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UV. W•APTBENT OF CWIEACEN TeMWo tatuad" Km O
AD-AQ27 958
CHAPACTERISTIC STEADY-STATE ELECTRON EMISSION
PROPERTIES FOR PARAMETRIC BLACKBODY X-RAYSPECTRA ON SEVERAL MATERIALS
MISSION RESEARCH CORPORATION
PREPARED FOR
DEFENSE NUCLEAR AGENCY
FEBRUARY 1976
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224208
DNA 3931T
CHARACTERISTIC STEADY-STATEELECTRON EMISSION PROPERTIES FOR
oC PARAMETRIC BLACKBODY X-RAY•2 SPECTRA ON SEVERAL MATERIALS
-,,e Mission Research Corporation
735 State Street
C' Santa Barbara. California 93101
e• February 1976
Topical Report
CONTRACT No. DNA 001-76-C-0086
' APPROVlD F0R PUGLIC RELEASE;Dt-SVTRIGUTIOM UNLIMITED.
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Prepared for
Director
DEFENSE NUCLEAR AGENCY
Washington. 0. C. 20305
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DNA 3931T A
S'LTLr Will %r-,hwfhft) 5 TyVP. Or REPORT ph Pt1IOC D QýOvv'rF
CHARACTERISTIC STEADY-STATE ELECTRON EMISSIONPROPERTIES FOR PARAMETRIC BLACKBODY X-RAY Topical Repor"SPECTRA ON SEVERAL MATERIALS 6 PERFO.RING O.G REPORT ,-jM9LR
14RC-N-221 Revised7 Au'I..~oR- A CO)%-ACT 1) GRANT NUM9(a-
Neal j. Carron DNA O0i-76-C-OC6
9 F`E7r'QMtN(. 004CANIIATIO.4 -40AF AND AODRfSS 10 PRO(.1AM V1 1,F4 ~OPrr" 'A~iiaAE A A *O08. UNT Nu" ofrp,
Mission Research Corporation735 State Street NWED Subtask F?'-QAXE-,BO, 9-' hSanta Barbara. CalJfornia 93101
II CONTQOLLI4G OFFICE NAME ANO ADORESS 12 REPORT DATE
Director February 197,Defense Nuclear Agen,,y ,, Nu.•F'-O O PA , %
Washington, D.C. 20305 _ _ _ _
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Approved for public release: distribution unlimitv-d.
' ,TPI1UTIOd STATFMFN ,I 1h. .h.f.- .. i,..14n If ,,e :', I I-ff11-1 I-ff, Nfq,,, '
8 91UPPLEMFN"A•v NOTES
This work sponsored by the Defense Nucl-ar AF,-ný:y under RDT&• !RSS Cod'B32u7,1,, h 4 R9)QAXEpO, 9'1h H2590D.
14 WFV N1CROS t,,ma, ,i-v.,,e .r~,itr nfre .- ,,¶n% mitt n h%- 111-b. -
Nuclear Weapons E.f.cts Space-charge LimitingSGcalP BlackbodiesPhotoelectric Yield
'0 *8SRPAC7 ta ,r.- - ,-r -npIF , r if . mitt ltdr,,th P, Pita, b nt.-F-)
We collect together in this note certain photoelectric efficiencies.electron energy spectra. electron boundary lay--r plasma Debye lengths.f-lectron number densities, electric rields. and plasma frequencies prevailingin steady state when blackbody photon sources are incident on Aluminum, Gold.and Silicon Dioxide. Only backscattered electrons are considered. Thefigures presented allow quick estimates of many boundary layer properties.
DO I r)AN 1, 1473 OUNCLASSIFIED
I . SECURITY CLASSIrICATION Or TWIl. PAGE (WIl'..t P.. V ,4)
IJ
TABLE OF CONTENTS
PAGE
ILLUSTRATIONS 2
SECTION I.-INTRODUCTION 5
THEORETICAI ASSUMPTIONS 6
SECTION 2-ELECTRON YIELDS AND ENERGY SPECTRA 9
SECTION 3-TIMES OF VALIDII( OF STEADY-STATE THEORY 20
A. Transient Build-Up to Steady State 21
B. Maintenance of Quasi-Steady State 26
SECTION 4-DEBYE LENGTHS 31
SECTION 5-ELECi'RON NUMBER DENSITY AT SURFACE 36
SECTION 6-ELECTRON NUMBER DENSITY PROFILE 40
INTEGRATED NUMBER DENSITY 43
SECTION 7-ELECTRIC FIELD AT SURFACE 45
SECTION 8-ELECTPIC FIELD PROFILE 49
SECTION 9-PLASMA FREQUENCY AT SURFACE 5!.
