A STUDY OF THE RADIATIVE CAPTURE
OF 4He BY "1C BELOW 4 MeV
RONALD JACK JASZCZAK
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
The author wishes to express his deepest and most
sincere gratitude to Dr. F. E. Dunnam. Without his direction
the experiment would have been an exhausting toil.
A special thank-you goes to Dr. J. L. Duggan for his
guidance concerning target preparation and to Dr. T. A.
Tombrello for the use of his calculations. Thanks go to
Dr. H, A. Van Rinsvelt for his help (including the design of
the target holder) and for checking portions of the manuscript;
to Dr. R. A. Dlue for valuable suggestions concerning
electronics; and to Professor A. E. S. Green for his cclments
on the theory section.
Acknowledgement is made to other members of the
laboratory, including K, E. Baker, R. E. Daniel, R. Stein
and especially R. C. Johns for his generous assistance
during design and construction of the acetylene cracking
Finally, the author expresses his gratitude to the
graduate students who assisted in the data acquisition and.
to his wife for typing the manuscript.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS s .
LIST OF TABLES . .
LIST OF FIGURES . .
INTRCDUCTION . .
Nuclear Astrophysical Interest
Prior Results . .
Intent of Experiment .
I. THEORY . .
Nuclear Processes in Stars .
Radiative Alpha Capture .
Radiative Capture of Alpha Particles
by 1 0 . .
Angular Distributions .
Absolute Yields . .
Radiative Width Determination .
Nuclear Reaction Rates in Stars .
II, EXPERIMENTAL METHOD ,
Energy Analyzing System .
Cold Trap and Target Assembly .
Detectors and Circuitry ,
. 0 0 a ii
. .. Y
* I 9 S 3
. .9 37
9 0 0 a 0
Background Radiation .
Target Preparation .
Synthesis of Barium Carbonate
Synthesis of the Carbide .
Synthesis of Acetylene .
The High-Frequency Cracking of
Acetylene . .
* 0 0
* 6 6
* 6 0
* 6 0
III. EXPERIMENTAL RESULTS .
Excitation Functions .
The 9.85 MeV Level in 10 ..
Cross Sections and the 9.59 MeV Level
IV. DISCUSSION . .
Resonances . .
Target Deterioration ..
Cross Sections . .
The C(a,y)16 0 Reaction Rate at
Effective Stellar Energies .
APPENDIX . . .
GAMMA RAY DETECTION EFFICIENCIES OF
NaI(Tl) CRYSTALS .
Total Incident Intrinsic Efficiency .
Source Full Energy Peak Efficiency
LIST OF REFERENCES . .
BIOGRAPHICAL SKETCH . .
. .' 72
* 0 6
* 6 6
* 0 6
* 6 0
LIST OF TABLES
1. Nuclear reactions in the proton-proton chain. 10
2. Attenuation coefficients. . 24
3. Radiative widths for ground state and cascal
radiations from the 9.85 MeV (2+) level in -0. 70
4. Possible causes of error. . .. 71
5. Radiative width for the 0 P-- 1-(9.59 MeV) EL 0+
transition. . .. .. 80
6. The 12C(a,y)160 reaction cross section 83
LIST OF FIGURES
1. "Smoothed" abundance curve of the elements. 8
2. Nuclear reactions in the helium
burning process .. . 12
3. Energy levels of 160. . 13
4. Theorethical angular distributiDns for
0+1--- 1--->- 0 and 0--- 2+-E- 0
radiative transitions. . 22
5. In-line cold trap assembly. . 39
6. Target chamber. . . 41
7. NaI(T1) crystal and lead shield. 43
8.. Schematic electronic circuit for NaI(Tl)
detector. . . 45
9. Electronic pile-up elimination circuit. 47
10. Ge(Li) detector electronic circuit. 48
11. Barium carbonate apparatus. . 53
12. Acetylene apparatus. . ... 56
13. High-frequency discharge apparatus. 57
14. Enriched 12C excitation function in the
region of the E. = 3.58 MeV resonance 61
15. Excitation function for enriched 12C target. 63
16. Excitation function for natural carbon target. 64
17. Excitation function for enriched 13C target. .. 65
18. Gamma ray spectrum from Nai(Tl) detector
at Ea = 3.58 MeV. . .
19. Gamma ray spectrum from Ge(Li) detector
at E. = 3.58 MeV. . .
20. Branching ratios of 9.85 MeV level in 160.
21. Gamma ray spectrum from NaI(Tl1 detector
at Ea = 3.24 MeV for enriched 2C target.
22. Gamma ray spectrum at E = 2.95 MeV for
enriched 13C target. . .
23. Gamma ray spectrum at E. = 2.95 MeV for
natural carbon target. .
24. Gamma ray1~pectrum at E = 2.95 MeV for
enriched C target. .
25. Gamma ray spectrum at E = 2.14 MeV for
enriched 1C target, .
26. The 1C(a,y) 160 reaction cross section values.
27. The parameters involved in the calculation
of NaI(Tl) crystal efficiencies. .
28. Gamma ray spectrum from the 13C(p,y)14N
reaction (Ey = 9.17 MeV).. .
Nuclear Astrophysical Interest
The 12C(a,y)160 reaction is of considerable interest
in nuclear astrophysics, especially in studies of stellar
nucleosynthesis and models of stellar interiors. This
reaction occurs in the helium burning sequence and has
a direct influence on the relative abundances of oxygen
and carbon. The helium is produced from hydrogen burning
by the proton-proton and carbon-nitrogen-oxygen cycles (1, 2).
After the hydrogen is consumed by these processes, the
core is heated due to gravitational contraction and at a
temperature of about 10 K (corresponding to an effective
thermal energy of about 200 keV), helium burning is initi-
ated. The parameters for the 3a-reaction leading to 12C
are known to a fair degree of accuracy, and hence the
rates of these reactions have been determined (3).
The 12C(a,) 160 reaction in the helium burning
chain is believed to proceed through the 7.12 MeV (1~)
level in 160 which is about 40 keV below the threshold.
Very little is known about this state, especially the
reduced alpha particle width which is necessary in the
determination of the 12C(a,y)160 reaction rate at stellar
temperatures. The reduced alpha particle width Oa2 in units
of the Wigner limit is defined here by Oa2 = ya2(3h2/2M2)-1
The quantity ya2 is related to the partial width Fa by the
expression F = 2y A2 where A2 is the penetrability. Alpha
particle model calculations (where the 160 nucleus is assumed
to be formed by four alpha particles at the corners of a
tetrahedron) indicate that the reduced width for the 7.12 MeV
state is near unity (4); Cluster model calculations by Roth
and Wildermuth indicate a larger variation for the reduced
alpha width (5). The value of the reduced width for states
obtained by assuming a 12C + He cluster (in their ground
states) is about 0.6. If the 12C nucleus is in the
4.43 MeV (2+) state then the reduced width is about 0.02.
Roth and Wildermuth assign the 7.12 MeV level to the 12C + 4He
cluster, indicating that the reduced width is nearer the
value 0.6. Since the final abundances of 12C and 0 will
depend critically on the 12C(a,y) 160 reaction rate (and hence
on 2), a more accurate determination of the reduced alpha
particle width of the 7.12 MeV level is required. It is
possible to obtain some information on ea 2 by examining the
12C(a,y)160 reaction cross section at alpha particle bom-
barding energies near 2 MeV. Because there were no accurate
estimates of this reduced alpha width, Deinzer and Salpeter
(6), and also Fowler and Hoyle (7), calculated the final
abundance of 12C in the core of a helium burning star for
various assumed values of 9 2
Prior results on the 12C(a,y)160 reaction have been
limited to work done by Allan and Sarma (8), Bloom et al, (9),
Meads and McIldowie (10), and Larson and Spear (11, 12). In
the preliminary work done by Allan and Sarma, alpha particles
(Ea = 1.6 MeV) were used to bombard thick carbon targets
while observing the high energy gamma rays with a 5.08 cm
thick NaI(T1) crystal. The yields were determined by comparing
the counting rates in the high energy region with the known
counting rate due to the 4.43 MeV gammas from the 9Be(a,ny)12C
reaction. They obtained the value 3 x l0-5 barns for the
integrated cross section at Ea = 1.6 MeV. It is now felt that
the correct cross section is much smaller than this.
Bloom et al. examined the 1" state in 10 at 9.59 MeV.
