Angular correlation measurements with a segmented clover detector in a close geometry

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Nuclear Inst. and Methods in Physics Research, A 1008 (2021) 165458

Contents lists available atScienceDirect

Nuclear Inst. and Methods in Physics Research, A

journal homepage:www.elsevier.com/locate/nima

Angular correlation measurements with a segmented clover detector in a close geometry

S.H. Mthembu

^a^,^b^,^c

, E.A. Lawrie

^a^,^c^,∗

, J.J. Lawrie

^a

, T.D. Bucher

^a^,^d

, T.R.S. Dinoko

^a^,¹

,

D.R.R. Duprez

^d

, O. Shirinda

^a^,^d^,²

, S.S. Ntshangase

^b

, R.T. Newman

^d

, J.L. Easton

^a^,^c

, N. Erasmus

^a^,^c

, S.P. Noncolela

^a^,^c

aiThemba LABS, National Research Foundation, PO Box 722, Somerset West 7129, South Africa

bPhysics Department, University of Zululand, Private Bag X1001, KwaDlangezwa 3886, South Africa

cPhysics Department, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa

dDepartment of Physics, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa

A R T I C L E I N F O

Keywords:

Angular correlations Mixing ratios

133cs

Segmented clover detector Close geometry

A B S T R A C T

Angular correlation measurements were carried out with a segmented clover detector in a close geometry where the segments were used as individual detectors. This allowed up to 35 unique angles when only front segments are used and up to 140 unique angles when both front and back segments are used. The angular correlation was tested with⁶⁰Co and¹³³Ba sources. The large number of angles and the close geometry, allowing increased efficiency, produced precise angular correlation measurements and allowed the determination of several multipole mixing ratios in¹³³Cs with a precision comparable to that of the presently adopted values.

1. Introduction

The determination of (i) the spin and parity of nuclear states, and of (ii) the multipolarity and mixing ratio of the emitted 𝛾 rays is important for nuclear structure studies. In particular they are needed to establish reduced transition probabilities which in turn carry direct information on the similarities of the wave functions of the initial and final nuclear states. For instance, the magnitude of the mixing ratios of the𝛥𝐼 = 1transitions linking two bands shows whether the excited band is generated by single-particle or by collective excitation with respect to the yrast band. This is of crucial importance when one needs to identify collective bands, such as wobbling or 𝛽 and𝛾 vibrations, as opposed to single-particle excitations (for the impact on the identification of wobbling bands see for instance [1]). Measured mixing ratios are also needed to determine possible𝐸0components in 𝛥𝐼 = 0transitions. Such a component is direct experimental evidence of co-existing nuclear shapes.

The angular correlation technique is a powerful technique for measuring multipolarities and mixing ratios of nuclear transitions. It had been extensively used during the 1960s and 70s, mostly with a limited number of relatively small detectors, initially Ge(Li) detectors [2,3].

A review of early angular correlation work can be found in Ref. [4].

In subsequent years there had been a significant advance in detector

∗ Corresponding author at: iThemba LABS, National Research Foundation, PO Box 722, Somerset West 7129, South Africa.

E-mail address: [email protected](E.A. Lawrie).

1 Present address: Department of Physical and Electrical Metrology, NMISA, Private Bag X34, Lynnwood Ridge, Pretoria 0040, South Africa.

2 Sol Plaatje University, Department of Physical and Earth Sciences, Private Bag X5008, Kimberley 8301, South Africa.

technology and for the last three decades a large number of Compton- suppressed high-purity Ge (HPGe) detectors have been utilized in a number of𝛾-ray arrays [5]. The angular distribution and Directional Correlation of Oriented nuclei (DCO) techniques [6] were commonly employed to measure𝛾-ray multipole order and assign spins to nuclear states. Often, due to the symmetric configurations of the𝛾-ray arrays, only a limited number of independent measuring angles were available, reducing the accuracy of the angular distribution and DCO techniques when applied to mixing ratios of the emitted𝛾rays.

A renewed interest in accurate angular distribution and angular correlation measurements followed the new developments in𝛽-decay studies, in particular applied to exotic nuclei following the development of radio-active beams. For instance, the GRIFFIN array [7], built of 16 large volume clover detectors, allows for 51 unique angles in terms of angular correlations if the Ge crystals in the clovers are considered as individual detectors [8]. Using the crystals as individual detectors allows a large number of measuring angles, but reduces the efficiency, as the addback in the clovers (where energies of coincident events in adjacent crystals are added) is lost. The latest-generation𝛾-ray arrays such as AGATA [9,10] and GRETINA [11] provide excellent (of the order of a couple of mm) position resolution of the𝛾-ray interactions due to their tracking capability. This allows a specific angle bin to be selected by the user depending on the statistics and the required number of measuring angles.

https://doi.org/10.1016/j.nima.2021.165458

Received 27 January 2021; Received in revised form 30 April 2021; Accepted 18 May 2021 Available online 24 May 2021

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Fig. 1. (a): Two consecutive 𝛾 rays for which angular correlation analysis can be carried out. (b) Sketch illustrating that the angle𝜃is energy dependent. The arrows point to the mean interaction positions for𝛾 rays with energies of𝐸₁ (red) and𝐸₂ (green), where𝐸₁< 𝐸₂.

In this work angular correlation analysis with a segmented clover detector was carried out in close geometry. The 32 segments of the four Ge crystals were used as independent detectors. It provides up to 140 unique measuring angles that, for a source-to-crystal distance of 40 mm, cover a range of angles between∼6^◦and∼80^◦. The large number of measuring angles in addition to the increased photopeak efficiency (due to the close geometry) result in high precision angular correlation data particularly suitable for accurate determination of mixing ratios.

