of these flows.

The present investigation deals with the problem of mea-suring the properties of viscous non-equilibrium flows of low density nitrogen when the gas has no significant electronic excitation, and is

thus only weakly radiative or non radiative. Nitrogen is of interest because of its importance in the earth's atmosphere. For a complete description of a non-equilibrium flow of nitrogen, the properties that must be deter-mined are;

1) translational temperature - no direct method of measuring this quantity but it rnay usually be inferred frorn the rotational temperature.

2) rotational te.mperature - method developed in this report may be used to measure the rotational temperature.

3) vibrational tem perature - m ethod developed in this report may be used to measure the vibrational temperature.

4) degree of dissociation - may be measured by method used in tl1is report or perhaps by the use of probes (see Refs. 7 and 8).

5) flow velocity - pressure probes may be used with caution. 6) static pressure - pressure probes may be used with caution. To measure these parameters in non-equilibrium viscous flows, it is

always necessary tb determine independently the temperature (1), and usually necessary to know (2) and (3), before the quantities (4), (5) and (6) can be obtained. The temperatures are also of fundamental interest

in determining the thermodynamic state of the gas. Consequently, the emphasis in this investigation was placed on developing a method, or methods, for measuring the temperatures (1), (2), and (3). For most flow conditions, the rotational and translational degrees of freedom are in equilibrium, since only a few collisions are required for rotation and translation to equilibriate (Ref. 9). Thus, a measurement of rotational temperature means that in many cases the translational temperature is also known. Accordingly, it is usually sufficient to m easure only the rotational and vibrational temperatures. The work reported here

describes the developrnent of a rnethod for measuring the rotational and vibrational temperatures of a weakly radiating or non-radiating flow of low density nitrogen. So.me work has also been done on the measure-. ment of the degree of dissociation of a flow of nitrogen molecules and

atoms, since the methods available for measuring this parameter have l'lot beèIlr~xtenàiv~lydeveloped (Refs. 7 and 8).

The proposed method for measuring the rotational and vibrational temperaturesjsthis. When a collimated beam of moderately high energy electrons (10 to 100 kev) is passed through nitrogen at.á low density, a relatively strong emission is observed from the nitrogen mole-cular ion N2+' If the beam electron energy is sufficiently large, the collision cross -section of nitrogen molecules for the beam electrons is small. Consequently, if the gas density is low, the beam electrons undergo few collisions in the nitrogen and as aresult the electron beam is practically unattenuated. Under these circumstances the beam appears as a thin luminescent cylinder, with the principal contribution to this

lum~escence being frorn the molecular ion N2+' The photograph in

.

.

of 1. 65 mm, as it traverses nitrogen at room temperature and at a

pressure of 0.200 mm Hg. There is no noticeable spreading or

attenua-tion of the beam. In the present experiments the condiattenua-tions were like

-wise such that the electron beam was not appreciably attenuated .

The N2

+

band system excited by the beam is prominent in

auroral spectra and is commonly called the first negative system of

nitrogen (Ref. 10). It was anticipated that the beam excited emission

could be described theoretically. It was hoped that the rotational and

vibrational structure of the emission spectra could be interpreted, with the help of the theory, to give the rotational and vibrational temperatures

of the ambient gas. This report describes the investigation, both

theoretical and experimental, of the use of an electron beam for making

such measurements.

In addition, the beam excited emission has been used to make measurements of the local partial density of the molecular nitrogen. From these measurements the degree of dissociation of a flow of nitrogen molecules and atoms could be estimated.

11. THEORETICAL

The rotational temperature of a molecular emission may be obtained by measuring the relative intensities of the rotational fine structure in the vibrational bands (Refs. 2, 10, 11). The vibrational temperature may be obtained by measuring the relative intensities of

various vibrational bands in the emission (Ref. 10). The accuracy of

the temperature determina.tion depends on the accuracy of the theoretical

description of the excitation and emission process and, of course, on the accuracy of the actual measurements.

This section gives the theoretical basis for the various measurements undertaken in the experiments. First, the excitation-emission path is established for the conditions encountered when using the electron beam. This is followed by the theoretical description of the excitation of nitrogen molecules by fast electrons with particular reference to the measurement of rotational and vibrational temperatures. Finally, an analysis is made of the factors affecting the measurement of gas density when using an electron beam.

2. 1 Excitation-Emission Path

The most prominent band system excited by an electron beam of moderate energy (10-100 kev) passing through low density nitrogen is the first negative system. This system corresponds to the

electronic transition B

1.~:

-

X a

.G;

in the nitrogen molecular ion

N2+ or . N₂ + [3 2 ~ _{-. N 2 +}

X

2 ~ The emission is excited by the

with neutral, electronic ground state nitrogen molecules (N2, X tI; ), and by some excitation path bring the molecules to N: B

2.2. ,

which is the upper state of the transition producing the first negative system. It is necessary to know the excitation path before a theoretical prediction of the emission intensities may be made.

There are several possible modes of excitation to the

N:

[3

'aZ

state. These are: direct excitation from

Ntx'E ,

cascade population of

N;

B?I,

from higher energy electronic states of N2+' excitation to some electronic state of N2 and then further excitation to

N:

B

2

L ,

excitation of ground state ions to

Ni

B

'I.,L by primary beam electrons, excitation of ground state ions by secondary electrons, and direct excitation of

N

₂ X

'L:..

molecules to

N:

B

2 ~ boY secondaries with energies in excess of the excitation potentialof

N

2

t

B

'2

2

.

The possibility of the occurrence of each of these modes is discussed in the following sub-sections.