They were investigating isotopic spin mixing of this T = 0
state, from which the ground state radiation should be inhib-
ited. The radiative width Fy for this state was found to be
0.006 eV corresponding to IMj2(E1) = 1 x 10-5 in Weisskopf
units. From this it was concluded that there was some T = 1
Meads and McIldowie studied the 2+ excited states in
160 at 9.85 MeV and 11.52 MeV in order to obtain the radiative
widths of these states. Using enriched 12C targets and a
10.2 cm x 10.2 cm NaI(Tl) crystal, the values 0.02 + 0.01 and
0.9 0.2 eV were obtained for these radiative widths,
respectively. The state at 9.59 MeV was observed, but the
radiative width was not found for this state,
Larson and Spear examined the region E. = 2.8 to 8.3 MeV
in some detail with enriched 12C targets using a 10.2 cm by
10.2 cmr NaI(T1) detector. The region of interest for the
present experiment includes principally the resonances at
Ea = 3.24 and 3.58 MeV corresponding to excited states in
160 at 9.59 MeV (1-) and 9.85 MeV (2+), respectively. The
radiative width for the 9.59 MeV state was given by Larson
as 0.022 + 0.005 eV, and the value of the radiative width
for the 9.85 MeV state was 0.0059 + 0.0006 eV. Also, the
cross section at E. = 3.24 MeV was given as 36 nb (10-9 barns)
(11). The disagreement of Larson with the results of
Bloom et al. was explained by assuming the latter had inad-
vertently lost data due to their background subtraction or
renormalization. Larson and Spear were in better agreement
with the expected isobaric spin impurity predicted by
Wilkinson (13). It was not possible for them to determine
the capture cross section for lower alpha particle bombarding
energies. Cascade radiation of about 7 MeV was observed
from the 9.85 MeV state; however, it was impossible to
determine whether this cascade proceeded through the 7.12 MeV
(1-) state or the 6.92 MeV (2+) state in 160. A width of
0.0012 + 0.0004 eV was given for this radiation, assuming an
A recent endeavor on the 12C(a,y)160 reaction was
presented at the 1968 American Physical Society Meeting in
Washington byAdams et al. (14). Using time-of-flight methods
to discriminate against the neutron background, a value of
10 nb was obtained for the capture cross section at
Ea = 2.75 MeV. Data accumulation was hindered somewhat by
low yields resulting from the detector being relatively far
from the target and by low beam currents from the tandem
accelerator employed in the experiment.
Intent of Experiment
At the time this experiment was first proposed it was
hoped that absolute cross section measurements could be
extended to an alpha particle bombarding energy below 1.5 MeVi
however, this did not prove feasible because of the extremely
low yield from the 12C(a,y)160 reaction, and the lowest
bombarding energy at which the cross section was determined
was 1.9 MeV.
Because of the discord between the results of Larson
and those of previous experimenters concerning the radiative
widths of the 9.59 MeV and 9.85 MeV excited states, it was
felt that these radiative widths should be measured. Also,
since Larson was not able to determine the state through
which the cascade radiation from the 9.85 MeV state in 160
proceeded, it appeared desirable to examine this cascade
radiation using a 20 cm3 Ge(Li) detector.
Nuclear Processes in Stars
In search of an understanding of the origin of the
elements in the universe, many theories have been formulated.
So far the theories fall into two classes. One class assumes
that the elements were formed in a primordial state of the
universe. Two examples of this class are the non-equilibrium
theory of Alpher and Herman (15) and the poly-neutron theory
of Mayer and Teller (16). The other type proposes that stars
are the principal instrument in element formation. A
relatively complete theory of this type has been formulated
by Burbidge, Burbidge, Fowler, and Hoyle (17). Primordial
theories distribute the elements on a cosmic scale; however,
the theories imply that the distribution should be independent
of time. This is contrary to what is observed since, for
example, anomalies (such as the presence of technetium)
have been observed in certain stars. Stars may distribute
material by ejection (such as the explosion of a supernova)
which would result in a distribution of the elements on a
The theories must all be able to explain the relative
abundance distribution of the elements. Suess and Urey have
constructed a "smoothed" abundance curve by employing data
obtained from terrestrial, meteoritic, and solar measurements
(18). It is impossible to say unequivocally that such a
curve is universal. Thus, one only attempts to explain the
origin of the material that is observed. The general
feature of the abundance curve is that it decreases exponen-
tially from A = 1 to about A = 100 (figure 1). This is
from Burbidge et al. (17). Other principal features include
the change in slope at A = 100, the scarcity of D, Li, Be and
B, the relatively high abundance of alpha particle nuclei
such as 160, 20Ne, 4Ti, large peaks centered at A equal to
about 85, 134, 202, and finally the small abundance of
proton-rich heavy nuclei. Any complete theory must be able
to explain all of these features.
Burbidge, Burbidge, Fowler, and Hoyle proposed eight
processes in stars that would account for these features in
the abundance curve (17). A few modifications to the theory
have occurred since the original article, and the number of
processes has increased. The processes are labeled hydrogen
burning, helium burning, carbon and oxygen burning, the alpha
process, the equilibrium process, the s-process, the r-process,
the p-process and the f-process. It will be shown that the
12C(a,Y)160 reaction is in the helium burning chain of reac-
tions. However, to better understand the motivation behind
this experiment it is desirable to examine the over-all
Figure 1. "Smoothed" abundance curve of the elements.
Burbidge et al. (17).
theory slightly. Also, all of the separate processes inter-
connect to form the complete theory. It is for these reasons
that a brief description of the individual processes will be
The f-process is introduced to explain the formation
of the elements lithium, beryllium, and boron. These elements
are rapidly reduced to helium through numerous nuclear reac-
tionswith protons in stellar interiors; thus, it appears
that they must be produced near the surfaces of magnetic
stars through spallation of carbon by highly energetic protons.
The p-process is employed to explain the existence of
certain proton-rich nuclides found in the heavy elements.
These proton-rich elements are formed by (p,y) or (y,n)
reactions on the elements along the nuclear stability line.
Hydrogen burning is the synthesis of helium from
hydrogen (table 1) (19). The alternate endings seem to
depend on the temperature of the region where the hydrogen
burning is occurring. The quantity So is the cross section
factor at stellar temperatures (19). The cross section
factor is directly proportional to the reaction rate which
determines abundances. The cross section factor S is
related to the cross section a by
S(E) = o(E)Eexp(31.28 Z1ZoA E ) keV barns,
where a(E) is the cross section in barns (10-24 cm2)
measured at the center-of-mass energy E in keV (17).
I o o
S000- 0 0 r- I
c ) > CM 0 0 l 0
4 ,M H M M 8 v H m
0 0 O r 4 1 O 1 B
? CO a H OC n I IP
A 03 G0 o
) o o o N
0 r0 0
0 H H
03 m O \O 0 0n O
0 H H H &
) I U CI
- r + 4 .0 N
H+ ) m +
C 0 +W
+ PQ +
N NC n 0 H0
I 0 2, 4
E- + + +
r-i C4 n Ell.- CO cO 4
aS 0f +e ?- +
E-4 0j11 I- +
Z1 and Zo are the charges of the interacting particles in
units of proton charge, and A = A1Ao(A1 + Ao)-1 is the reduced
mass of the system in atomic mass units. S results after
effects due to resonances and barrier-penetration have been
eliminated in the expression for the cross section.
In stars that contain enough carbon, nitrogen, or
oxygen, helium may be produced from hydrogen through the
carbon-nitrogen-oxygen cycle. The cycle starts with radiative
proton capture on 12C and progresses to the reaction
15N(p,a)12C. After the hydrogen in the core has been exhausted,
the core begins to contract due to gravity. As this occurs
the temperature rises from the 107 OK that was present during
the hydrogen burning process.
When the temperature is about 108 oK the core begins
to burn helium. The principal reactions involved are shown
in figure 2 (19). The 0+ ground state in 8Be and the 0+
state at 7.65 MeV in 12C allow the first two reactions to
proceed at a reasonable rate. The formation of 160 appears
to be caused mainly by the 1- state at 7.12 MeV (figure 2).
This is about 40 keV below the 12C + He threshold. Using
resonance fluorescence, Swann and Metzger found the mean
lifetime of this state to be about 1.0 x 10-14 s (as corrected
from their previously obtained value), corresponding to a
radiative width of about 0.065 eV (20). The alpha particle
width for this state remained unknown. If the reaction is
due to a single resonance (as is the case for 12C(a,Y)160),
then it can be shown that the cross section factor So is
( -0094 j -B-.
Nuclear reactions in the helium burning
!2 S5 c5 "
124 3 N t
!22 3 e7Z16 --J
V3 ':I -- He' -a
I j I _/I'_ .4 -
i -...~~--- ~~.
N, .-- .;
_2 ;-_ -- _-
9 27.1 2
ii J=^ a.4 -> .
0 d^o O x -10
0 I t
Si Cle i 1 ( -'1 I.- 1-2.
\_ ,_ .____
Figure 3. Energy levels of 160. Lauritsen and
proportional to 92 (the reduced particle width of that state)
(17). Thus, it is evident that a determination of the
reaction rate requires a determination of that reduced width.
Figure 3 shows the energy levels of 160 (21). It is
possible for the broad (Fcm = 645 keV) 1" level at 9.59 MeV
to interfere with the 1- level at 7.12 HeV. This inter-
ference will either enhance or decrease the capture cross
section in the region between these resonances, depending
on whether the interference is constructive or destructive,
respectively. If the relative sign of the amplitudes of
these states is minus, constructive interference will occur.
If the sign is positive, destructive interference will occur.
Thus, a dete7inration of 0 2 is possible by fitting the
observed cross section in the region Ea = 1.5 to 3 MeV to
calculated values of the cross section (for assumed values
of e 2, and computing both destructive and constructive
values), A calculation of this type has been done by
Tombrello (22). It is therefore necessary to measure the
capture cross section in this region in order to determine
the reduced alpha particle width ea2. Larson has measured
the cross section at an alpha particle bombarding energy
of 3.24 MeV, but this does not allow a determination of
the interference effects (11),
The relative abundance of 160 to 12C will depend on
the cross section of this reaction at stellar temperatures;
thus, the necessity for a better extrapolation forms an
incentive for the present experiment. The main difficulty
in performing the experiment, besides the extremely low cross
section values for the 12C(a,y)160 reaction, is the relatively
large cross section (20-100 mb) for the 13C(an) 60 reaction,
since NaI(T1) detectors are quite sensitive to neutrons.
After the helium has burned, the core again contracts,
and the stars will begin burning the carbon and oxygen.
In this way nuclides up to 281 may be formed.