2. Gamma–gamma angular correlation

The angular distribution function for a 𝛾 ray emitted from an oriented nuclear state (for instance a state with total angular momen- tum aligned perpendicular to the beam direction following a fusion- evaporation reaction) reflects the probability for this𝛾ray to be emitted at a specific angle 𝜃 (with respect to the beam direction) and is governed by the multipole order 𝐿and the multipole mixing ratio𝛿 of the𝛾ray and the spins of the initial,𝐼_𝑖, and final,𝐼_𝑓, nuclear states.

For randomly oriented nuclear states, as populated in 𝛽 decay, the angular correlation analysis of two consecutive𝛾 rays (seeFig. 1(a)) is appropriate.

The angular correlation function depends on the multipole orders 𝐿₁(+𝐿^′₁)and𝐿₂(+𝐿^′₂)(where𝐿^′=𝐿+ 1is the multipole order of the higher order admixture,𝛿²_𝑖 =𝑇(𝐿^′_𝑖)∕𝑇(𝐿_𝑖), and𝑇(𝐿_𝑖)is the transition probability) of both𝛾rays and on the spins of the initial (𝐼_𝑖), intermediate (𝐼_𝑚), and final (𝐼_𝑓) nuclear states. It reflects the probability that the second𝛾 ray is emitted at an angle𝜃with respect to the first. The theoretical angular correlation function of such consecutive 𝛾 rays is expressed as [4]

𝑊(𝜃) =

∑∞

𝑙=0,𝑒𝑣𝑒𝑛

𝐵_𝑙𝐺_𝑙(𝑡)𝐴_𝑙𝑃_𝑙(𝑐𝑜𝑠𝜃). (1)

𝐵_𝑙 describes the initial nuclear orientation and, for non-oriented states such as those following𝛽decay,𝐵_𝑙= 1.𝐺_𝑙are time dependent perturbation factors. For lifetimes of the intermediate states that are short (in this study the lifetimes are shorter than a few ns)𝐺_𝑙(𝑡) = 1.𝐴_𝑙 are the angular correlation coefficients that depend on the spins of the three states and the multipole orders and the mixing ratios of the two𝛾 rays. (The Krane–Steffen³phase convention was used in the definition of the mixing ratios [12].)𝑃_𝑙(𝑐𝑜𝑠𝜃)are Legendre polynomials of order 𝑙;𝜃is the angle between the two𝛾 rays, and𝑙≤4is used in this work as the transitions of interest are dipoles and quadrupoles, [13]. The angular correlation function is

𝑊(𝜃) =

∑∞

𝐴_𝑙𝑃_𝑙(𝑐𝑜𝑠(𝜃))

= 𝐴₀{1 +𝑎₂𝑃₂(𝑐𝑜𝑠(𝜃)) +𝑎₄𝑃₄(𝑐𝑜𝑠(𝜃)) +⋯}, (2) where 𝑎₂ = 𝐴₂∕𝐴₀ and𝑎₄ = 𝐴₄∕𝐴₀ are angular correlation coefficients, which can be calculated theoretically, [14]. The𝐴₀ coefficient represents the total𝛾-ray intensity.

3 The sign of 𝛿 in the alternative Rose–Brink convention is opposite, i.e.𝛿(Rose–Brink)= −𝛿(Krane–Steffen), [12].

When the angular correlation function is measured experimentally, various experimental factors play a role. For instance, due to the finite detector size and the irregular shapes of the detectors the angle defined by a pair of detectors covers a range𝛥𝜃(𝜃). Such a range in𝜃 causes some attenuating effects on the theoretically calculated angular correlation function. The attenuation can be described by attenuation coefficient𝑄_𝑙;

𝑊^′(𝜃) =𝐴₀(1 +

∑∞

𝑎_𝑙𝑄_𝑙𝑃_𝑙(𝑐𝑜𝑠𝜃)). (3)

The attenuation coefficients 𝑄_𝑙 depend on the geometrical prop- erties of the detectors, on the distance between the target and the detector, and on the energy of the 𝛾 rays of interest. For cylindri- cal detectors the attenuation coefficients can be calculated, see for instance [15,16]. Alternatively the attenuation can be simulated by Monte Carlo codes, for instance [8,17]. The last approach is better suited as it allows to consider the exact shape and tapering of each detector element and to derive the energy dependence of𝑄_𝑙.

Furthermore considerations should be given to the angle𝜃between the two detected gamma rays. Usually this angle is evaluated from geometrical consideration taking into account the placing of the detectors. In general it is affected by: (i) the position of the detector and whether its face is orthogonal to the direction of the source, (ii) the shape and tapering of the detector, (iii) the energy of the incident 𝛾 rays. These considerations are particularly important in the present application where the distance between the source and the detector is small, the detector elements are tapered and of different size and shape, and in general the detector face is not orthogonal with respect to the source direction. For instanceFig. 1(b) shows𝛾rays with energies of𝐸₁ (in red) and𝐸₂(in green), where𝐸₁< 𝐸₂, impinging on two segments 𝑠₁and𝑠₂ of a detector in a non-orthogonal geometry. On average, the higher-energy𝛾ray is absorbed deeper in the detector. InFig. 1(b) the arrows point to the average interaction positions. Consider the case where𝛾₂ is absorbed in segment𝑠₂. If the coincident𝛾 ray absorbed in𝑠₁ has energy of𝐸₁, the angle between the two𝛾 rays will be𝜃₁. However if the𝛾 ray has energy of𝐸₂, the angle will be𝜃₂ which is clearly smaller than𝜃₁. Therefore the angle between two detected𝛾 rays is in principle energy dependent. Note also that the angle𝜃₁when 𝛾₁is detected in segment𝑠₁and𝛾₂is detected in segment𝑠₂is different from the angle𝜃₃when the energies are inverted. In order to determine the angles𝜃for the𝛾-ray energies of interest and for the geometry of the segmented clover, and in order to evaluate the attenuation coefficients due to the non-zero opening angle of the detectors, we carried out GEANT4 simulations.

