Understanding Nuclear Physics: An Experimental Approach 9811984360, 9789811984365

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Nikit Deshmukh Nirav Joshi   Editors

Understanding Nuclear Physics An Experimental Approach

Understanding Nuclear Physics

Nikit Deshmukh · Nirav Joshi Editors

Understanding Nuclear Physics An Experimental Approach

Editors Nikit Deshmukh School of Sciences P P SAVANI University, Kosamba Surat, India

Nirav Joshi University of Barcelona Barcelona, Spain

ISBN 978-981-19-8436-5 ISBN 978-981-19-8437-2 (eBook) https://doi.org/10.1007/978-981-19-8437-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface I

This book provides a unique approach to understand the nuclear Physics especially from the experimental end which is not common in most of the books that are available in market. Many books available on the subject Nuclear Physics have the conventional approach of theory know-how and then the traditional description of the experiments. But this book on the contrary will provide the other approach covering experimental competence. The main highlight of this book is that a special care has been taken to provide more experimental information in the sense considering real experimental data, real experimental structure showcased from international/national laboratories. This book covers key areas in Experimental Nuclear Physics, namely offline gamma counting and online particle detection approach, overview, and developments in accelerators, detectors, and their associated electronics, data acquisition systems, and computers for data analysis. This book “Understanding Nuclear Physics: An Experimental approach” will provide an extra interest to the readers and open the doors for new opportunities in the youth to pursue research in the experimental nuclear physics field. Kosamba, India October 2022

Nikit Deshmukh

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Preface II

The atomic nucleus has proven to be an exceedingly interesting many-body system to study as it has brought up surprises over and over again. Nuclear physics prior importance lies in the study of atomic nuclei that has historically given us many of the first insights into modern physics. The future prospects remains very promising. The properties of nuclei at low excitation and low angular momentum have been extensively studied and possible understanding has been achieved. Thus current research is on nuclei at high interaction energies, high angular momentum, and far from the valley of stability. Further, nuclear physics is closely allied with numerous other active branches of physics where versatile research is pursued. To highlight the few: particle physics, in terms of the large overlap of interests in fundamental interactions and symmetries, and condensed matter physics, through the many-body nature of the problems involved. The book is aimed for students who want to pursue their career in the field of Nuclear Physics. It mainly focuses on experimental approach to study nuclear physics. The book provides information about the recent developments in accelerators, detectors and their associated electronics, data acquisition systems, and computers for data analysis. It will showcase the real experimental data which will give the reader actual understanding of the present problems in the fields and encourage to pursue the further research in the field. The topics covered in the book are of vital importance in a wide range of modern and emerging interdisciplinary research at the intersections of nuclear physics with related disciplines, such as particle physics, astrophysics, atomic physics, and condensed matter physics and consequently have great impact on our society. Kosamba, India Barcelona, Spain October 2022

Nikit Deshmukh Nirav Joshi

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Acknowledgements

The foremost acknowledgement goes to the authorities of P. P. Savani University, Kosamba, Surat, Gujarat, for giving me the space to accomplish my goal in bringing out this edition. I am grateful to my co-author Dr. P. K. Rath who continuously motivated and gave encouragement; along with my co-editor Dr. Nirav J. Joshi who stood behind in executing this task. It is my great pleasure to thank and appreciate my co-authors Dr. Balaram Dey and Dr. Sushil Kumar Sharma for their valuable suggestion and constructive criticism. My sincere thanks to all other co-authors of this book Dr. Srijit Bhattacharya, Dr. Pankaj K. Giri, Dr. Rudra Narayan Sahoo, Dr. Andrés Arazi, Dr. Daniel Abriola, and Dr. B. Kamys for providing all the materials related to chapters in a requisite time. A special Hug and thanks to my dear co-author Dr. U. Singh who helped in formatting all the chapters. Finally I can’t forget the inspiration and nurturing provided by my dear wife Ami and love by my parents. I thank all my collaborators in the field without whom it wouldn’t be possible for me to do a constructive research. At last want to thank God without whom I wouldn’t be able to do any of this. My sincere thanks are incomplete without acknowledging Springer Nature Singapore Pte Ltd and their editorial team for their thorough support throughout this journey and bringing out this edition in time.

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balaram Dey, Srijit Bhattacharya, and Nikit Deshmukh

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Basic Properties of the Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanvi Bhavsar and Nikit Deshmukh

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Experimental Details for a Typical Nuclear Physics Experiment . . . . . . . Balaram Dey and Srijit Bhattacharya

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Offline Measurements and Extraction of Fusion Cross Section . . . . . . . . . Pankaj K. Giri, Rudra N. Sahoo, and P. K. Rath

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Measurements of the Angular Distribution of Elastically and Inelastically Scattered Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrés Arazi and Daniel Abriola

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Proton Induced Spallation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 S. Sharma, U. Singh, and B. Kamys

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Contributors

Daniel Abriola Laboratorio TANDAR, Comisión Nacional de Energía Atómica, San Martín, Argentina Andrés Arazi Laboratorio TANDAR, Comisión Nacional de Energía Atómica, BKNA1650 San Martín, Argentina and Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina Srijit Bhattacharya Barasat Government College, Barasat, West Bengal, India Tanvi Bhavsar School of Engineering, P P Savani University, Dhamdod, Kosamba, Surat, GJ, India Nikit Deshmukh School of Sciences, P P Savani University, Dhamdod, Dhamdod, Kosamba, Surat, Gujarat, India Balaram Dey Bankura University, Bankura, Purandarpur, West Bengal, India Pankaj K. Giri Department of Physics, Central University of Jharkhand, Ranchi, India; India and UGC-DAE Consortium for Scientific Research, Kolkata Centre, Kolkata, India B. Kamys The Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, Kraków, Poland P. K. Rath Centurion University of Technology and Management, Odisha, India Rudra N. Sahoo Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel S. Sharma The Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, Kraków, Poland U. Singh The Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, Kraków, Poland

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Introduction Balaram Dey, Srijit Bhattacharya, and Nikit Deshmukh

Nuclear physics deals with nucleons (neutrons and protons) within the atomic nucleus and the nature of the interaction between those nucleons that glue them together within the minuscule nuclear dimension of a few femtometer or 10−15 m. Piggybacked on some important experimental discoveries, nuclear physics emerged as a new and separate branch altogether different from atomic physics. In 1896, the discovery of radioactivity by Henry Becquerel, albeit an accidental one, is considered as initial breakthrough for the separate existence of the nuclear physics field [1], where the keen observation was made for the well-wrapped photographic plates which were blackened when placed near the uranium ores. In the subsequent years, the phenomenon was widely investigated by Pierre Curie, Marie Curie, Ernest Rutherford and his collaborators [1, 2] and a conclusion was attained for the atoms being made up of smaller particles, and these can be rearranged. In 1899, Rutherford and Villard identified the radiation consisting of three distinct types—alpha, beta, and gamma rays [3, 4]. In 1907, Rutherford and Royds studied the nature of α-particles and identify α-ray as helium ion [5], which has a positive electric charge and has about four times the mass of a hydrogen atom. Ernest Rutherford, who was loosely called “Earnest” in his childhood, actually had tremendous contributions in order to clear up the obscure physics of atomic nucleus with his “earnest” endeavor. The series of experiments [6, 7] performed by Geiger, Mersdan, and Rutherford of α-ray scattering by thin gold foil are the examples of classic nuclear physics experiment that helped nuclear physics to leap forward. Thomson’s plum pudding model, preB. Dey (B) Bankura University, Bankura, Purandarpur 722155, West Bengal, India e-mail: [email protected]; [email protected] S. Bhattacharya Barasat Government College, Barasat 700124, West Bengal, India N. Deshmukh School of Sciences, P P Savani University, Dhamdod, Kosamba, Surat 394125, Gujarat, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Deshmukh and N. Joshi (eds.), Understanding Nuclear Physics, https://doi.org/10.1007/978-981-19-8437-2_1