As the temperature rises (2 x 109 OK) photodisintegra-
tion occurs producing alpha particles, protons, and neutrons.
Since the thresholds for (y,p) and (y,n) processes on nuclei
with A = 2Z = 2N are higher than for other nuclei, there
tends to be an increase in abundance for nuclides with
A = 2Z = 2N, These give the appearance of being composed
of alpha particles. This is known as the alpha process.
As the temperature rises to about 4 x 109 OK, many
nuclear reactions are occurring so rapidly that a statis-
tical equilibrium is created. This is known as the e-process.
Since electron capture can occur quickly at this tempera-
ture, there is a shift to nuclei that have N slightly larger
than Z in the vicinity of the "iron peak."
The terms "r-and s-processes" refer to rapid and
slow neutron capture, respectively. The s-process is the
capture of neutrons such that there is enough time between
captures to permit beta decay. The necessary neutrons for
the s-process are produced by (aCn) reactions on nuclei such
as 13C and 21Ne. Since certain elements beyond bismuth have
decay times that are short compared to the characteristic
time for the s-process, the elements heavier than bismuth
could not be produced by the s-process. But neutron capture
at a rapid rate (the r-process) could produce the heavy elements
such as uranium, etc. Since the cross section in the
r-process and s-process are small at neutron magic numbers,
the abundances are large at these points. The r-process
produces the peaks in the abundances curve at lower values of
A than the s-process, of course, since the r-process occurs
to the neutron rich side of the stability (N = Z) line; thus,
the formation of the double peaks observed in the abundance
curve may be explained by these processes.
As the iron-group nuclei increase, the star again
contracts and the core implodes; however, nuclear reactions
are still occurring in the outer part of the star. This
implosion creates the supernova. This also produces a rapid
increase in temperature, causing an emission of a large
part of the star; furthermore, neutrons are produced profusely
and are thus convenient for the execution of the r-process.
In this way evolution of the star is completed.
Radiative Alpha Capture
Radiative alpha capture has often been employed for
determining detailed information (such as the energy, spin,
parity, etc.) on excited states of nuclei (23, 24). These
parameters (along with others such as radiative widths,
branching ratios of gamma rays, mixing parameters, etc.)
are required in comparing nuclear models and extending
their usefulness. The alpha particle capture reaction is
different from proton capture in that the alpha particle
is spinless; thus, there is a simplification in angular
correlation meaurements. If the target nucleus is also spin-
less, as is the case for even-even nuclei such as 160, a
further simplification occurs. In this case only the relative
motion of the interacting particles may contribute to the
spin of the compound state. This requires that the possible
values of spin (J) are limited to J = 1, 2, 3. .
These are simply the eigenvalues (f) of the orbital angular
momentum operator. The concept of parity is determined by
the behavior of the wave function for the system upon
reflection of spatial coordinates (25). If upon this reflec-
tion the spatial part of the wave function does not undergo
a sign change, the parity (n) is even (+). If the sign
changes, the parity is odd (-). Furthermore, it is known
that the parity is determined by the orbital angular
momentum eigenvalue (f). If I is even (0, 2, 4, .),
than the sign of the wave function will not change upon
spatial reflection, and the parity will be even (+). If
I is odd (1, 3, ), the sign will change, and the
parity will be odd (-). Hence, in cases where the pro-
jectile and target are both spinless, the spin (J) and
parity (r) of the states that can be excited are limited
to J" = 0+, 1-, 2+, 3-. These are the so-called
states of "natural" parity.
For this experiment the concept of the compound
nucleus is also useful (25). Here the incident particle is
captured and a compound system is formed, with the energy
shared among all the particles of the compound system. A
relatively long time elapses before sufficient energy is
concentrated onto a particle to allow it to escape. This
time is long compared to the "characteristic nuclear time,"
which is the time required for the incident particle to
traverse the target nucleus without interacting. Since the
decay time is long compared the the "characteristic time,"
the compound nucleus "forgets" its method of formation, If
there is sufficient energy, the compound nucleus can decay
though particle emission in addition to gamma emission.
The states excited in this way are called virtual states.
If there is not sufficient energy to allow particle emission,
the compound nucleus can decay by gamma emission or by
internal conversion. In any case conservation laws are
In gamma ray emission, the angular momentum of the
gamma ray for the ground state transition must be equal to
the angular momentum of the compound system, which is equal
to the orbital angular momentum in the particular case where
the spin of the interacting particles is zero. A light
quantum has an angular momentum equal to hf(f + 1)i. The
maximum projection is Ph. The multiple order of radiation
is 2 ; thus, for example in the case where the interacting
particles are spinless, p-wave capture will result in
electric dipole radiation for the ground state transition.
Since the radiation is transverse, there cannot be an I = 0
multiple. This is the reason that 0+ ---> 0+ radiative
transitions are strictly forbidden.
Radiative Capture of Alpha Particles by 160
The two resonance levels in 160 that can be reached
by the present experiment are the 2+ level at 9.85 MeV and
the I1 level at about 9.59 MeV (figure 3). These occur at
incident alpha particle energies of 3.58 and 3.24 MeV,
The level at 9.85 MeV has a total width rem of 0.75 keV
(26, 27). This narrow resonance proved quite useful for
determining the target thickness. The thickness of the
target is essential in the determination of cross sections
and radiative widths. Also, because of the large cross
section for the 13C(a,n)160 reaction immediately preceding
this resonance, the 12C enrichment was indicated through a
comparison of the excitation function for a natural carbon
target with that from an enriched 1C target.
Unfortunately a broad peak due to the 13C(a,n)160
reaction is located in the same alpha particle energy region
as the 9.59 MeV peak (28). This makes the gamma radiation
from the 9.59 MeV level extremely difficult to observe.
The 9.59 MeV level is a broad resonance having a total
width rcm of 645 keV (26). Hence, the peak extends over a
considerable energy region, but the determination of the
off-resonance yield is hampered by the sensitivity of the
NaI(Tl) spectrometer to neutrons originating in the
Below about Ea = 2.9 MeV the main process is nonreso-
nant capture. Here the cross section for the 12C(a,y)160
reaction is extremely small (<10 nb), while the 13C(a,n)160
reaction remains relatively large (-20-100 mb). Thus, it is
even more difficult to determine the nonresonant reaction
cross section at the lower energies. Around E. = 2.0 MeV
interference effects between the 7.12 MeV and the 9.59 MeV 1-
states in 160 become important. The cross section will be
increased or decreased depending on whether this interference
is constructive or destructive. The magnitude of this effect
is dependent upon the parameters of the states involved.
Tombrello has calculated this effect assuming 2 = 0.1;
however, without knowing the relative sign of the amplitudes
it is impossible to determine whether this effect is con-
structive of destructive (22).
In the appendix the efficiency of the NaI(Tl) crystal
is derived assuming an isotropic distribution of gamma
radiation. It is necessary to determine the effects of an
anisotropic distribution on these results. Gamma radiation
resulting from transitions between natural states, i.e.
0+, 1-, 2+. will have an angular distribution given
by W(e) = anPn(cose), where Pn(cose) is the nth order Legendre
polynomial, and e is the angle between the emitted photon and
the incident particle. Since the ground state of 160 is 0+,
the observation of radiation Yo to the ground state resulting
from the capture of an alpha particle in most cases indicates
that the spin and parity of the resonance state is either 1"
or 2'. J = 0 would not decay to the 0+ ground state, and
J = 2 is more likely to decay to higher excited states which
would then decay to the ground state in 160. In particular,
for 0 d 2+ E2 0+ and 0+ P 1 El 0+ transitions the
-) ----4 -4- ---
theoretical angular distributions will be of the form
W(e) sin2e -1l-P2(cose) (J=l")
W(e)- sin22e0 -+(5/7)P2(cose)-(l2/7)P4(cose) (JT=2+).
From figure 4 it can be seen that these distributions are
entirely distinguishable. The simplicity of these distri-
butions results from the fact that there can be no radiation
mixing, channel spin mixing, or orbital angular momentum
mixing for these transitions. However, the distribution
resulting from the transition between the 2+ states at
9.85 MeV and 6.92 MeV in 160 (0_ d 2+ M2lE2 2+) is
complicated by the fact that Ml E2 radiation mixing is
.possible. In this case it is possible to obtain from the
0.0 0 0. 0.4 0.6 0.8 1.0
Figure 4. Theorethical angular distributions for
0-+ 1 i-- 0+ and 0--J 2 -2- 0 radiative
angular distribution the value of the mixing parameter X,
where X = S(E2)/S(M1),. S(E2) and S(M1) are the amplitudes of
the E2 and Ml transition probabilities, respectively (29).
In determining the total yield (and therefore radiative'
widths, cross sections, and branching ratios) it is essential
to have a knowledge of the angular distribution of the gamma
radiation. If a measurement is taken at only a single angle,
then inaccurate determination of the yield can be obtained
only with the aid of a prior knowledge of the angular distri-
bution, unless the detector is so close to the target that
the distribution appears isotropic.
The observed angular distribution Wobs(G) is then
"smoothed" by the large solid angle subtended by the detector.
This means that the larger the solid angle subtended (the
larger the detector or the nearer the detector to the target),
the smaller the angular variations will be. In other words,
the observed angular distribution will have a decreased
dependency on higher order Legendre polynomials. Wobs(q)
has the form
Wobs(O) = anbnPn(cos9), bnlS 1.