3. GEANT4 simulations for the segmented clover detector in a close geometry

The iThemba LABS segmented clover detector has a similar geometry to the TIGRESS (TRIUMF-ISAC𝛾-ray escape-suppressed spectrom- eter) detectors [18]. It comprises four electrically-segmented coaxial HPGe crystals in a single cryostat. The dimensions of each crystal are 60×90 mm before tapering. Each crystal has 8 segments (4 in front and 4 at the back) giving a total of 36 signals (32 for the segments and4for the core contacts). The longitudinal segmentation is at 35 mm from the front face of the crystal (31 mm for a standard TIGRESS detector). The four detector crystals are labeled anti-clockwise A, B, C and D, the front segments in each crystal are labeled1to4, and the back segments5 to8, as shown in Fig. 2. It also shows a schematic diagram of the segmented clover detector mounted on a scanning table where the radioactive source was moved to a desired(𝑥, 𝑦)position with a precision of∼ 0.2mm.

The GEANT4 simulations [19] performed for the iThemba LABS segmented clover were based on the geometry of a Compton suppressed EXOGAM detector [20] within the AGATA simulation package [9]. The geometry was extracted and modified to reflect the segmentation of the

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Fig. 2. (a): The crystals of the segmented clover detector as simulated in GEANT4.

Crystals are labeled A to D and segments numbered 1 - 4 for front segments and 5 - 8 for rear segments. (b): Schematic diagram of the segmented clover mounted above a scanning table.

Table 1

Absolute efficiency of the segmented clover at 1332 keV when the 32 segments are used as individual detectors, when the 4 crystals are used as individual detectors, and in addback mode.

Efficiency

Exp GEANT4

Segments 1.05(1)% 1.13%

Crystals 2.28(2)% 2.48%

Addback 3.01(3)% 3.42%

crystals and their actual dimensions, as depicted inFig. 2(a). Segments were modeled as individual detectors and energies were scored in each segment. The source was placed at 40 mm from the face of the Ge crystals.

When the segments are used as individual detectors a large part of the efficiency of the detector is not taken into account because the events where the 𝛾 ray is absorbed in two or more segments are not considered. However, the placement of the detector at close geometry increases the solid angle of the segments and thus increases the efficiency. The absolute efficiency of the segmented clover was measured as 1.1% at 40 mm when used in segment mode, while in crystal (addback) mode it was 2.3% (3.0%), seeTable 1. The agreement between measured and simulated absolute efficiencies, as listed in the table, are within expectation for GEANT4 simulations.

The important advantage when using the detector in segment mode is that it allows for a large number of unique angles for angular correlation analysis. The use of the 32 segments as individual detectors and the close geometry offered a number of measuring angles covering≈1/8 of the total solid angle of 4𝜋. The 32 segments define 1024 segment- pair combinations, however, several of the pairs correspond to the same angle because of identical geometry of the segments involved. For instance, the four crystals are segmented in the same way, thus segment 𝐴1 (the first segment is crystal 𝐴) is identical to segment 𝐵1, etc.

Furthermore, segments 2 and 4 of each crystal have identical geometry.

Therefore, the angle defined by the segment pair (𝐴1,𝐴2) is the same as that by the pair (𝐴1,𝐴4), and the same as (𝐵1,𝐵2), etc. Among the 1024 segment-pair combinations there are 140 unique angles,𝜃_𝑖, 6 of which correspond to cases where both𝛾rays are absorbed in the same segment (3 non-identical front and 3 non-identical back segments), 32 correspond to cases where both𝛾rays are absorbed in different front segments while the others involve absorption in front-and-back or back- and-back segments. Table 2lists the number of front–front segment pairs that correspond to unique angles. Three angles correspond to same-segment pairs and 32 to different-segment pairs.

Usually the angle between two𝛾rays absorbed in two detectors is calculated based on geometric considerations. Here the geometric angle is calculated with the assumption that the absorption of the two𝛾rays

Table 2

A list of front-front segment pairs which define unique angles𝜃_𝑖, where the notation A1A2 means that the first𝛾ray is detected in A1 while the second is detected in A2.

In general the pairs A1A2 and A2A1 define slightly different angles due the energy dependence of𝜃(for more details see the text andFig. 1). Such pairs are listed as A1A2/A2A1 and define identical angles only when the energies of the two𝛾rays are identical. The number of pairs that correspond to the same unique angle are also shown.

Angle pair Nr of pairs Angle pair Nr of pairs

A1A1 4 A2A4 8

A1A2/A2A1 8 A2B2 16

A1A3/A3A1 4 A2B3/A3B4 8

A1B1 8 A2B4 8

A1B2/A4B1 8 A2C2 8

A1B3/A3B1 8 A2C3/A3C2 8

A1B4/A2B1 8 A2C4 8

A1C1 4 A3A3 4

A1C2/A2C1 8 A3B2/A4B3 8

A1C3/A3C1 4 A3B3 8

A2A2 8 A3C3 4

A2A3/A3A2 8 A4B2 8

occurs on average at the geometric center of the segments, which we assumed to be the point midway between the front and back surface of the detector, along a line from the center of the front face of the segment to the center of the back face of the segment. In this work, rather than using the geometric angle, we simulated the average angle between the detected𝛾rays for each cascade and for each unique angle.