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decessor to Rutherford model, was pictorially described as randomly distributed “plums” or the negatively charged electrons within the positively charged “pudding” or the proton distribution. Rutherford’s experiment [7] showed that some particles are indeed scattered through very large angles even greater than 90 ◦ , which probed the existence of positively charged central nucleus responsible for the large deflection of electrons. According to Rutherford, the back angle alpha scattering is only possible if the incoming alpha particle could feel a large repulsive electric field. The large repulsive electric field is not achievable if the distribution of positively charged proton is assumed in accordance with the atomic model of Thompson. This contradiction leads to Rutherford’s visionary proposal of the planetary model of atom, which states that the positively charged proton must be concentrated in a very tiny place (he calculated the place of dimension ∼ 1/100000 the size of atom) because of which the incoming alpha particles in Rutherford experiment could experience a large repulsion and scatter in back angle. However, it seems extremely impossible for the protons to be concentrated in such a tiny place due to the electric repulsion between them making the whole nucleus highly unstable. Therefore, to explain the stability of the nucleus another force was assumed that is too highly attractive to hold all the protons together overcoming their electromagnetic repulsion. This assumption leads to introduce another kind of particle (should be charge less) which must exist with the protons. After the discovery of neutron by Chadwick in 1932 [8], it has been predicted that neutron could stay with the protons with strong short-range nuclear interaction. For the stability of electron, Rutherford also predicted that the electrons are in some kind of motion. This is how the concept of atomic nucleus or nuclear physics has been started. Thereafter, with the gradual advancement of technology, nuclear physics has become state of the art and highly popular research field, having several diverse applications such as in basic science to unravel the interaction of basic building blocks of matter, nuclear energy, medical fields, etc. To understand nuclear physics, we have to probe the atomic nucleus and interact with the family members (neutron and proton) of the nucleus. However, nuclear dimension is much smaller than the dimensions we can visualize. Therefore, we encounter a problem to get into a nucleus with our general ideas. Just as an example, let us suppose, a blind girl have been provided with an electronic stick that can sound to get a sense of the size of door, width of a pavement, etc. The girl can push the door or the pavement with the stick and listen the sound of the stick. In this way, she can get the idea of the size. But, if you give him a sewing needle, she faces difficulty as the needle is very much shorter in length than the stick. If you provide her a stick of similar length of the needle, her task becomes easier. Therefore, it is understandable that to explore any kind of object (quarks, nuclei, atoms, molecules, matter, etc.), you need a probe of dimension similar to the object itself. To get the clear idea, another example may be cited. We assume a chalk piece (a few centimeter length ∼3−6 cm) and by interacting with its constituents of chalk piece we could figure out its dimension. But, how do we figure out its dimension? We cannot get an idea about its size until the light waves fall on the chalk piece and interact with its constituents. After interacting with the particles of the chalk, the light gets scattered. This scattered light falls on our eyes and produces a signal. The signal is processed by our nerve

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system and then the processed signal goes to our brain and finally we can respond. This is how we can estimate the diameter of the chalk by receiving and analyzing the scattered light waves. On the other hand, if we close our eyes putting an audible sound source behind the chalk, we will not be able to guess its size. This is owing to the smaller wavelength of light wave than the diameter of the chalk piece. On the contrary, the wavelength of the sound source is typically more than the size that we want to analyze. The wavelength of visible light is λ ∼400−700 nm > size of the chalk. After Rutherford’s experiment [7], we get the essence of nuclear size (∼ 10−12 –10−14 m) and its constituents—neutron and proton. However, to know more about the constituents of those nucleons we need to move deeper, which implies the use of smaller probes. Fortunately enough, by increasing the energy of the incident particle we can fulfill our objective. On January 15, 1959, an experiment [9] was performed by Scientist R. Hofstadter that led to his Nobel Prize win. In this high-energy electron scattering experiment [9], he took 420 MeV electron as a probe and 16 O nucleus as a target. From the de Broglie relation, the wavelength of the MeV−fm) = 2π(197 ∼ 3 fm, which is comparable to the electrons was λ = hp = 2πηc ( pc) (420 MeV) dimension of nucleus and thereby is an excellent probe. From Quantum Mechanics, the de Broglie wavelength of the probe should be similar to the dimension of the object under investigation. Therefore, using this energetic electron beam one may be able to probe the atomic nucleus. However, to move deeper inside a nucleon, we need electron beam of wavelength lower than the radius of a nucleon (i.e., 0.5 fm). So the energy of the electron should be 2π (197 MeV-fm)/0.5 fm ∼ 2.5 GeV in order to see the inside of a nucleon. Therefore, this shows that the nuclear physics experiments become highly interesting albeit challenging and complex as we move deeper into the nucleus. In this book, we explain the experimental requirements of a typical nuclear physics experiment—starting from the particle beam, detector system, data acquisition, electronics system, etc. In addition, several specific topics in nuclear physics research are discussed. Moreover, the different experimental facilities for nuclear physics research existing in the world are also mentioned.

References 1. P. Radvanyi, J. Villain, The discovery of radioactivity. Comptes Rendus Phys. 18(9), 544–550 (2017). ISSN 1631-0705. https://doi.org/10.1016/j.crhy.2017.10.008, URL https:// www.sciencedirect.com/science/article/pii/S1631070517300786. Science in the making: The Comptes rendus de l’Académie des sciences throughout history 2. L. Gerward, Paul Villard and his discovery of gamma rays. Phys. Perspect. 1(4), 367–383 (1999). https://doi.org/10.1007/s000160050028 3. H. Sack, Pierre curie—a pioneer in radioactivity (2022a). http://scihi.org/pierre-curieradioactivity/

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4. H. Sack, Ernest Rutherford discovers the nucleus (2022b). http://scihi.org/ernest-rutherfordnucleus/ 5. E. Rutherford, T. Royds, Xxi. The nature of the α particle from radioactive substances. Lond., Edinb., Dublin Philos. Mag. J. Sci. 17(98), 281–286 (1909) 6. H. Gegier, E. Marsden, E. Rutherford, On a diffuse reflection of the particles. Proc. R. Soc. London. Ser. A, Contain. Pap. Math. Phys. Character 82(557), 495–500 (1909). https://doi.org/ 10.1098/rspa.1909.0054, https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1909.0054 7. E. Rutherford, Lxxix. The scattering of α and β particles by matter and the structure of the atom. Lond., Edinb., Dublin Philos. Mag. J. Sci. 21(125), 669–688 (1911) 8. James Chadwick, Possible existence of a neutron. Nature 129(3252), 312 (1932) 9. H.F. Ehrenberg, R. Hofstadter, U. Meyer-Berkhout, D.G. Ravenhall, S.E. Sobottka, High-energy electron scattering and the charge distribution of carbon-12 and oxygen-16. Phys. Rev. 113(2), 666 (1959)

Basic Properties of the Nucleus Tanvi Bhavsar and Nikit Deshmukh

Abstract This chapter gives a brief view of the basic structure of an atom and the initial experiments which gave the basis to explore the properties of the nucleus which lead to the independent existence of the Nuclear physics branch.

1 Introduction We must first know the basic structure of an atom before we can examine the basic experiments conducted to determine the properties of the nucleus. The structure of an atom includes a nucleus (central part), which consists of protons (positively charged particles) and neutrons (electrically neutral particles). Both particles have roughly equal masses. Negatively charged particles, called electrons, revolves outside of the nucleus. Figure 1 shows the structure of an atom. Let’s take a brief look at the evolution of atoms before learning about the experiments that reveal information about their structure. The history of atomic structure and quantum mechanics dates back to the time of Democritus. Democritus was a Greek philosopher who was the first person to use the term atom (atomos: meaning indivisible) [1]. He thought that if you take a piece of matter and divide it and continue to divide it, you will eventually come to a point where you could not divide it anymore. This fundamental or basic unit was what Democritus called an atom [2]. In 1800s , John Dalton was the first to apply Democritus philosophy to develop the theory of atomic structure [2]. All matter is made up of atoms, according to Dalton’s theory of the atom [2]. Atoms had been imagined as indivisible [2]. An element’s atoms were all identical, but atoms of various elements varied in size and mass. Numerous experiments carried out after Dalton’s Theory demonstrated that atoms can be further subdivided into subatomic particles. We shall describe each in detail. Before going into the structure and properties of each fundamental particle, we must T. Bhavsar School of Engineering, P P Savani University, Dhamdod, Kosamba, Surat 394125, GJ, India N. Deshmukh (B) School of Sciences, P P Savani University, Dhamdod, Kosamba, Surat 394125, GJ, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Deshmukh and N. Joshi (eds.), Understanding Nuclear Physics, https://doi.org/10.1007/978-981-19-8437-2_2

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Fig. 1 Atomic structure

examine the origin of nuclear physics. The study of nuclear physics begins with the discovery of radioactivity by Becquerel in 1896. Almost by chance, he discovered that when well-wrapped photographic plates were placed near certain minerals, they darkened. Two years after Becquerel’s discovery, Pierre and Marie Curie were able to extract radium (Z = 88), a naturally occurring radioactive element, from the ore (pitchblende) [3]. Soon later, it was understood that these interactions changed the chemical properties of an element. When a source was placed in a magnetic field, it was discovered that the trajectories of some of the “rays” emitted were deflected in one direction, some in the opposite direction, and some were not impacted at all. These rays were named α, β, and γ rays [3]. It was discovered that α-rays are made up of positively charged 4 He nuclei, β-rays are nothing but electrons or positrons, and γ -rays are electromagnetic radiation [4].

1.1 J. J. Thomson’s Experiment: The Discovery of an Electron Electrons are negatively charged particles that revolve outside of the nucleus, which consists of the proton and neutron and is located at the centre of an atom. While protons and neutrons are not fundamental particles and can be further divided into more fundamental particles known as quarks, whereas electrons are fundamental particles and cannot be further divided into more fundamental particles [5]. Electrons are denoted by the letters e or e− [6]. Electrons can move to higher energy states by absorbing some energy as they have different energy levels. Since electrons have a negligible mass of 9.1094 × 10−31 kg, or roughly 1/2000 the mass of a proton or neutron, they do not add to the overall mass of atoms. The charge of the electron is −1.6 × 10−19 C [6].

