The measrement of energy transfer in gas - solid surface interactions using electron beam excited emission of light

I 1

THE MEASUREMENT OF ENERGY TRANSFER IN GAS - SOLID SURFACE INTERACTIONS USING ELECTRON BEAM EXCITED EMISSION OF LIGHT

by

D. J. MARSDEN

NOVEMBER, 1964 UTIAS REPORT NO. 101

y

THE MEASUREMENT OF ENERGY TRANSFER IN GAS - SOLID SURFACE INTERACTIONS USING ELECTRON BEAM EXCITED EMISSION OF LIGHT

NOVEMBER, 1964

by

D. J. MARSDEN

UTIAS REPORT NO. 101 AFOSR64-2482

.

,

ACKNOWLEDGEMENTS

I wish to thank Dr. G. N. Patterson for providing the

opportunity to undertake th is investigation and to express my appreciation to the staff of the Institute for Aerospace Studies for their help and en-couragement. In particular the contributions made by my supervisor Dr. J. H. deLeeuw through discussions and suggestions and his continued interest, are gratefully acknowledged.

I would like to thank Mr. J. L . Bradbury and Mr. J. Leffers for their help in construction of the experimental apparatus.

This work was supported by the United States Air Force Office of Scientific Research Grant #AFOSR 276.64 and the Defence Re-search Board of Canada .

SUMMARY

This paper describes a new approach to the measurement of energy accommodation of a rarefied gas at asolid surface.

Gas molecules from a surrounding gas in thermal equilibrium and at rest are incident on a cooled metal target disc. Under conditions of -free -molecule -flow the incident molecules are not disturbed by the cooler

reflected molecules. A high energy electron beam is passed through the gas in front of the target disc a.nd parallel to it, exciting the gas to electro-luminescence. The rotational energy distribution and number density of -molecules can be determined from a spectrographic analysis of the light

emitted by the gas.

The contribution to the measured intensity of spectrum lines due to the incident molecules can be determined with the target retracted. This contribution is then subtracted from total light intensity measurements made with the target in position in order to get the rotational energy distri-bution and number density of the reflected molecules alone.

These measurements can be interpreted to obtain values of energy accommodation coefficients, both for the translational and rotational degrees of freedom of the gas.

Experimental results show these measurements to be practical, although a high degree of accuracy in the measured quantities is required.

Measurements made with nitrogen on a silver target showed that the translational and rotational degrees of freedom both had an

accommodation coefficient

eX..

= 0.85, and that the rotational energy dis-tribution of the reflected molecules corresponded to that of gas in thermal equilibrium.

• T ABLE OF CONTENTS NOTATION vi 1. 2. 3. 4. INTRODUCTION 1 THEORY 2

2.1 Relationships from the Kinetic Theory of Gases 2

2. 2 Accommodation Coefficients 6

2.3 Surface Interaction Models 8

2.4 Energy Accommodation of Polyatomic Gases 11

2.5 Surface Conditions 13

2.6 Electron Beam Measurements 15

2.6. 1 Density Measurements 15

2. 6. 2 Rotational Energy Distribution 17 2. 7 Measurement of Translational Energy .A,ccommodation 23 2.8 Measurement of Rotational Energy Accommodation 29 EXPERIMENT AL EQUIPMENT

3. 1 The Vacuum System 3. 2 The Electron Gun 3.3 The Spectrometer 3.4 Instrumentation

RESULTS AND DISCUSSIONS 4.1 Density Measurements 4.2 Molecular Trap Model

4. 3 Translational Energy Accommodation

30 31 32 33 35 35 35 36 37 4.3. 1 Estimation of Accuracy of Measured Translational 37

Accommodation Coefficient

4. 3. 2 Molecular Trap Target 39

4.3.3 The Silver Target 41

4.4 Rotational Energy Accommodation

4.4. 1 Rotational Temperature Measurements 4.4.2 Estimation of Experimental Accuracy of

Rotational Temperature Measurements 4.4. 3 Molecular Trap Target

4.4.4 The Silver Ta~get

42 42 44 45 46

5. CONCLUSIONS REFERENCES APPENDIX I TABLES FIGURES Page 47 48 50 - / •

Anm a(-y ) b c

c

_J

·

d E f

=f(1,.?5.'

t) f F G h I J k K NOTATION

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 state. unit vector normal to element of surface area dA

molecular speed

jth component of molecular speed, j = 1, 2, 3 most probable molecular speed

distance from the target surface to the measuring region (small volume P in Fig. 1).

energy flux carried by molecules passing through surface element dA.

molecular velocity distribution function focal length/ diameter standard terminology

an excitation parameter relating the rate of excitation to a particular energy level of the excited state to the number density in the neutral gas for given beam currend and voltage.

a function of rotational quantum nutnber andTR (see Eq. (36) Sec. 2.6.2)

=

~/4'1i)

a geometry factor Planck' s constant

intensity of radiation

rotational quantum number Boltzmann's constant

K m M n

=

n(x, t) n' N

N"

_K p p(v', V") P P(K', Kil) Pp q(v',v")

Maxwell's reflection coefficient (Sec. 2.3 only) mass of a molecule

molecular weight

molecular number density

number density of molecular ions in the N~ B22:: state contribution to density measured at point P due to mole-cules reflected from the target surface

contribution to measured density at point P due to back-ground molecules

number of molecules striking unit surface area in unit time number of molecules with rotational quantum number J total number of molecules considered

number of ground state molecules with rotational energy level K

ground state of the nitrogen molecules ground state of the nitrogen molecular ion

excited electronic state of the nitrogen molecular ion pressure in torr

vibrational transition probability

normal momentum flux carried across area element dA in unit time

rotational transition probability or Hönl-London factors rotation transition probability for the P branch

rotational transition probability for the R branch

Franck-Condon factors for transition between vibrational levels v' and v" of two electronic states.

, .,

q

₌

QR(TR) R As S t T u u· J VI x y 0(

Anm /2 Ó 2 ,j4'TtRT is a collision quenching parameter

rotation state sum

gas constant

spectrum line spacing, mm (Appendix I only) speed ratio

time, seconds

temperature in degrees Kelvin

macroscopic velocity of the gas jth component of ~ j

=

1, 2, 3

vibrational level of a molecular ion in the N~ B 2 ~ electronic state

vibration level of molecules in the N2 X'L: electronic state vibrational level of moleculaX' ion in the N;X2

2:

electronic state entra.nce slit width

exit slit width

effective exit slit width

geometric line profile width (see Fig. 1. 1) measured line profile width (see Fig. 1. 1) position vector

an integer

energy accommodation coefficient

collision dia.meter: collision cross section for quenching

collisions is

S

2 / 411

-8

wavelength of light emitted in Angstrom units (10 cm) average rotational energy carried per molecule

I

cr

wave number of emitted light

wave number of light from the rotational line (KI, K2)

=

(1, 0) in the N~(O, 0) band.

accommodation coefficient for momentum in the direction normal to the surface

accommodation coefficient for momentum in a direction tangential to the surface

half angle at P subtended by the target di sc

rate of excitation to rotational level KI of vibrational band VI of the

N2"B2I:

electronic state of the nitrogen ion angular position in polar coordinates

a collision quenching factor. See Sec. 2. 6. 1

~'2

'

X 3

'X.