To evaluate these average angles we simulated the corresponding angle distributions. For each segment-pair coincident𝛾 rays with en- ergies𝐸_𝑎and𝐸_𝑏were generated, uniformly distributed over the solid angle subtended by the segmented clover detector. The angle between two fully absorbed𝛾 rays, where𝐸_𝑎(𝐸_𝑏) was absorbed in segment𝑗 (𝑘), was recorded in individual histograms that were incremented for each segment pair (j,k) that corresponds to the same unique angle𝜃_𝑖. For the 1173 - 1332 keV 𝛾-ray cascade of ⁶⁰Ni about 4.5 × 10⁷ events/h/thread could be simulated on an Intel^®Core™i7-7700 3.60 GHz processor. A simulation of 10⁹ decays yielded approximately 30000 counts in a typical angle distribution histogram for an angle defined by two front segments. Examples of angle distribution histograms illustrating the distribution of𝜃for four unique angles𝜃_𝑖and for the 1173 - 1332 keV𝛾-ray cascade are shown inFig. 3. For instance, the distribution labeled𝐴1𝐶2corresponds to front–front segment pairs of (𝐴1,𝐶2), (𝐴1,𝐶4), (𝐵1,𝐷2), (𝐵1,𝐷4), (𝐶1,𝐴2), (𝐶1,𝐴4), (𝐷1,𝐵2), and (𝐷1, 𝐵4) and its centroid yields an average unique angle of𝜃= 68.4^◦. Note that the geometric angle in this case is 78.4^◦. The angle calculated from geometrical considerations for the case where both𝛾 rays are detected in the same segment is 0^◦. However, this does not take into account the effect of the solid angle that corresponds to angle 𝜃, (which is proportional to𝑠𝑖𝑛𝜃𝑑𝜃), for instance the probability for two 𝛾rays to be detected at 0^◦with respect to each other is vanishing. The angle distribution for such a case where both𝛾 rays are detected in segment𝐴2(or identical) is shown inFig. 3and corresponds to𝜃_𝑖= 7.9^◦. Each distribution inFig. 3corresponds to 8 identical segment-pair combinations. The different heights of the peaks represent the different coincident efficiencies for these segment pairs. The distributions are generally symmetric around the average angle 𝜃_𝑖, with a standard deviation of approximately 6^◦. The uncertainty in the mean simulated angle𝜃_𝑖is less than 0.1^◦.

As illustrated inFig. 3significant differences between the geometric angle and the simulated average angle can occur. This difference is further illustrated in Fig. 4 for the 1173 - 1332 keV 𝛾-ray cascade and for all 140 unique angles. The deviation from the dashed diagonal line in this figure represents the difference which is largest for large angles and for combinations involving two front segments where it can reach up to 11^◦. For combinations involving two back segments the differences are less than 2.5^◦. The range of the simulated angles𝜃_𝑖for different segment combinations and for two different𝛾-ray cascades are also listed inTable 3and compared with the corresponding geometric

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Fig. 3. The simulated distribution of the angle𝜃for selected segment pairs and for the 1173 - 1332 keV𝛾-ray cascade from a⁶⁰Co source. The labeling is as inFig. 2 e.g.𝐴1𝐶2refers to 1173 keV detected in crystal A segment 1 and 1332 keV detected in crystal C segment 2. The examples represent combinations of front–front, front–back and back–back segments, as well as for an absorption of both𝛾 rays in the same segment. The arrows represent the angles determined from the geometric centers of the segments.

Fig. 4. Geometric vs simulated angles for the 1173 - 1332 keV cascade and for all 140 independent angles. The deviation from the dashed diagonal line illustrates the difference between the geometric and simulated angle.

angles. For the 53 - 223 keV cascade only the front segments are active since, due to the low energy of both 𝛾 rays, the count rates in back segments are insignificant. The range of the simulated unique angles𝜃_𝑖 is a bit larger for𝛾rays with lower energy.

To investigate the energy dependence of the unique angles𝜃_𝑖sim- ulations were carried out for all segment pairs and for a fixed 𝐸₁ of 1000 keV, while several values in the range of 100 - 1500 keV were used for𝐸₂. The difference in the unique angle𝜃_𝑖for a given𝐸₂with respect to the case with𝐸₂= 1500 keV was calculated as𝛥𝜃_𝑖=𝜃_𝑖0−𝜃_𝑖 and is illustrated in Fig. 5 for several selected segment pairs. It is shown that the angle 𝜃_𝑖 for the same segment pair may change by up to∼3^◦depending on the energy of the second𝛾 ray. Therefore in our case where we used a segmented detector in close geometry the angle between two detected gamma rays shows a significant energy dependence. This is different from studies done with detectors at larger distances. For instance, this energy dependence had been investigated for angular correlation measurements with the GRIFFIN array and found to be small [8].

The GEANT4 simulations were also used to evaluate the effect of the finite size of the segments on the angular correlation function. When two𝛾 rays with energies𝐸_𝑎and𝐸_𝑏are detected in coincidence in the

Fig. 5. The effect of𝛾-ray energy on the angle𝜃_𝑖defined by two𝛾rays with energies of𝐸₁ = 1000 keV and varied𝐸₂ in four different segment pairs. The difference in the angle𝛥𝜃_𝑖=𝜃_𝑖 -𝜃_𝑖0 is shown as a function of𝐸₂ where the reference angle𝜃_𝑖0 corresponds to simulations with𝐸₂= 1500 keV. The values of the angles𝜃_𝑖0are listed in the legend.

segments𝑗and𝑘of the segmented clover, respectively, the measured number of coincidence events in the corresponding photopeak is 𝐴(𝑗, 𝑘, 𝐸_𝑎, 𝐸_𝑏) =𝜀_𝑗(𝐸_𝑎)𝜀_𝑘(𝐸_𝑏)

∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑊(𝜃)𝑑𝜃, (4) where the efficiency for detecting𝐸_𝑎in segment𝑗is𝜀_𝑗(𝐸_𝑎),𝑊(𝜃)is the theoretical angular correlation function,𝜃_𝑖is the unique angle that correspond to the segment pair (𝑗,𝑘), and𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)is the normalized angle distribution for the angle𝜃_𝑖. Note that to obtain𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏) the simulated angle distributions, such as those shown inFig. 3, were normalized to a total area of one. Therefore the angular correlation function,𝑊^′(𝑗, 𝑘, 𝐸_𝑎, 𝐸_𝑏), which takes into account the finite size of the detectors by considering the angle distributions𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)is 𝑊^′(𝑗, 𝑘, 𝐸_𝑎, 𝐸_𝑏) = 𝐴(𝑗, 𝑘, 𝐸_𝑎, 𝐸_𝑏)

