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IONIZATION AND RADIATIVE CORRECTIONSTO
ELASTIC ELECTRON SCATTERING
John Alexander Gordon
UnitedNaval Postar
StdtSSduate Sc
rin T T T7^
IONIZATION AND RADIATIVE CORRECTIONSTO
ELASTIC ELECTRON SCATTERING
by
John Alexander Gordon
June 1970
Thi6 document hcji b^e/i ap.yovtd ^OK public kq,-
tzci6e, and loJit; i/J> dAJitAi.bwtion iM ujtUjrUXe,d,
1 J4bU--
Ionization and Radiative Corrections
to
Elastic Electron Scattering
by
John Alexander pordonFirst Lieutenant, United States Air Force
B.S., University of Missouri, 1968
Submitted in partial fulfillr-.^nt of therequirements lor the dc-gree of
MASTER OJ^ SCIENCE IN PHYSICS
from the
NAVAL POSTCFvADUATE SCHOOLJune 1970
11 I ' ' »^—*^»^i^—^"l^—WW 0mmiimmmmmtimmmmtmm^mf^'^»''mi^mmmtif^'mmmm^
NAVAL POST..:;-.
MONTEREY, CALI.^ .
ABSTRACT
The theory of the ionization and radiative corrections to
electron scattering cross sections has been reviewed to determine
the formulas most valid for the experimental arrangement at the
NPGLINAC. Experimentally it has been determined that the cor-
rections most nearly account for undetected electrons when the
lower limit of the measured scattering spectrum is at least four
half-widths below the peak of the spectrum. It has also been
found that the corrections, most likely the Bethe-Heitler cor-
rection, induce a positive error of several percent to the cross-
section when thick targets are used.
TABLE OF CONTENTS
I. INTRODUCTION 9
II. IONIZATION CORRECTION 12
III. RADIATIVE CORRECTION -19
A. SCHU'INGER CORRECTION 19
B. BETHE-HEITLER CORRECTION -24
IV. CHOICE OF AE 28
V. EXPERIMENTAL CONSIDERATIONS 30
A. COUNTING SNSTE?4 30
B. EXPERIMENTAL METHOD 33
1. Basis of the Experiment 33
2. Inlernal Consistency of the Data 33
VI. DATA ANALYSIS - 36
A. DETERMINATION OF CROSS-SECTION 36
B. EXPERIMENTAL ERROR 37
VII. RESULTS --- 39
A. CHOICE OF AE 39
B. ACCURACY OF THE CORRECTIONS 45
VIII. CONCLUSIONS - - - 51
APPENDIX A: Bremsstrahlung 52
APPENDIX B: Secondary Emission Monitors 57
APPENDIX C: Scattering Within the Spectrometer 61
LIST OF REFERENCES--- 65
INITIAL DISTRirsUTION LIST 67
FORM DD I473-- --- ---69
TABLE OF SYMBOLS
A atomic weight
d^J/dCl differential cross section
E, incident electron energy
E^ energy of the elastic peak (most probable energy of thescattered electron spectrum)
E, E, + M - E„4 1 3t
E lower limit of the electron spectrum3
K. ionization correction1
M mass of the scattering nucleus
T target thickness in radiation lengths ("effective"thickness)
2X radiation length (g/cm
)o
atomic number
c velocity of light
2m,mc rest energy of the electron = 0.511 MeV
p^p^ = 2E^E^ sin^(Q/2)
2'^
P4 = (^4 - M)
q momentum transfer
2t target thickness in g/cm ("effective" thickness)
r classical electron radiuso
r jF half-width of the scattered spectrum
AE - E^ - e!
AE resolution of the incident electron
AO angular spread due to finite width of entrance slit
scattering angle
$(x) Spence function
a = 1/137.04
b Bethe-Heitler correction
b Schwinger correctionS ^
5 Wheeler -Lamb factor
17 recoil factor
77 jT) factors defining F as a function of AE^ B
3p target density (g/cm )
ACKNOWLEDGEMENTS
The author wishes to express his appreciation to Associate
Professor Fred R. Buskirk who conceived of this project and pro-
vided invaluable assistance throughout the experiment. The author
also wishes to thank Professor Franz A. Bumiller for the use of
unpublished data as w.-ll as several valuable discussions, and
Professor John N. Dyer for his continuing support.
LCE*?. Jake W. Stewart provided the author's initial instruction
on the operation of the LINAC and assisted immensely throughout
the experiment. CAPT. Louis Gaby provided many of the important
data reduction programs.
I . INTRODUCT ION
Linear electron accelerators (LINAC) are well suited for the
study of nuclear structure. The interaction between a nucleus and
a high energy incident electron, which is entirely electromagnetic,
is thought to be well understood. By momentum analyzing electrons
of initial energy E which are scattered through an angle into a
solid angle dCl, one can determine the differential cross section
for scattering, dcr/d^. This cross section can yield much infor-
mation about the charge structure of the target nuclei.
Hofstadter Ll,2] has written a full review of the theory and tech-
niques of elastic electron scattering.
It is well known that these high energy electrons radiate a
portion of their energy (br emsstrahlung) in the presence of nuclei
or other electrons. This br emsstrahlung leads to a broadening of
the scattered electron distribution such that there is a finite
probability of scattered electrons with energy approaching zero.
A broadening of the distribution is also due to energy loss by
ionization. Since in most experiments it is not feasible to mea-
sure the spectrum down to zero energy, there is a lower energy
limit to the measured spectrum. It is thereff-re necessary to cal-
culate and apply corrections to the experimentally determined cross
sections to account for the undetected low-energy scattered
electrons
.
As the precision and accuracy of scattering experiments
improve, these corrections become more and mor u significant. If,
for example, a ten percent correction were only known to ten
percent, this would result in only a one percent contribution to
the uncertainty in the experimental cross section. But obviously
such uncertainties are not acceptable when attempting to do an
experiment to one percent accuracy.
This paper is concerned with three distinct corrections. The
first of these corrects for energy loss due to Landau straggling
(ionization). The other two are properly called radiative cor-
rections. The Schwinger correction compensates for bremsstrahlung
in the field of the target nucleus (scattering center) while the
Bethe-Heitler correction deals with bremsstrahlung in the field
of atomic electrons and other nuclei in the target. The cross
section for the first process is proportional to the target thick-
ness while for the lattt •: it is proportional to the square of the
thickness
.
The following sections include a review of the theory for each
of the three processes, as best suited for application at the Naval
Postgraduate School LINAC (NPGLINAC) , that is, for scattering ex-
periments in the 30 to 100 MeV range where the scattered electrons
are detected and momentum analyzed while the recoil nuclei are not
detected. The NPGLINAC is described in detail in theses by Harnett
and Cunneen [ 3J and Midgarden L4].
It should be noted that in actual applications the corrections
are multiplicative corrections to the experimentally determined
cross sections. However, to be consistent with most publications
in this field, the sections concerned with the theory use the form
/exp
dfTl /dCT \
10
where K., S , S are the corrections,( "TTy ) is the corrected cross
section while / - /s-J
is the experimentally determined raw
cross section.