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During the 1880s and 1890s , scientists looked for the carrier of electrical properties in matter using cathode rays. Their efforts helped in the discovery of the electron in 1897 by J. J. Thomson. The discovery of an electron suggested that the atom is actually divisible in more fundamental particles. Research on Cathode rays started in 1854, when glassblower Heinrich Geissler and Julius Plucker ¨ improved the vacuum tube [7]. In 1858, Plucker ¨ observed cathode rays by sealing two electrodes inside an evacuated tube and passing an electric current between two electrodes [7]. He discovered a spot on the wall of the tube and concluded that these rays are coming from the cathode. Plucker’s ¨ student Johann W. Hittorf also observed a mark by an object placed in front of the cathode in 1869, using better vacuums [7]. The mark proved that the cathode rays came from the cathode. In 1878, William Crookes found that cathode rays were bent by a magnetic field; the direction of deflection suggested that they were negatively charged particles [7]. Because the luminescence was independent of the gas in the vacuum tube or the metal of the electrodes, he also concluded that the rays were a property of the electric current [7]. Vacuum tube which was used in Crookes’ work is called Crookes’ tube. The Crookes tube is more commonly known as Cathode Ray Tube (CRT) [7]. Crookes’ research did not resolve the question of whether cathode rays were particles or radiation similar to light, despite the fact that he thought the particles were electrified charged particles. The debate over the nature of cathode rays had split the physics group into 2 categories by the late 1880s [7]. Because cathode rays were impacted by magnets, the majority of French and British physicists were influenced by Crookes’ idea and believed that cathode rays were electrically charged particles. On the other hand, because the rays moved in straight lines and were unaffected by gravity, the majority of German physicists thought that they were waves [7]. Construction and Working of Cathode Ray Tube (CRT): CRT is a long glass tube filled with gas and sealed at both ends [8, 9]. It consists of two metal plates, which are served as electrodes [8, 9]. High voltage is further provided to these electrodes. The electrode which is attached to the negative terminal of the Battery is termed a Cathode and the electrode that is attached to the positive terminal of the Battery is termed an Anode [8, 9]. It was observed that when very high voltage was provided to the electrodes in an evacuated glass tube, the cathode produced a stream of particles and these particles travelled from the cathode to the anode termed cathode rays [9]. In the absence of an external magnetic or electric field, these rays travel in a straight line. Figure 2 shows the pictorial representation of the Cathode ray tube. J. J. Thomson, in 1897, demonstrated that atoms were not the fundamental building block of matter. He revealed that magnetic or electric fields could deflect or bend cathode rays, suggesting that they are made up of charged particles. More significantly, Thomson was able to determine the mass-to-charge ratio of the particles by determining the amount of cathode ray deflection in magnetic or electric fields of different strengths [10]. Particles which were emitted from the cathode (negative electrode) were repelled by the negative terminal of an electric field. From his

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Fig. 2 Cathode ray tube (CRT)

observation Thomson concluded that the particles had a net negative charge because like charges repel each other and opposite charges attract; these particles are now known as electrons. Thomson suggested that these negatively charged particles are unaffected by the type of metal used to make electrodes or the gas inside the tube. He also examined positively charged particles in neon gas. Thomson discovered that the conventional model of an atom did not account for negatively or positively charged particles. As a result, he proposed a plum pudding-like model of the atom [11]. The raisins in the pudding were represented by negative electrons, while the dough was positively charged [12]. Thomson’s model of the atom explained some of the electrical properties of the atom due to electrons, but it failed to understand positive charges in the atom as particles. In Fig. 3, we can see J. J. Thomson working on a Cathode Ray Tube, and Fig. 4 is the pictorial representation of Thomson’s Plum pudding model. Thus, the smallest first subatomic particle was identified as Electron by J. J. Thomson.

1.2 Millikan’s Oil Drop Experiment: Charge on Electron Robert A. Millikan and Harvey Fletcher conducted the oil drop experiment in 1909 to determine the elementary electric charge (the charge of the electron). The experiment took place at the University of Chicago’s Ryerson Physical Laboratory [13–15]. Millikan conducted the experiment by allowing charged tiny oil droplets to pass through a hole into an electric field. The charge over an oil droplet was calculated by varying the strength of the electric field, which always resulted in an integral value of “e.” The apparatus for the experiment was once developed by Millikan and Fletcher [16]. It includes a chamber containing an atomiser, a microscope, a light source, and

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Fig. 3 J. J. Thomson with cathode ray tube

Fig. 4 J. J. Thomson atomic model—plum pudding model

two parallel metal plates [16]. These metal plates obtain a negative and a positive charge when an electric current passes through them [16], as we see in Fig. 5 [17]. First, the atomiser was supposed to spray a fine mist of oil into the chamber [16]. Due to gravitational pull, the oil droplets would drift into the bottom half of the chamber (i.e. between two metal plates) [16]. The oil droplets become negatively charged after being ionised. The external power supply would then be used to give additional charge to the two metal plates, while these negatively charged droplets are being drawn down by gravity. In particular, the upper plate would create a positive charge and the bottom plate would generate a negative charge [16]. Gravity pulls the oil downward, while the electric field pulls the charge upward [17]. The strength of the electric field is controlled so that the oil droplet achieves

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Fig. 5 Millikan’s oil drop experiment apparatus

equilibrium with gravity [17]. At equilibrium, the charge over the droplet is calculated, which is dependent on the strength of the electric field and the mass of the droplet. By adjusting the field strength and careful measurements, Millikan was able to determine the charge on each individual drop [17]. Millikan found that all drops had charges that were 1.6 × 10−19 C multiples [18, 19].

1.3 Rutherford Experiment: The Discovery of a Proton E. Goldstein discovered the presence of positively charged particles in an atom in 1886, based on the idea that atoms are electrically neutral, meaning they have an equal number of positive and negative charges. J. J. Thomson’s plum pudding model could not explain certain experimental results related to the atomic structure of the elements. Ernest Rutherford conducted an experiment to explain the structure of an atom and also discovered the Proton. Rutherford carried out an experiment in which he bombarded α-particles on a thin gold sheet and then studied the trajectory of these particles after they collided with the gold foil [20, 21]. In this experiment, Rutherford directed high-energy αparticles on a ∼ 100 nm thin sheet of gold [20, 21]. He enclosed the gold foil with a fluorescent zinc sulphide screen to examine the deflection of the alpha particles [20]. Rutherford made some observations that contradict Thomson’s atomic model. Figure 6 shows the experimental setup of Rutherford.

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Fig. 6 Rutherford’s experimental setup

Observations of Rutherford’s Experiment Rutherford’s Alpha Scattering Experiment yielded the following outcomes: 1. He notices that the majority of the α-particles bombarded towards the gold sheet pass through the foil without deflection, indicating that the majority of the space is empty [22]. 2. Some of the α-particles were deflected through the gold sheet by very small angles, indicating that the positive charge in an atom is not distributed uniformly. The positive charge of an atom is concentrated in a very small volume [22]. 3. Only a small proportion of the α-particles (1–2 %) were deflected back, implying that only a small proportion of α-particles had a nearly 180 ◦ angle of deflection. This demonstrates that positively charged particles occupy a very small volume [22]. Based on these observations, he proposed an atomic model, which indicated the atomic structure of elements. Rutherford proposed the nuclear model, which states: • There is a positively charged middle of the atom called the nucleus, where all of the positive charge of an atom is concentrated. It is very small, approximately 105 times smaller than the radius of an atom. The atom’s entire mass is concentrated here [22]. • The negatively charged particles that surround the nucleus revolve at high speeds in fixed circular paths known as orbits [22]. • A strong electrostatic force of attraction holds positively charged particles (protons) and negatively charged particles (electrons) together. Figure 7 provides the pictorial representation of the scattering of α-particles by gold foil.

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Fig. 7 Scattering of α-particles by gold foil

Drawbacks of The Rutherford Atomic Model: The Rutherford Atomic Model has some constraints because it does not explain certain things: • According to Rutherford, electrons orbit the nucleus in circular orbits. Maxwell’s theory predicts that any charged particle in circular motion will accelerate and emit electromagnetic radiation. As a result, the orbit will contract as the revolving electron loses energy and collapses to fall into the nucleus [23, 24]. • Rutherford was unable to explain the stability of atoms because it contradicted Maxwell’s theory. The electron will collapse in less than 10−8 s, based on the calculations, making it highly unstable [23, 24]. • One major drawback is that Rutherford did not explain the arrangement of electrons in orbits [23, 24].