The coefficents bn are the attenuation coefficients described
by Rose (30). They are given by bn = Jn/Jo where Jn is the
nth order integral defined in the appendix. These integrals
were evaluated numerically through the use of a subprogram
(forming a part of a program that determines angular cor-
relation parameters) (31) (table 2). In general, the values
CO CO 0
vx V- o
of Jn and b. decrease rapidly with increasing n for large
solid angles. It is obvious that the proximity of the detector
to the target determines how rapidly the bn approach zero
with increasing n. If the detector is very close to the
target, the observed distribution will be quite independent
of higher order terms.
The absolute total yield of a nuclear reaction is
defined here as the number of disintegrations per incident
particle. In the following discussion the notation adheres
to that used in the appendix. The yield may be related to
Nobs(e) (the number of counts in the spectrum peak observed
at the laboratory angle 0 per incident particle, corrected
for background and dead-time) by employing the ideas derived
above and in the appendix. The number of counts in the
peak (here defined as the region (Ex 1.02 MeV) to 1.1Ex)
at angle e per steradian per incident particle, is given by
Nobs(e)/4T = np x (Y/4n) Wobs(e)/ao,
where rp = REt = R(fqT) = R(Jo/2)
R = peak-to-total ratio
R= solid angle subtended by detector
lp = source full energy peak efficiency
rt = source intrinsic efficiency
9T = source incident intrinsic efficiency
Jo = the zeroth-order integral as defined in the
The absolute total yield is denoted by Y; hence, Y/4n is the
average yield per steradian. Thus, from the above equation,
it is seen that the yield is
Y = Nobs(e)(R(Jo/2) x J(an/ao)(Jn/Jo)Pn(cose))'1
The expression Nobs(e)/4n may be defined as the "ob-
served differential" yield and denoted by nobs(e). Similarly,
the expression N(e)/4n may be defined as the actual differ-
ential yield and set equal to n(e). Using the expression
derived for the yield it is easy to show that the relation
between n(e) and nobs(e) is
n(e) = x
R(Jo/2) (an/ao) (Jn/J o)Pn
Radiative Width Determination
For a resonant radiative capture reaction the cross
section a is assumed to depend on the energy according to
the single-level dispersion formula (25). Explicitly for an
(a,Y) reaction this has the form
(E En)2 r2/
(E El)2+ F2/4
where u = statistical weight = (2J + 1)/((2I + 1)(2S + 1))
J = total angular momentum of system (spin of state)
I = spin of target nucleus
S = spin of incident particle
I = total width of resonance
y = radiative width
r = particle width
S= h(2mE)-2 = wavelenght of interacting particles
m = reduced mass of system = (AaAT/(Aa + AT))1.66x10-24g
Aa = incident particle mass (atomic mass units)
AT = target mass (atomic mass units)
E = center-of-mass energy = (T/(Aa + AT)) x Elab
Elab = incident particle energy in laboratory.
At resonance, a is given by
oR = T X2 x /(2/4).
Hence, a may be expressed in the form
UR( r 2/4)
(E ER)2 + r2/F
The relation between the yield and the cross section, assuming
the Breit-Wigner formula is applicable, has often been cited
in the literature (32, 33, 34, 35). The relation is given by
Y = -
where 6 is the target thickness in energy units and
e is the stopping cross section for the incident
particle per disintegrable nucleus in the target
The relationship between and the stopping power is
where -(dE/dx) = the stopping power of the target material
n = gpNAM-1 = number of disintegrable nuclei per cm3 of
p = the density of target material in grams per cm3
NA = Avogadro's number
M = the molecular mass of target material
g = the fraction of disintegrable nuclei in target
If it is assumed that 6 and e are independent of E over the
resonance, then 6 = (dE/dx)t = net, where t is the target
thickness in centimeters. The integral for Y may now be
evaluated analytically. The result is
2n X2 )o1 [ (E-ER) (E-ER )6
Y = 2 x arctan -arctan ,
e F 1F2 r/2
FoR a (E-ER) (E-ER-6)
or Y arctan arctan .
2 r/2 r/2
It can be shown that this expression has an observed width
r' given by
= (r2 + 62)
Thus, the thickness of the target tends to broaden the
observed width. When the target thickness is determined
(as is often the case) by examining the excitation function
across a narrow resonance, errors enter through such factors
as accelerator instability, energy-defining slits, etc. The
final observed width will thus be the square root of the
sums of the squares of all of these contributions. It is
evident from the above expression for the yield that a
maximum occurs at the energy E = ER + 6/2; thus, the observed
resonance energy is shifted to the higher energy side by
approximately 6/2. This is only approximate because in the
derivation it was assumed that the parameters in the formula
for the cross section were independent of energy over the
energy integration. This shift in resonance energy is one
of the reasons why it is very desirable to have an extremely
thin target for resonance spectroscopy measurements; however,
there are situations where it may be more desirable to have
a thick target (for example, to increase data acquisition rate).
The maximum yield is given by
Ymax = (aR rF/)arctan(6/r).
In terms of measured quantities the resonance cross section
OR has the form
aR = x
arctan(6/F) R(Jo/2) (an/ao) (Jn/Jo)Pn(cose)
where Nmax(e) is the maximum counts observed in the peak
at the laboratory angle e. Using the expression relating
aR with the radiative width Fy, it can be shown that Fy is
r Nmax( )
Iy = x
y 4TT rFaarctan(5/r) R(Jo/2) (an/ao) (Jn/Jo)Pn(cose)
This is the expression that was used to obtain the radiative
widths from the observed counting rates. For off-resonance
cross section measurements, a and eare assumed constant in
the integral over the energy interval (E 6) to E.
Nuclear Reaction Rates in Stars
At a temperature T the number of collisions (coll cm-3s )
between nuclei of types 0 and 1, having a center-of-mass
energy between E and E + dE is proportional to
nonlE2exp(-E/kT)dE, where ng and n1 are the number densities
of particles 0 and 1, respectively. The mean reaction rate
(reactions per cubic centimeter per second) of a thermonuclear
process is thus proportional to the integral (36).
where E is the center-of-mass energy, Eiexp(-E/kT) is the
Boltzmann factor (probability for the interaction energy E),
P(E) is the barrier penetration factor, and S(E) is the
cross section factor (defined previously in section 1). The
quantity S is a slowly varying function of the energy E for
off-resonant reactions. It can be shown that the product
of exp(-E/kT) and P(E) has a fairly strong maximum centered
at an energy Eo given by.
Eo = 26.29(Z12Zo AT82) 1 keV,
where Z1 = charge (in units of proton charge) of interacting
Zo = charge of particle 0
A = A1Ao/(A1 + Ao) = reduced atomic mass number of
T8= stellar temperature in units of 108 oK.
The quantity Eo is the effective thermal energy used by
Burbidge et al. (17). For reactions such as those normally
found in the helium burning process, the effective thermal
energy is much larger than the mean kinetic energy.
This means that the region of interest occurs in the high
energy tail of the Maxwell-Boltzmann distribution of velocities.
The number densities no and ni are related to the
abundances Xo and X1, expressed as the fractional amounts
(by weight) of the interacting nuclei:
ni = NAXi P/Ai*
where ITA is Avogadro's number, p is the density in g cm-3, and
Ai is the atomic mass number of species i. The mean reaction
rate Pl(0) per nucleus of type 0 (expressed in terms of So)
for the interaction with nuclei of type 1 is explicitly
1 o 01 pX1
Pl(0) = 1/r1(0) = 3.63 x 107S ofo ---
where 0 = 9.15(ZL2Z 2A/T8 .
The quantity T1(0) in the above expression is the mean
lifetime (seconds) of the nuclei of type 0 for interaction
with nuclei of species 1. The quantity fo is the correction
due to electron screening of the bare nuclei as explained
by Salpeter (37). This is a result of the spherical
polarization of the electron gas immediately surrounding
the bare nucleus.
If the reaction rate is due principally to a single
resonance at Er (not in the range of the effective thermal
energy), then an expression for So can be obtained by using
the Breit-Iigner single level formula for a. The result
3 01 1 aYE
So = 3.10 x 10 -
A [K21+ 1(Z)
Eo 2 ER
(E0 Er) + /4
where 012 = reduced particle width
z = 2(Bo/ER)
Be = Coulomb barrier height = ZZ1oe2/R
Eg = h2/2MR2
K21 + 1(z) is the modified Bessel function of order
21 + 1.
The quantity a, is the function of z defined by Burbidge
et al. (17). Values for the modified Bessel function have
been tabulated (38). For the C(a,Y) 0 reaction (assuming
the state at 7.12 MeV n 160) the expression for S
the 1 state at 7.12 MeV in 0) the expression for S0 is
1.17 x 105 a 2(7.12)
( 2 2
(5T38 + 1)
where the following values have been used
F2/4 << (Eo Er)2
r2 = fy = 0.065 ev
Er 40 keV
Eo = 200 T8 s.
The reaction rate is given by
p(12C) = (C(12C))-
1.69 x 1012Xo 2 12c p
T8 (5T8S+ 1)2
exp(-69.2 T8" ) s-.
Energy Analyzing System
Singly charged helium ions at energies up to 4 MeV
were available from the University of Florida Van de Graaff
accelerator. Beam currents for the present experiment were
between 1.5 and 3 ,A. Some preliminary experiments were
carried out using beam currents in the range of 4-7 pA;
however, the rate of target deterioration was intolerable
with these currents. The singly ionized 4He beam was
deflected through a 900 arc by the use of a momentum analyz-
ing magnet (of radius 45.72 cm). The magnetic field B
determined the energy of the He+ beam traversing the slit
system of the analyzer. This magnetic field was determined
by the usual nuclear magnetic resonance (NMR) methods (23, 39).