Since the Bethe-Heitler correction is proportional to the
square of the target thickness, its validity can be checked ex-
d aperimentally by measuring "T(T on targets of different thickness.
Such experiments wore performed and the results are reported. The
proper applications of the corrections, for example, the choice of
lower energy li..iit of the spectrum, have also been examined.
11
I I . IONIZATION CORRECT TON
As a monoenerget ic beam of electrons passes through a scat-
tering target the resultinr; spectrum of scattered electrons is
broadened, in part, by ionization loss, or Landau straggling.
This straggling induces an error in the cross-section determination
because the scattered spectvum is not normally measured to zero
energy. Thus a correction should be applied to account for these
undetected electrons. This correction has largely been ignored
at the NPGLINAC and other accelerators. For targets and energies
encouiitered at the NPGLINAC, the ionization correction is typi-
cally about one percent. Until recently the accuracy of cross-
section measurements was insufficient to warrant the inclusion of
such small corrections. Since many cross-section measurements are
made relative to another nucleus, the correction is even less
important. A review of this correction is provided for complete-
ness and possible future use.
Since in electron scattering experiments it is neither desirable
nor feasible to measure the scattered spectrum down to zero energy,
there is a lower energy limit of the detected spectrum, E , so that
AE = E„ - E„. This is illustrated in Fig. 1. Thus a fraction of
the spectrum, V(AE), is lost by ionization. Then the ionization
correction can be xpccted to have the form:
K. = 1 - V(AE) .
This is applied to thr- exper imc^ntal cross-section:
(da\ -1 _ daVdTT/ ^i dTI •
V / exp
12
Landau [5] fii^st calculated the energy distribution of
scattered electrons when monoenergetic electrons are incident on
2a target of thickness t(g/cm ). Following this theory, the
fraction lost can be calculated by integration of the Landau
function 0{ AE)
,
V_ (AE) = 1 - ^(AE)
E , / Eo / f o
^(AE) = \ 0(AE)dE/J
0(AE)dE.
A graph of v (AE) is given by Landau and also by Symon [6] and is
reproduced as Fig. 2.
V (AE) has been approximated by Isabelle and Bishop [7] who
have derived an expression for the correction, K ,
K = 1 - 1^21__aj2^L
^£,11
' ^^ ^^
where a = 0.1537-Z.p/A- p MeV cm g".
The above equation has regions of applicability which must be
observed:
a. When AE is of the order of a-t or smaller, the
approximation of v (AE) is unsatisfactory.
b. For very thin targets the Landau theory is only a
zero-order apj^r oximation . This treatment neglects
fluctuations due to distant collisions, in which the
atomic electrons cannot be treated as free electrons [8].
To improve the Landau cor re :tion, Breuer [9] used V (AE) as
given by Landau rather than the Bishop- Isabelle approximation. He
also incorporated the theory of Blunk and Leisegang [ lO] since this
13
Fig. 1.
Typical elastic spectrum
l.Or
0.8
V t 0.6l'
O.Zi
0.2
10
VL
Fig. 2. V as given by Landau.1^
14
shows better agreement with experiments than the Landau theory.
Since the calculations of Blunk and Leisegang introduce only a
broadening of the distribution, the Landau V (AE) can be used by
substituting the Blunk and Leisegang values for the full width at
half maximum (half-width), T Breuer has plotted this half-widthB
2as a function of b , where
b^ = q Q z'^'^V(a-t)
q = 20 eV
Q = Q,pt = mean energy loss for thickness t
— 2Q = mean energy loss per g/cm .
2The plot of r /(a*t) as a function of b is given in Fig. 3.
B
Q can be calculated '.j>y an empirical equation given by
Sternheiraer [ll]:
^*^ A r« ^/ Tin -^
Q^ = ~2 t Lb+0.43 +^ E^-P + C -a^(X-log^Q p/mc) jMeV. (2-2)
The empirical constants for this equation, as determined by
Sternheimer , are given in Table 1 for selected targets.
The abscissa values of Fig. 3 are defined as T) ,
?7 = AE/l.5l-a-t .
The Landau relation for the half-width,
can be substituted into the above relation to give
V^ = AE/(0.379 r^).
15
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wa
oro (/)
(T3
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-H
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CN] 'O•H^
^rHfO
o K
ro
•H(I.
(i.v;.^j
16
<
C\UQ)
CQ)
q
e
co•H
N•Hco•H
+->
{«
rH
oiH
U
o+->
TJQJ
W13
(fl
C
V)
cOu
u•HVI•Hae
vt
e•H
c
q;
+j
0)
X
uI
•HH
2
(N CM CN) rOrofOro<t-<rc^
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r- CO CMO 00 CM CM
ro CM ro <r <f 0> o o
On 00 OO VOrH r-l iH iH
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q;
H•H,c
arj
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00
COCM CM coco-crcofOrocM
vO-T CO rH o ON m m o ror-- rH m 0^ tH O ro o^ CO C7N
fO ^ in O rH lA r-< rH rH rf")
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<}- o 00 00 o <rt^ lA rH VO 00 CJN
CM
ro<fiAr\ONl>C7N-:ft^iA\0 "^ CMl^OO C^CJnvOOn
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t^ 00 OO VOVO vD \0 I>>
r^ r-- i> 00o o o o
uCMX X
CM
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CQ)
HQ)
4->
Q>
CM Oo a.
17
Now the Landau half-width should be replaced by the larger and
more correct Blunk and Leisegang half-width, F •
B
For relativistic energies and T} > 2, v(AE) falls off inversely
with AE. From Fig. 2 it is seen that V {rf = 10) = 0.08, and thus
1 - v(r/ = 10) = .92 .
Hence the ionization correction can finally be written as
0.8K. = 1 -
^B
~ ^ 2.65 AE• ^^^^
To apply the ionizai ion correction it is first necessary to
2 . . -n .
determine b for the particular exy^eriment. 1 is then determined
from Fig. 3 and substituted into the expression for K..
2In most experiments at the NPGLINAC, b -<.0.5 so that the
Landau relation can be used:
r^B/(a-t)= 4 .
Then
so that
T?g = .6625-AE/(a-t),
1 .66^5 AE \ ^J
18
III. RADIATIVE COR KT.CT IONS
The radiative corrections to elastic scattering are of two
types. The Schwinger correction, called simply the radiative
correction by some authors, accounts for the emission and re-
absorption of virtual photons in the field of the scattering
nucleus, nuclear br emsstrahlung , as well as the emission of soft
photons of energy less than a specified cutoff energy, E_. The3
Bethe-Heitler correction, also called the radiation tail, accounts
for the emission of real photons, by br emsstrahlung in the field
of atomic electrons and in the field of nuclei other than the
target nucleus.