SECTION 10-DIPOLE MOMENT PER UNIT AREA 55
SECTION 11-EXAMPLE 58
REFERENCES 62
[ . . . . .. . . . .. . . . . .
p.
LIST OF ILLUSTRATIONS
F IGURE PAGE
1 Total backscattered electron yield for incident blackbodyspectrum. 11
2 Electron spectra, 1 keV incident blackbody. 12
3 Electron spectra, 2 keV incident blackbody. 13
4 Electron spectra, 3 keY incident blackbody. 14
5 Electron spectra, 5 keV incident blackbody. 15
6 Electron spectra, 8 key incident blackbody. 16
7 Electron spectra, 10 key incident blackbody. 17
8 a. ts for Aluminum. 2
8 b. ts for Gold. 24
8 c. ts for Silicon Dioxide. 25
9 a. tret for Aluminum. 1,3
9 b. tret for Gold. 29
9 c. tret for Silicon Dioxide. 30
10 a. Debye lengths for Aluminum. 33
10 b. Debye lengths for Gold. 34
10 c. Debye lengths for Silicon Dioxide. 35
11 a. Surface eletron der.sity for Aluminum. 37
11 b. Surface ele,-.ron density for Gold. 382
'4. .
FIGURE PAGE
11 c. Surface electron density for Silicon Dioxide. 39
12 Normalized electron density profile. 41
13 Normalized electron density profile. 42
14 Fraction of electrons out to x. 44
is a. Surface electric field for Aluminum. 46
15 b. Surface electric field for Gold. 47
15 c. Surface electric field for Silicon Dioxide. 48
16 Normalized electric field profile. so
17 a. Surface plasma frequency for Aluminum. 52
17 b. Surface plasma frequency for Gold. 53
17 c. Surface plasma frequency for Silicon Dioxide. 54
18 Normalized dipole moment per unit area. 56
19 X-ray time history for illustrative example. 59
SECTION 1
INTRODUCTION
To estimate the properties of the electron boundary layer produced
when X rays fall on a material surface, it is helpful to have available
tabls and curves of' useful boundary layer properties when a parametric set
of photon spectra is incident on common materials.
This note collects some boundary layer properties for hackscattered
electrons for the cases in which blackbody photon sources of temperatures
kT = 1, 2, 3, 5, 8, and 10 keV are normally incident on aluminum, gold, and
silicon dioxide. For reasons discussed below the curves presented are for
the case of a steady-state boundary layer, that is, it is assumed that
conditions are not changing with time.
The eclectrc,i properties given 'are:
1. Photoelectric yield, and bckscattered electron energy
spectra (Section 2);
2. Conditions for which steady state theory is valid (Section 3);
3. Boundary layer Debye lengths (Section 4);
4. Electron number density at the surface (Section 5);
5. Electron number density profile and the integrated
density (Section 6);
6. Electric field at the surface (Section 7);
Pmrceding pap blank" 5~
I
7. Electric field profile (Section 8);
¶ 8. Plasma frequency at the surface (Section 9);
S9. Electric dipole moment per unit area (Section 10).
In Section 11 we give an example that illustrates the use of the
graphs.
The graphs given here are meant te provide the reader with a ready
reference for estimating boundary layer properties for incident photon
spectra approximating blackbodies when steady-state conditions are valid.
THEORETICAL ASSUMPTIONS
The exact conditions prevailing when an X-ray beam is incident on"
a material surface can vary widely. The incident photon pulse can have any
time history and any energy spectrum, and the energy spectrum can itself
vary with time. The angle of incidence and the physical size and shape of
the target can also vary widely.
It would clearly be a difficult and lengthy task to paraimetri:e
all of these variables, tantamount to solving the general problem. Hlere we
choose a more modest goal. We restrict attention to normal ang.es of
incidence, and targets that are flat surfaces with dimensions large compared
to the thickness of the boundary layer. This last assumption permits a one-
dimensional theoretical treoftment.
In addition, we choose for the energy spectrum a blackbody spectrum
independent of time. This is partly because of its universal availability
(theoretically) and similarity to some experimental sources of interest,
and partly because a blackbody photon spectrum produces a t)ackscattered
electron energy spectrum which is very nearly exponential (see Section 2),
6
I|
and the steady-state boundary layer structure for an exponential electron
spectrum is known,
A parametric study using monoenergetic X rays would 'not be particularly
useful unless one were actually interested in the response to monoenergetic
X rays. Since the boundary layer problem is a non-linear one, it is not pos-
sible to determine the response to a spectrum by superposing the responses
to monocnergetic photons.