𝜀_𝑗(𝐸_𝑎)𝜀_𝑘(𝐸_𝑏)

=∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑊(𝜃)𝑑𝜃. (5) As there are several segment pairs (𝑗,𝑘) that correspond to the same unique angle𝜃_𝑖the number of coincident events in the corresponding photopeak is a sum over all such pairs;

𝐴(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) =∑

𝑗,𝑘

𝜀_𝑗(𝐸_𝑎)𝜀_𝑘(𝐸_𝑏)

∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑊(𝜃)𝑑𝜃. (6) The distribution𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)is the same for all segment pairs (𝑗, 𝑘) that correspond to the same angle𝜃_𝑖, thus the integral is separable from the sum. Therefore the angular correlation function is

𝑊^′(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) = 𝐴(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏)

∑

𝑗,𝑘𝜀_𝑗(𝐸_𝑎)𝜀_𝑘(𝐸_𝑏)

= ∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑊(𝜃)𝑑𝜃, (7)

which can be written using the Legendre polynomials as 𝑊^′(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏)

=∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)

∑∞

𝐴_𝑙𝑃_𝑙(𝑐𝑜𝑠𝜃)𝑑𝜃

=

∑∞

𝐴_𝑙

∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑃_𝑙(𝑐𝑜𝑠𝜃)𝑑𝜃. (8) Therefore, the finite size of the detectors can be taken into account by using attenuated Legendre polynomials,𝑃_𝑙^′(𝑐𝑜𝑠𝜃_𝑖), defined as 𝑃_𝑙^′(𝑐𝑜𝑠𝜃_𝑖) =

∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑃_𝑙(𝑐𝑜𝑠𝜃)𝑑𝜃, (9)

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Table 3

The simulated angles𝜃_𝑖obtained from GEANT4 for the 1173–1132 keV and 53–223 keV𝛾-ray cascades in comparison with these angles calculated using the geometric center of the corresponding segments.

Coincident segments Number of angles

Geometric 1173–1332 keV 53–223 keV

Min angle Max angle Min angle Max angle Min angle Max angle

Front–Front 32 19.4^◦ 89.4^◦ 17.5^◦ 78.4^◦ 17.7^◦ 81.3^◦

Front–Back 70 5.8^◦ 74.7^◦ 9.5^◦ 68.3^◦

Back–Back 32 13.7^◦ 60.0^◦ 12.8^◦ 57.6^◦

Front: same seg. 3 0^◦ 0^◦ 7.1^◦ 8.8^◦ 7.9^◦ 10.4^◦

Back: same seg. 3 0^◦ 0^◦ 6.0^◦ 6.4^◦

Fig. 6. Legendre polynomials,𝑃_𝑙, and attenuated Legendre polynomials,𝑃_𝑙^′, of order 2, 4 and 6 for𝐸₁= 1173 keV and𝐸₂ = 1332 keV. The values of𝑃_𝑙^′ are plotted at the simulated𝜃_𝑖. In the inset the difference𝑃_𝑙−𝑃_𝑙^′is shown.

and an angular correlation function defined as 𝑊^′(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) =

∑∞

𝐴_𝑙𝑃_𝑙^′(𝑐𝑜𝑠𝜃_𝑖)

=𝐴₀(1 +

∑∞

𝑎_𝑙𝑃_𝑙^′(𝑐𝑜𝑠𝜃_𝑖)). (10) Alternatively, it is possible to use attenuation coefficients 𝑄_𝑙, see Eq.(3), defined as

𝑄_𝑙(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) = ∫ 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏)𝑃_𝑙(𝑐𝑜𝑠𝜃)𝑑𝜃 𝑃_𝑙(𝑐𝑜𝑠𝜃_𝑖)

= 𝑃_𝑙^′(𝑐𝑜𝑠𝜃_𝑖)

𝑃_𝑙(𝑐𝑜𝑠𝜃_𝑖). (11)

The angles𝜃_𝑖 as well as the attenuated Legendre polynomials𝑃^′

𝑙

depend on the 𝛾-ray energies. The values of 𝑃^′

𝑙 were determined for the discrete angle set𝜃_𝑖 using the normalized angle distributions 𝑁_𝜃

𝑖(𝜃, 𝐸_𝑎, 𝐸_𝑏) obtained in the GEANT4 simulations and for all 𝛾-ray cascades of interest. In Fig. 6 the attenuated Legendre polynomials calculated for the 1173 - 1332 keV 𝛾-ray cascade are shown and compared with the standard Legendre polynomials. The case for𝑙= 6 is added to illustrate the trend of increasing attenuation with increase in𝑙, although for the𝛾-rays of interest in this work𝑙_𝑚𝑎𝑥= 4. The figure suggests that the attenuation due to the finite size of the segments of the clover is small.

The calculated values of the attenuated Legendre polynomials ex- hibit a certain amount of deviation from a smooth dependence on angle, as is highlighted in the inset in Fig. 6 where the differences between the unattenuated and attenuated Legendre polynomials are shown. These deviations are not statistical and can be associated with the geometry and tapering of the segments that may be different for segment pairs that yield very similar angles. Furthermore,𝑃^′

𝑙 is only defined at the discrete angles that correspond to the geometries of the detector segments.

Fig. 7. A comparison between the angular correlation function𝑊(𝜃)and the attenuated angular correlation function𝑊^′(𝜃_𝑖) calculated with standard and attenuated Legendre polynomials, respectively. The inset shows the difference between the two functions.

Although the distance between the source and the detector is small, the physical size of the segments of the HPGe crystals is also small, causing the attenuation of the Legendre polynomials to be minor. This is in line with expectations as the opening angle of the segments has similar magnitude to that of the detectors of other𝛾-ray arrays such as Gammasphere [21], Jurogam [22], AFRODITE [23], Miniball [24] and others.