A. SCHWINGER CORRECTION
There are sev.:r 1 version: of the Schwinger correction, all
of which are improvements on the original correction first given
by Schwinger [12]. His calculations include the following
assumptions
:
a. stationary nucleus
b. Coulomb point field
c. first Born approximation
d. one-quantum emission
e. p = v/c ^ 1
with the restrictions
2a. AE<< E - m c
2b. E sin — / m c >>1.
19
The Feynman diagrams corresponding to Schwinger's calculation are
given in Fig. k. This correction, b , is applied as
dCT
dTTexp
dCT
dTT1 - b
where
{^ AE13wr ^^o . 1. 17 1 .20 ^,^,_^)(^. _. s.n-- -)+—-^^ sin -f(0)
This can be put into a more convenient fori;: [13]^
2X _ 2Q:
J //7 AE 13> ,.
^S -irn^-F - l2)(^ 2^ mi)*^-\fw\ ,
where
f(Q) =i&7?(sin — )^(cos —) +$(sin~) .
is the Spence function [ l4]
<f(x)r =^
_ toll-yl dy .
To simplify calculations this is usually seen as
2
S n (_^^ AE 1
3\ //?^ Z3. 1712) (^-r- 1) ^3^5
ra
(3-1)
$(x)
$(1)
$(x)
$(x)
12 13 / x"= ^ ^ 4
X ^ 9x^^--- -^y-
^ n '^
= -g n and $(-i) = - ^ tt
= -1^' |x| -in2-f(i)
if |x|^ 1;
if x > 1
if X < - 1.
20
4^ /N
\
(
A A /v
'"V-'-^v^-v/^ _{
Fig. 4. Feynman diagrams corr esyionding to theoriginal Schwinger correction.
A A A A
^:^Q'v^-
(
A
)
>(^
^
A A
Fig. 5. Feynman diagrams corresponding to theformulas of Meister and yennie and Tsai.
21
since this calculation was made for potential scattering, the
energy of the final and the incident electrons are the same, and the
formula is ambiguous as to the energies to use and the definition
of AE. If we neglect all energy losses except recoil, the most
probable energy of the elastically scattered electrons is
E3 = E^/r? (3-2)
where 77 is the recoil factor
-17] = 1 + EM (1-cos 0) . (3-3)
It can be shown that the maximum energy of a photon emitted
in the direction of the scattered electron is AE> while in the
2direction of the incident electron the maximum energy is r/ AE.
To account for this Tsai L13J gives an improved formula:
S TT )U ^2^^ 2 AE 12i0n (
m̂
)
1136 r
' (3-4)
Note that when AE ~* all of the above equations are infinite,
although we would expect the measured cross section to go to zero
as AE ~* 0. This is apparently due to the fact that multiple
photon emissions have been neglected. Hence Schwinger suggested
that (1 - b ) be replaced by e .It was later proved [iSl that
the term
20!
^ AE^
should be exponentiated. It is still uncertain [16] if the other
terras> iiould be exponentiated, but the difference is negligible
at energies obtainable at the NPGLINAC.
22
Meister and Yennie [l6] give an improved version of th
Schwinger correction:
.ll^(!y3j.i^2^.|
my ^ \ m .
2 \ m / 2 \ m(3-5)
where p is the step function: p(x) =^(x)0(l-x).
This calculation is described by the Feynman diagrams in
Fig. 5« The detailed assumption and restrictions for this
calculation are given b Breuer [l?], and are sii^ilar to those
of the Schwinger calculation.
The latest and most accurate version of the Schwinoer cor-
rection is given by Tsai [ l8] . This is essentially the Meisler
and Yennie formula without approximations for the Spence functions.
This formula takes into account both target recoil and a dynamical
effect due to photon emission. Two-photon emission still has not
been sufficiently extracted except in order to cancel infrared
divergence in real photon emission. The assumpt ic ns , restrictions
and Fc'vnman diagrc-ns are the same for the Tsai formulation as for
the Meister and Yennie formula. Tsai's expression for the
correction is:
23
b = ^/^ - l^r^fzR^uL :iH_ . i4.2z^T])|2572^ -3^T]AE
'^J±]-A ^
1 + P/ \ 2^ l+R, E, +M
+Z
-z
^^Elp^ 1-p^ P4(2 i-p^
M-E„\ /M(M-E„) \ /2E,^(M-E_)
C ^3 y v"^iV^^3) A'^iV^^3/
3 ^ 1
E^(M-2E^)^1
' M2E,
5^1
2E,E, -ME^1 H 3
E2(M-2E-j^)^H
2e\
-z
+z
M-E \ /M-E A /2(M-E )\^* "' L '+$1 L- + ^
1
M-E.
MM
2E -M
f M2E.
. ^3 / \ ^3 / ^ ^ /
(3-6)
B. BETHE-HEITLER CORRECTION
The Bethe-Heitler correction accounts for energy loss by
br emsstrah] ung in the field of nuclei other than the target
nucleus, as well as br emsstrahlung in the field of atomic
electrons. A review of br emsstrahlung , screening, and the
resulting cross-section formulas is given in Appendix A.
Bethe and Heitler [l9,20,2l] have shown that if the cross-
section for br cnsstrahlung were given by
dpd
A -1
^2 X No
[e^^Ce^/e)]'-", (3-7)
2k
the probability an electron of initial energy E being in an
energy interval dE at E, after traveling thi- pugh a target of
thickness T (in units of radiation length), would be
E^ [^(E^/E)]I(EET)=-^= . (3-8)^
r{T/0n2)
Integration of this formula, with the assumption that T is small,
leads to the original Bethe-Heitler correction, first given by
Hofstadter as
B ^2 AE
To include the effect of target recoil, this should be written
T ^1b = - r-i 2n i~~-^
. (3-9)B S/n2 ^ 3/2 - vo ^;
.onBetho and Heitler admit that Eq. (3-7) is a poor approximatii
to the correct cross section and was chosen for its simplicity.
To improve u; :>n the correction it is necessary to use a more nearly
correct expression for the cross-section. However, it is also
necessary to be able to perform the integrations.
Tsai [l8] has developed a unique method for the case of
complete screening. By combining Eq. (3-7) and Eq. (3-S) for
the case of small T, Eq. (3-8) becomes
Ie(El''^>T) =(-A VdEJ^TJ • (3-10)
T/2'2 2Tsai shows th:-, > the term {2n E,/E) " is a correction for
multipl'^^ scattering which is insensitive to the choice of the
cross-section formula. Hence it would seem appropriate to replace
25
dCT/dE in Eq. (3-10) by a correct expression, thereby immediately
obtaining the intensity I . By adding the exact cross sections
for nuclear and atomic electron br emss trahlung , the total cross
section for one-photon emission and complete screening can be
written (with a cross terra of < . 5% deleted)
dadE
dadE
nuc
dadE
elec
o N J \^ 1 /1 +
9E.
Z+1Z+§
1-1
^(l83Z
where X is the radiation length given by Bethe and Ashkin [22]
k' = (t)"'oZ(Z * ?)««(l83Z
Thus Eq. (3-8) becomes
I^(E^,E,T)= bT(E^-E)-1
(3-12)
where to a very good appr oxinationn -1
Z+l)/(Z+§) mi83Z ^ (3-13)
and 5 is the Wheeler-Lamb factor.