Even with these restrictions, the photon source can have an arbi-
trary time history. The solution for the time-dependent boundary layer
structure even in one dimension is very difficult, and until a thorough
stud%- of this problem is made we find it wise to restrict parametric studies'
to the time-independent, or steady-state, case. Many photon sources of
interest, while varying with time, vary slowly enough so that steady-state
theory is approximately applicable at every instant of time using the
instantaneous value of the X-ray flux.
Hence, in spite of the above stated restrictions, the graphs
presented in this report should be useful for estimating boundary layer
properties in many experimental situations of interest.
We have mentioned that a blackbody phuton spectrum produces'a
backscattered electron enerýy spectrum that is very nearly exponential. It
also turns out that the electron angular distribution is very nearly a cosO
distribut ion. The steady-state boundary layer theory for this case of
exponential energy spectrum and cosO angular distribution was presented in
Reference. I. The graphs in the present report were constructed from the
formulae given there.
7
In Figures 8, 9, 10, 11, 15, and 17, information has beencompressed by using the top and right scales as well as the, bottom and left
scales. The top scale should be used with the right scale, and the bottom
scale should be used with the left scale.
Bi
SECTION 2
ELECTRON YIELDS AND ENERGY SPECTRA
The backscattered electron yields and energy spectra depend only
on the X-ray energy spectrum and angle of incidence and on the surface
material. (For a thin surface they also depend on the material thickness.
Here we assume the material is at least one electron range thick, and
they are therefore independent of thickness.) They do not depend on the
X-ray time history, and hence can be discussed independently of any
dynamical assumptions about the boundary layer.
The photoelectric yields and backscattered electron energy spectra
were computed using the electron transport code QUICKE2 2 . The photoelectric
yields (or "efficiencies") are listed in Table 1 in units ofelectrons per
photon and in units of electrons per calorie of incident fluence. These
same backscattered yields are shown as a continuous function of incident
blackbody temperature in Figure 1 (in units of electrons per calorie only).
The backscattered electron energy distributions are shown in
Figures 2 through 7 for the six blackbody te-mperatures chosen. The
ordinates are in units of electrons/keV per calorie incident fluence.
Figures 2 through 7 are partially smoothed from the output of QUICKE2.
It is seen from these figures that, except for certain jumps at
low energies caused by Auger electrons (at 1.4 kcV in Aluminum and 8.25
keV in Gold), the electron energy spectra are nearly exponential with some
characteristic energy E1 ,
9
X -~ -ý -cl- -
V). Cai- 00J k5
4-J
4- U i C~l t2- tn m- if
I-,~CC
Gaw *Oco Cu (n LO. -P (
W oo (_ _ __ _ _ _ __ _ _ _
4 )0 41- -
..Cr C\J M'. CJ NJ N\ '%
Iti '. O Ifl CO '-v L I~) LCI j (~I C)
Ga w cu5- - - - .-EGa Ga
LO D n .'r.4-'l
CL 0 c'4j- u
-0 f
.10
I14 * j * . ... ... .. -.
.4J0
iQii
121
S. .. .. .. . . . . . . .
, . . . . .. .. . . . . . . . . .. . . .
.. ...... .. .i 2• .... ... ...... . . ..- --,-.. ., ... . . .. . . . . . . .. . . .. . ...
% u 10 3 il I I- -I"0 1 2 ... 4 5 6 7 8. 9 TO."2
Incden Blacbod.Temera.re...V
Figure ~ ~ ~ ~ ~ ~ -.. 1.Ttl--.ctee eeto ildfricdetbakod.pcrm
,• .• * 6 - --• -° . . .. . . .-.
i. . . . . . . . . . ..
13 - ... KT I keV
10
" A...il ; 'T * . -. I . . .. . . ..
"I-
., I . . . . . . . . . . . . . .. .. .
01210 .
0~
S22 : '. . . . . . • . . . . . .
4.)
CLJ
0u . . . . . . . . . . - . . . . . .•
Si A Au
109
10 - - - *- -- - -
0 5 1015 20 z 5
Electron Energy (key)
Figure 2. Electron spectra, 1 keV incident blackbody.
12
It
4 4 •4
.1. .
1 01 ... ... ... ........ ........ . .4 . . . . . . . . . . .
012 11'
Au
10 11A.... ......o ,I '4-
.~ .44 .... .4, . ., . 4. 4 .4 . .101 .... ' . .......... - - - -
1 o 1 ' 1 TTTk - 7 . 4 . -1
. .. .. . . .. . .
10 0 ' 1-
Electron Energy (keV)
Figure 3. Electron spectra, 2 key incident blackbody.
13
L
1013 "KT 3 kPV
io 2 I , - - - - - .
..... .. . . . . .
..u . .... ... .. . . .
. . . . . . . . .
12
L
Ail A Au I" I l
T o
00
- 4U
- 1
0 10 20 30 40 50
Electron Energy (keV)
Figure 4. Electron spectra, 3 ',V incident blackbody.