2. 1. 1 Cascade Populations of

Ni

B

~~

The population of

N;

8

i

_l

_{by an electronic transition from}

a higher energy level of N2 + does not appear likely. The transition popu-lating B~ would be accompanied by an emission comparable in intensity to that of the first negative system. No such emission has been reported in the literature. Two electronic states of N2 + other than

8 'lL

have been identified, but these states combine only with the ground state N~

X

~.

2. 1. 2 Double Excitation to

N:

B

2L:.

Double excitation involves the excitation by primary bearn electrons of a ground state nitrogen molecule to some electronic state of N 2 or N 2 + followed by the further excitation of this particle by beam

electrons to

N;

8

'aL

.

In the electron bearn, by far the most likely mode of double excition, is the excitation of ground state ions to

N;BZ

The ground state ion population comes from two sources: ions produced by direct excitation from

N

2

X

12:

and those formed as a

result of transitions from

N;

B'1..

L

.

The ground state ions move with the thermal motion of the gas and thus they take a finite time to diffuse from the space occupied by the electron beam. Accordingly, they are exposed to the possibility of collisions with beam electrons.

To discuss the effect of the ions (and later the secondary electrons) it is convenient to think in terms of a typical experimental situation. The following conditions are assumed: beam electron energy 17. 5 kev, beam current 250}L amps, nitrogen pressure O. 330 mm. Hg, the nitrogen temperature T

=

3000_{K, and the beam diameter O. 17 cm.}

A 250)J. amp. current represents an electron flow of 1. 57 x 10 15 electron/ sec. If the ground state ion concentration in the space occupied by the

beam is hr , and it is assumed to be constant within the beam, then roughly,

nr

must be such that the number of ions produced equals the number düfusing (due to thermal motion) through the sides of the O. 17 cm diameter cylindrical volume occupied by the beam. From data presented in Ref. 13 for the energy loss of high energy electrons in air, it can be estimated for the presently assumed conditions that the mean free path of the beam electrons is 3 cm. It is assumed for this estimate that the only inelastic collisions the beam electrons experience correspond to

18. 7 ev energy loses due to the excitation of Nt X

'L

to

Ni

B

\t .

This assumption neglects the fact that there are two

possible excitation paths to the ground state ion. These are, the indirect path which requires 18. 7 ev for the excitations of

N,l

X

'I.

to

Ni

B

'lol:

(the ground state ion is then formed by a spontaneous transition to N2t-

x

2.1. ),

and the direct path which requires only 15. 6 ev for direct excitation to

N;

X

~L from

N

2

X

'i. .

These two mechanisms result:in

approxi-mately equal rates of ground state ion population, thus the average electron energy loss for the formation of a ground state ion would be about 17. 1 ev. As aresult, the assumption of 18.7 ev energy losses introduces an error in the calculated ground state ion population rate. On the other hand,

this error is small compared to errors introduced by other approximations made in this analysis. These include, the assumption that the only energy loss experienced by high energy electrons in air is due to the formation of nitrogen ions from the nitrogen molecule. Also, the assumption that the ions diffuse from the beam column with their norm al thermal velocity is probably in error, since a type of ambipolar diffusion caused by the secondary electrons is more likely to take place.

Since the average electron in the beam experiences one 18.7 ev collision within a beam length of 3 cm there are 1. 57 x 10 15 ions produced every second in a 3 cm length of the beam. The number of ions düfusing through the wall of a 3 cm long portion of the beam cylinder is nI. ë'iT ('3)(0-17)

/4

,

where C is the mean random speed of the ions,

( ë

= 475 m/sec). Assuming no recombination, and equating the number of ions produced to the number diffusing through the wall) T'l_x

= 9 x 10 10

ions/cm 3 . This compares to the neutral nitrogen molecule number densityof 1. 1 x 10 16 /cm 3 . The rate of excitation (Cf>'t.) to

Ni

a1.g

from the

N;

x

2I:, ground state ions, will be proportional to the ion concentration and the excitation cross-section

(QJ:)

of

Nt

X2.g

for excitation by electrons to

Nt

B

2.Z

.

Similarily, the excitation rate

(((JN) to

Ni

B?oL

from Nt

X.

ILo

depends on the neutral m?lecule nllm!:>er density and the excitation cross -section (QN) for the

N

2 X

L

to ' . .

Ni

B2.~ transition.

There is no direct evidence to suggest what the cross-section Qr might beo However, an estimate of Qr may be obtained from the value of. Q:S J which is the excitation collision cross -section for the collision of slowelectrons (order of 50 ev) with ground state ions. Qr& has been estimated to within a factor of 10 3 (see Sec. 2.1. 3. 2) andis between 1. 5 x 10- 12 and

1. 5 x 10- 15 cm 2. For the fast primary electrons, Qr wil! be at least one order of magnitude less than

Qrs.,

and therefore

Qr

=

1. 5 x 10- 13 to 1. 5 x 10- 16 cm 2. The cross-section QN may be obtained approximately from the beam energy loss data given in Ref. 13 by making the same assumptions as in the previous determination of the mean free path of the beam electrons. For these assumptions, QN

=

0.3 x 10- 16 cm 2 for 17.5 kev electrons. Using these cross-sections it is found that ({)x. / (f)N is between

4 x 10- 2 and 4 x 10- 5 . The excitation of ground state ions by primary electron can thus be neglected.