4 are constants defined in the text

cA

TM Subscripts i r s T R

angle between macroscopie velocity vector u and a unit vector ?ormal to the surface (

Y'

~ -r2.)

-solid angle at point P subtended by the target disc

translational accommodation coefficient calculated assuming the reflected molecule stream has a velocity distribution function corresponding to a gas in therm al equilibrium and at rest.

translational accommodation coefficient calculated assuming the reflected molecules to have the velocity distribution function given in Eq. (55)

refers to inc ident gas

refers to reflected molecules

denotes conditions pertaining to the temperature of the target transla tional

1. INTRODUCTION

At normal atmospheric temperature and pressure a gas acts as a continuum and heat transfer between the gas and asolid surface may

be determined without taking into account the molecular structure of the gas. At much lower pressure, however, when the mean free path of the

gas molecules becomes large compared to the dimensions of the solid

sur-face the heat transfer to asolid sursur-face is best described by the kinetic theory of gases. Gas molecules incident on asolid surface have velocities,

rotational energy and vibrational energy determined by the temperature of

the surrounding gas. If the stream of reflected molecules leaving the sur-face had velocities and internal energy corresponding to those of a gas in equilibrium at the temperature of the solid surface, the energy transfer could be readily calculated. However, measurements of heat transfer under these conditions indicate that, in general, the molecules are not per-fectly accommodated to the temperature of the solid surface and that the

degree of accommodation depends both on the characteristics of the solid surface and those of the gas.

The most widely used and best known experimental method of investigating gas -solid surface energy exchange is the molecular heat transfer method. A measurement is made of heat transfer from an

electrically heated wire mounted concentrically in a glass tube and kept at

a constant temperature, usually about 200C above the temperature of the surrounding tube. Gas is introduced into the tube at a pressure low enough so that free-molecule flow conditions prevail. Heat is conducted away from the wire by molecular heat transfer, by radiation, and by conduction through the leads supporting the wire. The latter two are independent of gas pressure and can be measured in vacuum, to be subtracted from later total heat

trans-fer measurements in order to determine the energy carried away by mole-cular heat transfer. The wire temperature is determined from a resistance

measurement and the energy transfer from the current required to

main-tain the wire at constant temperature.

This method was introduced as early as 1910.

Improve-ments made since that time have been mainly concerned with providing a

truly clean metallic surface free from adsorbed gas layers. A clean

bakeable vacuum system is needed together with a method of purifying the

gas admitted to the system for experimental measurements so that when

the wire is cleaned by flashing it to a high temperature unwanted gases will

not be present in the system or introduced with the test gas to contaminate

the wire surface aftel' it cools.

J. K. Roberts (Ref. 1) made the first measurements of

accommodation on clean surfaces using this approach. L. B. Thomas and

his associates at the University of Missouri extended this clean surface technique further, using getters in the tube to clean up the chemically active gases sueh as oxygen, nitrogen and hydrogen. (See for example Ref. 2.)

The molecular heat transfer method gives only an overall measurement of ènergy transfer with no direct determination of such details as the velocity distribution function of the reflected molecules, or the de-gree of accommodation of the internal dede-grees of freedom in the case of a polyatomic gas.

A new approach to the measurement of energy transfer be-tween a rarefied gas and asolid surface is presented in this report. Use is made of spectrometer measurements of light emitted from the gas when it is excited by a high energy electron beam to determine separately the translational and rotational accommodation of nitrogen at a polycrystalline metal surface. This is an adaptation of the electron beam technique develop-ed by E. P. Muntz (Ref. 3) for the measurement of density, the rotational distribution of energy and vibrational temperature in a rarefied gas at a point defined by the position of the electron bea.m and the optical arrange-ment.

2. THEORY

2. 1 Relationships from the Kinetic Theory of Gases

As was noted in the introduction, energy and momentum ex-change between a rarefied gas and asolid surface is best described by the kinetic theory of gases.

As a practical exam ple of an application of kinetic theory we can consider a satellite in a low orbit, below 1000 miles say. The drag force due to the upper atmosphere will cause the satellite orbit to decay' slowly until at an altitude of about 80 miles the atmospheric drag will be large enough to cause the satellite to spiral into the earth's surface. At this altitude the molecular mean free path is of the order of 50 to 100 ft. and thus we see that in the region where atmospheric drag will cause a gradual change in its orbit the aerodynamics of the satellite is in the free molecular flow regime. We would like to be able to specify the forces and energy exchange on the satellite in terms of the incident gas velocity, tem-perature and composition, and the temperature and composition of the solid surface. The momentum and energy carried away by the reflected mole-cules are needed in these calculations but are not known unless they can be specified in terms of known conditions of the incident gas molecules and the solid surface. The necessary relationships to do this are w:dtten in terms of accom modation coefficients as we shall see in the following sections . Conversely if the accommodation coefficients are known we may be able to obtain inforrnation about upper atmosphere conditions by observations of satellite orbital decay.

In order to consider in detail the interactions taking place between a gas and asolid surface we will need some concepts and equations from the mechanics of rarefied gas. These equations can be found in ReL 4.

For convenience the definitions and equations needed here are introduced in the following paragraphs.

Let n(~, t) be the number of molecules per unit volume at

position ~ at time t. The number of molecules in a volume element,

d~

=

dXldx2dx3, is then n(~, t)dx. d~ is chosen to be small compared to the physical dimensions of a particular problem but big enough so that

n(~, t)d~ is a large number in a statistical sense.

All the properties of the gas depending on the translational motion of its molecules can be determined from a knowledge of density and velocity distribution function, as will be seen in the following sections. The velocity distribution function f(~,

f,

t) is defined to be the number of mole-cules in volume element dx, centred about a position indicated by position vector ~, having a velocity

1-

in the range

1

_{1 to 11 + d~l' ~2 to}

12 + d'f 2' 13 to 13 + d

T

_{3 ·} It follows immediately fr om this definition th at the integral of f(x,

1,

t) dx d ~ over all possible molecular velocities will be equal to n(x, t) dx, or

oe 00-00

-ij

f

f(

i, .":'

t)

d.": dr

1

d1

2

d

13 "

n(x, t)

d.":

(1)

-O()_oo_cO

For a gas in thermal equilibrium at rest, or in a frame of reference mov-ing with the macroscopic velocity of a flowmov-ing gas, the velocity distribution

function is given by the well known Maxwellian form as

where

f = n _e

n = number of molecules,

cm

=

,I2RT is the most probable molecular speed, R is the gas constant,

~. i: ~~e+jct~ component of random molecular velocity,

?