14
. . . ... . . .
I; 4 KT 5 keV ....
10l ,io Au , ,
19.104
4J '4 8
f o . I Ll i i
OJt
Fiur 5. Elcto spcr,5k icietbakoy
I I15
I 41 I1 I I" I " .1 1 t i l : I ! ,,
12 'KT 10 keV
, * - , , .* ,4
1010 - -
10
. . .... .. ..
!.. .
Au
10,S2.. A . ..........
S. .. ....... ' • . .4 ,4
0 10 20 30 40 50 60 0 I 90 100
Electron Energy (keV)
Figure 7. Electron spectra, 10 keV incident blackbody.
17
dn "E/EI
dE. , C electrons/keV/unit fluence
where E is the electron energy. By fitting a straight line to Figures 2
through 7, ignoring the Auger edges, we have determined values for E11Those are listed in Table 2. 'rho energies E 1 are not very different from
the temperature KT, and are nearly independent of target material. Using
electron energy spectral response to monoenergetic photons, and superposing
a blackbody spectrum, it is not hard to show that the electron spectrum
should indeed be nearly exponential with exponentiation energy approximately
equal to KT, as has been discussed by Higgins .
i !i• . 18
Table 2. Exponentiation energies, El, forbackscattered electrons.
Incident PhotonBlackbody Exponentiation Energy. E, (keV)
Temperature .. .
(keV) Aluminum Gold Silicon Dioxide
-1 1.2 1.01 0.98
2 9.09 2,05 1.97
3 2.9r, 2.99 2.86
5 4.77 4.A7 41.63
8 7.27 7.95 ',. ,..
10 8.69 10.07 8.36
19
SECTION 3
TIMES OF VALIDITY OF STEADY-STATE THEORY
For a given X-ray pulse there are two characteristic times that
determine whether or not steady-state theory is applicable.
The first electrons ejected by the arriving X-ray pulse arecertainly not in steady state. Some time, t., must pass after the X rays
first arrive before the boundary layer will have achieved a steady" or quasi-
steady state. There will be a transient build-up to steady state. We may
expect this time t to depend on the fluence and the rise time of the
incident pulse.
Once this transient build-up is over, we will achieve a steady
state and stay in a (quasi-) steady state only if the X-ray pulse is not
changing too rapidly in time. Each ejected electron takes a certain time,
t rt' to reach its maximum distance from the surface, turn around, and
return to the surface. Steady-state formulae will be applicable after time
ts only if the incident puisr changes only slightly during thL time trot
that it takes an electron of, say, average energy to return. Thus the
boundary layer will maintain a (quasi-) steady-state structure with the
instantaneous value of the flux only if the flux is nearly constant over
the lime trt.
Hence, for a given pulse time history, one must estimate both t
and t ret' We now discuss these two times more quantitatively.
20
A. Transient Build-Up to Steady State
Let c(t) be the X-ray fluence [cal/cmI2 that has arrived up to
time t. Then the X-ray flux is
= d- [cal/cm-/sec] (1)dt
Let Y [electrons/cal] be the photoelectric yield as given in Table I or
Figure 1. Then electrons are emitted at a rate
r 0 Y-) Ielectruns!cm-/;ec] , (2)
andt
N(t) fJr 0 dt [electrons/cm] , (3)
0
are emitted bN time t.
Now Reference 1 shows that, for an exponential electron energy
spectrum with exponentiation energy El and a cosO angular distribution, the
number of electrons required for steady state when the emission rate is r
is
Ns = (r= 0 V 3EI- .T e 2 ) 1/2 [electrons/cm] (4)
Steady state theory should begin to apply after the number of
electrons ejected, Equation 3, exceeds that necessary for steady state,
Equation 4. Hlence, equating N(t) and N will determine the time t sS 5
To simplify matters we assume the pulse is linearly rising,
= -- , (5)
at a rate p [cal/cm2/sec2], so that
21
N(t) y t~,2 ,(6)
2
and
"s " 3V'e 2 (7)
Equating these we obtain
1/3
2(
This is plotted as a function of [ [cal/cm /nanosec in Figures
8a, b, and c for aluminum, gold, and silicon dioxide.
For exampl , if a 3 keV blackbody falls on gold and has a rise
rate of 10"6 cal/cm /ns", then Figure 8b shows ts = 6 ns.
Actually some of the electrons emitted by time ts, Equations 3 or
6, will have returned so that the total number of electrons residing outside
the surface at t will be less than N(ts). Hence it will take somewhat
longer than ts as given by Equation 8 before steady-state theory is fully
applicable.