From this, Tsai evaluates the correction under the assumption
that the scattering event occurs midway through the target. He
also incorporates energy loss in an entrance window of thickness
T. and a final window of thickness 1" to derive the formulaiw fw
([6„ =< b T. + ^ bTB / w iw 2 HE^/l^hE) +
[ V^^+l bxl ^^^(E^/AE) |. (3-14)
26
In examining, for example, the proton scattering in a solid target
such as (CH., ) , br emss trahlunq from the carbon in the target can be^ 2' n' ^ ^
accounted for by substituting the effective thickness of the carbon
for the window thickness.
The expression for 5 as derived by Tsai is slightly moreB
accurate than Eq . (3-9) and certainly more versatile. It is,
however, derived on the assumption of complete screening, an
approximation which begins to fail at energies attainable by
the NPGLINAC. Unfortunately, the magnitude of the errors induced
is, as yet, unknown. A more correct theory would require use of
a cross-section formula which includes the effects of "irjtermediate"
screening. These formulas, which include atomic form factors, are
not known ei^ii-'licitly and determination of the correction would
require a double numerical integration, with no hope of an ex-
2plicit expression for the Bethe-Heitler correction . There was
insufficient time to pursue this idea further for inclusion in
this paper
,
2Incorporating the effects due to screening also means
re-evaluating the Wheeler -Lamb factor which in turn implieschanges in the radiat on length.
27
IV. CHOICE OF AE
The shape of an elastic spectrum near the peak is largely due
to the energy spread of the incident electron beam. A broadening
of the peak is also due to the finite width of the entrance slit
as well as the ionization and radiative corrections. The cor-
rections should be able to account for electrons lost by ion-
ization and by radiation for any reasonable choice of AE. However
AE must be chosen sufficiently large to include the energy spread
due to the tivo machine effects.
Tsai [13] gives requirements on the choice of AE. If the
incident beam has an energy spread of AE , the equation
E^ --• E^/17
is used to calculate the energy spread of the ^^cattered electrons:
bE.3 \
-2
1&e: I ^'h = '^^'^
77 is the recoil factor given in Eq. (3-3). Similarly Tsr i shows
that the energy spread due to the finite width of the entrance
slit is
(b?7 '^ =( ^ '
"" ^"^
Hence AE should be chosen such that
AE>>AE^rj" ,
and2
AE > (E^ / M) sin OaO.
28
In the development of the Tsai formulation of the Schwinger
correction there is an assumption
E(l + 2E, /M)<< E_.
This puts .n upper limit on the choice of AE, but is not a
severe restriction.
29
V. EXPER IMENTAL CONSI DERAT TONS
A. COUNTING SYSTEM
The design and operation of the NPGLINAC has been described
in detail in several theses L3>4]. The most significant recent
modification has been the installation of a ten-channel counting
system. Ten plastic scintillators have been positioned vertically
in the focal plane of the l6 inch double-focusing spectrometer.
The spectrometer is described in a thesis by Oberdier [23].
Electrons which enter the spectrometer with an energy (more
properly, momentum) corresponding to a given spectrometer setting
follow a path through the spectrometer such that they focus at
the intersection of the focal plane and an extension of the
central path of the spectrometer. Electrons of greater enerc:y
are focused above this point in the focal plane while less
energetic electrons are focused at a lower point. The distance
from the central point is given roughly by the dispersion formula
\-{f)-^- (5-1)
where
r = central radius of the spectrometer = 16 inches
D = dispersion constant = 3.92
I —J-\ = momentum difference from central focusing point.
{")
The ten scintillators are arranged above and below the
central point (channel five very nearly corresponds to the
central energy) so that each one "sees" a different energy.
30
Immediately behind the front counters is a large vertical backing
scintillator. To minimize the detected background due to the
intense radiation field in the accelerator end station, only
electrons which pass through both one of the front counters and
the back counter in coincidence are registered as scattered
electrons. The coincidence system is illustrated schematically
in Fig. 6.
To get a reasonable density of points defining a scattering
peak it is necessary to change the central energy of the spec-
trometer in small steps of one-third to one-sixth the average
counter separation (0.10 to 0.05 MeV at E = 90 MeV) , as well as
one or more large steps ( ^-^Z.S MeV at E, = 90 McV) to define the
radiation tail. This requires a rather involved data unfolding
procedure before a usable spectrum is produced. We follow
closely the procecl -e given by Suclzle l2Z+,25J. The energy of
each < ounter as a function of the central energy has been deter-
mined by plotting, at several incident energies, the elastic
peak as it occurs in each channel as a function of the spectro-
meter setting. The resolution of the i-th channel is defined as
1E1 2 ^ 1+1 1-1 '
The efficiency of each channel is determined for each run by
taking at least five overlapping data sets on a smooth portion of
the radiation tail and fitting these points to a second order poly-
nomial. The efficiency oi each channel is then found from the
difference of <he data points and the polynomial fit.
31
Counter Hous.
II
ChroneModelDiscr ii
ics01inator
EG&GC102ATvi'o-f
Scintillator
Photomult ipl ier
(Amperex XPlllO)
Preamplifier
HewliPackaScale]
EG. GFanout
Hewl i
PackaSea] e::
Fig. 6, Count iiig Systca
32
B. EXPERIMENTAL METHOD
1 . Basis of the Ex]oer iment
The purpose of this experiment is to examine the adequacy
of the radiative corrections as a function of target thickness, and
the importance of the cutoff energy, E , to the corrections. This3
leads to the straight-forward procedure of determining the cross
section for several thicknesses of the same target material at a
given incident energy and scattering angle. Since we are only
interested in the relative differences between targets, it is
sufficient to determine the area under the peaks. If the cor-
rections are correct, the ratio of tl^e corrected area to the
target thickness will be constant. Similarly, for any single peak,
the corrected area should be independent of AE, provided the
conditions discussed in Section IV are met.
These corrections have been examined for six thicknesses
of graphite, p = 1.441 g/cm (.028 to .250 inches) and seven
thicknesses of aluminum, p = 2.543 g/cra (.0104 to .1354 inches).
Incident energies of 35, 55 and 90 MeV were used. The scattering
angle was 90 degrees in all runs. The dependence of area on AE
was examined on several targets at different energies.
2. Internal Consistenc y of the Data
In this work it was necessary to measure the area
(counts) of an elastic peak but noi necessary to actually de-
termine the cross section. Therefore errors in the measurement
of the absolute val'-es of experimental constants (incident energy,
33
scattering angle, target angle, solid angle, integrator capacitance,
SEM efficiency and others) do not enter into these relative experi-
ments. However, it is important to insure that none of these fac-
tors vary during a data run.
To avoid effects which may be due to the tuning of the
accelerator, such as variations in incident energy, energy reso-
lution of the beam, and spot size of the beam, it has not been
attempted to compare different runs. A run, consisting of a
spectrum from each target, was completed without retuning the beam.