J J J '

Uj is the jth component of macroscopic velocity.

(2)

Note that this velocity distribution function refers to a gas in which the macroscopic quantities, n, Uj' T are the same at all points (but can vary with time, t).

Consider a gas with a Maxwellian velocity distribution func-tion moving bodily with a velocity u. Following Ref. 4 we can determine the number of molecules crossing an element of area dA in unit time. For convenience in integration a coordinate system is chosen such that the coordinate xl is antiparallel to a unit vector normal to the area dA, .J

denoted by b , and the coordinate x2 is chosen so that the velocity vector u is in the plane determined by xl and x2'

dA

---.---.-~~---Xl'

Tt

Then

(3 )

where cl> C2 and C3 are the components of random molecular velocity

(thermal motion) in the xl, x2, x3 directions respectively and ~ is the

angle between vector ~:and the normal to dA.

The number of molecules with velocity ~ which will cross

area dA in time dt in the direction opposite to b is equaTto the number in

the volume

1

4 ( -b) dAdt. The number of molecules in this specific class

is given by

f dx d ~

and since

dx

=

~.

(-!:)

dAdt

=

1

₁ dA dt

th,e number crossing dA from one side in the coordinate system chosep here cD 0 0 00

f J

J

~

1 f d

3

_{1 d}2 d ?3 dA dt .}

o _eO_oO

is

(4) Making use of Eqs. (3) and using the Maxwellian form of the velocity

dis-tribution function, Eq. (2), the number of molecules crossing area dA in

time dt in the direction opposite to

!:

is

77'3/2 0() ~oo .

f

J

J

(Cl +Scosljl)e -C 2 dC1 dC2 dC3,

-s cos

~ _00 - c.O

where C j

=

Cj/c m and S

=

u/cm

Carrying out the integration, expression (4) becomes

n

R

[exp(-S2 cos 2

'I' )

+

Jff'

S cos

4'

(1 + erf S cos

41)]

dA dt.

2 'TT' (6)

For a gas at rest S :: 0 and Eq. (6) reduces to the familiar expression nCm

2,ffl

(7 )

Mass carried across a plane area dA in time dt is obtained simply by multiplying Eq. (4) by m, the mass of a molecule.

The momentum normal to dA is given by the mass times the normal component of velocity, resulting in the equation

., oC oCoe

p , m

[J

J?12 f dA dt d rl d T2 d!3- (8l

o

_00_00

Again, for a gas with a Maxwellian distribution function moving with a mass velocity u at an angle

'I'

to the unit vector normal to area dA the normal momentum transferred across dA antiparallel to b in unit time is

0 0 _oe

p ,

m';~i22

J

dCI

J

+ S cos

tp)

2 -C2 e dC3 dA dt. _01:1

- S cos

0/

::

~

n m cm 28- (1 + 2 S2 cos 2

c.p)

(1 + erf S cos

l/I)

1

+ff

S cos

r

e -S2cos2

1# ]

dA dt. (9)

Similarly, the tangential momentum transported across area dA in time dt is givenby

è,mJfhl

12 fd

~ldr2

d1'3 dAdt 0 _ 0 0 _ 0 0 (10) oe 00 t>C ::

J

dCI

J

dC 2

f

(CI+SCos'l')(C2+ssin<.fle-c2dC3dAdt _00 _00 -Scos

-+'

1 2 S . ti) [ 1 - S 2 2 I U ( f f U )] d d

:: '2 n m cm Slll,

R

e cos T +S cos 1 +er S cos T A t.

The transfer of molecular kinetic energy across dA in time dt is given by c:>O oD eiO

E

:!m

f f f

~1

'P

f d }' 1 d

32

d ., 3 dA dt

=

(12)

o -<00 -~

[ (1 +

î

8 2 ) exp( -82c052 "") + 4 - 8 cos

'I'

(~

+82)(1 + erf 8 cos Ijl llAdt.

(13) In calculating the total energy transfer taking place it is con-venient to think of the solid surface as a window between two chambers, one containing the incident gas having a velocity distribution function fi and the other containing the reflected gas ha ving a velocity distribution function fr. The above equations describe the flux of momentum and energy carried to the surface by the incident gas. By simply changing the limits of integra-tion to include only molecules with negative values of the velocity compon-ent normal to the surface and using the velocity distribution function for flected molecules, fr , Eqs. (4) (8) and (10) can be made to apply to the re-flected gas stream as well.

The rotational and vibrational energy of the gas do not appear in these equations. Noting that the internal energy carried by a molecule is independent of its translational velocity the internal energy crossing dA in unit time could be obtained by multiplying Eq. (4) by the ave rage internal energy carried per molecule.

2.2 Accommodation Coefficients

Equations (9), (11) and (13) give the normal momenturn, tangential momenturn, and kinetic energy carried to an element of the solid surface dA by incident molecules under free molecule flow conditions,

assuming a Maxwellian velocity distribution function and uniform conditions in the incident stream of molecules. Given the velocity distribution function of the reflected molecules,the forces and energy exchange at surface ele-ment dA could be calculated directly. Unfortunately this reflected molecule stream velocity distribution function is usually not known.

For practical applications the normal pressure, tangential shear and energy transfer to the solid surface are the quantities of primary interest. Experimental investigations, with the exception of some recent molecular beam work, have usually measured these quantities directly and the accommodation coefficients given below have been introduced for

practical applications . Used in this way the accom modation coefficients

may be thought of as empirical coefficients. However, they can be expressed in terms of the distribution functions of the incident and reflected molecules.

'"

where

The energy accommodation is defined by the relation

0<

=

Ei - Er ₍₁₄₎

Ei - Es

Ei is the energy carried to the surface by incident molecules, Er is the energy carried away by the reflected molecules, Es is the energy which would be carried away if the reflected

molecules had a velocity distribution function correspond-ing to a gas at rest in thermal equilibrium at the tempera-ture of the solid surface.

Similarly, accommodation coefficients can be written for normal and tangential momentum rus,

(15)

and

cr=

(16 )

where P is the normal momentum,

è

is the tangential momentum,

and the subscripts have the same meaning as in Eq. (14). Although the coefficients 0(, (J" and

a-

are introduced to avoid the necessity of knowing the velocity distribution function of the reflected molecules directly, it wil! be seen that they are related to one another through the velocity distribution function. In general the velocity distribution function of the incident molecules is wel! known. Any chosen

~

or known form of f r wil! then give unique values for 0( ,

cr

and (J. If a form is assumed for fr , the inter-relationship of the three coefficients provides a possible method of checking this assumption experimentally. For example, as will be shown in Section 2.3, one particular choice of fr I '

makes

d.

