Chapter 1 Introduction
3.4 Experimental methodology for cross section measurements
The initial data analysis comprises of: (i) energy calibration procedure, and (ii) efficiency measurements. This section will also describe how the total cross sec- tion was calculated for two γ-rays with energies of 754.8 keV and 3736.5 keV. Details about the differential cross section calculations and the evaluation of the uncertainty are also given.
3.4.1 Energy calibration
The data for energy calibration was collected before the in-beam data was taken.
Three radioactive sources, i.e. 137Cs, 60Co and 152Eu were used for energy cali- bration. Each of these sources was placed at the center of the AFRODITE array.
These radioactive sources allow us to cover the energy range of 122 keV to 1408 keV, because:
• the 137Cs source emits a γ-ray of 662 keV
• the 60Co source emits two rays, 1173 keV and 1332keV
• and the152Eusource emitsγ-rays in the energy range from 122keV to 1408 keV
The centroids of the photo peaks for each source were obtained using gf3 radware program [Rad95]. gf3 has a peak fitting function using a least squares minimization algorithm. The code was designed primarily for use in the analysis of spectra of γ-rays measured with germanium detectors. The centroids obtained with gf3 were listed on an excel spreadsheet, then used to determine an energy calibration for each crystal, in the form:
E =a+bx+cx2 (3.1)
Chapter 3. Experimental Data and Results 33 where x is the channel centroid. The coefficients a, b, and c are gain matching coefficients, that map the channel to energy calibration for clovers.
Complex Gaussian-type form was used to fit each photo peak in each group of peaks. In gf3, a portion of the spectrum up to fifteen peaks can be fitted si- multaneously assuming a quadratic polynomial background. Each peak is com- posed of three components: (i) a Gaussian; (ii) a skewed Gaussian; and (iii) a smoothed step function to increase the background on the low-energy side of the peak [Rad95]. The free parameters are the position adjustment of the centriods, the amplitude of the most intense Gaussian component, as well as its relative width.
3.4.2 Efficiency calibration
The clover detector efficiency calibration measurements were performed using the above mentioned radioactive sources. When performing the efficiency calibration each source must be positioned exactly at the target position. All the events were recorded and sorted offline. The efficiency was fitted using the following equation:
ln(ε) =
A+Bx+Cx2−G
+ D+Ey+F y2−G −G1
(3.2) where
x =log 100Eγ
(3.3) y =log 1000Eγ
(3.4)
where the A, B and C coefficients describe the efficiency at low energies and the D,EandF coefficients describe the efficiency at high energies. Gis an interaction parameter between the low and high energy region; it determines the efficiency in the turn-over region. The effit program [Rad95] from the Radware software was used to calculate the relative efficiencies for all the curves. The clovers’ relative efficiencies drop after reaching a maximum of ∼150 keV. The decrease in the efficiency is caused by the decrease in the cross section of both the photoelectric effect and the Compton scattering as a function of the γ-ray energy.
3.4.3 Efficiency calculations
Relative efficiency is a general performance measure of a detector that reflects the efficiency for detecting the 1332 keV γ-ray from a standard 60Co source to the efficiency of a standard sodium iodide scintillation detector. The intention of the efficiency calculation performed in this work is to relate the peak area in the spectrum to the number of γ-rays emitted. The absolute photo peak efficiency is the ratio of the emitted γ-rays by the source and the number of counts in the corresponding photo peak. It depends upon the geometrical arrangement of the source and detector, as well as on the intrinsic efficiency of the detector. The absolute total efficiency relates the number of γ-rays emitted by the radioactive source to the number of counts produced anywhere in the spectrum. The intrinsic efficiency is defined as the ratio between the counts in the spectrum to the number of the γ-rays incident on the detector.
The detector efficiencies were calculated using the absolute photo peak efficiency formula:
ε= A
A(t)× 4t×I (3.5)
where:
• ε is the detector photo peak efficiency,
• A is the photo peak area of interest in the background subtracted source spectrum
• A(t) is the absolute activity of the radioactive source [Bq],
• I is the intensity for the γ-ray produced per decay
• 4t is the live time acquisition of the source spectrum.
The efficiency of the detector remains constant over the experiment.
Activity calculations
The average number of disintegrations of the radioactive source per second is called activity and is measured in units of Becquerel (Bq). One decay per second
Chapter 3. Experimental Data and Results 35 is equal to one Becquerel. The activity of a radioactive sources is defined based on the following equation:
A(t) = dN
dt (3.6)
where dN is the change in the number of undecayed nuclei and within the time intervaldt, measured in seconds.
The radioactive decay law is written as;
N =N0eλt (3.7)
where N is the number of undecayed nuclei at the instant t, N0 is the number of undecayed nuclei att = 0,λ is decay constant andtis time aftert = 0 in seconds.
3.4.4 Transition probabilities
A γ-ray is emitted by a nucleus during a transition from an initial state of spin Iiπ towards a final state with spin Ifπ. The photon emitted during the decay must carry an angular moment L which satisfies the relation:
|Ii−If| ≤L≤ |Ii+If| (3.8) in order for the transition to be allowed between these states. The multipolarity order of the γ-ray is given by L, which takes values of 1 for a dipole transition, 2 for a quadrupole transition and so on. The γ-ray can be of electrical or magnetic nature, with a parity of π(EL) = (−1)L and π(M L) = (−1)L+1. The γ-rays can link two nuclear states with parities πi and πf for the initial and final states if 4π =πiπf =π. Therefore,
4π = +1
corresponds to even electric multipoles and odd magnetic multipoles 4π = -1
corresponds to odd electric multipoles and even magnetic multipoles
The γ-ray selection rules and multipolarities are given in table 3.2. For example, the 3736.5 keV γ-ray 40Ca is a transition from the first excited state Iiπ = 3− to the ground state Iiπ = 0+. If we apply the selection rules (equation 3.8) to find the multipolarity of the γ-ray emission, the angular momentum of this γ-ray is 3 (L= 3), and there is a change in parity, therefore the multipolarity of the 3736.5 keV γ-ray emission is electric octupole (E3).