The attenuation coefficients𝑄_𝑙can also be calculated using Eq.(11).

This approach has a distinct disadvantage because𝑄_𝑙becomes unde- fined at the zeros of the Legendre polynomials and varies rapidly near the zeros. The average values, obtained at angles far from the zeros of the Legendre polynomials, are𝑄₂= 0.98,𝑄₄ = 0.95, and𝑄₆ = 0.80, which are similar to those found for large𝛾-ray arrays.

To evaluate further the effect of the attenuation, the angular correlation function was calculated with the attenuated Legendre polynomials (𝑊^′(𝜃)), and with standard Legendre polynomials (𝑊(𝜃)) for 𝑎₂=0.1005 and𝑎₄=0.0094, which are the theoretical angular correlation coefficients for the 1773/1332 keV𝛾-ray cascade, seeFig. 7. The small difference between𝑊(𝜃)and𝑊^′(𝜃), highlighted in the inset in the figure, reflects only the attenuation effect.

As a final evaluation of the impact of the attenuation the values of the simulated attenuated angular correlation function𝑊^′(𝜃_𝑖)were fitted with the standard angular correlation function 𝑊(𝜃_𝑖)and the coefficients 𝑎₂ and 𝑎₄ were extracted. The obtained values of 𝑎₂ = 0.0984 and𝑎₄= 0.0090 show only a 2% and 4% difference respectively, illustrating the magnitude of the attenuation. Therefore, the attenuation due to the finite size of the segments of the detector plays a minor role in the angular correlation function. However, the effect of simulated versus geometric angles is not negligible, as shown inFig. 8.

The values of𝑊(𝜃)are calculated with𝑎₂=0.1005 and𝑎₄=0.0094 and these 𝑊(𝜃) values are plotted at both the simulated average angles (blue filled circles) and the geometric angles (red open circles). A fit

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Fig. 8. The calculated angular correlation values for the 1173 - 1332 keV𝛾-ray cascade from a⁶⁰Co source with𝑎₂=0.1005 and𝑎₄=0.0094 plotted at simulated and geometric angles.

Fig. 9. The simulated angular correlation data for the 1173 - 1332 keV𝛾-ray cascade from a⁶⁰Co source using the theoretical values of𝑎₂=0.1005 and𝑎₄=0.0094. The solid line is a fit to the data with attenuated Legendre polynomials, while the dashed lines indicate the uncertainty on the fit.

to𝑊(𝜃)as a function of geometric angle (dotted red line) yields 𝑎₂ and𝑎₄values that deviate by more than 7% from the expected values.

A simulation where the 1173 - 1332 keV⁶⁰Ni𝛾 rays were emitted according to the angular correlation function with theoretical values of 𝑎₂=0.1005 and𝑎₄=0.0094 was also carried out. Coincident data were simulated and the areas of the 1332 - 1173-keV coincidence peak were measured for all segment pairs and corrected for the corresponding energy-dependent efficiencies. Results are plotted inFig. 9as a function of the simulated angles 𝜃_𝑖. The fit using the attenuated Legendre polynomials yields angular correlation coefficients of𝑎₂ = 0.1008(32) and𝑎₄=0.0109(32) in good agreement, within the uncertainty, with the input values.

4. Experimental data

The data reported in this work were collected in close geometry with a source at 40(2) mm from the front face of the HPGe crystals. No escape-suppression shield was in place. The 32 segments of the detector were used as individual detectors to measure both the single and coincidence events. Data with⁶⁰Co and¹³³Ba sources were collected.

The activity of the ⁶⁰Co and¹³³Ba sources was 1.1𝜇Ci and 1.0𝜇Ci, respectively. The output signals of all segments and cores were read

with four digital electronics XIA Pixie-16modules [25] with 100 MHz sampling frequency, and recorded using the Maximum Integrated Data Acquisition System (MIDAS) [26]. The count rates of the segments were approximately 1–2 kHz for the front segments and 300–500 Hz for the back segments. The four core contacts of the detector were fitted with cold Field Effect Transistors (FETs) while the segments had warm FETs which resulted in different energy resolution of≈ 2.35keV for the cores and≈ 3.5 − 4keV for the segments at 1332 keV. The data were sorted and analyzed using ROOT [27].

5. Data analysis

The data were recorded in triggerless mode, whereby each channel registers energy and time for signals above threshold independently from signals in any other channel. The energy of the segments was gain matched and corrected for gain drifts. The energy signals of the segments (from any segmented detector) experience proportional crosstalk, thus a proportional crosstalk correction as described in Refs. [28,29] was applied. The corrected and energy calibrated data were sorted into ROOT trees using a coincidence window of 1 μs, and time synchronization and alignment was confirmed by inspecting time-difference spectra. Direct spectra for each segment were extracted when a single segment was hit. These spectra were used to measure the relative efficiency of each segment.𝛾-𝛾coincidence matrices were generated for all possible segment combinations when 2 segments of the detector were hit. These matrices were used to measure the coincidence intensity for a given angle𝜃_𝑖. The⁶⁰Co matrices were constructed with a smaller time coincidence window of 100 ns, as in this case the narrow time gate is reasonable due to the similar high-energy𝛾- rays involved. The matrices for identical segment pairs were summed, resulting in a total of 134 matrices that correspond to unique angular correlation angles 𝜃_𝑖. For ¹³³Ba only the 32 matrices corresponding to the 32 unique angles defined by all different front–front segment combinations were used because the𝛾rays emitted by the¹³³Ba source hardly penetrate into the back segments of the detector.

A couple of examples of angular correlation matrices for the¹³³Ba data are shown in Fig. 10. The matrix in panel (a) corresponds to coincidence data between segment pairs of (𝐴1,𝐶1) type. When the unique angle is defined by segments lying close to each other the matrix shows Compton scattering ridges, see the matrix shown in panel (b), which corresponds to segment pair (𝐴1,𝐴3) and equivalent.

