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Direct simultaneous measurement of rotational and translational accomodation coefficients
Direct simultaneous measurement of rotational translational accommodation coefficients
and
received 25 April 1973
V Ramesh
and D .I Marsden,
Department
of Mechanical
Engineering, University of Alberta, Edmonton, Alberta, Canada
When molecules of a low density diatomic gas strike a solid surface both translational energy and the internal energy modes of rotation and vibration will contribute to the energy exchange that occurs. Theoretical studies indicate that the accommodation coefficient for rotational energy should be less than that for translational energy, and this is confirmed by experimental results. The experimental apparatus described in this paper uses the electron bream fluorescence detector to measure simultaneously both rotational and translational energy accommodation coefficients of room temperature nitrogen reflecting from a solid surface. A bakeable ultra high vacuum system was built to provide a clean vacuum environment for control of the solid surface properties. In addition to being the only known direct measurement of rotational accommodation coefficient the system offers an advantage over some previous methods of translational accommodation measurement in that there are few restrictions on solid surface temperature or composition.
1. Introduction Internal energy of a gas is divided between translation, rotational and vibrational motions in proportions that are kept in equilibrium by constant intermolecular collisions. This equilibrium can be disturbed, for example by the passage of a shock wave, and it is well known that vibrational and rotational energy modes require many more collisions to come to equilibrium again than the translational energy does. Since the energy transfer between a rarefied gas and a solid surface is also accomplished by means of molecular collisions, it might be expected that the transfer of energy from the rotational and vibrational energy modes of the gas would be less efficient than that from the translational energy mode. A theoretical study by Feuer’, based on a quantum mechanics model, indicates that the ratio of rotational accommodation coefficient to translational accommodation coefficient, Q/L+, would be expected to be less than 1 .O. This result is confirmed by experiments of Sasaki, Taku and Mitani2 who found cur = 0.48 and 0~~= 0.10 for hydrogen on tungsten. They also reported GL~= 0.393 and 0~~= 0.314 for nitrogen on nickel. Measurements by Schafer and Riggert3 of accommodation of a number of gases on a gold surface also show Q consistently less than Q. Knudsen4 found equal values for rotational and translational accommodation coefficients of hydrogen on platinum and Marsdens found equal values of C+ and c+ for nitrogen on silver. The apparatus described in this paper uses the electron beam fluorescence detector to provide a simultaneous direct measurement of translational and rotational accommodation coefficients for nitrogen at a solid surface. It is an extension of the technique described by Marsdens. The electron beam fluoresence technique was suggested by Schumacher and &tin6 for measurement of number density in a gas and further developed by Schumacher and Gadamer’ and Muntzs. Muntz extended the technique to the measurement of gas temperature from the distribution of rotational energy in nitrogens. Vacuum/volume
23/number
10.
Pergamon Press LtdlPrinted
In the present investigation the translational accommodation coefficient was obtained from density measurements, and rotational accommodation from measurements of rotation spectrum line intensities.
2. Experimental
apparatus
Figure 1 shows a schematic of the bakeable stainless steel vacuum system. Chamber A is connected to chamber B by a 2.08 mm diameter orifice as well as by a 37 mm dia by-pass tube. The solid surface target is in chamber A, just above the orifice. Pressure is measured by ionization gauges as indicated. A partial pressure gauge monitors the test gas pressure and composition. The temperatures of the test gas and the target are measured by thermocouples. Care was taken to see that cool spots did not develop on the target at attachment points of the thermocouple wires. Admission of the test gas and system pressure were controlled by a manually operated needle valve in series with an automatic
Figure 1. Schematic diagram of experimental
in Great Britain
apparatus. 365
V Ramesh
and D J Marsden:
Direct simultaneous measurement of rotational and translational accommodation
control
needle valve actuated
(Figure
I ).
A
by the signal from the ion gauge
type
electron
gun
producing
focussed beam of I.5 mm dia is attached which
passes through shown
allows
the beam
three other
in Figure
electrically
proper
with a temperature
to the system by a
to be aimed.
small openings
2, ensuring
an
The
beam
experiments.
Consider
near the orifice as
location.
The gun is capable
15 kV across its accelerators,
The
beam
is
tional
of being run at
and operates at a beam current of
I .O mA.