=

(f'

=

(j while another possible form makes

d"

=

1. 0 while

U' and 0( are given by two different functions of temperatures. (Eqs.

(26) and (27);).

An auxiliary equation relating the number of incident mole-cules to the number of reflected molemole-cules has to be satisfied by fr. In

the case of dynamic equilibrium the flux of molecules striking the solid surface will be equal to the flux leaving it. From Eq. (4) this condition can be written in the form 0 0()

j

j

jr1

fi d

f1

d

T2

d]3 dA d! =

f f

jr1

fr d

31

d

~2

d f3 dA dl.

(17)

2.3 Surface Interaction Models

Let us consider two types of interaction of gas molecules with asolid surface. First, if the surface is very rough or if the molecules are assumed to be adsorbed and later re-emitted, the reflected stream of mole-cules will have lost all memory of incident direction. Because of the ran-domness of the collisions taking place with the solid surface the reflected stream should have a cosine law number distribution such that the number of molecules reflected from the surface with velocity vectors making an angle

cp

with the normal to the surface is proportional to cos

cp .

This is the angular distribution law which is followed by mole-cules passing through an element of area from one side coming from a gas at rest and having a Maxwellian velocity distribution function.

We will refer to this model for reflection of molecules at a solid surface as diffuse reflection. If in addition the reflected stream of molecules is at the temperature of the solid surface we can say the reflect-ed stream of molecules has been completely accommodatreflect-ed to the tempera-ture of the surface.

In the other extreme, the incident molecules may suffer a mirror-like or specular reflection in which the velocity component normal to the surface is reversed in direction but unchanged in magnitude, while the velocity components parallel to the surface remain unchanged. In this case the velocity distribution function of the reflected molecules is the same as that of the incident molecules. Since the molecules still have the same speed and kinetic energy, no energy has been given to· the surface and we say there has been no accommodation, (although this is the case where norm al momentum exchange is a maximum).

Early experiments showed that there was some accommoda-tion of molecules with the solid surface but not complete accommodaaccommoda-tion. Maxwell suggested a model for the reflection process at asolid surface in which a fraction, K, of the incident molecules are diffusely reflected and the remainder are ,specularly reflected. This results in the reflected mole-cule distribution function,

(18) Energy càrried away by the reflected molecules in the direction of!: may then be written

Er ,tm

jjfis

2 ird

f1

d

T2

d 13

=imK

J

r

f~1~2

is d

f1

df2 d}3

- 0/) -dO - QO 0 - 0 0() _ 00 -00 -00

,

..

Note that in this model of the surface interaction

r

1 = -c 1 in the first integral on the right hand side of Eq. (19) because of the assumption here of diffuse reflection, and r1

= -

(c 1

+

u cos ~ ) in the second integral where specular reflection has been assumed.

Carrying out the integration gives the result Er

=

KEs + (1 - K) Ei

and hence

K

=

(20)

Normal momentum carried away by reflected molecules in the direction of b is given by

OoCJ~

Pr=

!

r

Jrr

rr

dlld72 d 73

=mKJf

J~~fsdAdtdrldr2dr3

-aO - 0 0 _oa _ 0.:> _~ _(;:JO

o C/O oe::>

+<l-K>mJf

f~~fidAdtdrld?2dr3

_oO_c:IO_oD

and from this

K =

/

- 0 . (21)

Similarly, the tangential momentum carried away from the surface in the direction ~ by the reflected molecules is

and

K =

o.

(22)

Thus, (1'" =

r:r '"

0( = K for the choice of velocity distribution function given in Eq. (18).

In general, experimental results show that <TI

I

(f"

I

0( . Molecular beam experiments (for example Refs. (5) and (6) ) indicate that for "technical surfaces", i. e. metal surfaces with oxide layers and adsorb-ed gas layers, the reflection of molecules from the surface is diffuse. A diffuse reflection results in t-r = 0 in Eq. (16), making

r:J

= 1. O. On the other hand d. is almost invariably found to be less than 1. 0, the value de-pending strongly on the condition of the solid surface and the type of gas.

The experimental evidence that 0-'

I

<r

I

0( indicates that the simple model described above is not adequate and other models should be considered .

. _{Another useful and simple model for f r , is one in which the}

velocity distribution functioh of the reflected molecules is Maxwellian corresponding to a temperature T_{r intermediate between the incident gas}

stagnation temperature and the solid surface temperature T s . This assumption results in (J

=

1. 0 and, in general, yields different values for (T' and 0(. In the special case that S

=

0, use of Eqs. (13) and (14) leads to or

0<=

rj

=

m n' 1 cm·3 1 ni cmr - nr cmr3 ni cmi3 - ns cms3

=

m ns cms

2ff

ni Tr/2 - nr T:/2 ni T ~

7

2 - ns T s 3

7

2 From Eqs. (17) and (7), it is found that

(23 )

and

Hence,

=

'0(= Ti-Tr Ti - T s Similarly, from Eqs. (9) and (15)

I

(j=

and, using Eqs. (24) and (25).

I

5;-

JTr

cr

=

JTi

.{Fs

(25 )

(26 )

(27 )

It would be possible to consider other models for the distribution function of the reflected molecules. However, the two simple forms given in this section wil! be sufficient for the purposes of this report.

2.4 Energy Accommodation of Polyatomic Gases

The expressions for momentum accommodation given for monatomic gases in Sec. 2.3 hold for polyatomic gases as weU since only mass and translational velocity are involved. However, the expression for energy accommodation wil! have to be changed to include the contribu-tions of vibrational and rotational energy carried by polyatomic molecules.

It is wel! established that in a polyatomic gas in thermal equilibrium there is no correlation between the velocity of a particular molecule and its rotational or vibrational energy le-yels. Hence, the ro-tational and vibrational energies caR, be ir.eated .as average quantities trans-ported by the molecules as suggested at the end of Sec. 2. 1.

The rotational accommodation coeJficient.may he defined as,

(28)

where the rotational energy carried by the incident stream is equal tothe number flux of molecules times the average rotational energy per incident molecule,

ê(,

or 00 00 0<)

ERi 0

[ij

J

J f i

f

1 d

~1

dJ 2 d

13

dA dt. (29)

a _ 0 0 - 0 0

Similar equations can be written for ERr and ERs.

We know from statistical mechanics that for a gas in thermal equilibrium and at rest the average energy per molecule: is 1/2 kT for each degree of freedom. Thus, for a diatomic molecule with three translational and two rotational degrees of freedom the average translational energy due to thermal motion is 3/2 kT and

f:

= kT, where k is Boltzmann's constant. The rotational energy carried by the incident stream is thus related to the translational energy through the gas temperature. The total energy accommodation coefficient becomes,

~= Ei + ERi - Er - ERr (30)

Ei

+

ERi - Es - ERs

Ei, Es, ERi' ERs can be calculated directly knowing the gas temperature, Ti, macroscopie velocity of the incident gas, and the surface temperature T s . A measurement of the molecular heat transfer between the gas and the solid surface will then give: an overall accommodation coefficient which is a weighted average of tr,anslational and rotational accommodation coefficients. For example, if we take the case of zero speed ratio, we have the following relationships, 5 Ei

+

ERi =

'3

Ei 5 Es

+

ERs =

'3

Es , E _{r l S 1}

= -

rX

_{T (E' - E )}

+

E .