There is some experimental evidence which supports this conclusion. Experiments haye been conducted by Langstroth (Ref. 12) under conditions similar to those existing in the present case. These show that the optical excitation functions (the curves of emission intensity versus electron energy), of the (0, 1) and (1, 2) vibrational bands of the first nega-tive system, when extrapolated to zero emission intensity come very close to

18. 7 volts. This is the potential energy of

Nt

B"L.

above the ground state

Nt X

'r.

.

This implies that there is no double excitation of ions formed directly from

N

2

X

'I.. ,

or of molecules in an upper electronic

level of N2. Tt.e optical excitation functions would still have a finite value at electron energies of 18. 7 ev if there were significant amounts of these types of double excitation taking place. The experiments exclude the possibility of double excitation to

N;

elI:

from some upper electronic level of the neutral molecule or from ground state ions formed directly from

N

2

X

'g

.

They do not exclude the excitation of ground state

i.ons which have been formed in the beam column due to the

N: 81.2.

to

Nt X"a

r..

transition. The rate of excitation from

N2.,X'

L.

to

N~X

t!

is expected to be approximately equal to the rate of excitation from

N1..X

'l:

to

N:

B"La

.

Since the double excitation of ground state ions formed directly from

N'2, X

1[. may be neglected (as shown by Langstroth's experiments), it seems plausible to assume that the double excitation of approximately twice as many ground state ions mayalso be neglected. Therefore,Langstroth's experiments support the results of the previous theoretical analysis.

by Secondary Electrons

The secondary electrons are more difficult to consider. For the secondaries both the energy distribution and the initial direction after ejection from a molecule are unknown. Assuming single collisions,

secondaries with energies below 3. 1 ev cannot produce any excitation. Secondaries with energies greater than 3. 1 ev can excite ground state ions to

Nzt-

8

'l.L.

.

Secondaries with energies greater than 18.7 ev can excite the neutral molecule directly to

N;

B~Z

.

.. 2.1. 3.1 Secondaries with Energies in Excess of 18.7 ev. First, secondaries with energies above 18. 7 ev wiU be considered. The maximum total ionization cross -section of nitrogen molecules to electrons is 2. 87 x 10-16 cm 2 , this is at electron energies of 100 ev (Ref. 14, p. 265). An approximate ionization coUision cross-section for 17. 5 kev electrons1from the energy loss data given in Ref. 13, is 0.3 x 10- 16 cm 2 . There is approximately one order of magnitude difference in these cross-sections. It is possible that if a significant number of the secondaries were 100 ev electrons, there would be a large contribution to the observed emission owing to the excitations caused by the secondaries. This would be of concern if the nature of the excitation by relatively slowelectrons is significantly different from the excitation

.~ due to fast electrons.

The possible presence of an effect due to emission excited by

secondaries with energies in excess of 18. 7 ev requires some experi-mental consideration. The work done regarding this point has been covered in Sec. 3. 3 of this report.

2. 1. 3. 2 Secondaries with Energies in Excess of 3. 1 ev

For secondaries with energies greater than 3. 1 ev, those that can excite ground state ions to

N.t

B

~

g

,the cross-section for the

excitation

N; X

'Lr

to

N:

8

tI.

is not known to better than perhaps three orders of magnitude. The value of the excitation coefficient (JfNtJ , for the

X

1.[ to

B

~ transition, may be obtained from an estimate made by Bates (Ref. 15, p. 249) of the ratio of the excitation coefficient

fI(N:')

to the nitrogen ion-electron recombination coefficient O(e (Ni) . Bates states that the ratio ()(N/)

/o<.e,(N:) "

probably lies between 1 and 103". From Loeb (Ref. 20, p. 562) o(e ~ 1. 5 x 10- 7 cm 3 /ion-sec at a

pressure of 5 mm Hg. Assuming this value for the present case {J(N:)

= 1.

5 x 10- 7 cm3 fion-sec

to

1. 5 x 10- 4 cm 3 fion-sec.

The secondaries are assumed to have energies of 3. 1 ev. It is noted that

P(Ni)

=

Ce

Ot.s,

where

ce

is the m ean speed of the electrons and

Qrs

is the excitation cross-section of the ions for the secondary electrons. Then,

Qrs

= 1. 4 x 10- 12 to 1. 4 x 10- 15 cm2 . The ion concentration in thé beam column for the assumed conditions is approximately 9 x 10 10 i?3ns / cm 3. The number of ions excited to

B

't~ from

X

tI..

per second per cm is then

'2

S-({)r s :: nu

Ol.

Ce

Q'!. s ::::

J.

3 x 10 ne.s t-o /. ']

x

10

fles ,

wj.1ere

fles

is the number density of secondary electrons in the beam capable of causing an excitation. The secondary electron concentration

ne..s

is

difficult to estimate because of the lack of information about the secnndaries.

It seems reasonable to assu.me that the secondary electron concentration wil! be of the same order of magnitude as the prirnary electron concentra-tion. Thus, for this case,

nes

= 107/ cm 3 and accordingly CfJz.s = 1 x 10 9 to 1 x 10 12 excitations / cm 3sec .

The number of transitions due to primary electrons excit-ing neutral molecules to

N/ l3

"-1:. ,

assu.ming the electron concentration to be 107Jcm3, the collision cross-section to be 0.3 x 10- 16 for 17.5 kev electron, and the molecule concentration to be 1. 1 x 10 16 / cm 3, is

~N

=

3 x 10 16 Jcm 3 sec.

It can be seen from this that {fÎzs / (fJN

~

0.3 x 10- 4 . Accordingly, the

secondary electron excitation of ground state ions may be neglected. This is of course contingent on the fact that the estimate by Bates of

(}tN)..t)

IO<el/IJ ...

l-)

is accurate to within the given limits.