Figure
The number through N(c)
2 shows a schematic
of the optical
is measured
spectrometer,
with an EMI
system. The rota-
using an j/6
point
of reflected molecules
dA with a velocity
P subtending
an angle
as shown in Figure 3.
in solid angle d(f) passing
c is:
= c;f; cos 8 dw dA/cos
Where
and at rest
r, and number density II,, above the orifice.
a small area dA about
H
(1)
dw is the solid angle subtended
and j; is the velocity
band spectrum
grating
It will be assumed that the
de at centre of orifice, at 0 to the vertical,
picked up by a receiver at the other end and the current is monitored during about
is Maxwellian.
reflected molecules come from a gas in equilibrium
television
mechanism
velocity distribution
coefficients
distribution
by the oritice at point P
function
of the reflected mol-
cules. Now
Czerny-Turner
9502 S photomultiplier
to
measure line intensities.
dw =
A’ cos3 H
and
dA = 2;
M’d61.
J2 where w is the width of the beam which is observed through the slit of the spectometer,
3. Theory
and A’ is the orifice area.
3.1 Geometrical considerations. A heated solid surface made up of the material to be tested is located in chamber A opposite a small orifice leading to chamber The pressure in chamber
B as shown in Figure
A is of the order of IO-”
I. torr, and
the target surface is placed in such a way that substantially the molecules reaching the electron come
directly
collisions. typically
from
the target
surface
The pressure in chamber
with
all
A have
no intermediate
B is kept as low as possible.
IOmh torr.
The valved by-pass connecting pumping
beam from chamber
to outgas
impurities
chambers that
may
A and B allows for be absorbed
on the
system walls. This by-pass is also used to reduce the pressure in A and thereby through density chamber
the
reduce
orifice
so that
the contribution
to a small
of molecules
fraction
the contribution
of
the
of the background
Figure 3. Details of the detector geometry.
gas in
B can be determined.
The number density of gas at a point P is II II,
coming
background
is the contribution
to
number
density
reflected from the target and issuing through is the contribution
of the background
Since the gas in chamber
II,,
The
I II,,, where
by the molecules
number
passing through
dA
in time df with velocity
c
is:
the orifice, and II,,
molecules
A is in equilibrium
in chamber
B.
and at rest, the The number
density of reflected molecules
ber passing through
dA divided
in dA is the num-
by cdAdt,
The number density due to reflected molecules is obtained
by integrating
equation
of all
velocities
(3),
(4) The number
density
dw/4p). ‘1s ( 1 The average number the optical
in dA
due to background
molecules
is
density in the length of beam viewed by
system is obtained
by taking
the integral
(5) ENTRANCE
i,iT
111
Figure 2. Schematic diagram of the optical system. 366
where viewed.
H,, marks
the ends of the portion
of electron
beam
V Ramesh and D J Marsden: Direct simultaneous
(n, - ttB) tl=
measurement
of rotational
-$ 8,+sF
Z = nk@, where @ =
accommodation
g
+(K’,T,).
coefficients
@ is a constant,
and Z is
K’ = 0
nB.
+
In tan 42 + e,/2
then proportional to n. The distribution of rotational energy, and the rotational temperature of the gas, can be obtained directly from a measurement of rotational line intensitiesa. Rotational energy in a diatomic gas has the distribution9 given by equation (9)
tan 3714- eo/2 This can be written as, n=Qn,+(l
and translational
-Q)n,.
(6) NJ =- yGB
The constant Q depends only on system geometry.
(2.Z + l)e’-Bhc.Z(J + l)/kT,
R
When the by-pass between A and B is closed, the orifice is the only connection between the two chambers. Let its conductance be Cr. Also let the speed of the triode ion pump be S, and the conductance of the bypass be Ca. Pressure in chamber A is related to that in chamber B by the relation pa = ps (S + Ci)/Ci when by-pass is closed, and since the temperature in the two chambers is same, nd = ns (S + C,)/C,. From conservation of mass at the target surface, n, = n,J(T/T,). Therefore, substituting in (6), n =
Ll
(F )
J(T/TJtzB + (1 - +I,
(7)
1
and similarly when the by-pass is open,
where NJ = number of molecules in rotational energy level J N = total number of molecules B = the rotational constant for the molecule h = Plank’s constant k = Boltzmann’s constant c = speed of light In the case of the Nz +(0,O) band of the nitrogen spectrum the line intensity distribution is given8 by equation (10). ZK, = (K’ + K” + 1)X4[G] c4 exp[BK’(K’ vo
+ l)hc/kT,]
(10)
where K is the rotational quantum number apart from spin (J = K f l/2). A single prime refers to the upper electronic state and double prime refers to the lower electronic state.