The Compton scattering in the segments of the same detector cannot be avoided, and when they are used as individual detectors it creates additional background. In many cases the Compton ridges are negligible, in particular when the segments are further away from each other as shown inFig. 10(a). In other cases they do exist but do not affect the coincidence peaks of interest, e.g. 384 - 81 keV coincidence peaks inFig. 10(b). However, there are cases where the coincidence peak lies above a Compton ridge, for instance the coincidence peak of 276 - 80 keV that is observed above the 356-keV Compton ridge inFig. 10(b).

In such cases the Compton background should be taken into account.

Two alternative ways can be used. Firstly, one can gate diagonally on the matrix. For instance, the E(𝐴1) + E(𝐴3) = 356 keV diagonal gate placed on the (𝐴1,𝐴3) matrix (panel (b) ofFig. 10) is projected on the 𝑌-axis, and is shown in panel (a) ofFig. 11. The background represents the Compton ridge of the 356-keV transition and the peak area for the transition of interest can be measured above it. Alternatively, one can use the standard gating on the x or y axis, but in this case the gate needs to be wide. For instance, consider studying the 303 - 81 keV cascade in the matrix (𝐴1,𝐴2), which involves pairs of segments that lie next to each other and thus corresponds to largest Compton scattering. A gate on 303 keV𝛾 peak shows a peak at 81 keV, which lies above the Compton ridge of the 384 keV𝛾ray. A narrow gate on the 303-keV peak may not make the Compton background underneath the 81-keV peak visible. Signs that the 81-keV peak is lying above a Compton background are visible with a gate width of 17 keV, as shown

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Fig. 10. Angular correlation matrices for two different unique angles defined by, (a):

segment pair (𝐴1,𝐶1) and equivalent; (b): segment pair (𝐴1,𝐴3) and equivalent.

in Fig. 11(b), but a gate width of 27 keV, see Fig. 11(c), shows the Compton ridge underneath the 81-keV peak well and makes it possible to measure the peak area.

The coincidence intensity of the𝛾 rays of interest were measured using gated spectra extracted from the angular correlation matrices for all unique angles. Background corrections were taken into account, including corrections for background under the gate, random coincidences, and for small angles background produced by a Compton ridge.

In the latter case a wide gate was used in order to visualize the height of the Compton ridge.

The low-energy peaks of¹³³Ba showed small low energy tails. As an example, the region near the 81-keV peak is plotted inFig. 12(a), (note the logarithmic scale). The peak near 62 keV is caused by the back-scatter of the 81-keV𝛾rays. In addition to this peak, however, a tail appears on the low-energy side of the 81-keV peak. To understand the origin of the tail GEANT4 simulations were used. The tail was found to originate from small angle Compton scattering of the 81-keV𝛾 rays from the surrounding material. The tail disappears if the material surrounding the Ge crystals is simulated as vacuum, see the red curve in Fig. 12(b), but is present if the surrounding material is taken into account (the black curve). Therefore when analyzing the experimental data the low-energy shoulders of such peaks were consistently excluded from the peak areas, as they do not correspond to full absorption.

The uncertainties on the peak areas included uncertainties from the peak area evaluation as well as the errors caused by background handling. These uncertainties were evaluated from the statistical errors on both the coincidence peak and background areas. In addition we added a systematic error on handling the Compton background (estimated at between 0.2% and 0.5%). The statistical uncertainties on the efficiencies were negligible. In the case of the⁶⁰Co data statistical errors on the peak areas were between 1 and 4%, the uncertainty due to the choice of the background contributed typically by 0.5%.

The angular correlation analysis for⁶⁰Co involved all 140 angles, including all combinations of front and back segments, and also data

Fig. 11. (a): Diagonal gate on 356 keV in the (𝐴1,𝐴3) matrix shown inFig. 10(b). (b) and (c): Spectra gated on303keV extracted from the angular correlation matrix for the segment pair (𝐴1,𝐴2), and equivalent, which corresponds to largest Compton-scatter background. The gate width is of 17 keV for (b) and 27 keV for (c).

Fig. 12. (a): The low-energy part of a spectrum from the¹³³Ba data around the 81- keV peak, (the peak at 62 keV is the backscatter peak) (b): GEANT4 simulations of the low-energy region (in black) and the same simulations assuming that the material surrounding the HPGe crystals is vacuum (in red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of same-segment events (both𝛾rays are detected in the same segment, using the summed-energy peaks in the direct spectra). The analysis in

133Ba involved the 32 unique angles defined by front–front segment pairs only.

The values of the experimental angular correlation function𝑤^′(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) were determined from the coincidence intensity,𝐴^𝑎,𝑏

𝑖 , for the cascade 𝐸_𝑎−𝐸_𝑏 and the coincidence efficiency 𝜖_𝑖 of the segment pairs that correspond to the unique angle𝜃_𝑖as follows,

𝑤^′(𝜃_𝑖, 𝐸_𝑎, 𝐸_𝑏) =𝐴^𝑎,𝑏

𝑖

𝜖_𝑖 , (12)

where𝜖_𝑖represents the combined coincidence efficiency at the angle𝜃_𝑖 and for energies𝐸_𝑎and𝐸_𝑏, given by

𝜖_𝑖= ∑

𝑗,𝑘∈𝜃𝑖

𝜖_𝑗(𝐸_𝑎)𝜖_𝑘(𝐸_𝑏), (13)

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where the sum is over all segments pairs𝑗and𝑘that correspond to the unique angle𝜃_𝑖.

The efficiency for each segment and for each𝛾-ray energy,𝜖_𝑗(𝐸_𝑎), were obtained from the corresponding direct spectra extracted from the same data set as the coincidence data. The efficiencies therefore included a small reduction caused by real coincidences of two 𝛾 rays in the same segment. To evaluate this effect the coincidences were simulated by building the decay paths of ¹³³Cs and⁶⁰Ni in the GEANT4 simulations. The reduction was found to vary for different 𝛾-ray cascades and different segments in the range of 1.3% and 6.7%.