2. 1. 3. 3 Effect of Flow Velocity on Secondary Excited Emission

The number of secondaries with energies above 18. 7 ev is not known. Assu.ming that there are some, there is the possibility that any gas flow-velocity would affect the emission caused by these secondaries.

If the average energy of the secondaries is assumed to 100 ev, the speed of these electrons upon ejection from their parent molecule is 5. 94 x 108 cm

J

sec. For the typical experimental case of p

=

330 ~ Hg and

T

=

300oK, the electron

Nteig

excitation mean free path for 100 ev electrons may be calculated approximately from the total ionization cross-section of 2.87 x 10- 16 cm 2 (Ref. 14, p. 265). The calculation gives L

=

3. 2 mm. To cover this distance the 100 ev secondary electrons require

o.

5 x 10- 9 seconds. Consequently, the transit time of the

electrons is negligible with respect to the mean life of

Nt B

7.2,

,which is 6.5 x 10- 8 sec (Ref. 17). Accordingly, any effect on the emission due to flow velocity, will not differentiate between emission caused by

secondary electrons and emission due to primary electrons. 2 . .1.3.4 Conclusions Concerning Secondary Excited Emission

Fro.m the previous discussions, it may be concluded that the secondaries will only cause difficulties if two conditions are satisfied. These are: that the secondary excited emission is different in character from the primary excited emission, and that there is a significant number of secondaries w:Lth energies greater than 18.7 ev so that the emission excited by them is relatively strong.

2. 1. 4 Self-Absorption

Another possibility for excitation of the first negative systern is self-absorption af the first negative system by

N/'I.:a.

L.

and its subsequent re-ernission. The possibility of self-absorption in this case would not seem to be very great because of the rather low ion dens ities which rnight be expected in the vicinity of the beam. On the other hand, the absorption would be of a resom~mce type which could be expected to be particularly strong. A simple check of the presence of self-absorption is possible in this case. In the rotational fine structure of the first negative system, every other rotational line should alternate in intensity

with an intensity ratio of 2: 1 owing to the nuclear spin of the homonuclear nitrogen molecule. If self-absorption occurred in the bearn, the observed intensity ratio of the rotationallines should not be 2: 1 but somewhat

less, with the amount less depending on the degree of self-absorption. Measurernents have been made of this intensity ratio and a typical measurement gives an alternation of 2. 04: 1; this is within the experirnental accuracy of a 2: 1 ratio. It can be concluded that there was no significant self-absorption in the observed emission.

2. 1. 5 Conclusions Concerning Excitation Process

From the discussions presented in the foregoing sections, it may be concluded th~ the prirnary excitation process is direct:excitation from

N

1

X

12:

to

N

1

8

1

.E .

Contributions to the ernission of the first

negative system by ground state ions being excited to

N:

B

1. I; may be neglected. If there are a significant number of secondary electrons with energies in excess of 18.7 ev, the number of excitations to

N:

B?~ due to these secondaries could be comparable to the number of excitations due to the primary electrons. This would only be of concern if the

character of the emission caused by the relatively low energy secondaries was different from that caused by the much higher energy primary electrons.

2. 1. 6 Excitation - Emission Process

The following physical situation exists for the excitation of the first negative system by the electron beam. The bearn electrons excite, by- collision of the !irst kind, the electronic state

Ni

BaL.

(18. 7 ev). The excited .molecular ion spontaneously emits radiation and falls to the

N! X

'l.

L

state (15.6 ev). The excited particle exp~riences no gas kinetic collision during the process of excitation and emission. The excitation-emission path is illustrated schematically in Fig. 2.

This description of the excitation-emission path holds good until the average free flight time of an excited particle cornpares with the mean life of the excited state. Bennet and Dalby (Ref. 17) have .measured the mean life corresponding to the (0, 0) band as 6.53

'!

0.22 x 10- 8 seconds.

For room temperature, number densities of the order of 1. 5 x 10 16/cm3

(pressure 470 p. Hg, the maximum existing in the experiments) and

assuming the diffusion collision cross-section of 4.8 x 10- 15 cm2, the

time required to traverse a mean free path at the mean speed of 475 m

Is

is

approximately 2 x 10-7 sec. Thus, there will be little interference

with the electron excited distribution in the N'l.~

B

'2.

L.

state, (for a

more detailed discussion of these points see Secs. 2.4 and 3.3.2).

N

"X\~

The excitation path is direct excitation from .. L.. to

N

"te'2. _{.. L..} ~ _{The emission is not disturbed by gas kinetic collisions.}

Knowing this, a theoretical prediction may be made of the emission

intensities as a function of temperature~in the vibrational and rotational

structure of the first negative system.

2. 2 Vibrational Excitation and Vibrational Temperature

The theoretical prediction of the relative intensities of the vibrational bands in the beam excited first negative system, as a function of vibrational temperature, gives the basis for making an experimental determination of the vibrational temperature of the nitrogen through which

the beam passes. It was assumed for the theoretical analysis th at the

process of direct excitation and re-emission went on unhindered by gas

kinetic collisions.