3.2 Electron beam measurements. The high energy electron beam excites and ionizes the neutral nitrogen molecules, which are initially in N2X’z ground state, to an upper state N,+B’&‘,+. The light emitted comes almost entirely8 from the first negative band due to the transition N2+B2Z,,+ --t Nz+X’&+ of the nitrogen ion. If the doublets are not resolved, the transition becomes ‘z--t ‘z and the applicable selection rule AK = *l, gives rise to two branches of emission P and R. The P branch has a vertex towards red on the Fortrat diagram and forms a bandhead. The R branch has peaks occurring at ever widening intervals towards the ultra-violet. A scan of the R branch of the N2+(0,0) band is shown in Figure 4. The two to one alternation in intensity between odd and even numbered lines is a characteristic of nitrogen. The number density is determined by a summation of all the light output from one branch of the N1+(O,O) band or alternatively by measuring the total light output from the unresolved band. Light intensity is given by the equation, Z = nk#, where n is number density, k is constant depending on the optical system and beam current and Q is a factor depending on the spectrum lines being observed. If the rotational spectrum is resolved 4(K’,T,) depends on rotational quantum number K’ and rotational temperature TR, n and k being constant. If the spectrum is unresolved and all light from the band is observed
El =
(2K’ + 1)
[Cl is a function of T,, x4 is a constant related to overall intensity, and v/v0 is approximately constant for a given band. Equation (10) is derived from equation (9) by going through the process of excitation (and ionization) of the nitrogen molecules by the electron beam, followed by a spontaneous decay to the ground state of the ion with emission of light. The temperature TR and rotational constant B appearing in both equations are the same. A plot of ln[Z,,/(K’ + K” + l)[G]x,(u/~~)~l against K’(K’ + 1) will result in a straight line with a slope (-Bhc)/(kT,) if the gas has an equilibrium rotational energy distribution. Thus the rotational temperature TX can be determined from a measurement of spectrum line intensities. Figure 5 shows some plots of measured line intensities used to determine rotational temperature. 3.3 Translational accommodation coefficients. The translational energy accommodation coefficient, Q, is defined as Ei - Er u. T - Ei - Es where Z$ is energy incident per unit area per second. E, is energy reflected. Es is idealized energy reflected.
19
17
15
13
11
9
7
5
3
I<'
Figure 4. Spectrometer scan of the R-branch of the N,’ (0,O) band.
If the incident gas is in equilibrium TuT-T-Ts
Tr =---T,IT - 1 TJT - 1
and at rest,
(11) 367
V Ramesh and D J Marsden: Direct simultaneous
measurement
of rotational
and translational
accommodation
coefficients
3.4 Rotational accommodation coefficient. The intensity of a measured spectrum line corresponding to rotational level K’ is I( K’) = .nl,( K’) + ( I - Q)I,( K’)
(IX)
where Q/,(/C’) is intensity due to reflected molecules and Q)l,(K’) represents the contribution due to background (I molecules. With the by-pass valve open the number density in 4 is St
Il,q = C
c, c,
+ Cl
+ c,
1 ‘lg.
If the target is unheated,
T, ~~r,
T.
f,( K’) = I,( K’)n,/t~, and
i(
(19)
Io,,,(K’) = Q L
0
100
For the system geometry used here, Q ~~ 0.001876. S C-Y100 l/s, Cz ‘v 20 I/s and C1 Pi 0.40 I/s.
200
K' (K’+l) Figure 5. Typical line intensities in background
and reflected mole-
I,( K’) = lopt.J K’)/
cules.
where T is the temperature of the incident gas, T, is the temperature of the reflected gas and T, is temperature of the solid surface. The temperature ratio T,/T is determined experimentally as follows: When the bypass is open, the number density measured at the electron beam is,
= 0.9889/,,,,(
K’).
Thus values of the spectrum line intensities f,(K’) are determined by a measurement of I,,,,(K’) with the by-pass valve open and the target unheated. Values of the desired spectrum line intensities f,(K’) are then obtained from equation (18) by measuring intensities I(K) with the valve closed and target heated, and making use of the measured values of I,(K). .nI,( K’) = I( K’) - ( I - Q)I,( K’).
and when the bypass is closed Q
112 =
(~ > s+c, Cl
/I,,‘(
the number
density
T/T,) + ( I - WI,.
measured
(13)
Measurements are made with the target temperature the same as that of the incident gas (T, =y T, -= T), and at a higher temperature resulting in T, > T. Then, for T, ~ T,
(112- /l,)r
=
aw, C,(C,
+ C,)
‘IL?