1.4 Chadwick Experiment: The Discovery of a Neutron In 1928, Walter Bothe, a German physicist, and his student Herbert Becker bombarded a thin sheet of beryllium with alpha particles released by a radioactive source of polonium [25]. They discovered that when the beryllium sheet was hit by alpha particles, it emitted electrically neutral radiation with tremendous penetrating strength [25]. Because it was already known that electrically neutral gamma radiation is emitted during natural radioactive decay, Bothe and Becker assumed that the chargeless radiation generated by beryllium foil was likewise gamma rays [25]. Irene Joliot

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Fig. 8 Sketch of the experimental setup used by Chadwick

Curie and her husband Frederic Joliot Curie performed the experiment carried out by Bothe and Becker in 1932, utilising a stronger and more intense polonium source [25]. For further study, the properties of this radiation of high penetrating power, Irene and Frederic allowed beryllium radiations to fall on a sheet of paraffin wax with a high percentage of hydrogen atoms. Joliot Curies were astonished when they noticed that penetrating uncharged radiation knocked out protons from the paraffin wax sheet [25]. Irene and Frederic attempted to describe the process by stating that penetrating radiation is high-energy gamma-ray photons that ejected protons from the atom of hydrogen, as electrons are ejected by photons in the photo-electric effect [25]. However, it was not conceivable due to the significantly greater mass of the proton in comparison to the mass of the electron [25]. Rutherford’s student James Chadwick presented the results of Irene and Frederic Curie’s experiment to Rutherford, who instantly stated, “I do not believe it.” Chadwick promptly recreated Irene and Frederic Curie’s experiment using the apparatus, a drawing of which is shown in Fig. 8. He tested not only paraffin wax to beryllium radiation, but also used other materials such as helium, nitrogen, and other elements. Chadwick measured the energy of emitted charged particles in each case and computed the mass of radiations emitted by beryllium using kinematical formulations [25]. In each case, he discovered that their mass was about equivalent to the mass of a proton. Chadwick’s research were reported in Nature on 17 February 1932, in which he discussed the discovery of a neutral particle of mass nearly equal to the mass of the proton, which he termed the neutron [26]. The nuclear reaction responsible for the emission of the neutron from beryllium foil is mentioned in Fig. 8. Chadwick was awarded the Nobel Prize in Physics for his work in 1935.

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1.5 Artificial Radioactivity Irene and Frederic Joliot Curie bombarded aluminium foil with 5.3 MeV alpha particles produced by the natural radioactive decay of polonium in 1934 [25]. They noticed that after irradiating the aluminium foil, positively charged particles began to emit from the foil, and that this emission remained for a while even after the polonium source of the alpha particles was taken away. Positively charged particle intensity was observed to drop exponentially after the alpha-particle source was removed, precisely like the decay of the intensity that occurs in the case of natural radioactivity. Based on the experimental data, they estimated the half-life of the observed activity to be 2.34 min. They explained this activity to the formation of a new unidentified isotope 30 P according to the following nuclear reaction: 4 2He

30 1 +27 13 Al →15 P +0 n

(1)

They then confirmed this by chemically separating phosphorus from irradiated aluminium foil within minutes of irradiation and observing that the activity comes from the separated phosphorus. The 30 P isotope was found as a result of radioactive decay. This reaction is given by [25] 30 15 P

0 →30 14 S ++1 e + ν

(2)

This was the first artificial production of a radioactive isotope and the first observation of positron decay in radioactivity. The Nobel Prize for chemistry in 1935 was awarded to Irene and Frederic Joliot Curie, who developed a technique for the rapid chemical separation of phosphorus from irradiated aluminium foil along with the experimental confirmation that the observed β + activity was attributed to phosphorus.

1.6 Strong Nuclear Force In 1932, Heisenberg and Iwanenko proposed the neutron-proton model of the nucleus in response to the discovery of the neutron [25]. The stability of this nucleus was attributed to a new force, the strong nuclear force [38], which was proposed in addition to the previously known classical gravitational and electromagnetic forces [25]. Gravitational force has no impact on the stability of the nucleus; nevertheless, the electromagnetic force of repulsion between protons counters the strong nuclear force and may break a heavy nucleus into two fragments [25]. Gamow (1930) constructed and explained a liquid drop model of the nucleus using a hint from the constant binding energy of nucleons. After Gamow, this model was also explained by Weizsacker, Bethe and Bohr [25]. The liquid drop model was important to understand the nuclear fission process [25].

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1.7 Nuclear Fission Before the discovery of the electron, proton, and neutron, Martin Klaproth discovered Uranium in 1789 and called it after the planet Uranus [39]. Wilhelm Rontgen developed ionising radiation in 1895 by passing an electric current through an evacuated glass tube and produced continuous X-rays [40]. Later, in 1896, Henri Becquerel discovered that pitchblende (a radium and uranium mineral) made a photographic plate darken. He conclude that this was due to the emission of beta radiation (electrons) and alpha particles (helium nuclei) [41]. Villard discovered a third sort of radiation from pitchblende, gamma rays, which were similar to X-rays. James Chadwick discovered the neutron in 1932 [26]. Cockcroft and Walton generated nuclear transformations by hitting atoms with accelerated protons in 1932, and Irene and Frederic Joliot Curie discovered that some of these transformations produced artificial radioactivity in 1934 [27]. The next year, Enrico Fermi discovered that using neutrons instead of protons allowed for the formation of a far wider range of artificial radionuclides. Fermi continued his experiments, usually synthesising not only heavier elements from his targets, but also some considerably lighter ones with uranium. Otto Hahn and Fritz Strassmann, two German scientists, first identified the fission reaction in 1938 [28–30]. In a series of tests, they had been bombarding various elements with neutrons. For instance, bombarding copper would result in the radioactive form of copper. A similar process made other elements radioactive. However, when uranium was exposed to neutron bombardment, it appeared that a completely other process took place. The uranium nucleus appears to have been greatly disrupted. Chemical analysis provided evidence for this reaction. Hahn and Strassmann presented a scientific study demonstrating that when uranium (Atomic number 92) was bombarded with neutrons, small amounts of barium (Atomic number 56) were generated [28]. They were puzzled as to how a single neutron could change element 92 Uranium into element 56 Barium [28–30]. A proper explanation for such a reaction was proposed by Lise Meitner [27], a colleague of Hahn. The uranium nucleus can be compared to a liquid drop containing protons and neutrons. The drop begins to vibrate when one extra neutron enters. If the vibration is strong enough, the drop may shatter into two parts. Meitner termed this process “fission” because it is analogous to the biological process of cell division [27]. It only requires a small amount of energy to start the vibration that causes a significant splitting of the nucleus. Scientists confirmed the idea of uranium fission, using other experimental facilities. For example, a cloud chamber is a device in which vapour trails of moving nuclear particles can be seen and imaged. A thin layer of uranium was placed within a cloud chamber in one experiment. Photographs taken after it was irradiated with neutrons revealed a pair of tracks moving in opposite directions from a common starting point in the uranium. Evidently, a nucleus in the process of fission had been photographed. Another experimental approach employed a Geiger counter, which is a small in size cylindrical tube that creates electrical pulses when a radioactive particle travels through it. A very thin layer of uranium was used inside a modified Geiger tube

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for this experiment. Large voltage pulses were recorded when a neutron source was brought close to it, far greater than normal radioactivity. When the neutron source was removed, the pulses vanished. The large pulses were clearly caused by uranium fission fragments. The pulse size indicated that the particles contained a significant amount of energy. Scientists conducted theoretical calculations based on Albert Einstein’s famous equation E = mc2 to better explain the high energy released during uranium fission. According to the Einstein equation, mass m can be converted to energy E, and the conversion factor is a large number, c, which is equal to the velocity of light squared. The total mass of the fission products remaining at the end of the reaction can be calculated to be somewhat less than the mass of the uranium atom + neutron at the start. This decrease in mass, expressed in the proper units and multiplied by c, explains numerically why fission fragments have high energy.

1.8 Nuclear Fission: Chain Reaction One of the modern applications of the fission process is nuclear chain reactions. Nuclear chain reactions, by using massive amounts of energy, can be employed for both constructive and destructive purposes. It is best to research the nature of fission reactions in depth to understand how a chain reaction works. Understanding fission reactions allows us to see that a chain reaction is simply an extrapolation of the excess particles created by fission for a big amount of an element. As we know, Lise Meitner (1878–1968) and her nephew Otto Frisch discovered atomic fission in 1939, which was described in her study Disintegration of Uranium by Neutrons. The term fission was invented to describe a new type of nuclear reaction. In 1942, Nobel Prize winner Enrico Fermi performed the first sustained chain reaction at the University of Chicago. His expectation was that the reactions would result in transuranium products. Otto Hahn and F. Strassmann repeated Fermi’s experiments and discovered them to be false. The first controlled chain reaction occurred in 1943, and the first atomic bomb was developed in 1945 [29, 30]. By 1951, nuclear chain reactions had evolved into a more feasible method of producing electricity. A nuclear chain reaction happens when the output of one nuclear reaction promotes the occurrence of further nuclear reactions. These chain reactions are nearly always a succession of fission events that produce too many neutrons. These extra neutrons can then generate more fission events, hence the name chain reaction. Nuclear chain reactions are required for nuclear power reactors to operate. Chemical reactions involve the recombination of various chemical species. Different types of nuclei (referred to as nuclear species) interact in nuclear reactions. Many chemical processes are chain reactions, and they share many characteristics with nuclear chain reactions. These analogies are as follows: • When chemical or nuclear species are ready to react, the reactions will sustain. When the nuclear species are removed or consumed, the chain reaction stops.