The theoretical description of the vibrational excitation of molecules by fast electrons may be made with the ai4 of the Franck-Condon principle in its wave mechanical formulation. For this analysis it is necessary to know: the shape of the potential curve for both states in the transition, certain of the molecular constants, and, it is necessary to make the assumption that the vibrational wave function is independent of

the electronic wave function. The analysis has been carried out by a

number of authors (e. g., Refs. 12, 15, 18). Generally, some model of

the potential curve such as the Morse function is assumed. The molecular constants are known from spectroscopic data. Using this information,

vibrational wave functions may be ca1culated (Ref. 15). From these,

the overlap integrals for the various vibrational transitions may be found

(Refs. 15 and 18). Bates (Ref. 15) has calculated the squares of the

overlap integrals, or Franck-Condon factors (q (VI, Vil», using a numerical

method for the integration. Jarmain, Fraser and Nicholls (Ref. 18) have

done the same calculation but have simplified the procedure by averaging the molecular constants of the two electronic states involved, to give a sort of average Morse potential curve. This simplifies the computation and in many cases gives answers very near those obtained by Bates. More recently, Wallace and Nicholls (Ref. 19) have given relative vibrational

transition probabilities (p(v~v"» for the

N:

8~!. to

N; X

a.l:

transi-tion. The transition probabilities (p(v!v"» take into account the variation of the electronic transition probability with vibrational quantum number. The relation between the Franck-Condon factors (q(v;v"» and the

vibrational transition probabilities (p(v', v"» may be expressed as

p

(V~

V")

=

(Reá~;ivI'J).~

q,

(V~

V")

where

R.

_e is the,eleclronic transition moment and 1;.",.. is the so-called

11 r centroid" (Ref. 20). The transition probabilities are available

for the

at.,

to ~

'1:..

transition and are given in Table I (from Ref. 19).

The Franck-Condon factórs are available for both the excitation and

emission. transitions. These are given in Tables II and III (from Ref. 15). Since the variation of Re is not great for the first few vibrationallevels

(compare Tables I and lIl), and since no transition probabilities are known for the excitation transition, the Franck-Condon factors were

used in the present case. That is, it was assumed that

(Ré(r;.",,,)y·

could

be represented by an average value

{Re

(r,,·v"

fl.

2.

To avoid any further confusion

{Re

(r",v"~

q.

(v!

v")

win be

called the relative vibrational transition probability or just vibrational

transition probability, and will be identified by p(v', v"). The

Franck-Condon factors will be identified by q(v' v"). It should be remembered

that while the spontaneous transition probability for absorption is proportional to p(v', v"), the spontaneous transition probability for

emission is proportional to . V' p(v', v"), where V is the wave number

of the emission.

Using the Franck-Condon factors calculated by Bates for

the excitation path

N

₂

X \

~ to

Nt+-

B

2.1..

(Tabie 1I), and the Franck-Condon

factors calculated by him for the first negative band system (Tabie lIl).

the relative intensities of the bands in the emif~sion may be obtained for an

arbitrary initial population distribution in the vibrational levels of the

Nt

X\~ state. If the distribution in the

X

'z

vibrational levels is a

Boltzmann distribution, with a vibrational tem perature Tv ' the relative intensities of the bands in the emission can be predicted as a function of the single parameter Tv'

In the present case it turned out for several practical experimental reasons, that it was advantageous to measure the intensity ratio of the (1,0) and (0, 1) bands of the first negative system. The variation of this ratio with vibrational temperature is shown in Fig. 3,

which was ca1culated wi th the help of the vibrational population distributions

given in Table IV, the Franck-Condon factors from Tables II and lIl, and the wavelengths of the vibrational bands from Table lIla.

It may be seen from the curve (Fig. 3) that areasonabie sensitivity to vibrational temperature is obtained for a temperature range

from about 5000_{K to 4000}0_{K and particularly between 500}0_{K and 2000}0_K.

The vibrational temperature was measured in the experiments by measuring the intensity ratio 10-1/11-0 and using the curve from Fig. 3 to obtain

The applicability of the theoretical analysis is rather difficult to judge. Comparison of the theory with the experimental results of Langstroth (Ref. 12) (see Sec. 2. 1. 2 for a more complete description of this experiment) shows that the value calculated by Bates for the band intensity ratio 10-1/11-2 agrees with the experiment to within the accuracy of Langstroth's work. This was apparently

±

five per-cent for these particular measurements. Langstroth's results, however, only

cover the electron energy range from 21 ev to 50 ev, although in other respects his experim ental conditions were very similar to the conditions encountered in the electron beam experiments. To draw conclusions from this experiment about electron collisions with nitrogen molecules at electron energies between 10 and 20 kev which correspond to the

electron beam energies, would appear to be unreasonable. However, the experiments reported here (Sec. 3.3.1 and 3.3.4) concerning the

excitation of

N

₁

X

I

L

nitrogen molecules to

N;

B'l.Z,

show th at for

beam electron energies from 12.5 to 20 kev there is no measurable interference by the electrons with the rotational motion of the nuclei

(only the nuclei are concerned since

ï::

states are the only ones taking part in the excitation). If any possible interference is considered to take place due to a simple momentum exchange between the high energy electrons and the nuclei, then the fact that the rotation is not disturbed means that the vibration is also not affected. It is perhaps of interest to note that while the electrons are very light, the energies they posess are large, so that in classical head on collision they would be capable of radically altering the rotational or vibrational energy distributions as a re sult of simple momentum exchange with the nuclei.

There is no interference with the nuclei at large electron energies (see above and Sec. 3.3.1 and 3.3.4). There also appears to be no interference for the 21 to 50 ev range (Bates' calculations (Ref. 15), which assume no interference, agree with the experimental results of Langstroth. Also, the measured intensity ratio 10-1/11-2 is constant for the energy range 21 to 50 ev in Langstroth's experiment). Thus, for the excitation conditions existing in the electron beam, it was concluded that the theoretical description of the vibrational excitation outlined in th,is section was accurate to within at least

±

5% (the accuracy of the experimental verification in the 21 to 50 ev electron energy range).