(20)
is, The pumping speed S and conductance of the by-pass Cz are only known approximately, but this does not affect the accuracy very much. An error of 100% in S would result in less than I ‘I,:, change in the value of I,(K). The averaging process of drawing a best fit line through the resulting points serves to further reduce the error in determining the rotational temperature of the reflected molecules.
(14) 4. Results and discussion
and for T, A T,
(112- n,).l.r =
QSC,
C,(C, + C,)
n,,’
( TIT,,
(ISI
Therefore
(16) Since I
kn@ and since k and @ are constants.
I,‘( C/T) =
(1, -I,),
t/z - /,)Tr.
From equation
368
(I I)
The technique outlined above has the advantage that no absolute calibration of pressure gauges or accurate determination of conductances is needed. A Bayart-Alpert type ionization gauge is used to monitor pressure and insure that it remains constant. The repeatability of such a gauge is much better than its reliability in measuring absolute pressure. 4.1 Translational accommodation coefficients. Temperatures T, and T in equation (17) are measured by thermocouples and are accurate to within _+ I K, The expected error in the expression T,/T is then less than 0.5%. The values of I, and lz with the valve open and closed respectively are obtained by summing the line intensities in the rotational spectrum, An analysis of the scatter in measured line intensities indicates that the sum would have a standard error of 0.3”& Using this figure, the probable error in “r is ::_O.O3/(7’,/T I). Some measurements of n,
V Ramesh and D J Marsden: Direct simultaneous measurement of rotational and translational accommodation
coefficients
the scatter of these spectrum line intensities, derived as indicated in equation (20) shows THr= 402 & 17 K for a typical case. Since the scatter in line intensities is independent of temperature and the slope of In[lK./[G](V/Y0)4x4(K’ + K" f I)]against K'(K'+ 1) is proportional to l/TR, the error in TH,AT, will be proportional to TR. If the above value of TR,= 402 f 17 K is typical, AT,, = O&IT,,. The rotational accommodation coefficient is then,
E
00
I PO0
I
300
700
500
TARGET
TEMPERATURE
TsoK
The limits of error given by the above expression are shown in Figure 6 along with some measured values of aH for nitrogen on stainless steel. The scatter in measured values falls easily within the predicted band of scatter, The value of translational accommodation at 385 K is 0.90, which is about the value expected from previous work. The rotational accommodation is considerably lower, c+ = 0.48. Heating the stainless steel target to 800 K has the effect of reducing c+ dramatically, probably due to the evaporation of adsorbed gas layers from the surface. It is interesting to note that L-Qdoes not change much with increasing T,.
Figure 6. Variation of accommodation coefficient with target temperature.
5. Conclusions
for nitrogen on stainless steel, shown in Figure 6, appear to fall within this expected probable error. 4.2 Rotational accommodation coefficient. TRY- TR
aR = T, _ TR '
TR = l-.
Again, T, and T are measured by thermocouple with an accuracy of *1 K. The rotational temperature of background molecules, TR,can be determined directly from the measured rotational spectrum. An analysis of the scatter of these measured intensities about a least squares best fit line shows that the standard error in the slope of the line is 13.2% giving a probable error of f9” in measuring TR. These measured values of TR agree with the temperature measured by thermocouples on the walls of the apparatus, usually within *5 K. In subtracting two sets of measured line intensities, one of which is approximately 1.5 times the other, the scatter in individual line intensities is considerably increased. An analysis of
A technique for the simultaneous direct measurement of both translational and rotational energy accommodation coefficients of nitrogen at a solid surface has been developed. This method has inherent accuracy and reliability since no absolute calibrations are needed to obtain the results. The technique has some advantages over previous methods for the measurement of translational accommodation in that wide range of target compositions and temperatures can be used. The results of some preliminary measurements made in developing the apparatus indicate that a= can be measured with a probable error of *O.O3/(T/T, - 1)and aR with an error of fO.O4[a, + T/T,- Tl. References
1
P Feuer, J Phys, 39, 1963, 13 1 I. * N Sasaki, K Taku, K Mitani, Memoirs of the College of Science, University of Kyoto, A, XXV, (1949). 3 K Schafer and K H Riggert, Z E/ek/rochen~ie, 57, 1953, 751. 4 M Knudsen, Ann Physik, 6, 1930, 129. 5 D J Marsden. Rarefied Gas Dynamics. Academic Press, New York (1966). 6 B W Schumacher and A E Grim, German Pat Appl. June (1955). 7 B W Schumacher and E 0 &darner, Can JPhys, 36, 1958, 654. s E P Muntz, Report No 71, University of Toronto, UTIA (1961). 9 G Hertzberg, Speclm of Diafondc Ma/ecu/es, p. 125, Van Nostrand, New York (1950).