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• The chain reactions are started, speed up, slowed down, and stopped by adding or eliminating chemical or nuclear species from the chain. • Energy is frequently released as the reactions take place. • The released energy is usually produced as thermal energy, resulting in heat that can be captured by heat engines to do useful work such as electricity production. Despite a few similarities, there are also some significant differences. Nuclear processes provide approximately one million times the energy of chemical reactions. As a result, chemical chain reactions occur significantly more often than nuclear reactions. Fire is a chemical chain reaction. Nuclear chain reactions need precise engineering, as a natural nuclear chain reaction has only occurred once. Nuclear chain reactions demand a great deal of preparation. When they do occur, there is significantly more energy released, resulting in nuclear fuel with higher energy density. To sustain a nuclear chain reaction, each fission event must be followed by exactly one additional fission event. The most practical nuclear species to use in nuclear chain reactions is a fissile isotope of uranium, 235 U. On average, 2.5 neutrons are released with each fission event of 235 U. It takes careful engineering to make those neutrons continue to cause other fission events. Contrary to expectations, issues emerge when obtaining sufficient neutrons to maintain a sustainable nuclear reaction rather than having many reactions. The nuclear chain reaction is said to be essential if each fission event causes exactly one more fission event. The fission chain process is simplified in Fig. 9. Nuclear fusion and nuclear fission are two types of energy-releasing processes that occur when atomic bonds between particles within the nucleus break. The major distinction between these two processes is that fission involves dividing an atom into two or more smaller ones, whereas fusion involves fusing two or more smaller atoms into a bigger one.

1.9 Nuclear Fusion Nuclear fusion is the process in which two light atomic nuclei form a heavier one, transforming a small amount of matter into massive amounts of energy. The stars, including the sun, get their energy from nuclear fusion, which allows them to produce light. The sun provides the enormous energy that Earth receives, and without it, life on earth would be impossible. Arthur Eddington proposed hydrogen-helium fusion as the primary source of stellar energy in 1920 [31]. Friedrich Hund discovered quantum tunnelling in 1927 [32, 33] and soon after, Robert Atkinson and Fritz Houtermans used the measured masses of light elements to demonstrate that large amounts of energy could be released by the fusion of small nuclei [34]. Mark Oliphant achieved laboratory fusion of hydrogen isotopes in 1932, building on Patrick Blackett’s early experiments in artificial

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Fig. 9 A nuclear fission chain reaction of uranium-235 atoms

nuclear transmutation [35]. Hans Bethe worked on the theory of the main cycle of nuclear fusion in stars for a decade. Self-sustaining nuclear fusion was first carried out on 1 November 1952, in the Ivy Mike hydrogen (thermonuclear) bomb test [36]. Research into developing controlled fusion inside fusion reactors has been ongoing since the 1940s, but the technology is still in its development phase [36].

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Fig. 10 A nuclear fission of uranium-235

1.10 Difference Between Nuclear Fission and Fusion Nuclear Fission • Fission reactions are unusual in nature. • Fission generates a large number of radioactive particles. • The energy released by fission is a million times larger than that released by chemical reactions, but it is less than that released by nuclear fusion. • A fission bomb, often known as an atomic bomb or atom bomb, is one type of nuclear weapon. • The splitting of a large atom into two or more smaller ones is known as fission. • High-speed neutrons are required. • A typical nuclear reaction containing 235 U and a neutron would look as shown in Fig. 10. Nuclear Fusion • Fusion occurs in stars like the sun. • The energy generated by fusion is three to four times that released by fission. • The hydrogen bomb is one type of nuclear weapon that employs a fission reaction to “start” a fusion reaction. • Fusion is the fusing of two or more lighter atoms into a bigger one. • A high-density, high-temperature environment is required. • It requires an enormous amount of energy to bring two or more protons together close enough for nuclear forces to overcome electrostatic repulsion. Nuclear fusion is the reaction that occurs when two or more nuclei unite to produce a new element with a higher atomic number. The energy released in fusion is associated with E = mc2 . • A typical fusion of deuterium with tritium is shown in Fig. 11.

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Fig. 11 Fusion of deuterium with tritium

1.11 The Benefits of Nuclear Fusion over Nuclear Fission • Fusion reactions cannot sustain a chain reaction; they will never break up like fission reactions. • The fusion reaction generates extremely small or no radioactive waste if the appropriate elements are used. Fusion produces only radioactive waste with a short halflife. • Large amounts of radioactive waste are produced during nuclear fission, and disposal of radioactive waste is a difficult task. • Fusion produces three to four times the amount of energy that fission produces. This is because the amount of mass converted into energy in a fusion process is substantially more than in a fission reaction. • Fusion is a seemingly unlimited, low-cost fuel that is available worldwide. • There is no creation of greenhouse gases, smoke, or acid rain in fusion, and there is no chance of an explosive eruption or meltdown that could harm public safety.

1.12 Ghosal Experiment Rutherford demonstrated the first artificial transmutation by irradiating a thin gold foil with α-particles produced from radioactive sources in 1919 and with it, a new era

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of scientific research in nuclear physics started. There are two major goals to study nuclear physics. The first is to learn about the characteristics of the nucleus and the second one is to understand the behaviour of the constitution of the nucleus. When a high-energy projectile goes towards the nucleus within the range of nuclear forces, a nuclear reaction happens which leads to the release of electromagnetic radiations or fundamental particles. The properties of a nuclear system in a nuclear reaction are well known before and after the reaction has taken place. But what actually occurs during the nuclear reaction is still poorly understood [25]. As it is a quantum mechanical process, it cannot be directly visualised. Consequently, the nuclear theory or model has been put forward [25]. Bohr proposed the compound nucleus (CN) model to understand the mechanism of nuclear reactions, in 1936. Bohr claims that a CN is produced when the incident particle interacts with the target nucleus, and it will share its energy and angular momentum among all nucleons via different collisions until the thermodynamic equilibrium is established [25]. The lifetime of the compound nucleus is sufficiently long ≈ 10−16 s. Bohr’s hypothesis suggested that the decay of CN is entirely defined by its good quantum numbers, including the spin, angular momentum, parity and excitation energy, etc. S. N. Ghoshal experimentally validated Bohr’s hypothesis in 1950 [42]. Ghoshal examined the formation of compound nucleus (CN) Zn64 via α-bombardment of Ni60 and proton bombardment of Cu63 [42]. The α-reaction channels and p-reaction channels are given in Chapter 4. The stacked foil method was used to determine the excitation curves [42, 43]. The α-excitation curves were obtained using the 40 MeV α-beam from cyclotron [42, 43]. In the Nickel experiment, thin Ni foils of enriched Ni60 were prepared by electroplating the nickel onto copper and then dissolving the copper in Ag N O3 solution. More than 85% of the Ni60 was present in the sample [42, 43]. The same procedure was used to determine proton excitation curves in this case, employing a 32-MeV proton beam from the Berkeley linear accelerator [42, 43]. The setup for the stacked foil method is shown in Fig. 12. Experimental results of Ghoshal’s experiments are discussed in a more detailed manner in Chapter 4.

Fig. 12 Setup for stacked foil method

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

14.

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30.

http://www.abcte.org/files/previews/chemistry/s1_p1.html S. Joshi , U. Sudhir, The Story of Atomic Theory of Matter S.S.M. Wong, Introductory Nuclear Physics, 2nd edn N.T. Khan, J. Phys. Chem. Biophys.7, (2017) E.J. Eichten, M.E. Peskin, M. Peskin, New tests for quark and lepton substructure. Phys. Rev. Letters. 50(11), 811–814 (1983) J. Coffey, “What is an electron?”(2010), Archived from the original on 11 November 2012. Accessed 10 Sept 2010 https://www.britannica.com/science/atom/Discovery-of-electrons https://collegedunia.com/exams/Discovery-of-electron-characteristics-jj-thomsonexperiment-discovery-of-cathode-rays-and-solved-questions-chemistry-articleid-501 https://www.yaclass.in/p/science-state-board/class-8/atomic-strucuture-12278/introductionto-atomic-structure-8507/re-958192b2-e76d-41db-9492-771b66f9006d J. Joseph, J.J. Thomson, (Science History Institute, 2016). Accessed 20 Mar 2018 K. Hentschel, Atomic models, J.J. Thomson’s “plum pudding” model . (2009), https://doi.org/ 10.1007/978-3-540-70626-7_9 Project: light quanta (photons): history of a complex concept and mental model https://flexbooks.ck12.org/cbook/ck-12-middle-school-physical-science-flexbook-2.0/ section/3.13/primary/lesson/thomsons-atomic-model-ms-ps/ “American Physical Society to commemorate University of Chicago as historic physics site in honor of Nobel laureate Robert Millikan at University of Chicago” (2006), https://news. uchicago.edu/ Accessed 31 July 2019 E.H. Levi Hall, Avenue Chicago, The University of Chicago, 5801 South Ellis; Us, Illinois 60637773 702 1234 Contact, “UChicago Breakthroughs: 1910s”. The University of Chicago. Accessed 31 July 2019 “Work of physicist Millikan continues to receive accolades”(2007), chronicle.uchicago.edu. Accessed 31 July 2019 https://www.wikiwand.com/en/Oil_drop_experiment https://chem.libretexts.org/Courses/Rutgers_University/Chem_160%3A_General_ Chemistry/01%3A_Atoms/1.06%3A_The_Discovery_of_the_Electron “2018 CODATA Value: elementary charge”. The NIST Reference on Constants, Units, and Uncertainty. NIST (2019). Accessed 20 May 2019 R. Millikan, APS Physics. Accessed 26 April 2016 E. Rutherford, The scattering of α and β particles by matter and structure of atom. Philos. Mag. 6, 21 (1911) S.N. Ghoshal, Nuclear Physics, (S chand and Company LTD) Evolution of atomic theory, Mrs. Baldessari, talk presentation A. Dahal, N. Parajuli, Outlines of Rutherford’s α-particles scattering Experiment, J. St. Xavier’s Phys. Counc https://testbook.com/learn/chemistry-rutherford-model-of-atom/ B.P. Singh, M.K. Sharma, R. Prasad, book chapter: Pre-equilibrium Emission in Nuclear Reactions,Fundamentals, measurements and analysis J. Chadwick, Possible existence of a neutron Nature 129 312 Chadwick J 1932 The existence of a neutron. Proc. R. Soc. A 138, 692 (1932) https://en.wikipedia.org/wiki/Nuclear_fission O. Hahn, F. Strassmann, Nachweis der Entstehung aktiver Bariumisotope aus Uran und Thorium durch Neutronenbestrahlung; Nachweis weiterer aktiver Bruchstücke bei der Uranspaltung . Naturwissenschaften. 27 (6), 89-95 (1939). Bibcode:1939NW.....27...89H. https://doi. org/10.1007/BF01488988.S2CID 33512939 “The Discovery of Nuclear Fission”. www.mpic.de “Hahn’s Nobel was well deserved” (PDF). www.nature.com