It is felt that the deflection magnets, which define the incident
energy, drift less than 0.1% over several hours, while short term
variations are undetectable.
To avoid errors due to hysteresis in the sp otrometer, the
magnet current, and hence the central energy, is monotonically
decreased during the run to span the peaks. Because of drift in
the current regulated supply or temperature change in the magnet,
the field of the magnet is not exactly proportional to the current
in the windings. Hence the field is sensed by a r otat ing-coil
fluxmeter , and maintained at a specified value by a balancing
circuit. The field is held constant to approximately ^.005 MeV
in the central energy. The precision potentiometer used in the
balancing circuit to define the field value is linear to 0.1%
throughout its range.
It is critical that the SEM ef a iciency and its associated
beam current integrator be constant over the time of the experi-
ment. Since the beam intensity must vary during a run to account
3^
for the difference in target thickness, the efficiency of the
SEM must be independent of beam current, or a correction must be
known. This is fully discussed in Appendix B.
There is some question as to the validity of the efficiency
determination for the individual counters. To minimize such errors
all peaks have been taken so that the top of the peak is defined
entirely by channel seven. Thus the efficiencies, even if in-
correct by several percent, are applied similarly to all peaks.
With the prt?vious one-channel system it was determined
that a counting rate cc^rrection of the form
C = G [l.O + 0.003C/t]
where
C = corrected counts
C = observed counts
t = integration time in seconds
should be applied to the observed counting data. This same cor-
rection has been applied to data taken on the ten-channel system,
although it is now apparent that this is too large. However,
roughly the same counting rate, ten counts per second, was main-
tained when running each peak, so that errors will again cancel
out. It is estimated that accidental coincidences do not con-
tribute more than a 0.1% error to the cross sections.
The most persistent problem encountered during this experi-
ment was the detection of an excessive number of electrons in the
upper channels when counting on a radiation tail. This problem
was partially solved by simply deleting obviously bad points before
the numerical integration of the peak. A discussion of this problem
ill be found in Appendix C.
35
VI. DATA ANALYSIS
A. DETERMINATION OF CROSS SECTION
The data from an electron scattering experiment is in the form
of a counts versus MeV (p^roperly MeV/c ) spectrum of the scattered
electrons. This spectrum is unfolded from the raw data by the
procedure mentioned in Section V. To find the actual number of
scattered electrons it is necessary to first divide out the
momentum acceptcxnce of each counter, Ap . To a very goc J approxi-
mation this is given by the dispersion formula
AP = 1^ (6-1)
where now a is the effective height of each counter, approxi-
mately 7/32 inch. The resulting spectrum is thus one of counts
per MeV versus MeV, so that if the area of the peak is measured
it has units of counts, the number of scattered electrons. The
number of scattered electrons per millivolt integration of the
incident beam is proportional to the cross section.
Determination of the area was by numerical integration of the
unfolded spectrum. A three-point method, similar to Simpson's
Rule, was chosen. Since the data points were not equally spaced,
it was necessary to exactly fit each set of three points to a
parabola, and to explicity perform the integration under the
curve. It is felt th' t this method is superior to the cominonly
used trapezoidal rule.
Once the areas have been determined, it is a simple procedure
to apply the corrections and normalize the areas to a given tar-
get thickness. All the corrections except the most complicated
36
of the Schwinger corrections, Eq. (3-6), lend themselves to
computation by hand or with a calculator
.
The data is presented in two forms. To examine the validity
of the Bethe-Heitler correction the ratio of the corrected area to
the target thickness versus target thickness has been plotted for
each run. Straight line least squares fits have been included.
To examine the choice of AE, the corrected area versus AE has been
graphed for several targets at different energies.
B. EXPERIMENTAL ERROR
The statistical error in the cross section due to the random
nature of scattering reactions is very nearly
V Sn._ ^ 1
En .
1
where 0" is the standard deviation and En. is the total number of1
counts in a peak. Typical statistical uncertainties vary from
one-half of one percent to one percent for the experiments re-
ported in this paper
.
A significant contribution to the uncertainty of the measured
cross section is introduced by the numerical integration. By
adding or deleting points on a peak and repeating the integration,
this uncertainty has been estimated to be about one percent
(slightly greater for the thin targets which have fewer points
defining the peak and slightly less for the thick targets).
37
The target thickness is known to one percent (with smaller
errors for the thicket taigets). For the experiments which com-
pare the cross section as a function of target thickness, these
errors propa9ate to a total uncertainty of about 1.5%. Experiments
to determine the proper choice of AE are not subject to these
uncertainties since only one peak is integrated, and the successive
integrations over the peak are performed only with variations in
the lower cutoff energy. The relative uncertainty in thes- suc-
cessive integrations is estimated to be less than one-half of one
percent
.
Because of the relative nature of all the e:.,.)er iments,
systematic errors are thought to be insignificant.
38
VII. RLSULTS
A. CHOICE OF AE
The data :/'.': om the experiments to determine the appropriate
choice of AE are shown in Figy . 7 to 13, and summarized in Table II.
Figure 7 is a plot of the cross section versus AE for carbon data
taken on the Mark III linear accelerator at Stanford University [26]
The cross section does not approach an symptotic value even for a
AE of 20 half-widths. Figures 8 to 13 show typical experimental
results obtained with the NPGLINAC. In almost all cases an
syr . totic value was reached for a AE of four ha If-widths (six
half-widths in the worst case, E, = 35.0 MeV) . No significant
difference was noted V7hen the energy defining slits were opened
from the normal position of .0920 inches to .l840 inches, al-
though the resolution of the incident beam, AE,/E , changed from
0.4% to 0.8%. Note that these values are less than two half-widths
of the elastic peak, so that this is the expected result.
3Cross sections were calculated by the procedure outlined forthe NPGLINAC; however, Bumiller and Dally [26] report similarresults using the same data.
^AE, /E, ^ x/r11 o
X = opening of the slits in inchesr = effective radius of the deflection magnet, 23 inches
39
MM(I)
-JCQ<
<]
O
Q)
U•Ho
u
C•H
BHQJ
S
->
03
a
;>
•H»->
4->
C(U
wQ)
Ua
to
(X '
D CO 2;
w
2 3:
2 •
J
s
C
w
•HE
DQ
BOuVi
a
•9
r-^
co
6
tn (/)
(y Q)
£ £u Uc C•H •H
o oCM <^C^ ooO iH
o oo oiH C\J
II II
U) (/)
+-> M•H •Hi-( iH0) C«
00 ON oiH
oON
oOn
OON
OON
CO
nV
co
un3
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oOn
ro
(NJ <f 00 in in <t-m 00 o iH r-^ rH oin CNJ fH rH CO ro rH
o O o rH o O O•
1—
(
!-( (H <J- O CO rHvn CN] NO NO r^ r- 00o O (M O CM Cvj rvj
o iH iH iH iH rH r-i
40
Fig. 7.