In addition, electrons emitted with energy less than E1 will be
in steady state before ts since their return time is less than that for
electrons of energy E . Similarly, more energetic electrons than E will
take longer to achieve steady state. Thus if one is interested in certain
details of the boundary layer that depend on high energy electrons, such as
the local electron number density far from the surface, one must wait
22
*-a. S,
ý4ca1/cm2 /nanosec2
100
0.i0
0 00V
10,
10-1 -
100
ý[ca 1/cm2 nnsc
Figure 8a, for Al uminum.
Numbers on curves are blackbody temperatures in keV.
23
$tfcal/m 2 /nanosec2]
100
11
1 0
ý4ca1/cm2 /nanosec2 I
Figure 8b. tsfor Gold.
Numbers on curves are blackbody temperatures in keV.
24
longer than ts before trusting steady-state expressions. Hence the precise
value of t as given in Figures 8 must be used with care.
Fortunately many X-ray pulses last mt,!'h longer than t, so that
even with the above two caveat-; Figures S are useful.
In Figures Sa, b, and c, and all later figures marked with two
pairs of arrows, the left-hand scale should be read with the lower scale.
and the right-hand scale should be read with the upper scale.
B. Maintenance of Quasi-Steady State
Steady-state theory with thci instantaneous value of X-ray flux ,il!
remain applicable if the flux is nearly constant during the turn around
time, t, of an electron of average energy. From Referen,.: 1, we find
that an electron emitted with normal component of energy 1/2 m-, where v• X X
is its normal velocity, equal to the average energy E,
1 2m v= E)
will return in time
t ret V .
= \2mVl/- e- r 0 0(lt)
where is the surface Debve length and vi = t. 11sing Luation.
this can be written
"ret - .1- 8 10 3 (C1 i ) -e aIl
26
BEST AVAILABLE COPY
This• is shown as a function of the flux 1 d:/Jt Ical/cm'/nanosecl in
Figures 9a, h, and c.
For example if a S keV blackbody is illuminating aluminum at an
instantaneous flux of 10) cal/cm /nanosec, Figure 9a shows that steady-state
exprtssions should he %jlid if the flux does not change imch over a time
t *O. 0.4 nanosec.
Since electrons emitted with energy greater than EI will have a
longer return time, one must use a larger t ret if one is interested in
detailed steady-state quantities that depend on these higher energy
electrons, such as the local electron number density far from the surface.
If. however, one is interested in a qiiantity that is determined primarily
by lower energy electrons, such a% the local numher density less than a
l•,,e length from the surface, one may use a smaller t.
For "global" or integrated quantities such as the surface electric
field, (or, what is the same thing, the surface charge density or the total
number of electrons outside the surface) the time tret plotted in Figures
9a, b, and c should he adequate.
From the given properties of the incident X-ray pulse one can
determine t5 and tret from Figures A and 9. The graphs in the following
sections will he valid for times greater than t if the flux is nearly con-
stant over a time t
ret•
27
X-ray Flux, ýIcal/cm2I/nanosecl
Aluminum -: 0
~10 101
10
X-ray Flux, lcl/n 2/aosc
Figure 9a. t ret for Aluminum.
Numbers on curves are blackbody temperatures in key.
28
X-ray Flux, ý[cal/cm2/nanosecI
io0 10
2: :~ 1.F
10
Figur 10 rtfo od
Numbers on curves are blackbody temperatures in keV.
29
X-ray Flux.i, Ncal/cm 2/nanosec]
S3ilicon Dioxide 101
I-
X-a lufca/m/nnsc
Fiu0 9c1rtfo iionDoie
Nubrtncre r lckoytmeaue nkV
30
SECTION 4
DEBYE LENGTHS
The L)ebye length is the characteristic distance over which the
properties of the boundary layer vary significantly. It is a measure of
the "thickness" of the boundary layer. If w is a characteristic energy
(ergs) of a plasma, and N is the electron number density (cm-3) then the
I)ebyc length isw
z D cm (12)
It is equal to a characteristic plasina velocity divided by the plasma
frequency.
In the present context we use for w the exponentiation energy E.
'rhis is also the average electron energy. For N we take the value at the
emission surface in steady' state when the electron energy distribution is
exponential and the angular emission distribution is cosý'• In this case
N = o (13)
where .v = Y21-E/m and r 0 is given by Ecuation 2. Thus
(16 Ti/2 e 2
1 (cm ns
31
These Debye lengths are those appropriate for the plasma at the
material surface in steady state including the returning electrons. Since
the local average electron energy divided by the number density increases
with increasing distance ftam the surface, Equation 12 shows that the locally
computed Debye lengths increase with distance from the surface.
These surface Debye lengths are shown in Figures 10a, b, and c.