The NMR fluxmeter (Varian Model F-8A) was supplied by
Varian Associates of Palo Alto, California. Its upper limit
of measurement is 52 kG. This easily surpasses the upper
limit requirements of 12.6 kG for 4 MeV alpha particles.
The NMR unit was calibrated using the 2.4374 MeV and
3.1998 MeV resonances in the 2Mg(a,y) Si reaction (40, 41).
The calibration has also been examined using the 7Li(p,n)7Be
threshold at 1.8807 NeV. The calibrations were accurate to
within 1 keV (112). A Fortran program for tabulating NMR
frequencies versus alpha particle energies in 5 keV intervals
was employed in this experiment (39). Energy variations
due to the finite apertures of slits introduces some error
into the determination of alpha particle energies. It was
assumed that this error was + 0.2 per cent. The magnitude
of the error is not important for off-resonance yields
(since thick targets were used) and enters mainly through
the determination of the target thickness using the 9.85 MeV
level in 160. Current integration was accomplished through
the use of an Elcor current integrator (Model A309B) (calibrated
for an accuracy of 1 per cent or better). The target was
cooled with "refrigerated" water that was demineralized,
thereby minimizing the possibility of current leakage to
ground. This precaution appeared desirable due to the high
mineral content of the water normally supplied to the
Cold Trap and Target Assembly
Initial experiments with natural carbon targets
were performed in order to test target preparation tech-
niques. Data were also later compared with enriched 12C
spectra. The initial target chamber was a 9.1 cm inside
diameter (I. D.) glass "T," separated from the main vacuum
system by an in-line liquid nitrogen cold trap. This
apparatus has already been described in some detail by
Bruton (39). It essentially consisted of a holder (cooled
by water) with accommodations for 3 targets plus a quartz
viewer. It was possible to slide the holder (on Viton 0-rings)
in a vertical direction (permitting the positioning of targets
without opening the system to atmospheric pressure). Pressures
were around 106 torr on the oil diffusion pump side of the
in-line cold trap. "Teflon" gaskets were used throughout
except on the target holder and in-line cold traps, where
Viton 0-rings were employed. Carbon deposition from organic
vapors was not apparent with this system, but several improve-
ments were devised to insure better results with the enriched
The improved in-line cold trap (designed by Dr. F. E.
Dunnam and Dr. H. A. Van Rinsvelt) had several features
which would minimize contamination of the target region by
organic vapors (figure 5). This consisted of an in-line
liquid nitrogen cold trap from Sulfrian Cryogenics, Inc.
(Model 338-1), to which a collimating system and a 36 cm
long polished copper tube have been added. The collimator
holders were mounted on the inside of copper couplers that
connected the cold trap to the vacuum system. Except for
the Viton 0-rings on the in-line cold trap the vacuum
gaskets for the beam transport system were of indium, which
is inherently free of organic contamination. The long
copper tube (2.8 cm outside diameter) was in good thermal
contact with the flanges on the cold trap. The collimating
system consisted of three 20 mil Ta discs, one on the diffusion
pump side of the cold trap and two on the target side. The
initial collimating aperture was 0.5 cm in diameter, followed
by 0.4 cm and 0.3 cm apertures on the target end. This system
made it impossible for the beam to strike the long copper
tube; furthermore, the maximum diameter beam spot on the
target was limited to about 0.3 cm by the collimators.
The target holder was designed by Dr. H. A. Van Rinsvelt
and is based on one used at Utrecht (figure 6). "Refrigerated"
water flowing between the outer and inner copper cylinders
cooled the copper endplate (0.02 cm thick) which then cooled
the target backing by conduction. The endplate wias at 550
with the beam axis. The outer cylinder was 4.8 cm in
diameter, the inner was 2.15 cm. A Ta insert fitted into
the inner copper sleeve. This insert had a circular hole
in one end that was slightly smaller in diameter (1.77 cm)
than the target backing. Extremely thin Ta fittings on this
end held the target in position. This insert (and therefore
the target backing) was pressed against the copper endplate
by tightening the bushing (lined with Ta on the beam side).
A greaseless Viton 0-ring sealed the target holder to the
glass reducer. An annulus of thin "Teflon" was placed
between the glass and the inner surface of the flange. This
limited the flow of any organic vapors (originating from the
Viton 0-ring) into the target region. The cold trap and
target holder assembly have the following characteristics
0d 4) 4
S0 pl.p m
c -, +s
C0 d H
Se |E t4 0
0 I3 >
/0 0 0 q
\o 0% O
a) The beam traverses a relatively long path through
an effective cold trap before reaching the target
b) The target holder is easily and quickly removed
from the system.
c) The target backings are effectively cooled to
insure maximum target life with relatively
large beam currents.
d) The detector may be located close to the target
in taking excitation runs (about 0.1 cm).
e) It is possible to outgas the interior of the target
chamber by heating the surfaces prior to the
performing of an experiment.
This assembly effectively minimized carbon deposition on the
target. Pressures on the diffusion pump side were typically
2-4 x 10-7 torr. The enriched 12C data were acquired with
Detectors and Circuitry
The NaI(T1) crystal (12.7 cm x 12.7 cm) was enclosed
in a 28 cm diameter lead shield in order to minimize room
background (figure 7). This was mounted on a table that
allowed the detector to be rotated about the target in a
horizontal plane. When the Ge(Li) and NaI(T1) detectors
were used simultaneously this rotation was not allowed. In
this case the detectors were set at fixed angles of 550
with the beam axis.
The detector used for most of the measurements was
a 12.7 cm x 12.7 cm NaI(T1) crystal mounted on an RCA-8055
photomultiplier tube. A 20.1 cm3 Ge(Li) detector (Ortec
Figure 7. NaI(Tl) crystal and lead shield.
Model 8101-20) was used in determining the 7 MeV cascade
radiation. The resolution of the IHaI(T1) spectrometer was
12 per cent for the 1.33 MeV gamria rays from 60Co. For this
experiment, resolution is defined as the full width (in MeV)
at half-maximum divided by the energy (in MeV) of the peakl
position. The Ge(Li) detector had a resolution of 0.5 per
cent for this gamma ray.
The NaI(T1) spectra were measured with a Nuclear
Data 512 channel pulse height analyzer (Series 130) (figure 8).
The signal from the photomultiplier went through a pre-
amplifier and then to an Ortec Selectable Active Filter
Amplifier (Model 440). The first two single channel analyzers
(Ortec Model 413) were set to sum over the 9 MeV and 7 MeV
radiation. The third single channel analyzer was used
(in conjunction with the gate generator (Ortec Model 406)
and the linear gate (Ortec Model 409)) to eliminate the
lower portion of the spectrum. In this way dead time
corrections (from the multi-channel analyzer) to the spectra
were less than two per cent. A pulse stretcher (inserted
after the linear gate) was employed on some of the runs.
The calibration and linearity of the analyzer was checked
often during the experiment using a pulser and several
known gamma rays. The following gamma rays were employed
for this purposes
a) 0.662 MeV photopeak frcm 137Cs
b) 1.17 and 1.13 MeV photopeaks from 60Co
c) 2.50 MeV sum peak from 60CO
C Z r
d) 2.75 MeV photopeak from 24Na
e) 2.225 and 4.432 MeV peaks from a Pu-Be neutron
source and its paraffin shield
f) 8.66 MeV (9.17 MeV-e) escape peak from the
C(p,y) IT reaction
g) 9.34 MeV (9.85 MeV-e) escape peak from the
C(a,y) 0 reaction.
The base of the photomultiplier tube was modified to
obtain a fast signal (from a circuit obtained from Dr. R. A.
Blue). A negative fast signal (rise time of about 40 ns)
was obtained from the anode of the 8055 tube. The slow
signal was taken from dynode 9 and therefore was positive.
The purpose of the fast signal was to eliminate electronic
pile-up problems using an EG & G Pile-Up Gate (Model GP100).
The fast circuit components (all EG & G) were set-up to
eliminate all pulses coming within 2 (s of one another
(figure 9). Although this circuit was employed for most
of the measurements, the count-rate was small, and thus this
circuit was not essential for the execution of this experiment.
The signal from the Ortec Ge(Li) detector (20.1 cm3
active volume) went into its matched Ortec Model 118A preamp-
lifier. The signal was then amplified by an Ortec Selectable
Active Filter Amplifier (Model 440) and processed by 1024
channels of a Technical Measurement Corporation 4096 channel
analyzer (figure 10). In addition to the calibration points
used for the NaI(T1) spectrometer the Ge(Li) spectrometer was
also calibrated using the second escape peaks of gamma rays
from the reaction 24Mg(a,y)28Si. The following gamma rays
11.72 MeV (12.74 MeV(Yo)-2e)
9.95 MeV (10.97 MeV(yl)-2e).
The ADC of the 1024 channel analyzer was not linear for
the larger pulses; however, itdidappear to be linear up
to at least 10 MeV. Another problem was a slight drift with
time in the gain of the ADC of the analyzer. This resulted
in a slight broadening of peaks in the spectra obtained from
long runs. The results should still be valid, even though
theseproblems were present.