ERr

=

-

~

[o(R(Ei - Es) - Ei] Substituting these into Eq. (30) we have

rX=

There are very few experimental results to show how

0<

R compares with

d...,

or with the translational energy accommodatioI1~ coefficient 0( T. Knudsen, (Ref. 7) was able to show that for hydrogen striking a platinum ribbon the overall energy accommodation coefficient

measured by the molecular heat transfer method was equal to the trans-lational energy accommodation coefficient

0<

T. If

0<

T

=

0( then

0(

T must also be equal to

0(

R for this zero speed ratio experiment.

,

-'-o'

Force measurements were made on a heated platinum ribbon, bright on one side and blackened on the other with platinum black. By mak-ing the assumption that the velocity distribution function of reflected mole-cules was Maxwellian, Knudsen was able to relate these force measurements

to the difference in translational energy accommodation coefficient

0<

T on the two sides and he found

o(T Black - o(T Bright = 0.415

Overall accommodation coefficients measured using the molecular heat trans-fer method with a ribbon blackened on both sides and with one bright on both sides gave the following results,

0( == 0.735

Black O(Bright == 0.315

o(Black o(Bright == 0.420

From this Knudsen concluded that it was very probable that C>( R = o(T . In support of his assumption that the velocity distribution of the reflected molecules was Maxwellian, Knudsen made similar measurements using helium and the results showed good agreement between

A

O<T == 0( TBlack - o(TBright derived from normal force measurements and Ao(T measured directly using the molecular heat transfer method.

In a similar experiment and also making the same assump-tion about the distribuassump-tion funcassump-tion of the reflected molecule, Saski, Taku and Mitani (Ref. 8) found 0( T == 0.393 and O<R == 0.314 for nitrogen on a

nickle ribbon.

Although a considerable number of theoretical investigations on thermal accommodation of monatomic gases have been carried out (see for example the latest of a series of papers by F. O. Goodman, Ref. 9) very little theoretical work has been put forward for the accommodation of rotational or vibrational internal energy of polyatomic gases.

2. 5 Surface Conditions

One of the most important factors in the energy and momentum transfer at asolid surface is the condition of the surface. For example, on a metal surface having th in layers of oxide and adsorbed gas layers as is likely to be the case in a practical application, experiment shows the reflection of molecules to be diffuse. On the other hand, if the surface is smooth and free from adsorbed gas layers, as was the case in the careful experiments described in Refs. 5, 6, 10 and 11, a considerable arnount of nearly specular reflection is observed. Measurements of energy accommodation by the molecular heat transfer method have also shown that adsorbed gas layers can change

d-.

by a factor of nearly 10-. Wachman reported in Ref. (12) an observed change from

cl

== O. 02 for helium on

clean tungsten to

0(

=

O. 185 for helium on tungsten with an adsorbed layer of oxygen. With these effects in mind any experimental apparatus to deter-mine quantitative values of accommodation coefficients or constants connect-ed with a particular theoretical model of the interaction taking place at a solid surface will have to provide a good means of determining the composi-tion of the solid surface. Essentially this means that the capability must be provided for producing and maintaining a clean, gas-free surface. Once a clean surface is obtained, variations due to adsorbed gases ,or an oxide layer can be studied by allowing these layers to form on the clean surface in a controlled manner.

In practice these ideal surfaces are very difficult to obtain because most metals react chemically with some of the residual gases which are found as background in vacuum systems. Gases such as N2, H2' 02, H20, CO, C02, C2H2, C2H4 and others are found adsorbed on surfaces or dissolved in parts of a vacuum system. They are gradually released from the system walls as "outgassing" and form a limiting gas load which at some pressure will just equal the capacity of the vacuum pumps used. If asolid surface is to be maintained in a clean condition for a useful period of time while experiments are being conducted, some means must be found to prevent the background gases from forming an adsorbed layer during this time. The most obvious way of doing this is to reduce the pressure of the background gases to the point where the time needed to form a monolayer is much longer than the time needed for the experiment. This can be done by careful choice of materials and care with the construct-ional details of the vacuum system' combined with a thorough bake to outgas all components.

The vacuum system is outgassed by baking it while under good vacuum. The clean surface is produced by flashing . to a high tem-perature under high vacuum conditions or by some other means such as deposition of metal from a vapour source or ion bombardment. Finally, chemical pumping with a selective getter inside the system is used to re-move traces of impurities from the test gas when it is admitted and to pump away any sma:ll amounts of impurities which m ay subsequently be evolved from the system walls.

Using this approach, Thomas and his associates at the

University of Missouri developed such good control over surface conditions that Wachman (Ref. 12) was able to use the energy accommodation as a method of studying adsorption of gas layers on tungsten.

In order to get an estimate of the reduction of background pressure necessary, let us assume that a useful testing time after cleaning our surface is 30 minutes. The number of molecules necessary to form a monolayer is approximately 10 15

I

cm 2 and the number of molecules striking the solid surface

Icm

2

I

second is given by, N

=

3.52 x 10 22 x pi MT, where p is pressure in torr, M is the molecular weight of the gas and T is

the gas temperature in oK. Consequently, if we want to form a monolayer of a gas with a molecular weight of 30 in a time of 30 minutes, when the gas temperature is 300oK, and assuming that each molecule that strikes the surface is adsorbed, we find that the pressure has to be

11 30x300 _ -9

P

=

5.5 x 10 x 22 - 1.5x10 torr. 3.52x10

While this figure for the residual pressure is within the capabilities of modern vacuum technique, careful construction and operation of the apparatus is required to achieve it.

A further problem is encountered with the purity of the gas to be admitted to the system. If the working pressure is to be 10- 3 torr, the impurities in the gas have to be less than one part in a million. This purity can only be achieved by having some process such as selective getter pumping to further purify the gas af ter it is admitted to the system.

An alternative method has been used by Smith and Salzberg (ReL 10), who have shown that it is possible to maintain a clean metal sur-face in a vacuum system in which the background pressure is much higher than 10-9 torr. by the continuous deposition of a metal film from a vapour source, provided the number of metal atoms striking the surface / cm 2 / second is much larger than the number of contaminant gas molecules. In their experiments, a rate of deposition of gold atoms as high as 10 17 /cm 2 / sec. was used. lf we could allow 10 16 background gas mOlecules/cm 2 /sec. to strike the target, the background pressure could be as high as 3 x 10- 5 torr.