For example, if a 10 keV blackbody is incident on silicon dioxide
at 0.8 cal/cm /ns, Figure lOc shows 9D % 0.1 cm. In this case the thickness
of the boundary layer, as nearly as it can lie defined, is at most a few
millimeters. The number density and electric field profiles, shown in later
sections, exhibit the actual structure of the layer.
32
X-ray Flux, *[cal/cm2 /nanosec]
100
10"1
107 0-6 10-5 10-4 10-30"
X-ray Flux, $(cal/cm2/nanosec1
Figure lVa. Debya lengths for Aluminum,
Numbers on curves are blackbody temperatures in keV.
33
X-ray Flux, ýp[ca1/cm2/nanosecj
101
~iol 1010
4j 4
.j 4L
20-
X-ray Flux, cý[Ca1/cm2/nanosec]
Figure l0b. Debys lengths for Gold.
Numbers on curves are blackbody temperatures in keV.
34
-- 9
SECTION 5
ELECTRON NUMBER DENSITY AT SURFACE
We treat the electron number density N(x) as a function of distance
x from the surface in two parts. We discuss first the density at the surface
N s N(x=O) [electrons/cm , (15)surface
and secondly the normalized profile N(x)/N(O). This permits some simplifica-
tion since the profile, when x is scaled to the Debye length, is a universal
function. The profile is given in Section 6. The density at the surface is
N 4. 4# o-surface v1
y(elec) calSCal
3.78 cmns [cm3 (16)VE1 (keV)
where vi a -E This is shown in Figures Ila, b, and c. It includes
both the emitted ard returning electrons.
For example if a 2 keV blackbody is incident on aluminum at a
flux of 3 X 104 cal/cm2/ns, Figure Ila shows Nsurface 9.9 x 109
electrons/cm.)
36
J /
SECTION 6
ELECTRON NUMBER DENSITY PROFILE
As mentioned in the last section we treat the electron number
density N(x) as a function of distance x from the surface in two parts. its
surface value, Equation 15, and its profile normalized to its surface value,
N(x)/N(O). If distance x is scaled to the Debye length, then N(x)/N(O) is
a universal function of x/Zo-
The density profile is shown in Figure 12 on a linear scale out to
10 Debyelengths, by which point it is about 0.01 of its surface value,
which is unity. This figure shows the true shape of the steady state
number density for an exponential emission electron energy spectrum and '
cosO angular distribution. It is taken from Reference 1. Its slope at
the surface is divergent.
The number density drops to 1/e of its surface value ia about
0.34 Debye lengths. At two Debye lengths it is less than 0.1 of its surface
va lue.
The profile is plotted again on a semi-log scale in Figure 13 out
to 50 1)ebye lengths to show the larger distance bchavior. As x
I
x (1.-)N(O) x"0
40
1.0
~ L L Linear Plot
NsuraceSee Figure 11, p. 371
.7 2 -See Figure 10,P. 33.. .... .
.6 V ~~ ~
.1... .t . .....
.41
:. hi ... ... ...
. . ... .. . Semi- og Plot
" • r • ; i ! • I ' ! • : J . . . I . .. , . . . . . . ..
0-l
Nsurface " See Figure 11 p. 377:, l Ii I:-See Figure 10, p. 33f ....
42
10
,. 1.1 . . .. ..
0 5 10 20 30 40 50
X/t
Figure 13. Normalized electron density profile.
42
INTEGRATED NUMBER DENSITY
The integral of the number density out to x, NT(x), gives the total
number of electrons out to x,
x
NT(X) = JN(x) dx [electrons/cm] (18)
0
Gauss' law
aE- = 4 = -p 4rcN(x) , (19)
where c(>O) is the mngnitude of the clectron charge, 1 is the electric field,
and p = eN is the charge density, can be used to express NT(X) in terms of
E(x):
N.r(x) = ECiI (1 - (x) (20)r4-,re E(O)
Here, E(O) is the surface electric field, and E(O)/4,r is the surface charge
density. NT() = E(O)/47c. The fraction of electrons out to x, NT(X)/
NT(oJ, is shown in Figure 14.
It shows, for example, that one half of the electrons are contained
in the first 1.5 Debye lengths. The first Debye length contains about 41
percent of all the electrons. About 88 percent are contained in the first
11 I)ebye lengths.
The surface electric field itself, E(O) = Esurface' and the electric
field profile E(x)/E(O) are shown in the next two sections.
43
I
61 1.0
'V. ~ T.T . . ~. .. ee Fi ur.1..
IJ; 4ý 4.
it 460 r-Figure 14. Froct ~iono~eetonuu ox
... ... .. ...... .44..