2.3. Rotational Excitation and Rotational Temperature

The theoretical prediction of the relative intensities of the rotational lines in the beam excited first negative system, as a function of temperature, gives the basis for making an experimental determination of the rotational temperature of the nitrogen through which the beam passes.

It was again assurned that the process of direct excitation to N~

B

2.L:.

and the subsequent re-emission went on unaffected by gas kinetic collisions. This assumption is va lid (see Sec. 2.1) for the

experi-mental conditions used here and for any projected use in a low density wind tunnel where the molecule number density is not likely to exceed

1. 5 x 10 16 Icm3 (p = 470)) Hg at room temperature). It was also assumed that the number or population distribution of molecules in rotational levels of the

Nt

X'~ state was a Boltzmann distribution which could be characterized by the ternperature T R . This assumption is not necessary in order to predict the relative intensities of the

rotationallines as any arbitrary distribution may be used. In this case the interest was in temperature measurements. Thus a Boltzmann distribution was assumed.

To predict the relative intensities of the rotational lines of a particular vibrational band in the emission, it was first necessary to find the relative population distribution in the rotational levels of a particular vibrational level of the

N;

B'll

state, owing to direct excitation from the

N'l. X

'L

state.

'CÇ'+

In the transition which occurs in the excitations

N2.X

Log

to

N; 8

2

L.: '

the change of multiplicity of 1, which implies a change in total electron spin of 1/2, can be accounted for by the ionization pro-cess. This is because the secondary electron will carry away with it an electron spin S equal to 1/2.

The spin coupling in the

B

22:

state is very weak so it was assumed that the doublet states of

8

2~ were unresolved. It was

further assurned that the two states X'2, and

82.2:

belong to Hund's case (b) (Ref. 10). The rotational energy levels are then designated by K. The selection rules applying to Hund's case (b) for ~-

L

transitions are

.6 K

=

:t

1,

A

K

=

0 being forbidden. It was assumed that these slection rule~ apply under the conditions existing for the excitation

N

₂

X'

2:

to

N

₂

B!

.

In this case the selection rule

A

K :;;:

:t

1 corresponds to

~:

=:t

1

i"

the spin change (1/2) due to the removal of an electron in

the ionization process. Satellite branches (~ K =1=

D.

J = 0) are not excited because of the singlet nature of the lower state quantum levels. The selection rules above predict the presence of a Pand an R branch in the excitation. Schematically, the excitation process appears below.

K'

₌

₀ I 2

3

.

..

B'l.L+

IA. _{0 00 00 00}

.

. .

'-""-' ...-.... ...,

X

'2:+

_Kil 0

~o

I !.. .3 9 ₁

₌

0

In the excitation process, the upper state level Kl is populated by the Rand P branches from the lower state rotational levels

Kl - 1 and KI

+

1. The number of molecules per second excited to the

KI level of a particular vibrational level VI of N~

B

22 is made up of the sum of the contributions from all of the vibrationallevels V"I of the lower state. The subscript 1 designates the NozXIL:. state. Say that in each vibrationallevel vï of

N

2

X '2

there are

N"II

molecules, and in each

. 1 f b . 1 VI

N

1I

rotatlOnalleve 0 a vi ratlOna level there are KIl moleOlles. Assume

that the relative rotational transition probabilities for the Pand R

branches are those normally associated with optically allowed

12

to'~ transitions. Then, if it is also assumed that the rotational and vibrational eigenfunctions are separable (Ref. 10, p. 203), the population rate to a

KI of a particular VI of

N; B

~L is

({J[v\

'

K

I

J

=

F.?

!r(N~'H),;e

_t-

(N~,_~?

P

R ]

..

Cj.(V:

v,")1

v,

<o",~·.·' r~J.l.~.K"')PP

t

(NKL,)PR]

)

(1) where q(vl

, VIi> is the Franck-Condon factor. The expression

~,~.

".:]

is a normalizing factor. Pp and PR are the relative rotational trab2sition probabilities or Honl-London factors for the Pand R branches in the excitation. F(v) is the excitation function of the electrons. If N is the number of N2XI~ molecules, then

N

Ffv) is the number excited to

N; 82..2

per second. For high energy electrons Ffv) may be assumed to be a constant, F, independent of the wave number.

The electronic transition moment is assumed to be independent of the vibrational levels involved in the transition, If it is desired to include the dependence of the electronic transition moment on vibrational quantum number, there is no fundamental difficulty as

p(v'v") may be substituted for q(v\ v"), and F redefined, with no other changes being necessary in Eq. 1.

The rotational transition probabilities or Hönl-London factors are those corresponding to a

'1: -

IL,

transition (the doublet state is not resolved). Written in terms of K' they are Pp

=

(KI+1)/(2KI+3)

and PR

=

K' /(2K'-1) (Ref. 10, pp. 20, 22 and 208 and Ref. 11, pp.

147-155). Since the rotational distribution in the NtX'~ vibrationallevels is assumed to be a Boltzmann distribution

11

I'JK,"

and then

N

u ( 11 ) '!'!y'"

2K

l

tJ

(QR

(r

_R) )\/,11 I (2)

.,

-

F

p

N#(.'t

I (3) and I1 " ,

P

R NK!...I -

..Hv."_

K

(QRm)~11

(4)

Where

(QRfrR))v,11

is the rotational state sum of the vï vibrational level and the rot1ational energy is given by ER

=

8...,

1«Kt I)hc.