Run 51
45E^ = 117.4
Carbon
44 -
i k
43 -
G y.
42 X
41 -
/.n 1 J L-.
typical X
error 1
10 12 l4 16 l8 20
AE
(half-widths)
f
G
•o
74
73
72
71
70
Fig. 8.
Run 1021E^ = 35.0
Carbon
-L.
X K X
10 12 14
AE>>
(half -widths)
41
a
b
40 r
38
36
34
Fig. 9.
PRun 1261
E =90.0 MeVAluminum
^ X ^ ^ ^ X
X
-X
1 1I
23 r
22
21
Fig. 10.
Run 1264E = 90.0 MeVAluminum
AE
10
10
I
12 14
A .J
12 i4
AE
42
a•v
bV
80
79
Fig. 11.Run 1270 slits = 100.E = 90Aluminum
X X X X X X X
78 I
[ L
AE
10 12 14
Fig. 12.
Run 1270 slitsE = 90Aluminum
= 200
a
b
80
79X X
X X
78
2
AE
10 12 14
43
G
82
80
Fig. 13.Run 1281E = 90.0 MeVAluminum
10 12
-J.
14
^..
Fig. 14.Run 1030
\ = 90.0
54 - bon
'<,
- X
G 52-
b-
JL^-""•0
50 ^^
48 -
1
,002 .004
i
006 .008 .010 .012
thickness (x ) >.
014
44
•i^
B. ACCURACY OF THE CORRECTIONS
The data from the exj)er iments to determine the validity of
the corrections to the cross section are shown in plots of cross
section versus target thickness in Figs. l4 to 22, and summarized
in Table III. In all cases the experimental cross section is con-
siderably larger for the thicker targets. No systematic errors
are known which could account for this fact. In fact, the possible
errors, change in SEM efficiency with beam current (APPENDIX B) or
multiple scattering events, would tend to magnify the observed
effect
.
45
HC
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rt
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25
Fig. 15.Run 1040E^ = 55-55 MgVCarbon
t
24
G-a 23
22
J-«.
002 .004 .006 .008 .010 .012 .014
56
54
thickness (x )
58r
Fig. 16.Run 1060E = 90.0 MgVCarbon
52
-L
002 .00/ .006 .008 .010 .012 .014
thickness (x ) >^ o'
47
120 r
Fig. 17.
Run 1220
E = 35.0 MgVAluminum
110
G 100
90 I
±
.002 .004 .006 .008 .010 .012
thickness (x ) ^
r.
.014
G\b
90
85
Fig. 18.
Run 1280E^ =90.0 MeV
uminura
.002 .004 .006 .008 .010
thickness (x )
.012
^
48
Fig. 19.Run 1230
48r E = 90.0 MeVAluminum
46
5 44
42
.i...« J. ^.002 .004 .006 .008 .010 .012 .014
thickness (x ) >^ o'
48
Fig. 20.
Run 1240E = 90.0 MeVAlummum
46\ i^i
b•D
44
42-
1
TX
i
002 .004 .006 .008 .010 .012 .014
thickness (x ) *
49
44
Fig. 21.
Run 1260 (41 )
E = 90.0 MeVAluminum
I-
^ 40
38
L 1 X,002 .004 .006 .008 .010 .012 .014
thickness (x )
44r
Fig. 22.
Run 1260 (VF)
E = 90.0 MeVAluminum
42
^ 40
38
1.002 .004 .006 .008
thickness (x )
.010 .012
—
>
.014
50
VIII. CQNa..USIONS
The ionization and radiative corrections are most accurately
given by Equations (2-3), (3-6), (3-l4). Thus the cross section as
determined by a scattering experiment is
dCT _/da\ _ -1
JTT i^lT - ^b ^^PV /exp
5+5S J
By integrating the experimental spectra for several successively
lower cut-off energies and applying the corrections, it was deter-
mined, within experimental error, that a AE of four half-widths is
more than adequate, even for thick targets (0.01 radiation lengths)
even with the energy defining slits opened more than usual
(slits = 200. = .184 inches). The consistency of this data in-
dicates that the unfolding procedure and the numerical integration
was done properly. Data taken on the Mark III accelerator does
not converge even when the integration is carried out to 20 half-
widths. This discrepancy has not been explained.
For thick targets these corrections lead to cross sections
which are too large by several percent. This may be due to the
fact that the assumption of complete screening in the Bethe-
Heitler theory begins to break down at energies in use at the
NPGLINAC. This error has not been estimated.
51
APPENDIX A. BREMSSTRAHI.UNG
A complete theory of br erasstrahlung has been developed by
Bethe and Heitler [l9,20,2l]. Other authors have developed
similar theories, but they lead to essentially the same results.
Since knowledge of the bremsstrahlung cross sections is basic to
the Bethe-Heitler radiation correction, it is appropriate to
review the basic features of the Bethe-Heitler theory.
When an electron of initial energy E, passes through the
electromagnetic field of a nucleus, the interaction of the two
fields results in an acceleration to the electron so that it must
—emit radiation. The emission of a quantum of momentum k/c leads
to a final electron state. of energy E:~
k=hV=E -E .
The problem is thus to find the transition probability, or cross
section to the final state E (momentum p)
.
The analysis involves the interaction of the electron with the
radiation field , H, and the nuclear field , V, both of which are
treated as perturbations. There are thus two intermediate states,
one caused by H where the quantum k is emitted, so that the electron
has momentuii _P = P - K,
and a competing intermediate state caused by V, where the electron
has mor.entura
p = p + K
5 -- - -•.
H = - ettA, where A is the vector potential, Oi the velocity
6 . 2 ,Assume a Coulomb field, V = Ze /r
.
52
and k is emitted with conservation of momentum in the transition
from intermediate to final states. By appropriately summing over
the intermediate states one can determine the matrix elements for
. . 7 . . . .
the total transition, K . This leads directly to the differential
o
cross section . This has been integrated over all angles by
Bethe [21] to derive the cross section (differential in energy)
for the case of no screening:
dadE
2 2Z Oir
o
[e^- e]4 1 +
2 _E
3 E,m
2E^E
nc [e -e]
- i . (A-1)
This result was derived on the assumption of a pure Coulomb
field, but in certain cases the screening of the nuclear field
by the atomic electrons necessitates significant changes. The
charge distribution changes the original potential
2
v = ;exp [ i(q-r )/fic] ^TTh c 2
r 2 ^
q
where F is the atomic form factor of the Fermi-Thomas model
dT,F(q) =J'p(Oexp ^ (q-^)
withp(r) the density of atomic electrons. Hence screening may be
accounted for by replacing Z in the original differential cross-
2section formula by [z-F(q)] .
Equation (25-8) in Reference 20.
Equation (25-13) in Reference 20
53
For screening to be significant, the field must be screened
appreciably for "impact parameters" which contribute most to the
bremsstrahlung process. F(q) becomes comparable to Z for
^ < "^c 1/3^ I37 ^
which is -fie times the reciprocal of the atomic radius. It can
also be shown that there is a minimum value of q,
2 2(pic ) hv
*^in ~ 2E E
Then screening is effective if
1 2 -iE^E/hV>- 137 mc Z 3
Numerical integration over the angles involved by Bethe leads
to the general formula
dadE
2Z 0!.