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31. A.S. Eddington, Nature 106, 2653 (1920) 32. F. Hund, Zeitschrift für Physik (in German) 40, 742 (1927) 33. Tunneling was independently observed by Soviet scientists Grigory Samuilovich Landsberg and Leonid Isaakovich Mandelstam. J. Russ. Phys. Chem. Soc. Physics Section (in Russian) 60 335 (1928) 34. R.D.E. Atkinson, F. G. Houtermans, Zeitschrift für Physik (in German). 54 656 (1929) 35. M.L.E. Oliphant, P. Harteck, E. Rutherford, Proc. R. Soc. A. 144, 692 (1934) 36. https://en.wikipedia.org/wiki/Nuclear_fusion 37. A. Das, T. Ferbel, Introduction to Nuclear and Particle Physics, 2nd edn. (World Scientific Publishing Co. Pte. Ltd) 38. A. Das, T. Ferbel, Introduction to Nuclear and Particle Physics, 2nd edn. (World Scientific Publishing Co. Pte. Ltd) 39. https://en.wikipedia.org/wiki/Uröanium 40. M. Tubiana, Wilhelm Conrad Rntgen et la découverte des rayons X [Wilhelm Conrad Röntgen and the discovery of X-rays]. Bull Acad Natl Med. 180(1), 97-108 (1996). French. PMID: 8696882 41. https://www.aps.org/publications/apsnews/200803/physicshistory.cfm 42. S.N. Goshal, Phys. Rev. 80, 939 (1950) 43. U. Mahanta, Nuclear Physics, Department of Physics, Bhattadev University, Bajali

Experimental Details for a Typical Nuclear Physics Experiment Balaram Dey and Srijit Bhattacharya

Abstract This chapter gives the essence of the experimental set-up which is required for performing nuclear physics experiment that is necessary for exploring properties of the nucleus. The chapter accordingly focuses on the possible instrumentation and tools required to perform a nuclear physics experiment. The chapter gives the attention to ion sources and describes how an ion is produced and accelerated to perform a required nuclear reaction. It features all the categories and features of the accelerators. It gives the glimpse of the targets meant for the nuclear physics experiments. The chapter describes about the detection systems which are key ingredients of the nuclear physics experimental set-up as it is must to identify and study the characteristics of the final product after nuclear reaction. The chapter also highlights about the nuclear electronics, vacuum systems, Faraday cup and beam dumping and its shielding. In the end of the chapter, some standardized laboratory radioactive sources are also presented along with a schematic presentation of the experimental set-up.

1 Introduction The progress in nuclear physics experiments sharply depends on the development in instrumentation. The advancement in accelerator, detectors, and other electronic equipment leads to the rapid growth in the field of nuclear physics and provides the technical underpinning required to try for quenching our thrust in this field. Experimental instrumentation is required extensively in order to conduct nuclear physics experiments in today’s era. In order to perform nuclear physics experiment, we should first go through a nuclear reaction, where a beam of projectile with kinetic energy approaches toward the target. An accelerator system is required to provide the kinetic energy of projectile. B. Dey (B) Bankura University, Bankura, Purandarpur 722155, West Bengal, India e-mail: [email protected] S. Bhattacharya Barasat Government College, Barasat 700124, West Bengal, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Deshmukh and N. Joshi (eds.), Understanding Nuclear Physics, https://doi.org/10.1007/978-981-19-8437-2_3

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The beam of projectile with some kinetic energy will interact with the target nucleus and share its kinetic energy with the target nucleons till an equilibrated system (compound nucleus) is formed. The compound nucleus is formed at certain excitation energy with broad angular momentum distribution depending on the initial kinetic energy and impact parameter of projectile. Then, several particles (gamma, neutron, proton, alpha, and heavy charge particle) are emitted from the equilibrated system. In addition, the projectile could also be scattered electromagnetically through Coulomb interaction with the target and make Coulomb excitation of the target nucleus; the projectile could pick one or few nucleon from target; one or few nucleon could be transferred from projectile to target; and many other possibilities. Those emitted particles from compound system or scatter particles or outgoing particles contain the information of nuclear properties of either compound system or target nucleus. Therefore, we need to detect those particles with a good detection system. There are several detection systems for several particles. When the emitted particle is incident on the detector, it can produce an electrical signal at the end of detector. The electrical signal needs to be processed to reject the background and select the proper signal, which contain all the information of incident particle and thereby containing the properties of nucleus. Therefore, a suitable electronic system (either analog or digital) is required with a data acquisition system. The nuclear reaction has to be done in a target chamber. The projectile beam will come from accelerator system through beam pipe and enter into the target chamber through slit and make the reaction and finally the beam will be stopped by some kind of Faraday cup. If the beam pipe is too long, then the projectile beam could defocus due to Coulomb repulsion. In order to get the focused beam on target, several quadrupole focusing lens are placed in between the extraction point of accelerator and target chamber. Vacuum system is another integral part in nuclear physics experiment. Without vacuum in beam pipe, the projectile cannot reach to the target nucleus. Therefore, several vacuum systems (rotary pump, diffusion pump, etc.) are used in between the accelerator hall and experimental hall to keep a constant vacuum. After the nuclear interaction, the beam has to be stopped by something at somewhere for the radiation safety, which is known as beam dump. It is very important to make a proper shielding of beam dump from the detection systems. A Faraday cup is used in the beam to know the intensity of beam coming from the accelerator. And finally, we need to have some laboratory standard radioactive sources for detector characterization. Here, all the required elements for a typical nuclear physics experiments are tabulated as follows: 1. 2. 3. 4. 5. 6. 7.

Making beams of projectile. Target system. Detection system. Nuclear electronics and pulse processing. Vacuum systems. Beam dump and its shielding. Faraday cup.

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8. Laboratory standard radioactive sources for detector characterization. 9. A schematic experimental set-up. All the above requirements for a typical nuclear physics experiment are discussed in the subsequent sections.

2 Making Beams of Projectile In order to perform a nuclear physics experiment, we need a projectile having de Broglie wavelength comparable to the dimension of atomic nucleus as discussed earlier, as a probe. There are mainly two types of projectile beams, which are generally used in a nuclear physics experiment: 1. Charged particle beams (light and heavy ion beams). 2. Uncharged particle beams (γ and neutron beams). Here, we will be largely concentrating on charged particle beams and also show glimpses of the uncharged particle beams. As we know most of the nuclear physics experiments are performed using beam of projectile produced by a machine, known as accelerators. In addition, the accelerator requires the sources of projectiles or, alternatively, ion sources. Here we will discuss two of the most important requirements for a typical nuclear physics experiment, namely, (i) ion sources and (ii) accelerators.

2.1 Ion Sources In an accelerator, one of the most essential parts is an ion source for the production of high-energy particle beams of various polarized and unpolarized atomic species of different charge states [1]. We will however not discuss polarized beams here. Ion sources create the initial beam that is accelerated by the rest of the machine. The interesting thing about ion source is that in majority of the cases the type of the ion source decides the target–projectile combination and the parameters of the accelerator. The much-needed qualities of an ion source are long life time, quick change time, and high reliability.

2.1.1

Basic Components of an Ion Source

The basic components of an ion source [1] are as follows: (i) Main chamber: A vacuum chamber is indeed important where ionization takes place and leads to the creation of ions. This chamber is made of ceramic or metal and should be well vacuumed, otherwise the ions will be lost.