SECTION 7
ELECTRIC FIELD AT SURFACE
We treat the electric field E(x) as a function of distance x from
.the surface also in two parts as we did the density. The surface electric
field
Esurface = E(x=O) , (21)
is given here. It also determines the surface charge density
Esurface (22)
and the total number of electrons
NT = surface (23)
to which Figure 14 was normalized.
The surface electric field [volts/meter] is shown in Figures 1Sa,
b, and c.
For example, a I keV blackbody incident on gold at 1 cal/cm/
nanosec produces a steady stat- field of about 7.4 10' volts/meter
(Figure iSb).
4 S
UX-ray Flux, $(cal/cm 2/nanosecJ
10 Z1
ITTT
4J
10 10
4. 4-8I
* LAJ ~10' wo
X-ayF~~ $ca/cZnao3c
Fiur INb SufcUlcrcfl o od
Numersoncuresarebakoytmeaue nky
V1 547
X-ray Flux. O4ca1/cm /naoflsecj
-1-141
4 ..
10 1
X-ray Flux, $(ca1/cm2/nanosec;j
Figure 15c. Surface electric field for Silicon Dioxide.
Numbers on curves are blackbody temperatures in keV.
48
/
SS A
SECTION 8
ELECTRIC FIELD PROFILE
The electric field as a function of distance from the surface,
normalized to its surface value, E(x)/E surface, is shown in Figure 16.
As Equation 20 shows, this is one minus the total number of
electrons. It drops to l/e of its surface value in about 2.8 Debye lengths.
Together with the surface field from Figures ISa, b, or c, and the
Debye length from Figures 10a, b, or c, Figure 16 will give the electric
field at any distance from the surface in steady state.
49
S• /
SECTION 9
PLASMA FREQUENCY AT SURFACE
If N is the local electron number density (cm3), the local plasma
frequency is defined asIV= -- N [radians/sec] (24)
We work instead with the frequency
p 27T
/_ Hz . (25)
We use the surface value of N, Equation 16, to obtain the surface plasma
frequency
elec • cal 1/2
fp a 1.746 X 1 4O ( lcal t cmanls). Hz (26)V4E I(keV)
This is shown in Figures 17a, b, and c.
For example an 8 keV blackbody on silicon dioxide at 10- cal/cm2 /
ns produces a surface plasma frequency of 9 x 1 Hz.
"The local plasma frequency at a distance from the surface drops off
as the square root of the number density, that is, as the square root of the
ordinate in Figures 12 or 13.51
/ •.
X-ray Flux, ý[cal/cm 2/nanosec1
10 1
10 1
X-ra + lx ~a/c 2 nns
Fiur lc Srac p~rn reuec fr iicn ixie
Numbrs n crvesareblakbod teperture ink06
54
SECTION 10
DIPOLE MOMENT PER UNIT AREA
In Gaussian cgs units, the electric dipole moment of the layer
contributed by all electrons out to x is
xfP(x) = fxo(x) dx esu-cm/cm- (27)
0
Reference 1 shows this to be proportional to a universal dimensionless.function, .9(x/Z ) ), of x/2.D
1W(x) : e "(x/;[) esu/cm , (2)
where :1 (ergs) is the electron exponentiation energy w.hich was given in
kex in Table 2. If F1 is written in kcV, and l 1(x) in .IKS units, Equation
2S becomes
Coulombsl(x) = S.S54 0 E- I(keV) iP(x/)) meter (29)
Hence the quantity
r Coulombs0I (ke, ) L mter-keV3)
is a universal function of x/W. It is shown plotted in Figure 18.
To use Figure 18 for a certain blackbodv spectrum on a particular
material, first obtain EI- from Table 2, and the l)ebye length from Figure
1Oa, b, or c. Then to obtain the dipole moment out to x, multiply EI(keV)
by the ordinate in Figure 1. at the correct value of x/Zý.
55
-83xl 0 ...
I0 4J L 1
* 1 __ i 1[ 4
it " _ ..._
.1F - See Table 2, P. l10..
- ,.:>: :2: - See Figure 10, p. 33:-K
S....... . . . . . . . . . . ,. , -_i~ i
0 5 1 0 15 20 2b 30 35 40 45 50
Figure 18. Normalized dipole moment per unit area.
For example for a 10 keV blackbody incident on gold at 101I cal/2
cm /nanosec, Table 2 shows E1 = 10.07 keV, and Figure l0b shows 9.. = 0.07 cm.
Then Figure 18 shows that electrons out to, say, x = 1 cm (x/9.D = 14.3),
-8 D
where the ordinate is 1.5 x 10-8, contribute a dipole moment of
P(x=l cm) = 10.07 x 1.5 x 018
U-7
= 1.51 x 10 Coul/meter . (31)
Estimates such as this for the dipole moment are useful for deter-
mining distant electric dipole field3 produced by the layer.