This method is particularly useful for molecular be am experiments because the gas load from the beam itself usually brings the background gas up to a pressure of the order of 10- 6 torr.

2. 6 Electron Beam Measurements

2.6. 1 Density Measurements

The use of electron beam excited emission of light as a measurement of molecular number density in a rarefied gas has been investigated, among others, in Refs. (3), (13), (14) and (i5).

Under the conditions listed below it can be shown that the light output from the gas is proportional to the molecular number density.

1. The upper electronic state of the observed transition is populated directly by excitation due to collisions with high energy beam electrons. This establishes a direct proportionality between the number of molecules excited to the upper state and the number density of molecules in a unit volume of the electron beam.

2. The observed emission should be due to an allowed electronic dipole transition with a transition probability in emission which is large, typically of the order of 107 / sec. The resulting short lifetime of the ex-cited molecules is necessary for spatial resolution in a high speed flow and to prevent loss of light owing to the diffusion of the excited molecules out of the region of observation before emission takes place.

3. The density must be low enough so that the time between molecular collisions of the type that result in non-radiating de-excitations is much greater than the mean lifetime of the excited state.

The electron beam method may still be used at densities high enough to violate condition 3 because the non-linear effect due to collision quenching can be weU represented by the following equation,

where

I

=

constant x n

<p

(31 )

<}=

1 1

+

n/q

I = light output,

n

=

number density of the gas,

q is a parameter which can be determined experimentally, and which is characteristic of the gas and of the observed electronic transition.

The foUowing derivation of Eq. (31) is essentially the same as that presented in ReL (3). Let F be an excitation parameter defined such that F. n is the number of molecules raised to the excited state per second, where n is the number density of neutral molecules. An equili-brium will be established, with a number density of excited molecules n' corresponding to n.

The number of molecules spontaneously emitting radiation per second is n'Anm where Anm is the Einstein spontaneous transition probability for emission. The number of excited molecules giving up their energy due to collisions with neutral molecules is given by the kinetic theory of gases (ReL 16, p. 37), as 2n'n

S

2 J41'(RT where

S

2/4

7T

is the molecule cross sectional area for collision quenching. Hence,

F.n

=

n'Anm + 2n'nS 2 J41rRT and consequently

n'

₌

F.n (32)

The emission intensity is given by, (see for exam ple Ref. (17) )

1

=

rtlhc

-z/

Anm'

where h

=

Planck l s constant,

c

=

speed of light,

11

=

the wave numb~r of the emitted light,

so that the intensity finally can be written in the form:

F.nhc

I

=

The parameter, q, of Eq. (31) is then

and ha.s the dimension of number density.

(33)

The emission for the case of nitrogen is almost entirely due

to the N~B2r--+ Ntx2

r:

transition of the nitro~en ion for which .

Anm

=

1. 53 x 10:.7 Tsec. At a temperature of appr.oxirnately 3000_{K, a .}

value q

=

10 t7 /cin 3 was found in Ref. 14. For nitrogen at room temperature

and a pressure of 10-2 torr, the number density n

=

3.3 x 1014/cm 3 ," and

the factor ~, from Eqs. (32) and (33) can be seen to be

,

~

=

1

+

3. 3

1

x _{10- 3} = 0.997

In this investigationthe gas used in all measurements was nitrogen, the

working pressure was always less than 10-2 torr and the temperature was

in the range 300 to 5000_{K, Hence t11.e light intensity is proportional to}

den-sity in very good approximation. 2,6.2 Rotational Energy Distribution

The intensity distribution of the lines in the rotational fine

structure of the nitrogen spectrum can be directly related to the

distribu-tion of rotadistribu-tional energy in the gas before excitadistribu-tion"by the electron beam.

A homonuc1ear diatomic gas sueh a,s rtitrogen iri thermal equilibrium will

have the following number distribution among rotational energy levels. (See for example Ref. (18».

(2J

+

1) [

where NJ /N_{o is the ratio of the number of molecules with rotational}

energy level J to the total number of molecules, QR is the rotational state sum,

B is the molecular rotational constant.

As mentioned in Sec. 2. 6. 1 light emitted when nitrogen is excited by a high energy electron beam com es almost entirely from the first negative band system due to the transition N~ B 2

L

---+N~ X2

L

of the nitrogen ion. The neutral nitrogen molecules are initially in the N2 X1L

ground state and are excited to the N~ B 2

L"

excited st'ate of the ion by collisions with high energy beam electrons. The electron spin coupling in the N~ B2

L

state is very weak and all electronic states of the transitions, both in excitation and in emission are known to be governed by Hund's case (b) (Ref. 17). If the doublets are unresolved the transitions become equiva-lent to

12:

---+

lL:

transitions with rotational levels designated by K, the rotational quantum number apart from spin (J

=

K

:t

i).

The selection rule applying to Hund's case (b) fortI:

~IL

transition is AK

=

i"

1.

This selection rule results in the formation of two "branches" in the emission spectrum, the P branch corresponding to Ll K = KI - K" = -1 and the R branch corresponding to 11K

=

+ 1.

In the R branch the wave length of emitted light decreases with increasing KI so that the lines are spaced toward the violet end of the spectrum and spacing between lines increases with KI. The P branch lines are initially spaced toward the red end of the spectrum but the spacing de-creases with increasing KI until the P branch doubles back on itself forming a band head.. Because of this only the R branch lines can be resolved with the spectrometer used in this investigation.

Making the assumption that the gas before excitation has a thermal equilibrium distribution of rotational energy as given by Eq. (34)

and that the electronic transitions, both for excitation by the electrons and for emission, are governed by optical selection rules and transition probabilities. Muntz in Ref. 3 derived the following expression for the pre-dicted distribution of line intensities in the rotational structure of the bands of the first negative system of nitrogen in equilibrium, . valid for temperatures below 8000K, 4 (I KI K") I

,,=

(KI +K" + 1) X4,

[GJ(:: )

, 2 v, v2 1'0 [ hc ] exp -Bvï (KI + 1) kT R (35)

where )(4 is a constant for a given band at a given temperature,

(K' +1) exp [-2 Bvï (KI +1)

~~J+

K' exp

~

Bvï

~RK']

[G]

=

(2KI

+

₁₎

One prime indicates the upper electronic state; a double prime indicates the lower electronic state and the initial ground state of the neutral mole-cule is indicated by subscript 1 to distinguish it from the ground state of the ion, labelled subscript 2. (For example Bv'! is the rotational constant for a molecule in the vibrational level v of the ground state of the neutral mole-cule before excitation. )

If the assumption that optical selection rules are obeyed is correct, the rotational temperature, TR, can be obtained from measured line intensities since all other quantities in Eq. (35) are known, except for

/<..4

which is a constant affecting only the intensity level and

[G]

which is a function of TR.