Each individual channel has a dead time of13 μs. Thus two gamma rays can be registered within13 μsin two different segments (and will thus contribute to the measured segment efficiency), however if they impinge on the same segment the event will be rejected. This 13 μs segment dead time influences the efficiency for the same-segment coincidences in comparison with that for different-segment coincidences.

Therefore, for same-segment coincidences (measured only for the⁶⁰Co data) an additional dead-time correction was applied. We estimated this dead time effect by analyzing the coincidence peak areas generated with1 μsand with13 μscoincidence time windows. The spectra with a longer coincidence time window showed a linear decrease in the coincidence efficiency. This reduction was extrapolated to a coincidence window of 100 ns. The effect of dead time was thus measured to be about 2.2% per segment, which led to a correction in the coincidence efficiency for single segment of about 4.4%. In¹³³Cs the sum peak for the cascades of interest is not unique, for instance the sum peak of the 356 - 81 keV cascade of 437 keV coincides with the sum peak of the 53 - 384 keV cascade and of the sum peak of the 276 - 161 keV cascade.

Therefore, same-segment data could not be used for the¹³³Ba decay.

The uncertainty of the measured𝑤^′(𝜃_𝑖)values were evaluated taking into consideration the uncertainty in the area of the coincidence peak and corresponding background as described above. The statistical uncertainties on the efficiencies were negligible. These uncertainties (taken in quadrature) were increased when calculating the experimental angular correlation values𝑤^′(𝜃_𝑖)by 20% to account for any other unaccounted systematic error. The overall uncertainty on 𝑤^′(𝜃_𝑖) was between 1.3% and 5%.

The experimental values of the angular correlation function were fitted with Eq.(10), and the coefficients𝑎_𝑙were obtained from a𝜒²fit. In determining the uncertainties of the𝑎_𝑙coefficients the uncertainties in the experimentally measured angular correlation data were considered, while the uncertainties in the attenuated Legendre polynomials and in the simulated average angles were assumed to be small. The mixing ratios were calculated using the code DELTA [30], based on calculating 𝜒²as a function of the mixing ratio𝛿of one of the transitions, while the experimentally measured𝑎₂and𝑎₄coefficients and the spins of the nuclear states involved were input parameters.

6. Results and discussion 6.1. ⁶⁰Co results

The angular correlation function for the4⁺→2⁺→0⁺1173 − 1332 keV cascade in⁶⁰Ni is well known and often used for testing [31,32].

The high energies of the two𝛾 rays makes it possible to use all front and back segments of our detector, although the count rates in the back segment are significantly lower than in the front segments. This cascade has no cross-over transition with notable intensity and no Compton- scatter ridge at 2505 keV was observed in our data. Following that, the summed-energy peak at 2505 keV could be extracted from the direct spectra to be used as same-segment coincidence data, allowing a minimum angle of 6.0^◦to be reached. Thus data could be obtained for 140 unique angles. Although many angles are near identical, the exact geometries and the attenuated Legendre polynomials of Eq.(9) are not, thus for this test all angles were analyzed independently. The analysis was performed for an energy gate on the1173-keV4⁺ →2⁺

Fig. 13. Angular correlation data for the 1173 - 1332 keV cascade in⁶⁰Ni for all 140 unique angles. The solid line represents the least squares fit to the data from which the𝑎₂ and𝑎₄ values were obtained. The dashed lines represent the uncertainty in𝑎₂ and𝑎₄. The inset illustrates the calculated𝜒²as a function of𝛿using the measured 𝑎₂and𝑎₄. Only one solution for𝛿was found.

transition. The subtracted background due to random 1173 - 1332 keV coincidences was small (0.5% on average) while all other background corrections were typically less than 3.5% of the total peak area. The experimental angular correlation data for this cascade are presented in Fig. 13. The uncertainties are obtained from a combination of counting statistics and the uncertainties on the background components.

Angular correlation coefficients of𝑎₂ = 0.1030 ± 0.0092and𝑎₄ = 0.0352 ± 0.0086 were obtained from a 𝜒² fit to the data, as shown in Fig. 13. The mixing ratio was extracted assuming that the 1332- keV transition is a pure 𝐸2 transition and the 1173-keV transition has an𝐸2 +𝑀3multipolarity. The results yielded only one solution, 𝛿= −0.009 ± 0.041. This value is in good agreement with the previous work [33,34] which reports a value of𝛿 = −0.0025 ± 0.0022and is, within the uncertainties, consistent with zero suggesting that the 1173- keV transition has negligible𝑀3contribution. The high precision of the measured mixing ratio highlights the power of this angular correlation technique, which uses the segments of the clover as individual detectors.

6.2. ¹³³Ba results

The𝛾rays emitted by¹³³Cs following𝛽-decay of¹³³Ba are very well studied (seeFig. 14). They afford the opportunity to test the accuracy and precision of the newly developed angular correlation technique, in particular for several𝛾 rays with mixed 𝑀1 +𝐸2 multipolarity. The mixing ratios of the 53-, 80-, 81-, 161-, 223- and303-keV transitions had been determined from numerous angular correlation and internal conversion measurements, see the adopted values in [35], and were used to compare with our measurements.

Due to their relatively low energies the 𝛾 rays of the decay of the ¹³³Ba source hardly penetrated into the back segments of the segmented clover, thus only front segments were used in the measurements. Data for the 32 unique angles defined by the front segments were analyzed.

6.2.1. The 356 - 81 keV cascade

The coincidence spectra for the1∕2⁺→5∕2⁺→7∕2⁺356 - 81 keV cascade were obtained by gating on the 356-keV pure E2 transition in the 32 angular correlation matrices. Background corrections were performed taking into account the background under the gate and random coincidences. The 81-keV peak showed low-energy tails of the order of 6% of the peak area that extends over a region of about 20 keV