Table 3.2: Gamma-ray selection rules and multipolarities Multipolarity
state of the gamma
Name L = 4I 4π
E1 electric dipole 1 -1
M1 magnetic dipole 1 +1
E2 electric quadrupole 2 +1
M2 magnetic quadrupole 2 -1
E3 electric octupole 3 -1
M3 magnetic octupole 3 +1
E4 electric haxadecapole 4 +1
M4 magnetic haxadecapole 4 -1
3.4.5 Angular distribution
Angular distribution is the distribution of the intensity of the γ-rays that were emitted by a nucleus as a function of the angle. The γ-ray angular distribution was explicitly determined by measuring the yields at several angles (for each beam energy and for each Ge crystal) and fitting them with a series of Legendre poly- nomials of even order; e.g. order of 4 and 6 for the 754.8 keV and 3736.5 keV transitions in 40Ca, respectively. Taking such measurements for each crystal re- sulted into a better angular distribution, since it allowed us to cover two angles per clover detector rather than one. The angular distribution measured can be expressed in terms of the Legendre polynomials Pl(cosθ);
W(θ) =
l=lmax
X
l=0
alPl(cosθ) (l−even) (3.9)
where, al are the angular distribution coefficients and lmax is the smaller of the following two quantities: i) twice the spin of the decaying state, and ii) twice
Chapter 3. Experimental Data and Results 37 the multipolarity of the γ-ray [Dye81]. The angular distribution function can be rewritten as:
W(θ) = a0+a2P2(cosθ) +a4P4(cosθ) +a6P6cosθ+... (3.10) The angular distribution function for some multipolarities are written below [Kra87].
For an E1/M1 transition:
W(θ) = a0+a2P2(cosθ) (3.11) For an E2/M2 transition, for instance the 754.8 keV transition of 40Ca, which is a 5− →3− transition with an E2 multipolarity:
W(θ) =a0+a2P2(cosθ) +a4P4(cosθ) (3.12) For an E3/M3 transition, for instance the 3736.5 keV transition of 40Ca, which is a 3− →0+ transition with anE3 multipolarity:
W(θ) =a0+a2P2(cosθ) +a4P4(cosθ) +a6P6(cosθ) (3.13) where;
P2(cosθ) = 1
2(3 cos2θ−1) (3.14)
P4(cosθ) = 1
8(35 cos4θ−30 cos2θ+ 3) (3.15)
P6(cosθ) = 1
16(231 cos6θ−315 cos4θ+ 105 cos2θ−5) (3.16) For the case of the 3736.5keV γ-ray, there are four terms in the angular distribu- tion function in equation 3.13, so at least four data points are required to obtain the angular distribution coefficients, while for anE2 transition at least three data points are needed. The four high purity germanium crystals of the clover detectors are placed at 85◦and 95◦for the clovers at 90◦, and at 130◦ and 140◦for the clovers
at 135◦ while for the clover at 168◦ the high purity germanium crystals are placed at 163◦ and 173◦. The angular distribution function W(θ) is symmetric with re- spect to 90◦. Data points at detection angles of 85◦ and at 95◦ are symmetric and therefore can be considered as one data point. However in this work we have used these two data points independently.
3.4.6 Differential angular distribution
In order to find the area under the photo peaks of the 3736.5 keV and the 754.8 keV transitions, the counts above background in each channel of the photopeak were added together using radware. Once the areas representing the number of photopeak events for eachγ-ray are available, they are used for the determination of the differential cross section for the production of these γ-rays. The differential equation can be calculated using the following equation:
dσ
dΩ = A
Q×ε×X (3.17)
where:
• A is the area of the peak of interest measured in the background-subtracted
40Ca spectrum
• ε is the efficiency of the Ge crystal at this energy,
• X is the target thickness
• Q is the total number of incident particles
The differential cross section values depend on the efficiency of the detector used.
The differential cross section uncertainty 4
dσ dΩ
can be written as:
4dσ dΩ
= dσ
dΩ
s (4A
A )2+ (4X
X )2+ (4Q
Q )2+ (4ε
ε )2 (3.18) Where 4A is the uncertainty in the number of events, 4X is the uncertainty in the target thickness, 4Q is the uncertainty in the number of incident particles and 4ε is the uncertainty in the efficiency.
Chapter 3. Experimental Data and Results 39 It is worth noting that the intrinsic spin of the photon is equal to 1. It follows that a γ-ray transition with L = 0 is prohibited, and a γ-ray transition from a Ii = 0 to a If = 0 state is prohibited [Bro05]. This type of transition is carried out by internal conversion and / or a creation of a pair.
3.4.7 Total cross sections
The total cross section is defined as an effective area that quantifies the essential likelihood of an interaction when an incident particle strikes a target nucleus. By definition, the total cross section for the production of particularγ-ray is obtained by integrating the entire angular range of the differential cross section. It is given by the zero order (a0) angular distribution coefficient in equation 3.11, namely:
σ = 4πa0 (3.19)
The cross-sections carry information about the total intensity of the γ-rays pro- duced in the nuclear reaction processes.
Chapter 4
Geant4 AFRODITE modelling
This chapter explains details about the Geant4 AFRODITE modelling that was used in this study. We also tested the AFRODITE Geant4 model with experimen- tal data. In addition, the methodology for determining the absolute efficiency for the AFRODITE clover detectors within the simulations is described.