Although there are obvious advantages in using the
spinless alpha particle as the projectile, there still
exists an important disadvantage. This disadvantage is
the prolific neutron background produced by the bombardment
of 13C by alpha particles. The carbon is usually in the
form of organic deposits on the vacuum system, walls, on
slits, on collimators, and on the target itself. Neutrons
produced by the 13C(a,n)160 reaction are captured by the
NaI(Tl) crystal, resulting in an exponentially decreasing
background spectrum extending to about Ey = 9 MeV. The
neutron background from the NaI(T1) detector results from
the radiative capture (by such reactions as 23Na(n,y)24Na
and 127I(n,y)128I) in the crystal (1). This is probably
the principal difficulty in performing an (a,y) experiment
using a NaI(Tl) detector. The cross section for the 13C + a
reaction in the region E. = 2 to 5 MeV ranges between 20 and
100 mb (43, 44, 45, 46, 47, 48, 28). This is indeed an
extremely large cross section compared to the 12C + 4He
capture cross section, and it was thus necessary to avoid
13C contamination. Methods employed included the preparation
of enriched (- 99.4 per cent) 12C targets and minimizing
natural carbon contamination of the system.
Room background is also a problem. For resonance
measurements the room background was not too important;
however, when off-resonant values were obtained, it increased
in significance. Shielding the detector or using an anti-
coincidence annulus will minimize this source of background
(49-). A plastic anti-coincidence annulus surrounding a
12.7 cm x 12.7 cm NaI(T1) detector had been constructed at
this laboratory by D. R. Wulfinghoff, but was not operational
when this experiment was performed. Lead shielding was
provided for the detector, and it was felt that room back-
ground did not negate the results of this experiment.
Gamma rays experience little attenuation in passing
through matter; therefore, self-supported targets are unneces-
sary in gamma spectroscopy. This is a considerable advantage
in target preparation and handling. However, because of the
prolific neutron background resulting from the 13C(a,n)160
reaction it was necessary to make targets enriched in
12C. The cost of enriched 12C isotope limits the total amount
available; thus, methods of preparation were desired which
would require a very small quantity (of the order of 10-20 mg)
of enriched 12C isotope. Two methods were investigated.
These were the thermal decomposition and gaseous (high-
frequency) discharge methods. In the thermal decomposition
process (50, 51) the target backing (10 mil Ta) is heated by
a flow of current through it. When the foil has heated and
the pressure reaches a steady value of about 5 x 106 torr,
the system is isolated and methyl iodide (CH3I), or acety-
lene (C2H2), is allowed to enter the system until the pres-
sure is about 50 torr. The carbon forms as a thin, uniform
film on the Ta backing. The thickness of the film is a
function of thedecomposition time and CH3I (C2H2) pressure.
The temperature of decomposition is quite different for the
two gases. The methyl iodide cracks readily when the foil
is heated to a very dull red, while the acetylene requires
a bright yellow glow of the foil to crack properly. The
latter high temperature causes the Ta backing to become
brittle, and the carbon film tends to peel from the backing
unless handled carefully. Targets thus could be prepared
using acetylene, but extreme caution had to be observed.
Targets could be easily prepared with the methyl iodide by
this method, and many of these (along with a few prepared
from thermally cracked acetylene) were used in preliminary
experiments on natural carbon.
It later became apparent that acetylene would be
easier to synthesize than methyl iodide (52). The gaseous
discharge method allows the efficient use of acetylene in
carbon target preparation (53). This method was finally
chosen in preparing the enriched 12C targets and, for this
reason, will be discussed in detail. Many of the techniques
involved (such as the preparation of carbides, the synthesis
of acetylene and the high-frequency discharge method) were
suggested by Dr. J. L. Duggan of Oak Ridge National Labora-
tory during a visit (supported by Oak Ridge Associated
Universities, Inc.) to the laboratory by the author,
The target preparation method is conveniently
subdivided into four separate steps which may be labeled
1) synthesis of barium carbonate (BaC03)
2) synthesis of the carbide
3) synthesis of acetylene
4) and the cracking of acetylene onto the target
Synthesis of Barium Carbonate
The enriched 12C (99.94 per cent) used in the prepara-
tion of the targets was obtained from Oak Ridge National
Laboratory in the form of graphite. The apparatus involved
in the production of BaCO3 from elemental carbon is shown
in figure 11. This mainly consists of a quartz combustion
tube coupled to a fretted glass bubbling tube. The carbon
(about 20 mg) is contained in an Al-foil capsule placed in
Figure 11. Barium carbonate apparatus.
the "plugged Teflon" stopcock at the top of the system.
Ta wire mesh in the center of the combustion tube holds
the Al capsule after it is released from the stopcock.
Oxygen flows into the system through the glass "T." The
carbon is heated forming C02 that then reacts readily with
the Ba(OH)2 contained in the test tube at the bottom of
the apparatus. The barium carbonate is formed as a white
precipitate in this test tube. This is then recovered by
vacuum filtration and dried. The efficiency (by weight)
of this process was typically greater than 70 per cent.
Synthesis of the Carbide
The carbide is prepared by placing equal amounts by
volume of the barium carbonate and clean calcium filings in
a small quartz test tube. Glass wool is positioned above
the mixture in order to confine the contents during the
reaction. The quartz tube (containing the BaCO3-Ca mixture)
is connected to a vacuum pump and heated slightly with a
Bunsen burner. The system is then isolated and heated
vigorously with an oxygen-acetylene flame. The carbide
forms a black deposit inside the quartz test tube,
Synthesis of Acetylene
Acetylene is produced by the addition of water to
the carbide (54). This process is carried out employing
the apparatus shown in figure 12. This consisted of a
waterreservoir (upper beaker), a reaction chamber and a series
of cold traps. The first two cold traps (A and B) were
alcohol-dry ice traps. The purpose of these traps is
to limit the flow of water vapor from the reaction chamber,
Acetylene (subliming at -830C) would not be removed by these
traps (having a temperature of -780C); however, it would be
quickly recovered by trap C (a liquid nitrogen trap at about
-1960C) (55). The final liquid nitrogen trap D served to
limit the possibility of contamination from the vacuum pump.
The procedure consists of breaking the quartz tube
containing the carbide (obtained from the previous process)
and immediately placing the pieces in the reaction chamber
which is then evacuated and isolated from the vacuum pump
by closing the stopcock farthest from the reaction chamber.
Distilled water is then allowed to enter the reaction
chamber. Acetylene is rapidly produced and captured as
a white solid by trap C.
The High-Frequency Cracking of Acetylene
The production of carbon targets by a high-frequency
(h. f.) discharge is described in the references (53).
The apparatus used in the present experiment was designed
so that it would rest on the base of a conventional
evaporation system (thereby using the diffusion pump, cold
trap and associated electronics of the evaporator) (figure 13).
Figure 13. High-frequency discharge apparatus.
The target backings (10 mil Ta) were placed in the 34.5 cm
long by 3,8 cm I. D. "Pyrex" tube. The tube could accommodate
four backings (three resting in the horizontal tube and one
attached to the endplate) without affecting the cracking
process. A Tesla coil furnished the h. f. discharge, The
system is evacuated to a pressure of about 3 x 10- torr,
then isolated from the diffusion pump. The Tesla coil is
activated, and acetylene is admitted into the cracking
chamber through the "side-arm" from trap C (which was removed
from the apparatus used to produce the acetylene). The C2H2
was allowed to sublime by removing the trap from the liquid
nitrogen and placing it in an alcohol-dry ice slush, thereby
allowing acetylene to enter the cracking chamber. To make
uniform targets required careful control of the C2H2 vapor
pressure. This was done by using a stopcock to limit the
flow of C2H2. Experience showed that the most efficient
pressure could be determined by the color of the glow dis-
charge. This optimum pressure is indicated by a pale green
glow with no visible striations. After the acetylene has
completely cracked, the color of the glow discharge
changes to blue, characteristic of nitrogen. At this point
the process could be halted (producing very thin targets)
or repeated several times (producing thicker targets). It
was observed that the target attached to the endplate was
thicker than the ones resting in the glass tube. It was
found that by starting with 20 mg of enriched 12C it was
possible to produce eight targets. Four were thin, three
were of intermediate thickness (15-30 keV for 3.6 MeV alpha
particles), and one was relatively thick (about 55 keV for
3.6 MeV alpha particles).
Targets prepared by this procedure were able to with-
stand beam currents of about 3 1A; furthermore, the spectra
indicated a significant enrichment in 12C (see next section).
The advantage of this procedure was that after the method
was perfected a sizable number of enriched 12C targets
was available at minimal cost. Of course, the above procedure
could also by employed to produce targets enriched in isotopes
of carbon other than 12C.
The yield curves which follow were obtained with
the 12.7 cm x 12.7 cm NaI(T1) crystal fixed at a 550 angle
with respect to the alpha particle beam. This detector was
usually 0.5 cm from the target; however, some long runs were
obtained with the detector at 2.5 cm from the target. Since
the detector was quite close to the target, summing of
cascade gamma rays may occur. A 20.1 cm3 Ge(Li) detector
was also employed in examining the 7 MeV cascade radiation
from the 9.85 MeV level in 160.
The target thicknesses were determined by employing
the narrow level (Fcm = 0.75 keV) in 160 at an excitation
of 9.85 MeV (26, 27). This part of the excitation function
for the target used for the 9.85 MeV measurements is shown
in figure 14. No corrections have been made to this curve.