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(ii) Material: According to the requirement of ion, materials must be supplied into the source. The substance used as material should be either of solid, gas, or liquid. (iii) Ionization energy source: To lead ionization of neutral atoms, energy is required that may be provided via light, electrical power, etc. (iv) Extraction system: After creation, the ions will be emerged from the extraction hole. But to extract and accelerate the ions, electric field is needed. (v) Collimator: Finally, it collimates the particles into a parallel beam and also injects the beam into an accelerator to accelerate for increasing the energy of particles in a nuclear physics experiment. These (i)–(iii) components constitute the ion production region. The remaining (iv)– (v) denote the ion extraction region.

2.1.2

The Dynamics of Charged Ions

To control the dynamics of the charged ions, external electric and magnetic fields are applied to the ion source. It is well known that a charged particle will make a circular trajectory if uniform magnetic field is applied perpendicular to the plane of the trajectory. The circular motion will have a radius (cyclotron radius) R = 2m E/qeB with frequency (cyclotron frequency) ω = qeB/m, where m is mass of the particle or ion, e is electronic charge, i.e., 1.6 × 10−19 C, q is the charge of the particle, E is the electric field applied, and B is the applied magnetic field. Again, charge particle shows cycloid motion if it is subjected to electric and magnetic fields perpendicular to each other. The drift velocity, given by Vdri f t = E × B/B 2 , is independent of charge and mass of the particle.

2.1.3

Different Types of Ion Source

Some of the highly popular ion sources are discussed below: (i) Ion source using thermionic emission from cathode: As example, electron impacts ionization source. Electrons coming from cathode due to thermionic emission collide with the gas ion, atom, or molecule (target) and electrons are removed from the target atom or molecule that converts the atom or molecule into ion. Sometimes, the charge state of the ion also increases in this way. (ii) Ion source using Plasma discharge: In thermionic emission, the charge density of the source is generally small. By increasing the density of the source gas, copious number of ions and electrons are produced. The accelerated ions toward the cathode and accelerated electrons toward the anode, in turn, increase the number of secondary and tertiary electrons as well as ions. The enhanced number of ions and electrons can be considered as plasma. One example of this is Penning or Phillips Ionization Gauge (PIG) ion source [2]. The Penning ion source is based on Penning discharge (ionization of a target molecule due

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to collision with neutral gas molecule). The PIG ion source is well known as an internal ion source inside a cyclotron. It derives its name from the vacuum gauge invented by Penning [3]. The source is constituted via a cylindrical anode and two cathodes (one heated cathode and another anticathode or reflector) at each cylinder end. A typical magnetic field of ∼ 0.1−0.3T is applied to parallel axis of the cylinder and pressure of the gas of the source is kept at ∼ 0.001 to 1 mbar [4, 5]. The plasma ions can diffuse to the slits at the anode walls through which these have escaped. Applying different configurations of the electrodes and magnetic field, the movement of electrons over a long distance within the source may be varied by increasing the ionization probability. The Plasmatron, Duoplasmatron, Magnetron, and Penning sources are different types of plasma discharge sources. In all of these sources, the constant sputtering of cathode by the ions reduces the lifetime of the cathode. Therefore, using radio-frequency ion source, electron cyclotron resonance ion source, laser ion source, etc., in which electrodes are not required, the problem of cathode sputtering has been overcome. (iii) Electron Cyclotron Resonance Ion Source (ECRIS) The ECR ion source makes use of the electron cyclotron resonance to ionize a plasma [6]. The ECR discharge is sustained in an RF electric field with static magnetic field. The ECR ion source is based on plasma heating at the electron cyclotron frequency in a magnetic field, and ECR condition is given by fr = eB/2π m, where fr denotes the frequency of the RF electric field, B is the magnetic field strength, and e and m are electron charge and mass, respectively. For electrons, the ECR frequency is 28 GHz T−1 . Microwaves are injected with same frequency in the plasma, thereby accelerating or decelerating the electrons, being the microwave electric field component in resonance with electron orbit. As a result, electrons are “heated” or energized to further ionize the plasma. The cyclotron at the Variable Energy Cyclotron Centre, Kolkata, India uses indigenously developed ECR ion source of 6.4 GHz (Fig. 1).

Fig. 1 Schematic of the PIG (SC 200) internal ion source

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2.2 Accelerators So far we have learnt about ion sources. But ion sources alone cannot accelerate or energize particles. Ion sources create the initial beam that is accelerated by the rest of the machine. The accelerator, in nuclear physics experiments, does this work by applying electrostatic force or electromagnetic force on the particles emitted from the ion sources. In such experiments, particles such as electrons, protons, and heavy ion beams are required for a wide energy range. These are primary particle beams. The secondary particle beams like pions, muons, neutrinos, and radioactive ions can also be produced indirectly from intense proton and heavy ion beams. Initially, such as in the alpha-ray experiment of Rutherford, energized alpha particles were directed to the gold foil from radioactive sources. Natural radium was one of the well-known sources. However, first particle accelerator was built by Cockroft and Walton in 1932 [7] and completely changed the perspective of nuclear physics experiments. Particle accelerators have the advantage of producing a variety of beams (of a large number of particles between proton and Uranium of periodic table) of much higher energies and beam currents or intensities in comparison to radioactive sources. The Cockroft–Walton accelerator [7] was electrostatic in nature and used voltage multiplier circuits to energize the particles. Gradually different accelerators came into existence with the advent of technology to explore “terra incognita” of nuclear physics. During the 1950s and 1960s, the requirements for advanced nuclear physics experiments led to major improvements in this sector and a number of accelerators were constructed, such as Van de Graaff, Pelletron (modernized version of Van de Graff), Cyclotron, LINAC (linear accelerator) [8] and used for research purposes at university laboratories. Meanwhile, LINAC beam was used for cancer radiotherapy in 1953 at London. With time, larger facilities grew to quench the thrust of the nuclear as well as medical physicists.

2.2.1

Cockroft–Walton Accelerator

Cockroft–Walton accelerator [7, 8] was the simplest of all the accelerators as shown in Fig. 2. The charged particles are accelerated between two terminals by the application of 200kV of accelerating voltage. The voltage multiplier circuit has been used for higher voltage generation. Interestingly, this device was used first time for artificial nuclear transmutation by using proton of energy on Lithium target (Fermilab Accelerator Report No. 1) p(800keV ) +7 Li →4 H e +4 H e. The energy gain in this accelerator by the ions is qV, where q is the charge state of the ion. Maximum voltage could be reached up to 200kV at Fermilab. This accelerator is, at present, used only as an injector to large accelerators. In Fermilab, such use of Cockroft–Walton accelerator can be seen.

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Fig. 2 Left—Cockroft–Walton accelerator (www.mikepeel.net). Right—A Cockroft–Walton voltage multiplier circuit (2 stages) (By Chetvorno—Own work, CC0, https://commons.wikimedia. org/w/index.php?curid=39545397)

2.2.2

Van de Graaff (VdG) Accelerator

Robert Jemison Van de Graff invented the very simple electrostatic accelerator in 1929, named as Van de Graff accelerator [9, 10]. The basic principle of Van de Graff is, when a charged conductor is brought internally in contact with externally hollow metallic conductor, then all charges can be transferred to the outer surface of metallic conductor and this process happens further and further via a rubber charged conveyor belt, which is connected at a high DC voltage. This leads to a huge potential difference to accelerate the charged particles. Maximum voltage could be reached up to 25 MV at Fermilab. The simplest kind of VdG accelerator can be seen at different museums. At Kolkata this is found at Birla Museum and Science City. But those are open air machines used for educational purposes only. The ionization of air surrounding the conductor or in other words “Corona discharge” limits the energy gain. Machines kept in vacuum may achieve voltage of even 25 MV. A schematic diagram of VdG accelerator is shown in Fig. 3.

2.2.3

Tandem Accelerator

The problem with VdG accelerator is the limitation of achieving maximum voltage. This problem can be removed with the tandem principle [11, 12]. The basic difference between tandem and VdG lies in its basic principle. The negatively charged particles (like H − , H e− , etc.) are at first accelerated toward positively charged conductor and those negatively charged ions are charge stripped via a stripping material. The negative ions are, therefore, converted to positively charged ions and they are

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Fig. 3 A schematic diagram of Van de Graaf accelerator (www.thebigger.com/ physics/electrostatics/ explain-van-de-graffgenerator/)

further accelerated by the terminal voltage. So, basically a tandem accelerator can provide twice (or more than twice) the amount of energy, if negatively charged ion becomes doubly (or greater than that) ionized positive ion after getting stripped than a conventional VdG generator. The final energy E of the emerging positive ions with charge state q in a tandem is E(MeV ) = V0 + (q + 1)VT , where V0 is the negative ion energy after getting injected from ion source. VT is the terminal voltage.