369
and
received 25 April 1973
V Ramesh
and D .I Marsden,
Department
of Mechanical
Engineering, University of Alberta, Edmonton, Alberta, Canada
When molecules of a low density diatomic gas strike a solid surface both translational energy and the internal energy modes of rotation and vibration will contribute to the energy exchange that occurs. Theoretical studies indicate that the accommodation coefficient for rotational energy should be less than that for translational energy, and this is confirmed by experimental results. The experimental apparatus described in this paper uses the electron bream fluorescence detector to measure simultaneously both rotational and translational energy accommodation coefficients of room temperature nitrogen reflecting from a solid surface. A bakeable ultra high vacuum system was built to provide a clean vacuum environment for control of the solid surface properties. In addition to being the only known direct measurement of rotational accommodation coefficient the system offers an advantage over some previous methods of translational accommodation measurement in that there are few restrictions on solid surface temperature or composition.
1. Introduction Internal energy of a gas is divided between translation, rotational and vibrational motions in proportions that are kept in equilibrium by constant intermolecular collisions. This equilibrium can be disturbed, for example by the passage of a shock wave, and it is well known that vibrational and rotational energy modes require many more collisions to come to equilibrium again than the translational energy does. Since the energy transfer between a rarefied gas and a solid surface is also accomplished by means of molecular collisions, it might be expected that the transfer of energy from the rotational and vibrational energy modes of the gas would be less efficient than that from the translational energy mode. A theoretical study by Feuer’, based on a quantum mechanics model, indicates that the ratio of rotational accommodation coefficient to translational accommodation coefficient, Q/L+, would be expected to be less than 1 .O. This result is confirmed by experiments of Sasaki, Taku and Mitani2 who found cur = 0.48 and 0~~= 0.10 for hydrogen on tungsten. They also reported GL~= 0.393 and 0~~= 0.314 for nitrogen on nickel. Measurements by Schafer and Riggert3 of accommodation of a number of gases on a gold surface also show Q consistently less than Q. Knudsen4 found equal values for rotational and translational accommodation coefficients of hydrogen on platinum and Marsdens found equal values of C+ and c+ for nitrogen on silver. The apparatus described in this paper uses the electron beam fluorescence detector to provide a simultaneous direct measurement of translational and rotational accommodation coefficients for nitrogen at a solid surface. It is an extension of the technique described by Marsdens. The electron beam fluoresence technique was suggested by Schumacher and &tin6 for measurement of number density in a gas and further developed by Schumacher and Gadamer’ and Muntzs. Muntz extended the technique to the measurement of gas temperature from the distribution of rotational energy in nitrogens. Vacuum/volume
23/number
10.
Pergamon Press LtdlPrinted
In the present investigation the translational accommodation coefficient was obtained from density measurements, and rotational accommodation from measurements of rotation spectrum line intensities.
2. Experimental
apparatus
Figure 1 shows a schematic of the bakeable stainless steel vacuum system. Chamber A is connected to chamber B by a 2.08 mm diameter orifice as well as by a 37 mm dia by-pass tube. The solid surface target is in chamber A, just above the orifice. Pressure is measured by ionization gauges as indicated. A partial pressure gauge monitors the test gas pressure and composition. The temperatures of the test gas and the target are measured by thermocouples. Care was taken to see that cool spots did not develop on the target at attachment points of the thermocouple wires. Admission of the test gas and system pressure were controlled by a manually operated needle valve in series with an automatic
Figure 1. Schematic diagram of experimental
in Great Britain
apparatus. 365
V Ramesh
and D J Marsden:
Direct simultaneous measurement of rotational and translational accommodation
control
needle valve actuated
(Figure
I ).
A
by the signal from the ion gauge
type
electron
gun
producing
focussed beam of I.5 mm dia is attached which
passes through shown
allows
the beam
three other
in Figure
electrically
proper
with a temperature
to the system by a
to be aimed.
small openings
2, ensuring
an
The
beam
experiments.
Consider
near the orifice as
location.
The gun is capable
15 kV across its accelerators,
The
beam
is
tional
of being run at
and operates at a beam current of
I .O mA.