The dipole moment diverges for large x. This is because the
charge density p in Equation 27 drops off as x -2 so the integral diverges
logaritemically. The electrostatic potential also diverges. The divergence
is due to the exponential spectrum which contains electrons of arbitrarily
high velocity, and is a theoretical artifact.
56
miigdsateeti ioe ilspoue ytelyr
Th ioemmn iegsfrlrex hsi eas h
II
To obtain a meaningful estimate for a real situation the following
procedure is suggested. At the time P(x) is desired, estimate the maximum
distance the most energetic electron traveling at less than the speed of
light could have gone. Determine P at this distance from Figure 18. Since
P increases slowly with distance, remaining of order a few times 10"a
Coulombs/meter for many I)cbye lengths, the exact distance x is not too
critical for rough estimates.
To convert P(x) to esu, use
I Coulomb ,3 1 0 7 e.umeter cm
57
BEST AVAILABLE COPY
SECTION 11
EXAMPLE
We discuss an example here to illustrate the use of all graphs.
We consider as a. example a S ke\' blacuhody s5wctrum incident on
aAhjinum with a time history a- shown in Figure 19, antl a fluenc, of I
eal/c'. This information, the incident spectrum, the target material. the
time history. and t'he fluence, we asuin•e given.
Irrespective of the time historyv and fIuence, rahbl I (or Figure I)
shows the hackscattered electron yield to he
iY .elctrun.sY * 2.$7 • J •'calorie (32)
Figure 5 shows. the approximate electron energy spectrum (for electron% of
energy greater than I kv\'. and i'ahle 2 shou- Chat the spectrum is nearly
exponential with exponent iat ion energy.
I. 1a -4.7' keV . 433)
The emission angular spectrum (per steradianl is approximately propoirtional
to coA, where -1 is the emissiun angle mcasured from the normal.
To estimate the time t after which we may use -tteadcy state theory
(and therefore all the graphs in this revtort,, we need at early times.
This is ohtained from Figure 09.
52
o00
The maximum flux, 0mis determined from the fluence by
-f ; dt0
1*1 •*mt2
so that
a " a 6.67 x 10-2 cal/c,2 ns . (34)t 2
Then
m 1 - - 6.67 x 10 cal/cm 2ns (3S)ti
From Figure 8a we determine
t 0.9 ns . (36)
Thus after about one nanosec we can expect to be in quasi-steady state.
To see if we sensibly remain in instantdneous steady state, we must
have the flux change only slightly in the time t et, determined from Figure 9a
and an average $ of, say, *m/ 2 • 3 x 10. 2 cal/rnm2 /ns. Figure 9a shows
tret a 0.28 ns . (37)
During these times, Figure 19 shows the flux does change only slightly, so
we may expect sensible estimates from all remaining graphs.
The Debye length, Figure 10a, shortens in time as * increases, and
then lengthens again as $ decreases after its peak. Its smallest value is
when • ' Equation 34, for which Figure 10a shows
60
.. /4
mD,inimum 9.5 x 1-2 (38)
Th.e surface number density increases with * to a maximum of
Nfae,max 3 X 1011 elec/cm3 (39)
when . M , as shown by Figure Ila.
The number density profile is shown in Figure 12 or 13, and the
fractional integrated number in Figure 14. Figure 14 shows 75 percent of
the electrons inside 5.2 Debye lengths, or within about 0.5 cm at time
t 10 ns.
T?.e surface field increases as the square root of • to a maximum of
E = 4 x 10 v/m, (40)max
at 10 ns, as read from Figure 1Sa.
The electric field profile is shown in Figure ib.
The surface plasma frequency increases as the square root of the
flux reaching a maximum at 10 ns of
f xa 5 X 09 HN, (41);, ,max
as determine,' from Figure l7a.
The dipole moment out to 50 Debye lengths (v 4.8 cm using the
minimum Z, Equation 38) is shown in Figure 18 to be about
P % El(keV) x 3 x 10.8
- 1.4 - 10.7 Coulombs/meter , (42)
which is ahout 4.3 esu/cm.
61
BEST AVAILABLE COPY
REFERENCES
1. Carton, Neal J., and C. L. Longmire, On the Structure of the Steadyv-State, Space-Charge-Limited Boundary Layer in One Dimension, MissionResearch Corporation, MRC-R-240, November 1975.
2. Dellin, T. A., and C. J. MacCallum, QUICKE2: A One-Dimensional Code forCalculating Bulk and Vacuum Emitted Photo-Compton Currents, SandiaLaboratories, SLL-74-0218, April 1974.
3. Higgins, D. F., X-ray Induced Photoelectric Currents, Mission ResearchCorporation, MRC-R-81, June 1973.
62
Recommended