By determining the full width at half-maximum, the target
thickness was found to be 25 keV for 3.6 MeV alpha particles
incident at 550 to the surface. Using standard methods the
thicknesses were extrapolated to the lower alpha particle
energies as required. A thicker target (55 keV for
3.6 MeV alpha particles incident at 550) was used for most
I I I I
3.50 3.55 3.60 3.65
Figure 14. Enriched 12C excitation function in the
region of the Ea = 3.58 MeV resonance.
off-resonance cross section measurements. Some off-resonance
cross sections were also determined with a 30 keV (for
3.6 MeV alpha particles) target.
The excitation functions in the region Ea = 2.7-3.7 MeV
for targets of enriched 12C, natural carbon and enriched 13C
are shown in figures 15, 16 and 17, respectively. The
ordinate in figures 15 and 16 represents the number of counts
per 300 p Coulombs of singly charged helium ions within the
region Ey = 8.7 MeV to 10.8 NeV. In the excitation function
for the enriched 13C target the ordinate represents the
number of counts (in the same sum region as for the natural
carbon and enriched 12C targets) per 150 Coulombs. Beam
currents were usually between 1.5-3 pA. The detector was at
a distance 0.5 cm from the target and at an angle of 550
with the beam axis. The enriched 12C target was 15 pg/cm2
thick, the natural carbon target was 22 pg/cm2 thick and
the enriched (probably 15-25 per cent enrichment) 13C target
was 14 pg/cm2 thick. The enrichment of 12C is indicated by
comparing the resonance at E. = 3.58 MeV in the three figures.
The broad peak in the enriched 1C spectrum at E, = 3.24 MeV
is in the region of the 9.59 MeV excitation energy in 160;
however, most of this yield is probably due to the 13C(an)160
reaction. It is difficult to determine the actual enrichment
of 12C, since some of the background is due to neutrons
0 00 0
0 0 0 0
0 0 0
o 0 0 0 0
O O O O
O O O O
-1-o o<002 /Si
o 0 0 0
0 0 0 0
(0 N cO V,
originating at collimators, etc. Certain runs on tantalum
backings (at Ea = 3.24 MeV) indicate that this source of
background may be as much as 30 per cent of the total back-
An indication of a resonance due to the 12C(a,y)160
reaction at Ea = 3.48 MeV was observed in the excitation
functions for the enriched 12C and natural carbon targets.
This structure was not observed in the excitation function
for the enriched 13C target. No attempt was made to determine
conclusively whether this structure could be attributed to
the 12C(ty)160 reaction.
The 9.85 MeV Level in 160
The level at E = 3.58 MeV corresponds to a level in
160 at 9.85 MeV (2+). This level has been observed through
the 12C(ay)160 reaction by Meads and Mclldowie, and Larson and
Spear (10, 12). A spectrum taken at E = 3.58 MeV is shown
in figure 18. No corrections have been made. Larson had
observed 7 MeV cascade radiation in the spectrum from his
thick target (11). This 7 MeV radiation is also visible in
figure 18. Because of the resolution of the NaI(T1) detector
it is difficult to determine the level in 160 through which
this cascade proceeds. For this reason the level was
examined using a 20.1 cm3 Ge(Li) detector. A spectrum from
the 1024 channels of the TMC analyzer is shown in figure 19.
This is a point-by-point sum of individual runs of
Figure 18. Gamma ray spectrum from NaI(T1) detector at
Ea = 3.58 MeV, The three arrows correspond to the positions
of the photopeak and the two escape peaks,
., ____ -.... ---,----
- --~-?-- ---T
N L4. .
0.09, 0.05 and 0.12 Coulomb. The (9.85 MeV-2e) peak is
clearly visible. The smaller peak is from the (6.92 MeV-2e)
radiation. The peak corresponding to the 2+_ 2+ transition
((9.85 6.92) MeV-2e) is also evident. The other peaks
apparently correspond to various background gamma rays. From
this it appears reasonable that the cascade radiation proceeds
through the 2+ level at 6.92 MeV in 160. The results for
the radiative widths (determined from NaI(Tl) data) for the
9.85 MeV ground state and cascade transitions are given in
table 3. The value Fy = 0.0050 eV corresponds to
IMI2 = 2.76 x 102, where IMI2 is the ratio of r to the
radiative width calculated from an extreme single particle
model (F'). Fy is the so-called Weisskopf limit. In deter-
mining the radiative width for the cascade radiation, Larson
assumed an isotropic distribution since he was not able to
determine the identity of the intermediate level (11).
The sources of error are given in table 4. Errors
due to current integration, extrapolation of stopping cross
sections to different energies, and geometry have been
combined and listed in the column labeled "other". The total
error is the square root of the sum of the squares of the
individual errors. From table 4 it can be seen that no error
is listed under "target thickness" for the 9.85 MeV radiations.
Due to the extremely narrow total width of this state
(Fcm = 0.75 keV) the factor arctan(6/r) in the expression
for the radiative width reduces to n/2 and is not a source
of error. The difficulty in extracting the 9.85 MeV cascade
< 0\ o rN f^ 3 cO r p
SO O VI O H v c O
Sr4 rr C N rN1
co o o\ o\ o\ o\ 0\ o\ o\ %o\ "
Hn H 4 N C 4 N 4 N C 'O
S-I H H H H H H r-H r- i-
0 N N N C0-N 0\N N- C- o I I
N 4 0 N 4 0c 0
n N H 0O C NEl U% n' H O\ WN V%
* 4 4 r A
C C^ CO f N N N N N H lCM C^
radiation in the presence of the strong ground state
radiation caused a 10 per cent error for "background sub-
traction" for the cascade radiation. The brancing ratios
for the 9.85 MeV level are given in figure 20.
Cross Sections and the 9.59 MeV Level
The 1" state at 9.59 MeV excitation in 160 was
examined with the 29 pg/cm2 thick target. This level is
very broad, having a center-of-mass width of 645 keV (26).
Figure 21 shows a spectrum from the NaI(T1) crystal at
E = 3.24 MeV. There were no corrections made to this
spectrum. The detector was 0.5 cm from the target and
at an angle of 550. The total amount of charge collected
was 5.5 x 10 o Coulombs of singly ionized helium. The arrows
indicate the expected positions of the photopeak and escape
peaks. For comparison purposes a spectrum at E. = 2.95 MeV
obtained from the enriched 13C target (14 pg/cm2) is shown
in figure 22. Spectra obtained at Ea = 3.24 MeV with this
target have the same general appearance. The spectrum
from the 13C is exponentially decreasing under the radiation
from the 9.59 MeV level. This same feature is also observed
in the spectra from a natural carbon target (figure 23).
The 9.59 MeV yield was obtained by subtracting
fractional parts of the 9.85 MeV spectrum from the 9.59 MeV
spectrum. The resulting spectrum was then compared with a
13C(,n)160 spectrum to determine ts fit with the background
C(cxn) 0 spectrum to determine its fit with the background.
9.85 M eV Z
Figure 20. Branching ratios of 9.85 MeV level in 160
The energy levels are not drawn to scale.
Figure 21. Gamma
Ea = 3.24 MeV for
ray spectrum from NaI(T1) detector at
enriched 12C target.
200 300 400
Gamma ray spectrum at Ez = 2.95 MeV for enriched
-1 9.37 MeV
10- I I I
200 300 400 500
Figure 23. Gamma ray spectrum at E = 2.95 MeV for natural
E,= 2.95 MeV
104 45000 pCOULOMBS
h 9.37 MeV
0 b r
150 200 300 400 500
Figure 24. Gamma ray spectrum at E. = 2.95 MeV for enriched
10 -I ..I .I,
200 300 400 500
Figure 25. Gamma ray spectrum at E = 2.14 MeV for enriched
This same method was also used in determining the off-
resonance yields. This procedure gave a value of
0.021 + 0.004 eV for the radiative width. A comparison with
previous results is given in table 5. The present work
agrees well with the value given by Larson (11). The value
Ty = 0.021 eV gives a value of 5.5 x 10-5 for IM12.
Although there is some indication of 7 MeV cascade
radiation in the 12C spectrum near channel 280, this feature
is also seen in the 13C spectrum. It is felt that the
spectrum peak in channel 280 is the result of the capture
of thermal neutrons by the 23Na in the NaI(Tl) crystal.
When a paraffin shield was installed between the detector
and the collimators, the spectrum peak became more pronounced.
The curves shown in figures 21, 22 and 25 were obtained
with this shield, while the data shown in figures 23 and 24
were obtained without the paraffin shield. There was some
indication that although this shield increased the 7 MeV
background, it decreased the background in the 9 MeV region.
It has been observed that 9 MeV radiation (resulting
from the 1C(d,y) N reaction) may also present a background
problem in a 12C + a experiment (11). The reason presented
is that if deuterons exist in the ion beam, the beam energy
analyzing magnet will permit the target to be bombarded with
deuterons having one-half the energy of the alpha particles,
since molecular D2+ ions will have nearly the same path as
singly ionized He. The bombardment of C by 1.65 MeV
deuterons produces 9.4 MeV cascade radiation to the 2.31 MeV
first excited state in 1N. The r. f. source bottle used
for this experiment has never been used to ionize any
hydrogen isotope; hence, it was assumed that the contribution
from deuteron-induced reactions was negligible in the present
The off-resonance cross sections were determined from
spectra such as the one shown in figure 24. The 52 Ag/cm2
thick target was used for this data. The ground state
radiation is clearly visible in channels 355-390. A
comparison with figure 22 shows that the number of counts in
the corresponding channels for the 13C spectrum is decreasing
almost exponentially. For the spectrum at E = 2.95 MeV
(as for the other cross section measurements) the 12.7 cm by
12.7 cm NaI(T1) detector was at a 550 angle with respect to
the beam axis, and was located 0.5 cm from the target. The
ground state radiation is more visible at EC = 2.95 MeV
(figure 24) than at the peak Ea = 3.24 MeV (figure 21). Two
factors contribute to this appearance. First, the average
number of counts per channel in figure 21 is about 10 times
that in figure 23; hence, on the logarithmic scale the peak
is not as evident. Also, the 13C(a,n)160 cross section is
about 5 times as large at Ea = 3.24 MeV as at Ea = 2.95 MeV
(43). Thus, the ground state radiation is more difficult
to observe at the resonant energy because of the increased
The yield spectra for the other data points are
similar to the spectrum shown in figure 24. As the alpha
particle bombarding energy decreases, the spectra resemble
the 13C background more closely. For example, at Ea = 2.14 MeV
there is only a slight indication of the ground state radiation.
