.,J; 2kT
(2. 1 )It is important to note that the Maxwell velocity distribution is in fact directly derived from the Maxwell - Boltzmann distribution of molecular velocities [76]. The Maxwell - Boltzmann distribution describes the distribution of molecular velocity, while the Maxwell velocity distribution describes the distribution of the velocity of the molecules irrespective of their direction of motion (based upon a three dimensional spherical expansion). The Maxwell -
Boltzmann is defined by equation (2.2) where the molecules having a velocity
v
z ' in thez direction, are described as a function of that velocity
f{vz ) .
1
(
m)i
-mv/f{v ) = - - e 2kT z
.,J; 2kT
(2.2)The most probable speed
v
w that a molecule has is determined by the maximum of the Maxwellvelocity distribution, this maximum value is derived from equation (2. 1 ) as:
(2.3)
A plot of the Maxwell velocity distribution using equation (2. 1 ) is presented below in graph 2. I .
The distribution in graph 2. 1 is generalised for any temperature and mass value. This is done by presenting both the speed
v
of a molecule and the number of molecules having this speedf{v)
as ratio values of the most probable speedv",
and the maximum value of the distribution2.2. 1 : Maxwell velocity distribution
(respectively). As observed in graph 2. 1 most of the molecules travel at speeds somewhere
v
3v
v
between � and , with the majority (over 94%) lying in the region between � and
2v
W •2
2
4
Graph
2. 1:
Plot of the Maxwell velocity distribution in terms of VI v w '
where the distribution has beennormalised with respect to the maximum value of the distribution.
The Maxwell distribution can be used to determine the longitudinal translational temperature of a supersonic molecular beam [77], by using a superimposed stream velocity Vs (which is the average velocity of the beam with respect to the stationary lab frame) with a normalized velocity component (in the direction of the translational velocity). The normalized velocity component is derived by dividing the given velocity
v by the velocity at the maximum value of
the beam intensity, giving a dimensionless quantity. The normalized Maxwell distribution using a superimposed stream velocity can be expressed as [77]:-(v-v, f
f{vnorm ) =
C3 (2.4)In equation (2.4) C3 is a normalization constant and the variable a is equal to
vw '
Because C3is an arbitrary constant, if a graph is plotted of
In(f{v norJI v2 )
as a function of{v
-VsY
the- 1
gradient of the graph will be equal to -2 ; the longitudinal temperature is then directly obtained a
from this gradient value. This method is applied to a selected data range consisting of data values from both the left and right hand sides of the normalised Maxwell distribution until the error in a linear least squares fit of the data is minimised (refer to [78] and [79] for details on the least squares method for linear fits). Generally it is a case of trial and error selecting the data
2.2.2: Free jet expansions
2.2.2: Free jet expansions
A continuum jet expansion is the expansion of a high pressure gas, from a nozzle, into a low pressure background. The physical properties of a continuum jet expansion are described using two properties the Mach number M defined by equation
(2.5)
and the specific heat ratio r . Inequation
(2.5) c
is the speed of sound in the gas (of molecular weight W ) at a given temperature T . The speed of sound is defined by equation (2.6) in terms of r .M = �
(2.5)
c
(2.6)
In a free jet expansion the gas starts at the source with a small velocity value, called the stagnation state, at a pressure
Po
and temperature To . Because of the pressure differencebetween the source and the background pressure
Pb
(wherePb
<Po
), the gas is accelerated intop
the low pressure region. If the pressure ratio � exceeds the critical value of
G
, defined by�
equation (2.7), the flow will reach supersonic speed; typically the value of
G
is less than 2. 1 for all gases.G
(2.7)If the pressure ratio is less than this critical value the gas will exit at subsonic speeds with an exit pressure almost equal to the background pressure and will not undergo any further expansion. As the pressure ratio increases beyond the critical value the Mach number will equal
p
one at the exit of the source and the exit pressure becomes equal to � (which is approximately
G
half of the value of the source pressure).
In this research project the source pressure was always above 300KPa and the background
pressure (in the high vacuum region) was approximately 1 3 .3 x l O-6 Pa (refer to section 2.4), giving a large pressure ratio (in excess of the critical value) of approximately
22.6
x 1 09 ; hence all the flows (presented in this thesis) are supersonic in nature.2 .2.2: Free jet expansions
A supersonic flow has two distinct characteristics. Firstly, it increases in velocity and Mach number as the flow area increases; hence the Mach number exceeds one beyond the source exit. And secondly, the supersonic flow is independent of conditions downstream from it, because while the flow is moving at a velocity less than the speed of sound information within the flow propagates at a velocity in excess of this. Because of this, boundary conditions will be created between the different regions, in which shock waves [68] provide a mechanism of meeting the boundary discontinuities. In particular there will be a shockwave in front of the free jet expansion called the Mach shock disk [67]. The Mach shock disk location given by equation (2.8) is defined in terms of the nozzle diameter and has been found to be insensitive to the specific heat ratio.
I
xm =
0.67(PO J2
d
Ph
(2.8)In equation (2.8) xm is the Mach shock disk location and d is the nozzle diameter. The actual width of the Mach shock disk is a difficult feature to measure and estimate, the width can usually be taken to be between three quarters and one half of the value derived for the Mach shock disk location; however its actual value is dependent upon the pressure ratio and the specific heat ratio [67].