An iterative procedure can be adopted starting with a guessed value of TR to calculate

[GT '

leading to a value of TR from Eq. (35).

[G] is approximately unity and does not vary strongly with TR so usually only one iteration wiU be necessary.

Equation (35) can be written,

Log

(IK' _, K2") , _{, v, V2} "

=

-Bvï hc

K'(K' + 1) + Log.

X.

4. (37 )

A plot of log (IK' K'2') , " , v v2 (K' +K"

+

l)rGlv_{~ 'J} 4

V o'4

against K' (K' + 1) wiU give a straight line

with slope -Bv'ihc kTR

-Bv'ihc 1

and hence TR

=

k x , C

-slope

If the rotational distribution is not one corresponding to thermal equilibrium, as might be the case for molecules reflected from a solid surface in this investigation, the plot of 10g(IK', K" /

(K'+K"+1)[Glv"lv'Q~

against K'(K'+1) wiU not give a straight line. This

devi~tion

will indicate qualitatively the extent to which the rotational distribution has a non-equilibrium form.

As will be shown below, the measured line intensities do not give enough information for an explicit determination of the rotational

energy distribution unless the gas can be assumed to have an equilibrium distribution of rotational energy. However, in the numerical analysis of a specific measured distribution of line intensities it may be possible to make a.pproximations which will allowan energy distribution to be calculated.

Following Muntz (ReL 3, page 13) we assume that the rela-tive rotational transition probabilities of the Pand R branches are those normally associated with optically allowed lL' to

12:'.

transitions and that the vibrational and rotational eigenfunctions are separable (ReL 17, p. 208).

The population rate to an excited rotational level, KI, of a particular upper vibrational level, VI, of N~ B2

L

is given by

L ( [

<N"'KI+1)PP+(N"KI-1)PR] , , )

:: F . q (VI, v 1 )

ill

N "K'+l)PP + (N"K'· _{-1) P R}

" - 0 1 2 . K I - 0 1 2 .. . ( 3 8)

vI - I .' , . . - , J • •

where q(y) vï ) is the Franck-Condon factor for the vibrational transition and

L:

[<N"K'+I) Pp + <N"K' 1-1) PRJ

KI :: 0, 1, 2 .. .

is a normalizing factor.

In Eq. (38) N"KI+1 is the number of molecules in the ground state of the vï vibrational level with rotational quantum level K" :: KI + 1 and similarly N"K I -1 refers to level K" :: KI - 1. Because of the selection rule 6K

=

!

1 the level KI of the excited molecules can only be populated from the levels K" :: KI

i"

1 of the ground state. For KI

=

0, N"K I -1

=

0 and the KI :: 0 level is populated from the K"KI+1 ground state level only.

Pp and PR are the relative transition probabilities or Hönl-London factors for the Pand R branches in the excitation. F is the excitation function depending on electron beam energy and the electronic transition being considered. If N is the number of N2 X 1

L

molecules, then N. F is the number excited to the N~ B2

L

electronic level per second.

The rotational transition probabilities are those correspond-ing to a

lL:

~

1

_L

transition (the doublet state is not resolved). Written in term s of KI, they are

Pp

=

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

and P_{R ::} KI / (2K'

+

1)

For emission of light the intensity may be written as

I :: a

(-V )

hc

7J ,

(39)

where a(

y )

is the rate of photon emission. The relative rate of photon emission in a rotational line (KI, K

₂₎

of the (VI, v~p band in the first nega-tive system is given by

~ [v~K'

!

] q(vl_,

v~p

_{P(KI, K}

2

)")I3 ::

.L:

[L(q

(VI,

v~

) P (K', K

₂

)V

3)J a(v

kl,

K" (40) v2:: 0,1,2 .. K ::O, 1,2 .. .

relating the rate of transitions giving rise to a particular line (K', K~) to the rate of population of the K' excited state. Subscript 2 denotes the

N2X2L

ground state of the ion.

For a given level, K', the sum

[

~

(41)

V " -2 - 0 1 2 , , ... K

2

=0, 1, 2 ...

is a constant. Hence Eq. (39) becomes

(42)

where

X

_{2 is a constant for any selected vibrational band.}

For the R branch of the band, which was used in this. in-vestigation

P(K', K~) _{= P R = K' /(2K' + 1) •}

If the vibrational temperature is less than 8000_{K essentially}

all molecules are in the lowest vibrational level, i. e. they have vï

=

0 and Eq. (38) becomes

~

[v',

K'J

=

F. q (v' 0)

l

-==-=:..:....-+~---

N'~,

1 Pp + N'k'-1 PR

j

,

~

r(N"K'+1)Pp-t{N'k'-1'PR} K -0,

1.

2 ...

and Eq. (39) can be written in the form

where

q (v', v2) hc F. q(v', 0)

L

[ Iq

(vl

, V2) P (K', K2)1J3]x

L:

[<N"K'+l)PP -t(N'k'-1)P_R]

v

₂

=0, 1,2 .. K'=O, 1,2. .. K'=O, 1,2 ... is a constant for a given vibrational band.

I

It is a characteristic of the nitrogen molecules th at the num-ber of molecules having Kil even is twice the numnum-ber having K" odd. This characteristic is basically determined by the nuclear spin of the molecule

and is due to symmetry properties of the electronic, vibrational, rotational, andl nuclear spin eigenfunctions.

Because of the selection rule ~ K =

!

1 the K' odd levels are

populated only from K" even levels and K' even levels are populated only from K" odd levels. Hence the alternation is retained through the transitions and appears as an alternation in intensity of rotational lines in the measured spectrum. This divides the sequence of line intensities naturally into two sets, a strong set corresponding to lines with K' odd and a weak set

corresponding to K' even.

In an experiment,measured values of IK', K2 for K' = 0 to

K' = 22 are obtained (higher lines are overlapped by the N2+(1, 1) band in

the spectrum) and the problem is to find values of N"K" from K" = 0 to

K"

=

23 from Eq. (43). This equation can also be divided into two sequences

relating to weak or strong lines, having the general form

I ,

_{(K =2y)}

= X (

2y )

[NI I

(2Y

+

1)

"

(2Y )]

3 4y + 1 2y+1 _{4y+3 + N 2y - 1 4y+1} '

(43a)

I -

x.

(2Y + 1)

r

N" (2 Y + 2 ) , , ( 2y + 1)1

(K'1=2y +1)- 3 4y + 3

L

2y + 2 4y + 5 _{+ N 2y 4y + 3}

T

(43b)

where y is an integer. In general,for .2n measured intensities I(K!.,:K" )each of

Eqs. (43a) and(43b) wi11 contain n + 1 unknown values of N" and so they are indeterminate.