This is shown in figure 25, where the data were obtained with
the 28.8 jg/cm2 thick target. Because of this the error
caused by extracting the yields from the backgrounds are
relatively large for the cross section meaurements made at
low bombarding energies (table 4). The amount of singly
ionized helium collected was typically between 0.045 and
The yield at each of the experimental points was
normalized to that at 3.58 IeV by multiplying the total number
of counts in the capture spectrum peak by the ratio of the
number of channels enclosing the peak to the number of channels
in the standard spectrum peak. The values of the cross
section are listed in table 6. The errors shown were computed
from the values given in table 4.
The 12C(ay) 160 reaction cross section.
He energy Cros- section
(MeV) (10-' barns)
3.37 32.0 + 6.1
3.24 35.0 + 6.1
3.22 35.0 + 6.1
3.10 29.0 + 5.8
3.07 19.8 + 3.8
2.95 15.2 t 3.0
2.92 11.9 + 2.4
2.76 8.0 + 1.8
2.54 4.3 + 1.3
2.35 3.5 1.2
2.14 1.5 + 0.5
1.90 0.3 + 0.1
In this experiment the radiative capture of alpha
particles by 12C below Ea = 3.6 MeV has been examined. The
7 MeV cascade radiation from the 9.85 MeV (2+) level in 160
was determined to proceed through the 6.92 MeV (2+) level.
It would be of interest to examine this (0+_ d 2+ MIE2 2+)
transition using angular distribution methods to determine
the mixing parameter X, where X = S(M1)/S(E2). S(M1) and
S(E2) are amplitudes for dipole and quadrupole emission,
respectively (29). Further study of this state would
probably best be accomplished through angular correlation
methods. At the time this experiment was performed, a new
correlation table was being designed and constructed by
members of the laboratory. This table will be able to make
the above measurements accurately. For this reason no
determination of the mixing parameter X was attempted using
the present apparatus.
The radiative width measurements agree in general
with the results of Larson (the errors overlap); however,
the values tended to be slightly smaller than the values
obtained by Larson. This may be due in part to the method
used in determining the peak-to-total ratio for the present
Target deterioration was present only at higher beam
currents (about 4-7 ;A).- At these beam currents the carbon
was dissipated, and the tantalum backing became quite pitted.
Carbon deposition was not observed on the target, even after
the relatively long bombarding times. In order to observe
this more carefully, several relatively long runs (greater
than 0.04 Coulomb) were made on clean Ta backings using
both low (1-3 ~A) and high (4-8 1A) beam currents. No
deposition of organic material was observed for either of
these currents. However, the higher beam currents did cause
the tantalum to change color at the spot which had been
bombarded by the beam. This spot was gold in color on both
sides of the backing. This is believed to be caused by the
tempering of the tantalum by the concentration of heat on the
beam spot. This same effect was observed during preliminary
experiments in target preparation by thermal cracking methods.
When the Ta backing was outgassed by heating the backing to
a bright yellow glow in a vacuum, the Ta backing would have
the characteristic gold color and be very rigid when removed
hence, it seemed that the backing had been tempered by the
high temperature used in the outgassing process. It might
be noted here that there was some evidence (using the natural
carbon target made by h. f. cracking) to indicate that
backings that had been outgassed in a vacuum produced targets
which exhibited greater resistance to deterioration (by
high beam currents) than targets produced from backings that
had not been outgassed. Enriched 12C targets were made using
Cu backings, but these were not used in this experiment,
These may be able to withstand higher beam currents than
those made with tantalum backings.
The cross section for the 12C(a,y)160 reaction at low
alpha particle bombarding energies is necessary for an accurate
extrapolation to effective stellar energies. The Coulomb
barrier Bc for the 12C + 4e can be approximated by
Bc = Z1Zoe2/ ,
where R = 1.4 x (A13 + Ao3 ) x 10-13 cm.
The prescription gives the value 3.18 MeV for B ; hence,
below 3 MeV barrier penetration becomes increasingly important.
Tombrello has computed the effect of interference
between the 1- states at 7.12 MeV and 9.59 MeV in 160.
Using a value a2 = 0.10 for the reduced width of the 7.12 MeV
state, he has calculated the values of the capture cross
section for both constructive and destructive interference
(22), Since the signs of the amplitudes were unknown, it
was impossible to determine whether the interference was
constructive or destructive.
The values for the 12C(a,y)160 reaction cross section
obtained from the present experiment, and those computed by
Tombrello (assuming ea2 = 0.1) are plotted in figure 26.
The value obtained for the capture cross section at
Ea = 3 MeV is slightly smaller than that calculated by
Tombrello; while at Ea = 1.9 MeV it is about 1.5 times as
large as the calculated value. The results of this experi-
ment indicate that the interference is constructive and that
Oa2 is between 0.1 and 0.3,
The 12C(a,)160 Reaction Rate
at Effective Stellar Energies
The reduced alpha particle width 0a2 of the 7.12 MeV
level in 160 is essential in the determination of the
12C(a,y)160 reaction rate at stellar temperatures (see the
section on theory). This reaction rate is important in
stellar model calculations since it determines the relative
abundances of 12C and 160 and hence the further evolution of
This experiment indicates the cross section factor So
at a typical stellar temperature of 108 OK has the value
So(12C(a,y)160) = (0.65 0.32) x 103 keV barns,
o 09.59 MeV LEVEL
/$ Ad DESTRUCTIVE
S 4 FROM- TOMBRELLO
C= CALCULATED (22)
Q / /* PRESENT WORK
o0 / aREF. (11)
/ / REF (14)
/y/ (-cE 12)
S/ o'/(E) u S(E) e
c = 31.28 Z ZoA 1/2
,01-2 I -TI l-I
2.0 2.5 3.0
E a (lab) M eV
Figure 26. The 12 16
Figure 26. The 12(ay) 0 reaction cross section values.
The solid line has been drawn through the calculated points.
using 8 = 0.2 + 0.1. Similar (using the equations derived
in the section on theory), the present experiment yields
the following value for the 12C(a,y)160 reaction rate at a
stellar temperature of 108 OKI
p (12C) = (0.84 + 0.42) x 10-20X Pf2C s-1
where Pis the density of the core in g cm-3 (-105), X is
the abundance (-1) by weight of He and f12 is the screening
factor (37). Although the above errors on So and p may
appear large, it should be noted that prior to this experiment,
0 could only be estimated to within a factor of 10 at best.
Also, it is quite probable that the errors will be further
reduced as revised fits are obtained to the measured values
of the capture cross section.
A determination of 12C + He capture cross sections
at lower bombarding energies than those employed in this
experiment would also produce a better value for the reaction
rate. Experimental procedures that could reduce the
13C(a,n)160 background would be necessary. Using-a more
highly enriched 12C target, and time-of-flight methods to
eliminate neutron background originating from the collimators
may be the most useful procedure since it appeared that same
of the neutron background resulted from the beam striking
collimators and slits on which carbon had deposited.
GAMMA RAY DETECTION EFFICIENCIES OF NaI(T1) CRYSTALS
Total Incident Intrinsic Efficieny
The "total incident" intrinsic efficiency, 7T(Ey), of
a NaI(Tl) crystal in the form of a right cylinder for gamma
rays of energy E is defined as the probability for the
detection of a gamma ray that is incident on the face of the
crystal (56, 57). This is equal to the ratio of the total
number of gamma rays detected per second to the total number
of gamma rays incident on the face of the crystal. In the
form of an equation, this is
1T(E) = -
where NT is the total number of gamma rays of energy E
detected per second, No is the total number of gamma rays
of energy E emitted per second by an isotropic source, and
f is the fraction of 4v that the solid angle of the crystal
face subtends. Stated differently, f is defined by
where 9 is the solid angle subtended by the crystal.
It is evident that NT/I'o is simply the fraction of the
gamma rays detected, and f is simply the probability that a
gamma ray from an isotropic source will strike somewhere on
the face of the crystal. The geometry for the case of an
isotropic point source of gamma rays of energy E located
on the axis of a right cylinder of radius R and thickness t
is shown in figure 27.
The parameter x2(e') is the path length in the NaI(T1)
crystal. The parameter xl(e') is the path length in any
absorbing material preceding the crystal. The quantities
x2 and x1 have the following explicit dependency on e':
xl(e') = d sece', for 0 <
x2(e') = t sece', for 0 e < e = arctan
x2(e') = R csce' d sece', for ei < eo = arctan -d
The intensity of the source will be attenuated by a
factor exp(- L1(E)xl(e')) in traversing the absorbing material
preceding the crystal. 1 is the total linear absorbing
coefficient for a gamma ray of energy E in the absorbing
material (58). In traversing the NaI(T1) crystal the gamma
ray intensity will be attenuated by a factor exp(- L2(E)x2(e')),
where I2(E) is the total linear absorption coefficient for
a gamma ray of energy E in NaI(T1).
N /4rT is the intensity per unit solid angle (the number