E (E -E)o^ o '
y s)C^(y)-
3^2^ - 1 -^oE (02^y)-j^j (A-2)
wher e
y = 100 mc hV
E E 2o
, are given in Fig. 23. y determines the effect of screening
If y = the screening is said to be complete while for y > >
1
screening may be neglected.
In the case of complete screening, Eq. (A-2) reduces to
1^2 2
dg _ ° /
dE " Le^-eJ^ 1 + i
ME_\ 2 _E_
^1J
3 ^l^ (183 z ^N^l (A-3)
54
21 1
Fig. 23. Influence on Screening by Atomic Electrons(high energy). ^ij^o ^^^^ i" ^q. (A-2).
0.25
5 6 7 8 9 10 15
Fig. 24. Influence on Screening by Atomic Electrons(lower energy). C(y) used in Eq. (A-5).
55
Two different expressions may be written for the intermediate
easels :
y < 2:
^,(y) 1
4 3-^ ^- z
2 _E_
3 E,
02(y) 1
4 3- — ^ z (A-4)
2<y<l5:
2^ 2^rr Z O^rdCT o
dE Le^-eJ4 1 +
2 _E_
3 E.
2E E
mc h V(A-5)
\ i; " "1
C(y) is plotted in Fig. 24.
Wheeler and Lamb [2?] have developed an expression for the
br emsstrahlung cross section for all the electrons in the atom.
In the limit of complete screening this is
'dadE
1 _ ^"^'o
, ^ [e, -eJelectrons 1
1 + i
2 ^3 E,
S/n (l440 Z 9E^ . (A-6)
This leads to the \ heeler-Lamb factor ^
:
-(If) f\ /electrons ~ ^ il440 Z
§ =
\ /nucleus ^183 Z
a term which occurs in many equations.
56
APPENDIX B. SECONDARY EMISSION MONITORS
The secondary emission monitor (SEM) was first described by
Tautfest and Fletcher [28]. An SEM consists of a set of thin
metal foils placed in the beam so that the electrons are normally
incident on the foils. Alternate foils are negatively biased to
several hundred volts, so that secondary electrons ejected from
the foils by the beam are collected on the positive plates. The
resultant current, which is proportional to the beam current, can
be monitored directly with an ammeter or collected on a capacitor
used in conjunction with vibrating reed electrometer to integrate
the current. To determ5 ;e the efficiency of an SEM, it must be
compared to an absolute monitor such as a, Faraday cup. Because
of the high gamma and neutron radiation produced by the Faraday
cup it cannot be used to monitor the beam during an experiment.
In most experiments performed at the NPGLINAC, a small,
three-foil SEM was positioned inside the target chamber immedi-
ately downstream of the target, with a larger SEM outside the
target chamber, but under continuous vacuum. The latter is used
to integrate the beam current, while the small SEM monitored the
beam current with a Beckman ammeter
.
It was found that this arrangement is not satisfactory for
experiments involving targets of different thickness. As the
thickness of the target was increased, the current of each SEM
was observed to increase (an increase of about 15% was noted for
57
9a .25 inch carbon target for an incident energy of 35 MeV) . To
avoid this problem, the small SEM was moved upstream of the
scattering target and used to integrate the current, while the
large SEM was used with a Beckraan ammeter
.
The upstream SEM, composed of three aluminum foils .0025
inches thick, is actually a scatterer and served to increase the
beam spot size. But even with the upstream SEM, and at low
energies (35 MeV), the spot size never exceeded 3/l6 inch in
diameter. The primary disadvantage of this arrangement was that
the target chamber, and thus the SEM, was occasionally exposed
to air . However , the vacuum was attained at least eight ho\a's
before beginnii-g the experiment and maintained below 2 x 10 mm Hg
.
It has been reported by several authors [29, 30] that the
efficiency of these monitors changes significantly when first
exposed to the beam. A gtaph of the SEM efficiency as a function
of time is shown in Fig. 25* On these runs the beam current was
- 8held constant at 8 x 10 amp. The efficiency settled down to a
constant value (+^ 1.5%) after, at most, two hours exposure to the
beam.
More important is the SEM efficiency as a function of beam
current, since the beam current had to be changed considerably
from thi.. to thick targets during each run. Burailler and Dally [30]
9 . . .
This increase is thought to be due to additional secondaryelectrons produced in the target which are collected by theSEM.
58
.0318 r
.0314
.0310
X «
Fig. 25.SEM efficiency vs.exposure to beam
TX typical errori
ocQ)•HO•HVt
.0306
0302
60 120 180 240
Exposure to Beam (minutes) >
300
0318
.0314
fX
OCQ)
•Ho
Mhvh
to
0310
0306
0302
typical error
1x10-10
Fig. 26.
SEM efficiency .s
beam current
1x10-9
Beam Current (amp xt.06)— >-
1x10-8
59
have reported a decreasing efficiency of 10% for one of their
SEMs as the beam current decreased two orders of magnitude. We
apparently see a reverse dependence on beam current, as shown in
Fig. 26.
It is still not clear if a correction should be applied to
the data to include this c fect . This was wholly unexpected,
and if correct, would tend to increase the slope of the cross
section versus thickness curves described previously.
60
APPENDIX C. SCATTERING WITHIN THE SPECTROMETER
The most persistent problem encountered during this experiment
was a "ghost" of the elastic peak seen on the radiation tail of
the peaks. The manifestation of this effect is the detection of
extraneous electrons, primarily in channels eight, nine and ten,
when the spectrometer is driven at least two percent below the
energy of the elastic peak. A typical example of this effect is
the thick target spectrum illustrated in Fig. 27. This problem
can be minimized by the use of thin targets, where, in most cases,
the peak can be measured to four half-widths without a major shift
of the spectrometer
.
It was first thought that this effect was due to scattering
through the upper portion of an aluminum flange holding a vacuum
window to the "snout" of the spectrometer. Initially a 1-1/8 inch
block of lead was placed at the top of the snout to attempt to
absorb the undesired electrons. This served only to increase the
overall background. Then the flange was milled to one-half its
original thickness with sharp edges filed so as not to interfere
with the beam. This also showed no noticeable effect. The snout
is illustrated in Fig. 28.
The rotating coil of the fluxraeter extends about two inches
into the vacuum chamber of the spectrometer. It was removed for
a single run to examine its contribution to the "ghosts". No
change was m ced.