From this we must conclude that, in general, the distribution of rotational energy in the non-equilibrium case is not directly determined by the line intensity measurements. However, it may be possible in a practical case to make some approximation to get starting values of N" from which to evaluate the whole sequence. For example, NIK" as a function of K" may become àpproximately linear at higher values of K" as it does for the case of an equilibrium energy distribution. Then making the a pproxim a tions K' + 1 -~----:-- ~ 2K'

+

3

K'

- - - - , - - - -:::::: 2K'

+

1 1/2 for K' larg~, Equation (43) becomes

[N

I! K'+1

+

··

N"

K' - 1

]

(IK' K")' " = 1/2 _{, 2 v, v2}

X

_{3 2} (44)

The factor [N"K' + 1 + N"K' _ IJ /2 in Eq. (44) looks like an ave rage value

and, if values are taken at a point where NIIK II as a function of Kil is nearly linear, LN'k'

+

1

+

NIk, _

IJl

2 may be approximated by NilK'

r.

The factor

r i s introduced to account for the alternation of intensities since if NilK'

belongs to the ~trong series (K' odd) NilK'

+

1 and NIK _ 1 will belong to the weak series. The required value of

r

would be 2.0 if K' is odd and 0.5 if K' is even. Equation (44) now becomes:

( I K' Kil ) , 11

~

1.)( NII •

r

2 v v2 2 3 K' •

This approximate equation shows that line intensities in the rotational spectrum have an approximately one-to-one relationship with the rotational

energy distribution and so measured intensities will tend to indicate where to make the approximate analysis suggested above. Wh en one value of NilK'

has been evaluated for each set of lines the rest can be determined directly from Eq. (43).

The foregoing analysis has been presented to indicate the problems which will be encountered in attempting to determine a

non-equilibrium rotational energy distribution from line intensity measurements and to suggest a possible approximate solution.

2.7 Measurement of Translational Energy Accomrpodation

Consider a system as shown in Fig. 1, where the target is a water cooled metal disc 12.7 mm in diameter and the surrounding cham-ber is maintained at a temperature of 5000K by an oven. The electron beam is parallel to the surface of the metal disc and a short distance in front of

it. The gas in the chamber is at a pressure low enough so that the mole-cular mean free path is large compared with the disc diámeter and larger than thè radius of the chamber.

Light is gathered from a small volume in the electron beam centred on the axis of the target disc as is indicated by P in Fig. 1. The volume selected by the optical arrangement is 1 mm deep, the beam dia-meter, by 2 mm high by 0.03 mm wide, the latter two dimensions being fixed by the spectrometer slit size. The spectrometer views the electron beam in a plane parallel to the surface of the disco The solid angle sub-tended by the target disc at any point within this volume is very nearly the same as the solid angle subtended at any other point, so the approximation is made that all light comes from a point P on the axis of the disc at the centre of the small volume.

Vnder conditions of free molecular flow the t otal light intensity measured at P corresponds to a number density, n, given by

(45)

from the surrounding gas and na is the number density at P due to reflected molecules coming from the solid surface .

In Sec. 2.1 we found the number density at a point was given

by Ol' C>C:I C>d

n (x,t),

J

f

f

_{f d f1 di2 df3} _ oe _00 _DO

In the experimental arrangement considered here the target disc prevents some of the molecules from the background gas from reaching point P, and supplies in their place the contribution na due to reflected molecules. These reflected molecules can be thought of as coming from a body of gas located behind the surface of the target disc, having a velocity distribution function, fr and a corresponding number density nr.

In calculating nb and na it is convenient to use polar coordinates. The elemental volume in velocity space then becomes

and the limits of integration are as indicated in Eq. (46) below. Since the gas is at rest, fj

=

Cj and Eq. (1) becomes

e>O 7f' 27(

n (x, t ) ,

1

e 2 f de

1

sin~

d

~

[ dG (46) We can write with the aid of Eq. (46)

~

%

~~

na'

f

e2 _{f r de}

J

sin

~

d

rPf

dG , (47)

o

0 0

where 1>2 is the half angle at P subtended by the solid surface as indicated in Fig. 1. ~

'I'~ 2.:1'(

The quantity

11

sin

~

d

~

dS is the solid angle

J1

_{2 at point P subtended} by the target

di~c ~

Hence,

(48)

(49)

Similarly (50)

where ni is the number density in the background gas and

111

is 41T -J).2

The value of ni may be directly determined with the cold target retracted far enough to make (nr - ni)

n

2 / 4 j( negligible. The tar-get is then moved into position and a density n is measured. Hence a value is obtained for na '

na = n

- [11

n. ₁

47r (51 )

The auxiliary condition for dynamic equilibrium given in Eq.

(17) of Sec. 2.2 can now be introduced to equate the number flux of

mole-cules striking the surface to the number flux of molemole-cules reflected from it:

(52)

where cl has been substituted for

J

1 since u

=

0 in this case and cl repre-sents the components of molecular velocity normal to the solid surface be-ing considered.

We can also show that

=

-

cl (r) , (53)

where -c 1 (r) is the average xl component of velocity of molecules crossing the surface in the -xl direction coming from an imaginary gas with distri-bution function fr located behind the surface.

The density integral

o Q() QC)

J

J

J

frdqdc2dc3 of Eq. (53) _00 _00 -00

is equivalent to putting

1>

2 = '11/2 in Eq. (47) with the result that

27f

na = _{n r 4}

1'1'

= _{n r / 2}

Provided f r is isotropic,as was implicitly assumed in going from Eq. (47) to Eq. (48) above (i. _{e. fr is not a function of} ~ or 8) ,

(54)

7(/2 by the simple relationship expressed in Eq. (49),

Dividing the right hand side of Eq. (52) by the integral expres sion equal to nr/2 in Eq. (54) _{and the left hand side by n r} /2 we have

ni The left hand side of this equation can be rearranged to be

cl

(i)

-nr

we finally have - - . ni c1(r)

=

-C 1(1)(-) _nr so that

The density ni is measured directly and nr is determined from Eqs. (49) and (51).

From the foregoing analysis we see that the. basic quantity determined directly by the density measuremei1ts ni and n is the m'ean velocity of molecules normal to the solid surface.

In order to relate the density measurements made here to translational energy accommodation coefficient, which is of more immediate interest than the mean normal velocity of the reflected molecules, we will have to make a further assumption about the velocity distribution function of the reflected molecules (in addition to the one made above, that fr is iso-tropie).

We will consider two simple models for f r which were dis-cussed in Sec. _{2.3. In the first model f r} = Kfs

+

(1 - K) fi where fs and fi are Maxwellian velocity distribution functions for a gas at temperatures Ts and Ti respectively. Subscript i refers to the background or incident gas, subscript r refers to the reflected gas and T s is the target temperature. In the second model fr is a Maxwellian velocity distribution function at a temper_{ature T r , where T s} ~ _{T r}

4

Ti·

Taking the first assumed model we found in Sec. 2.3 that Maxwell' s reflection coefficient K

=

0( •

Therefore (55)