Figure
The number through N(c)
2 shows a schematic
of the optical
is measured
spectrometer,
with an EMI
system. The rota-
using an j/6
point
of reflected molecules
dA with a velocity
P subtending
an angle
as shown in Figure 3.
in solid angle d(f) passing
c is:
= c;f; cos 8 dw dA/cos
Where
and at rest
r, and number density II,, above the orifice.
a small area dA about
H
(1)
dw is the solid angle subtended
and j; is the velocity
band spectrum
grating
It will be assumed that the
de at centre of orifice, at 0 to the vertical,
picked up by a receiver at the other end and the current is monitored during about
is Maxwellian.
reflected molecules come from a gas in equilibrium
television
mechanism
velocity distribution
coefficients
distribution
by the oritice at point P
function
of the reflected mol-
cules. Now
Czerny-Turner
9502 S photomultiplier
to
measure line intensities.
dw =
A’ cos3 H
and
dA = 2;
M’d61.
J2 where w is the width of the beam which is observed through the slit of the spectometer,
3. Theory
and A’ is the orifice area.
3.1 Geometrical considerations. A heated solid surface made up of the material to be tested is located in chamber A opposite a small orifice leading to chamber The pressure in chamber
B as shown in Figure
A is of the order of IO-”
I. torr, and
the target surface is placed in such a way that substantially the molecules reaching the electron come
directly
collisions. typically
from
the target
surface
The pressure in chamber
with
all
A have
no intermediate
B is kept as low as possible.
IOmh torr.
The valved by-pass connecting pumping
beam from chamber
to outgas
impurities
chambers that
may
A and B allows for be absorbed
on the
system walls. This by-pass is also used to reduce the pressure in A and thereby through density chamber
the
reduce
orifice
so that
the contribution
to a small
of molecules
fraction
the contribution
of
the
of the background
Figure 3. Details of the detector geometry.
gas in
B can be determined.
The number density of gas at a point P is II II,
coming
background
is the contribution
to
number
density
reflected from the target and issuing through is the contribution
of the background
Since the gas in chamber
II,,
The
I II,,, where
by the molecules
number
passing through
dA
in time df with velocity
c
is:
the orifice, and II,,
molecules
A is in equilibrium
in chamber
B.
and at rest, the The number
density of reflected molecules
ber passing through
dA divided
in dA is the num-
by cdAdt,
The number density due to reflected molecules is obtained
by integrating
equation
of all
velocities
(3),
(4) The number
density
dw/4p). ‘1s ( 1 The average number the optical
in dA
due to background
molecules
is
density in the length of beam viewed by
system is obtained
by taking
the integral
(5) ENTRANCE
i,iT
111
Figure 2. Schematic diagram of the optical system. 366
where viewed.
H,, marks
the ends of the portion
of electron
beam
V Ramesh and D J Marsden: Direct simultaneous
(n, - ttB) tl=
measurement
of rotational
-$ 8,+sF
Z = nk@, where @ =
accommodation
g
+(K’,T,).
coefficients
@ is a constant,
and Z is
K’ = 0
nB.
+
In tan 42 + e,/2
then proportional to n. The distribution of rotational energy, and the rotational temperature of the gas, can be obtained directly from a measurement of rotational line intensitiesa. Rotational energy in a diatomic gas has the distribution9 given by equation (9)
tan 3714- eo/2 This can be written as, n=Qn,+(l
and translational
-Q)n,.
(6) NJ =- yGB
The constant Q depends only on system geometry.
(2.Z + l)e’-Bhc.Z(J + l)/kT,
R
When the by-pass between A and B is closed, the orifice is the only connection between the two chambers. Let its conductance be Cr. Also let the speed of the triode ion pump be S, and the conductance of the bypass be Ca. Pressure in chamber A is related to that in chamber B by the relation pa = ps (S + Ci)/Ci when by-pass is closed, and since the temperature in the two chambers is same, nd = ns (S + C,)/C,. From conservation of mass at the target surface, n, = n,J(T/T,). Therefore, substituting in (6), n =
Ll
(F )
J(T/TJtzB + (1 - +I,
(7)
1
and similarly when the by-pass is open,
where NJ = number of molecules in rotational energy level J N = total number of molecules B = the rotational constant for the molecule h = Plank’s constant k = Boltzmann’s constant c = speed of light In the case of the Nz +(0,O) band of the nitrogen spectrum the line intensity distribution is given8 by equation (10). ZK, = (K’ + K” + 1)X4[G] c4 exp[BK’(K’ vo
+ l)hc/kT,]
(10)
where K is the rotational quantum number apart from spin (J = K f l/2). A single prime refers to the upper electronic state and double prime refers to the lower electronic state.