Also, h is Planck's constant, c is the speed of light, k the Boltzmann constant, T R the absolute temperature characterizing the rotational population distribution in

NiX

I

L ,

and

Bv."

is the rotational constant of the vï vibrationallevel. Using Eqs. (3) and (4), Eq. (1) may be written as

(5)

[e)

=-

[~oJ~N~'t-I)V;")

Pp

t

((N~LI)",,)

PRJ)

For

N2X~

N

II _1\1

_-6

_o

(v,")hclkTv

₍₆₎

v,"

=

rO:&))x:

. if there is a Boltzmann distribution in the vibrational levels of XIx. . No is the number of

N~X

'l

molecules and

(O...,(Tv

))x'Z

is the vibrational state sum _of tne X~ state. The vibrational energy of the level vï is

hc Go(vï ) when zero vibrational energy is assumed to correspond to vï

=

0 .. From Eqs. (5) and (6)

(7)

which is the number of molecules per second brought to the rotational level K' of the v' vibrational level.

-X,

is a cons~ant.

For the emission, the intensity may be written as

I .::

a.(v)hc

11 , where o..{v) is the rate of photon emission and 11 is

the wave number of the ernission. The relative rate of photon emission in a rotational line (K', K2") ot;. the first negative system is, if v'2 represents vibrationallevels of the

N2,

X

~

L

state, and the rotational and vibrational eigenfunctions are assumed to be separable,

,

(8) since the rate of photon emission from any rotational level Kt must be equal to the rate of excitation to that level. The subscript 2 designates the

N; X

2!

state. The rotational transition probability for the transition (~t, K'2) is represented by P(K', K'2)' For a given level Kt the surn

(9)

Frorn Eqs. (9) and (8) and the expression for emission intensity, the intensities of the rotational lines (KI,

Kï )

in a vibrational band in the first negative system may be written as

,

(10)

where X2,ls a constant for any selected vibrational band of the system.

KI::.

o

₃

₄

o 00 """"" ~ 00

--

00

o

•

o

J

2.

4

• •

'

.

: " .

A schematic diagram of the emission process is shown in the accompnaying drawing. In the present experiments the R branch is of interest as it is the only branch that the spectrograph used in the experiments could resolve. For the R branch P(K', K'2)

=

PR

=

K' / (2K'+1) (Ref. 10, pp. 20, 21 and 208), where it is again assumed that the doublet states are not resolved. Using P_R and Eqs. (10) and (7) the intensities of the rotationallines in the

(v', v'2) vibrational band are

(I

I 11)'

,,1K~k'~1)

=

X

7/4-2

(q.1Y:v,")[U~K/t'U

·

e-Ga(v,'·)hc/kTvl

(11)

K.

K~

v)

V7.V'

"t 3 V, 11

((Q,aJTR.))v,"

[eJ

j

where V is the wave number of the emission. It should be noted that (K' + K'2

+ 1)

=

2K' and that

X

₃

=

()(,

1(2. )/2

=

a constant.

There is one important detail concerning the line intensities

w~ch should be mentioned. In the

N2,

X'

~ state the nuc1ear spin of the nitrogen molecule causes the odd numbered rotational levels (K'l = 1, 3, 5 . . . ) to have exactly half the population of the even numbered levels (Ref. 10, p. 209). lf the selection rule ~ K

=

:!"

1 holds rigourously in the excitation, then the rotational lines in the em ission corresponding to the odd numbered rotational levels of

Ni

B

~ wiU have twice the intensity of the lines corresponding to the even numbered rotational ~evels of N2.~B22 The R branches of the vibrational bands should thus consist of two groups of rotaHonal lines with one group having exactly twice the intensity of the other. In the branches, the lines wiU alternate in intensity corresponding to whether they originate from even or odd numbered rotational levels of

N'l+

B

'4r

Any relaxation of the selection rule

L:::.

K =

i'

1 would tend to equalize the intensities of the two groups of lines. It should be noted that Eq. (11) applies to either group of lines.

Equation (11) is best dealt with by considering two cases. First, when Tv is low (<. 8000_{K) and there is no significant vibrational} excitation, and secondJwhen Tv is high and vibrational excitation becomes significant.

2.3. 1 Case of Low Vibrational Temperature

When Tv is low there is no excitation of vibrational levels of other than vï

=

0 (see Table IV). In this situation Eq. (11) rnay be written as

where

[G]

~

f(,<l

_fl

)e-:-

2

(3v..I(K

'H)hclkiR

+

KI

t

2

Bv,u /('hC/kTR]

l~

(2/<ltl )

and

)(t

=-

~q&'

0)

ë~o(o)

helk

Tv

(QR{TR)J

_O

[e]

= a constant for a given v'

that depends only on Tv and Tn. Taking logarithms., Eq. (12) may be put in the form

If the theoretical description of the excitation process is correct, the rotational temperature may be obtained by measuring the

relative intensities (1/1₀) of the rotationallines in a vibrational band of

the first negative system and plotting , ,)

- loge

f

[(I~~ I<~I)~~ V~I/rJ

I

(I<+I<~~

1)ft;J1I

4)

versus K'(K'+l). Thi,s is done by drawing a straight line through the

experimental points, measuring the slope (b) of the line, and then

T

R' -=

8v.'

1

he / k b .

_{The quantity}

CG

J1I4 _{has been evaluated up to}

K' = 21 for a range of rotational temperatures. Table V gives these values

in a convenient form for use with the log plots when obtaining temperatures.