With regard to this research project the Mach disk and any associated shockwave effects [68] are irrelevant, because the Mach disk location is located beyond the boundaries of the vacuum system volume. For example, using equation (2.8) and the experimental parameters given in section 2.4.8 results in a Mach disk location of 3.5m (in which the backing pressure on the valve
was
6.52
x1 05
Pa, the operational pressure within the main and source chambers (refer tosections 2.4.2 and 2.4.3) of
1
x1 0-5
Pa and the effective orifice diameter of O.0204mm (refer tosection 2. 1 . 1 » . This distance is well beyond any of the boundaries of the vacuum system volumes (Le. the main expansion of the gas from the valve into the main chamber) discussed in section 2.4. Because shock waves do not have any influence on any of the free jet expansions covered in this thesis, the free jet expansions can be represented by diagram 2.4, rather than a general free jet expansion [68], [67] (which consists of an isentropic region that is bounded by a shock wave region).
The temperature of a gas in a free jet expansion can be separated into two components, which
describe the velocity spread either parallel
111
or perpendicular TL to the flow direction. A2.2.2: Free jet expansions
2.4 is the "quitting surface" model. In the "quitting surface" model the free jet expansion is divided into two regimes, a continuum isentropic regime where
111
= Tl. that is followed by acollision free regime in which
1/1
remains constant "frozen", because the gas has stopped expanding (transitional relaxation). The most probable speed that a molecule will be travelling at within this "frozen" region will be the terminal velocity, which is a fixed quantity [67]. This velocity will be obtained at the point at which the continuum region changes to that of the collision free regime, which is known as the quitting surface. The quitting surface is the minimum point that a skimmer [68] should be placed in order to extract a molecular beam from the isentropic region (also known as the zone of silence).It is the collision free region that is of most importance to this thesis, in particular the value for the terminal velocity, since it is the maximum mean velocity value that a gas pulse produced from the piezoelectric valve (designed in section 2. 1 . 1 ) will obtain.
Expansion fan
Diagram 2.4: Diagrammatic representation ofafreejet expansion in which shockwave effects are irrelevant (modifiedfrom [68])
The continuum region is described using the Mach number and the speed ratio
S ,
which is defined by equation (2.9 ) below.(2.9)
In the continuum region the collisional frequency is sufficient to give an equilibrium where
111
= Tl. ' The most frequently used velocity distribution model in this region is that of the2.2.2: Free jet expansions
ellipsoidal drifting Maxwell model [67] (which is a variation of the distribution described in section 2.2. 1 ), given by equation (2. 1 0).
I
() ( m J2( m ] -
(vl-v�- V.l'! v = n
-- e I .l21lkTII
27TkT1.
(2. 1 0)I f this distribution is used to derive the moments of the Boltzmann equation [67] the resulting energy equation will give a constant enthalpy ho condition of:
(2. 1 1 )
At equilibrium (where
111 = T1. )
this equation reduces to:(2. 1 2)
So an effective temperature might be defmed as [67]:
(2. 1 3)
The reason that
11,
is weighted is that the "flow work" term is included along the streamline[67].
The collision free regime is usually described by only one variable (within this region), the terminal translational speed ratio SlIoo ' which is defmed by using the terminal velocity in
equation (2.9). There is no simple empirical equation that can be used to describe the terminal speed ratio [67] for polyatomic gases in which vibrational
(C02 )
or rotational(H 2 )
relaxation occurs in the continuum region long before translational relaxation. Instead experimentally determined values are used, such as the values given for hydrogen in graph 2.2 [67].The location of the quitting surface is determined from the terminal speed ratio [67] and can be
2.2.2: Free jet expansions
For example given a value of r =
� Cl
is equal to 3 .232 [67]. In terms of the work conducted3
in this thesis using hydrogen gas, the quitting surface occurs a few centimetres from the (nozzle of the) valve; hence the expanding gas can essentially be considered to be in the collision free
"frozen" regime.
SI�H
Xs = dCl
� "0 c;.> 'J.l Q. • 'l 0 0- I - r-I . , • , 1 1 1.0 , HEL11..! •• Te : 77 K " I I .-,Y C R O j E rJ I , . To ' 30c r� ; o ·300K I , '� I
Po d ( torr cm) I i , 1 1 I I I : '1 1
LOOO:
(2. 1 4)Graph 2. 2: Terminal speed ratio as afunction afthe source pressure and nozzle diameter for hydrogen and helium [67]
In the case of a polyatomic gas such as hydrogen (in which rotational relaxation occurs [67]) the terminal velocity can be determined once the internal energy relaxation has been calculated. The simplest l inear relaxation model that describes the rotational energy state populations (such as the Boltzmann distribution of particles with a temperature Tr coupled with a translational
temperature T ) is that given by equation (2. 1 5).
(2. 1 5)
2.2.2: Free jet expansions
In equation (2. 1 5) " is the relaxation time, which is equal to the collision number divided by the local collision frequency. The rotational energy can be included in the overall energy equation, through the use of the equation of state