3.2 Electron beam measurements. The high energy electron beam excites and ionizes the neutral nitrogen molecules, which are initially in N2X’z ground state, to an upper state N,+B’&‘,+. The light emitted comes almost entirely8 from the first negative band due to the transition N2+B2Z,,+ --t Nz+X’&+ of the nitrogen ion. If the doublets are not resolved, the transition becomes ‘z--t ‘z and the applicable selection rule AK = *l, gives rise to two branches of emission P and R. The P branch has a vertex towards red on the Fortrat diagram and forms a bandhead. The R branch has peaks occurring at ever widening intervals towards the ultra-violet. A scan of the R branch of the N2+(0,0) band is shown in Figure 4. The two to one alternation in intensity between odd and even numbered lines is a characteristic of nitrogen. The number density is determined by a summation of all the light output from one branch of the N1+(O,O) band or alternatively by measuring the total light output from the unresolved band. Light intensity is given by the equation, Z = nk#, where n is number density, k is constant depending on the optical system and beam current and Q is a factor depending on the spectrum lines being observed. If the rotational spectrum is resolved 4(K’,T,) depends on rotational quantum number K’ and rotational temperature TR, n and k being constant. If the spectrum is unresolved and all light from the band is observed
El =
(2K’ + 1)
[Cl is a function of T,, x4 is a constant related to overall intensity, and v/v0 is approximately constant for a given band. Equation (10) is derived from equation (9) by going through the process of excitation (and ionization) of the nitrogen molecules by the electron beam, followed by a spontaneous decay to the ground state of the ion with emission of light. The temperature TR and rotational constant B appearing in both equations are the same. A plot of ln[Z,,/(K’ + K” + l)[G]x,(u/~~)~l against K’(K’ + 1) will result in a straight line with a slope (-Bhc)/(kT,) if the gas has an equilibrium rotational energy distribution. Thus the rotational temperature TX can be determined from a measurement of spectrum line intensities. Figure 5 shows some plots of measured line intensities used to determine rotational temperature. 3.3 Translational accommodation coefficients. The translational energy accommodation coefficient, Q, is defined as Ei - Er u. T - Ei - Es where Z$ is energy incident per unit area per second. E, is energy reflected. Es is idealized energy reflected.
19
17
15
13
11
9
7
5
3
I<'
Figure 4. Spectrometer scan of the R-branch of the N,’ (0,O) band.
If the incident gas is in equilibrium TuT-T-Ts
Tr =---T,IT - 1 TJT - 1
and at rest,
(11) 367
V Ramesh and D J Marsden: Direct simultaneous
measurement
of rotational
and translational
accommodation
coefficients
3.4 Rotational accommodation coefficient. The intensity of a measured spectrum line corresponding to rotational level K’ is I( K’) = .nl,( K’) + ( I - Q)I,( K’)
(IX)
where Q/,(/C’) is intensity due to reflected molecules and Q)l,(K’) represents the contribution due to background (I molecules. With the by-pass valve open the number density in 4 is St
Il,q = C
c, c,
+ Cl
+ c,
1 ‘lg.
If the target is unheated,
T, ~~r,
T.
f,( K’) = I,( K’)n,/t~, and
i(
(19)
Io,,,(K’) = Q L
0
100
For the system geometry used here, Q ~~ 0.001876. S C-Y100 l/s, Cz ‘v 20 I/s and C1 Pi 0.40 I/s.
200
K' (K’+l) Figure 5. Typical line intensities in background
and reflected mole-
I,( K’) = lopt.J K’)/
cules.
where T is the temperature of the incident gas, T, is the temperature of the reflected gas and T, is temperature of the solid surface. The temperature ratio T,/T is determined experimentally as follows: When the bypass is open, the number density measured at the electron beam is,
= 0.9889/,,,,(
K’).
Thus values of the spectrum line intensities f,(K’) are determined by a measurement of I,,,,(K’) with the by-pass valve open and the target unheated. Values of the desired spectrum line intensities f,(K’) are then obtained from equation (18) by measuring intensities I(K) with the valve closed and target heated, and making use of the measured values of I,(K). .nI,( K’) = I( K’) - ( I - Q)I,( K’).
and when the bypass is closed Q
112 =
(~ > s+c, Cl
/I,,‘(
the number
density
T/T,) + ( I - WI,.
measured
(13)
Measurements are made with the target temperature the same as that of the incident gas (T, =y T, -= T), and at a higher temperature resulting in T, > T. Then, for T, ~ T,
(112- /l,)r
=
aw, C,(C,
+ C,)
‘IL?
(20)
is, The pumping speed S and conductance of the by-pass Cz are only known approximately, but this does not affect the accuracy very much. An error of 100% in S would result in less than I ‘I,:, change in the value of I,(K). The averaging process of drawing a best fit line through the resulting points serves to further reduce the error in determining the rotational temperature of the reflected molecules.