In the table,V has been normal~~ed with

110

t which is the value of 71

corresponding to the line (KI, K2) =(3,2). Since[qJ depends on the

rota-tional temperature TR and on K, it is necéssary to assume a temperature

and th en obtain the appropriate value of 10910

[[G](lJ11Io)+

J

from

Table V (for convenience logarithms' to the base 10 rather than base

e

were used in the log plots). Since 1.0910[[6](11/110)4 J is small for most

temperatures and K's, one iteration is usually sufficient to obtain a

satisfactory temperature measuremenf, If the variation in

[GJ(lIIYo~

is neglected, the measured "temperature" will tend to be too high. Some constants which are useful in making the rotational temperature measur..e-ments are given in Table VI.

The accuracy of rotational temperature measurements made by this method will depend on the accuracy of the assumptions

regarding the applicability of the selection rules and rotational transition probabilities to the excitation transition. An experimental verification of these assumptions may be made by measuring the rotational temperature of nitrogen, which is at a known rotational temperature, and then com-paring the measured and known temperatures.

2.3.2 Case of High Vibrational Temperature

When Tv> 8000K there is significant excitation of the upper vibrational levels of

N

_l

X'L.

From Eq. (11), assuming that the emission band observed is the (0, 0) band of the first negative system

(II<~

11

),,;v." ::

v4-X

3

[O/q,o)ëGofo/)hc/kT.t e-

BolJ (l<f/)l<'

hC!kTR..[G]

(1<'+1<;+,

t-

q.t, ..

0)

ë",(I,lhclkT.

e-

B,:, K

'(I<i-,)hclk

\[G]

of

q.l2/

j

o)ë

Go

_(?.)hc/kTv

e

_-B'l':

_/<

_'(K

_'tI)

hc/kr~[G]

t

•

J

(14)

(2.

I< '

+-1

The vibrational temperature may be measured and will be known independ-ently of the rotational temperature (see Sec. 2.2). If the vibrational

excitation corresponds to öther than a Boltzmann distribution, the

relative populations of the vibrational levels of

N

_{2 Xl} ~ may be found by measuring the relative intensities of all the first negative system bands in the emission. From these rblative intensities the corr~esponding

Ng,X

'L

vibrat~onal

population distribution may be

ca~culated

by using the Franck-Condon factors given in Tables U and UI and the wavelEmgths. in Table lIL a

_,

. In this case, it is assumed th at the vibrational levels of _'

NtX

L

have a vibrational temperature Tv' The q(v', VIi.) values are known (Tables U and UI and Ref. 15) so that for the known vibrational temperature Tv the terms

q.(V:v,")

e-fio(v,")hclkT., may be calculated. Then, from Eq. (14)

- A

-Bol' kl(KI+J)hc/~TRr.G]

8

-el:.!<'I(/<I+/)hclhlRr:

G

]

e ·

.L\

+

e

·

.Ll

(15)

+. ., .

'

.

Because of the near equality of the equilibrium internuclear distances of N2X'L, and

NrBR.L:.

the q(v', vï ) values other than q(O, 01) q(l, 11), q(2,21), are all quite small. This is indicated in Table II.

By referring to Table IV, it mayalso be seen that the number of molecul~s

in vIl

= 1,

-2, . . . is, except for very high! vibrational temperatures, ' always small compared to the number of molecules in vï

=

O. For the above reasons Band C etc. will for this partlcular molecule be generally small compared to A. For the

N

₂

X'L

state B v· 1-1" shows a very weak

dependence on v"1 (Bv"

=

2.010 - 0.0187 (vï

+

1/2) (Ref. 10». In view of the situation describeJ above it was possible to write Eq. (15) as

or

Ir )

IT'

-8;((

K'(k '

+l)hclkT

R ( .... K,K~

0,'1"

J.o -

e

(I<

1+

K~

+n

114[6]

(17)

where

I~"=:

><3

[A

/(QR(TR))o[eJ

t · . . . ] ,

where B"eff is an average rotational constant determined by weighting the

Bv:~

.

in the average with the appropriate

rA

/bo)/(A/~

-f

B/Ó,l . . . )

t

(8 /

d, ) /

(A

I

Ó

_o

t

B /

cS, + . • .. ) ,

(C /

Ó 'J. ) /

(A /

~

t

8 /

~,

+ . .• . ) ,

Kv,"

:=

(QR(TR))v,,1

re] .

The errors introduced by such a procedure are not significant. As an example cons~d.~:. the cas':(?~)T,Y

=

4000oK. Then. for TR = 4000oK, the exact sum

A e (

)/~o

+

Be

Và.

+ .•.

is proportional to 0.48 x 10- 6 for the rotational line (KI, K"2)

=

(21, 20). The approximate approach also gives 0.48 x 10- 6 for the same rotational linea

Accordingly, in the case of high Tv it is possible to obtain a measure of the rotational temperature from Eq. (17) in the same way as from Eq. (13) in the lower temperature case.

Wh en making high rotational temperature measurements. care should be taken in the determination of the intensities of the R branch lines since the underlying P branch will have a significant strength when TR> 800oK. It is usually quite obvious from the log plots when the P branch interferes with the R branch lines as the points for the R branch lines will be noticeably off a straight linea The maximum deviation occurs for the lowest numbered rotational linea An example of this

interference is shown in Fig. 4. For the rotational temperature measured (TR = 8900_{K) the interference of the P branch with the R branch is quite}

pronounced for the lower rotational quantum numbers. It dies away very r.apidly as K increases.

Another consideration when making high rotational temperature measurements is, that as the rotational temperature rises the sensitivity

of the slope of the log plot to temperature decreases. Consequently, at high rotational tem peratures the experimental errors cause greater per-centage errors then at lower temperatures. For constant error in the slope, the percentage error in the .measured rotational temperature varies directly