61
ooo
s:i.unoo
Ooo
OOO
OoO
oo
OoCO
C
E
wPoQ)
CU0)
•p
03+J
u•H
P
INCM
>
<U Oa •
inO CO•H-H II
V)
W
o
•HH
M <tn
o r
>(U
2
oo
CO
y o
X
X —oo
CO
62
Fig. 28.Spectrometer Snout Assembly
window flanges ^CentralPath
1 g" Pb block
( temporary)
Snout
backing counter
front counter
3 rail Al window
3" extension
spectrometer vacuum chamber
magnet windings and pole faces
Scale 1: .25
63
It is now believed that these extraneous electrons are due
to a small angle scattering of electrons which comprise the
elastic peak, as they graze the walls of the snout and a small
overhang at the top of the spectrometer . Order of magnitude
calculations can be made of the position of the elastic peak by
the dispersion formula
f = ^ • (5-1)
For 35 MeV electrons to be shifted three inches up the focal
plane (where they first hit the flange) a shift in the central
energy of the spectrometer amounting to
^ = |^ = 4.7« = 1.5 MeV.
is required. This corresponds to what is observed. Since this
effect is observed as the spectrometer is stepped down as much
as six percent more, it is unlikely that the ghosts are due to
scattering from a single point, but rather scattering over a
large region such as the entire top wall of the snout.
It would seem reasonable to construct a new snout of the
greatest possible vertical dimension. This would serve to move
scattering areas as far from the counters as possible. Any
scattering from the walls of the spectrometer cannot be corrected;
however, lowering the counters would help remove them from the
worst region.
64
LIST OF REFERENCES
1. R. Hofstadter , Rev. Mod. Phys . , 28, 2l4 (1956).
2. R. Hofstadter, Ann. Rev. Nucl . Sci . , 7, 231 (1957).
3. M.T. Barnett and W.J. Cunneen, Naval Postgraduate SchoolThesis, 1966.
4. P.N. Midgarden, Naval Postgraduate School Thesis, 1967.
5. L. Landau, J. Phys. USSR , _8, 201 (1944).
6. K.S. Symon, Harvard University Thesis, 194S.
7. D.B. Isabelle and G.R. Bishop, Nucl. Phys . , 45 , 209 (I963)
8. B. Rossi, High Energy Particles,
(Prentice-Hall, New York,
1952), p. 35.
9. H. Breuer, Nucl . Inst. Meth . , 33 , 226 (1965).
10. O. Blunk, S. Leisegang, Z. Physi , 1^^, 500 (1950).
11. R.M. Sternheimer, Phys . Rev . , IO3, 511 (1956).
12. J. Schwinger , Phys. Rev ., 76, 790 (1949).
13. Y.S. Tsai, Phys. Rev . , 122 , 1898 (I96I).
14. K. Mitchell, Phil. Mag . , 40, 351 (1949).
15. D.R. Yennie and H. Suura, Phys. Rev ., 105, 1378 (1957).
16. L.C. Maximon, Rev. Mod. Phys . , 4l , 193 (1969).
17. Saskatchewan Accelerator Laboratory Report No. 4, TheRadiation Correction a Review of the Theory and SomeCalculated Values , by H. Breuer , December I964
.
18. L.W. Mo and Y.S. Tsai, Rev. Mod. Phys . , 4l , 205 (1969).
19. H. Bethe and W. Heitler, Proc. Roy. Soc . (London) , Al46,83 (I93A).
20. W. Heitler, The Quantum Theory of Radiat ion (OxfordUniversity Press, London, 1954); 3rd ed., p. 224 andp. 378.
65
21. H.A. Bethe, Proc. Cambr idge Phil . Soc . , 30 ,52l- (1954).
22. H.A. Bethe and J. Ashkin, in Experimental Nuclear Physics,
E. Segre, Ed. (John Wiley & Sons, Inc., New York, 1953),p. 265.
23. L.D. Oberdier, Naval Postgraduate School Thesis, 196?.
24. L.R. Suelzle, Stanford University Thesis, I967.
25. H.L. Crannel and L.R. Suelzle, Nucl . Inst. Meth . , 44 , 133(I966)
.
26. F.A. Bumiller (private communication, 1970).
27. J. A. Wheeler and W.E. Lamb, Phys. Rev . , 55 , §58 (1939).
28. G.W. Tautfest and H.R. Fletcher, Rev. Sci. Inst . , 26,
229 (1955) .
29. L.A.L. Report No. IO33, Factors Influencing the Stabi l ijty
of Secondar y Electron Monitors , by D.B. Isabelle andPh. Roy, 1902.
30. F.A. Bumiller and E.G. Dally, Proceedings of theInternational Conference on Instrumentation in HighEnergy Physics (Berkeley I96O ), (Interstate Ed.,Nev/ York, I96I, p. 305.
66
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 2Cameron StationAlexandria, Virginia 223l4
2. Library, Code 0212 2Naval Postgraduate SchoolMonterey, California 93940
3. Assoc. Professor Fred R. Buskirk 10Department of Physics (Code 6l Ps
)
Naval Postgraduate SchoolMonterey, California 93940
4. LT John A. Gordon, USAF 1
225 Hazel StreetWest Plains, Mo. 65775
5. AFIT-CI 1
Wright-Patterson AFB, OH 45433
67
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68
UnclassifiedSeovintv riassification
DOCUMENT CONTROL DATA -R&D{Security etas silicnl ir>n ol titio, holly of flhsfr«rf and indexind Hnnotalion nnjsl be entered when the overall report is cliiisilied)
I ORIGINATING ACTIVITY (Corporate author)
Naval Postgraduate SchoolMonterey, California 93940
2a. REPORT SECURI TY CLASSIFICATION
Unclassified2b. GROUP
3 REPORT TITLE
Ionization and Radiative Corrections to Elastic Electron Scattering
4 DESCRIPTIVE HOT ES (Type of report and.inclusive dates)
Master ^s Thesis]_ June 19705 AUTHORIS) (First name, middle initial, last name)
John A. Gordon
6 REPOR T DA TE
June 1970
in. TOTAL NO. OF PAGES
67
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b. PROJEC T NO.
9a. ORIGINATOR'S REPORT NUMBER(S)
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10. DISTRIBUTION STATEMENT
This document has been approved for public release and sale; itsdistribution is unlimited.
n- SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate SchoolMonterey, California 93940
13. ABSTRACT
The theory of the ionization and radiative corrections to electron scatteringcross sections has been reviewed to determine the formulas most valid fcr
the experimental arrangement at the NPGLINAC. Experimentally it has beendetermined that the corrections most nearly account for undetected electronswhen the lower limit of the measured scattering spectrum is at least fourhalf-widths below the peak of the spectrum. It has also been found thatthe corrections, most likely the Bethe-Hcitler correction, induce a positiveerror of several percent to the cross-section when thick targets are used.
DD/r:..1473S/N 01 01 -807-681 1
(PAGE 1)
69Unclassified
Security Classification1-31408
UnclassifiedSefuritv Cla';?iil'iration
nscgM
KEV WORDS
Bethe-rieitler correction
Br ems s tr ah 1un g
electron scattering
ionization correction
Landau straggling
radiative corrections
Schwinger correction
JD .ir..1473 (BACK)
/N 01 01 - 607- e82J
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70 UnclassifiedSecurity Classification