(14) 4. Results and discussion
and for T, A T,
(112- n,).l.r =
QSC,
C,(C, + C,)
n,,’
( TIT,,
(ISI
Therefore
(16) Since I
kn@ and since k and @ are constants.
I,‘( C/T) =
(1, -I,),
t/z - /,)Tr.
From equation
368
(I I)
The technique outlined above has the advantage that no absolute calibration of pressure gauges or accurate determination of conductances is needed. A Bayart-Alpert type ionization gauge is used to monitor pressure and insure that it remains constant. The repeatability of such a gauge is much better than its reliability in measuring absolute pressure. 4.1 Translational accommodation coefficients. Temperatures T, and T in equation (17) are measured by thermocouples and are accurate to within _+ I K, The expected error in the expression T,/T is then less than 0.5%. The values of I, and lz with the valve open and closed respectively are obtained by summing the line intensities in the rotational spectrum, An analysis of the scatter in measured line intensities indicates that the sum would have a standard error of 0.3”& Using this figure, the probable error in “r is ::_O.O3/(7’,/T I). Some measurements of n,
V Ramesh and D J Marsden: Direct simultaneous measurement of rotational and translational accommodation
coefficients
the scatter of these spectrum line intensities, derived as indicated in equation (20) shows THr= 402 & 17 K for a typical case. Since the scatter in line intensities is independent of temperature and the slope of In[lK./[G](V/Y0)4x4(K’ + K" f I)]against K'(K'+ 1) is proportional to l/TR, the error in TH,AT, will be proportional to TR. If the above value of TR,= 402 f 17 K is typical, AT,, = O&IT,,. The rotational accommodation coefficient is then,
E
00
I PO0
I
300
700
500
TARGET
TEMPERATURE
TsoK
The limits of error given by the above expression are shown in Figure 6 along with some measured values of aH for nitrogen on stainless steel. The scatter in measured values falls easily within the predicted band of scatter, The value of translational accommodation at 385 K is 0.90, which is about the value expected from previous work. The rotational accommodation is considerably lower, c+ = 0.48. Heating the stainless steel target to 800 K has the effect of reducing c+ dramatically, probably due to the evaporation of adsorbed gas layers from the surface. It is interesting to note that L-Qdoes not change much with increasing T,.
Figure 6. Variation of accommodation coefficient with target temperature.
5. Conclusions
for nitrogen on stainless steel, shown in Figure 6, appear to fall within this expected probable error. 4.2 Rotational accommodation coefficient. TRY- TR
aR = T, _ TR '
TR = l-.
Again, T, and T are measured by thermocouple with an accuracy of *1 K. The rotational temperature of background molecules, TR,can be determined directly from the measured rotational spectrum. An analysis of the scatter of these measured intensities about a least squares best fit line shows that the standard error in the slope of the line is 13.2% giving a probable error of f9” in measuring TR. These measured values of TR agree with the temperature measured by thermocouples on the walls of the apparatus, usually within *5 K. In subtracting two sets of measured line intensities, one of which is approximately 1.5 times the other, the scatter in individual line intensities is considerably increased. An analysis of
A technique for the simultaneous direct measurement of both translational and rotational energy accommodation coefficients of nitrogen at a solid surface has been developed. This method has inherent accuracy and reliability since no absolute calibrations are needed to obtain the results. The technique has some advantages over previous methods for the measurement of translational accommodation in that wide range of target compositions and temperatures can be used. The results of some preliminary measurements made in developing the apparatus indicate that a= can be measured with a probable error of *O.O3/(T/T, - 1)and aR with an error of fO.O4[a, + T/T,- Tl. References
1
P Feuer, J Phys, 39, 1963, 13 1 I. * N Sasaki, K Taku, K Mitani, Memoirs of the College of Science, University of Kyoto, A, XXV, (1949). 3 K Schafer and K H Riggert, Z E/ek/rochen~ie, 57, 1953, 751. 4 M Knudsen, Ann Physik, 6, 1930, 129. 5 D J Marsden. Rarefied Gas Dynamics. Academic Press, New York (1966). 6 B W Schumacher and A E Grim, German Pat Appl. June (1955). 7 B W Schumacher and E 0 &darner, Can JPhys, 36, 1958, 654. s E P Muntz, Report No 71, University of Toronto, UTIA (1961). 9 G Hertzberg, Speclm of Diafondc Ma/ecu/es, p. 125, Van Nostrand, New York (1950).
369