2.3 Pelletron Accelerator There is no difference between Tandem and Pelletron accelerators except the design of conveyeor belt. Here in Pelletron the conveyor belt is basically a chain which is constructed with small metallic tubes connected via insulating material like nylon. That’s why it is charged quickly than VdG or Tandem. In India, a 14 MeV Pelletron is situated at TIFR Mumbai [13] and another at IUAC, Delhi (15 MeV Pelletron accelerator) [14]. This type of accelerator needs initially negative ions. However, negative ions are difficult to be achieved. Besides, the maximum potential difference also puts limit to the energy of the particles.

Experimental Details for a Typical Nuclear Physics Experiment

2.3.1

33

Linear Accelerator or LINAC

Linear particle accelerators can accelerate charged particles to very high velocities in linear direction and eliminates the disadvantages of Pelletron. It uses a number of cylindrical metallic drift tubes, which is placed at a certain constant distance in linear fashion (as length of these tubes are increasing in forward direction) and drift tubes are connected via a radio-frequency oscillating electric potential in such a way that the consecutive tubes are in opposite polarities. The oscillating electric potential is applied to create an alternative electric field to accelerate charge particles. A charged particle traveling within the tube is accelerated in the gap between the electrodes when the voltages are applied in appropriate phases. The length of the tubes gradually increases along the tube length to keep particle in phase with the electric field. Tata institute of Fundamental Research (TIFR) has set up a Pelletron accelerator. To increase the energy of the projectile, a superconducting LINAC has also been established in 2007 [14]. A schematic diagram of LINAC is shown in Fig. 4.

2.3.2

Cyclotron

Cyclotron [15] is a particle accelerator, which is constructed via two “D” (spelled as “Dees”)-shaped hollow metallic cylinder, separated by a small distance and, being hollow metal, they are electrically insulated. A constant magnetic field is applied perpendicular to the plane of “Dees” for the circular motion of the charged particle coming from ion source and an oscillating radio-frequency (RF) electric field (oscillating voltage) is applied along the two “Dees”. These two Dees are connected to two terminals of an alternating voltage source so that if one Dee is in the positive potential then the other will be in exactly opposite negative potential at the same time. A schematic diagram of cyclotron is shown in Fig. 5. The magnetic field gives rotation to the particle and the electric field increases its velocity after each rotation and

Fig. 4 A schematic construction geometry of LINAC

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Fig. 5 A schematic diagram of cyclotron

hence the energy of the charged particle. The charged particle (mass m and charge q) rotates in circular path (radius r) under the action of magnetic field (B). Therefore, we can equate the centripetal force with magnetic Lorentz force as follows: mv 2 = qv B r qr B Or, v = m v Or, ω = = q Bm, r where v is the velocity of the charged particle. The particle rotates in a circular path inside a Dee (as inside the Dees, there is no effect of electric field) due to the magnetic field. The frequency of RF oscillator is made in such a way that when the particle reaches just at the edge of one Dee, the polarity of the Dees gets reversed and the particle is attracted to the other Dee. Hence, the velocity of the particle increases and the energy also increases. As a result, from the above equations, the particle rotates in an orbit of higher radius within the other Dee. Therefore, in this way, the radius of the semi-circular orbit increases in each half-cycle. For this, the resonance condition should be achieved, that is, the time taken by the particle to complete one complete cycle should be T =

2π m 2π = . ω qB

(1)

When the radius of the orbit becomes same as that of the Dees, the energy of the particle becomes maximum and it is deflected from the circular path and directed

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toward the fixed target. India has K130 cyclotron (224 cm radius) for producing light and heavy ion beams and K500 superconducting cyclotron at the Variable Energy Cyclotron Centre, Kolkata, India [16]. For non-relativistic case, cyclotrons are characterized by K-value. K is given by 

K E max A

2

 =K

Q A

2 ,

(2)

where K E max is the maximum kinetic energy achieved by the particle and Qe is the charge of the particle accelerated. The K130 cyclotron can accelerate protons from 6 to 60 MeV and alpha from 25 to 130 MeV at the VECC [16]. The magnetic field of superconducting cyclotron could be increased up to 5 T, which is almost five times higher than the K130 cyclotron. Therefore, superconducting cyclotron can impart much higher energy (80MeV/A for light ions and 5–10 MeV/A for heavier ions) to the particles. If the charged particle could reach velocity to the relativistic region, then the value of T changes, owing to the relativistic increase in mass as 2πm , where m = √ m 0 2 2 is the relativistic mass and c is given in equations. T = (q B) 1−v /c

the velocity of light in free space. So, the resonance condition could not be fulfilled and the particle could not be accelerated. This problem is seen for very light particles like electron.

2.3.3

Synchrotron

Cyclotron cannot accelerate the electron because of its relativistic nature. The problem due to relativistic increase in mass in cyclotron can be overcome using synchrotron. In a synchrotron [17–19], the radius of the particle orbit is kept fixed by utilizing a magnetic field that increases with time as the momentum of a particle increases. A synchrotron machine accelerates electrons at extremely high energy and then makes them change direction periodically.

2.3.4

Betatron

It is also a cyclic accelerator specially designed to accelerate electrons [20]. Here a variable magnetic field is applied perpendicular to the plane of the motion of the electron at torus-shaped vacuum tube. Due to the changing magnetic field, it creates an electromagnetic induction and induced emf at electron’s motion (here electron’s motion acts like current loop or secondary coil) and gives the acceleration. It gives more energetic electrons in comparison to an electron gun.

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Neutron Beam

Neutron is a very important particle as a beam or projectile for nuclear physics experiment [21]. We know neutron is a chargeless particle. So, it cannot be accelerated through any type of accelerator. So how to produce neutron beams is a big question. Scientists have used a simple technology to achieve this. To understand this, first of all we should know the neutrons are classified according to energy as (i) thermal neutron (∼0.025 MeV), (ii) epithermal neutron (∼1eV), (iii) slow neutron (∼1 KeV), and (iv) fast neutron (100 keV–10MeV). In accelerators, such as LINAC electron or proton accelerators, pulsed or repetitive electrons are produced with energy ∼50 MeV or so, which hit heavy ion metallic targets. As a result, sub-thermal, thermal up to fast and relativistic neutrons (of energy up to 50 MeV) are originated using the reaction (γ , n). This is how neutron beam can be produced and further used in nuclear reactions. Energy of the neutron beam can be controlled by the energy of produced pulsed beam. Fast neutrons can be slowed down using moderators like heavy water. Neutron produced in the fission reactor can also be used as neutron beam. Spallation reaction [22] is another technique to produce the neutron beam, in which a light projectile (proton or light nucleus) with the kinetic energy from several hundreds of MeV to several GeV interacts with a heavy nucleus (e.g., lead) and causes the emission of a large number of hadrons (mostly neutrons) or fragments.

2.4 Targets for Nuclear Physics Experiment A target used in nuclear physics experiment can be defined as a substance or object subjected to bombardment by projectile such as elementary particle, proton, alpha, and heavy ions. The projectile could originate from the spontaneously decaying radioisotopes or can be delivered by accelerator discussed in previous chapter. There are different kinds of targets used in nuclear physics experiment—solid target, gas target, etc. As, for example, the thin metallic foil targets were used in Geiger and Marsden experiment in 1909 [23] by using the alpha beam coming from a radioactive source and the results introduce a new concept of atomic nucleus by Rutherford in 1911 [24]; a gas target (a glass tube filled with nitrogen gas) was bombarded with 7.83 MeV alpha particle emitted by 214 Po by Rutherford and his team in 1919 [21] for the study of nuclear transmutation; the use of target for accelerator-based research started with the 1932 experiment on 7 Li target by using 0.45 MeV proton beam delivered by a linear accelerator, known as Cockroft–Walton cascade generator [7]; and so on. Earlier, the targets were prepared by simple way such as filling containers with gas, pressing powders into cavity in backing plates, drying a solution of target material on backing plates, etc. [21]. However, the development of accelerator technology providing the high-energy beam increased the interest to prepare the high-quality target for obtaining the reliable data, which trigger the development of techniques used for target preparation.

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Target preparation is often crucial for the success of an experiment. It is therefore of the utmost importance that the target conforms to what is required regarding purity and composition. So, targets used in nuclear physics experiments are mostly prepared from the enriched isotopes. A good target should have a constant and satisfactory yield during the experiment. Each experiment has some specific purpose and thereby needs a specific target, which depends on various parameters. The target could have different dimensions (square, circular, rectangular), thickness (typically 1µg/cm2 to 20mg/cm2 ), physical (solid, liquid, gaseous), and chemical (elemental, compound, or alloys) forms. They can be prepared on a backing or self-supporting. The target should have a relatively uniform thickness distribution, good mechanical strength, and stability under the projectile beam and high chemical purity. The chemical purity mainly depends on the enrichment of the available material. Because of these large parameters, it is very difficult to decide which technique is the most suitable for a specific target. In addition, the method should not only be effective (assure the production of the final target with the required parameters), but also be efficient in terms of minimizing the costs. In general, the techniques to prepare the target to be used in the nuclear physics experiment can be roughly classified as mechanical, physical, and chemical: 1. Mechanical—rolling (reshaping), powder processing (sedimentation and sintering), etc. 2. Physical—vacuum vapor deposition, electron bombardment, electron beam gun, levitation heating and melting, ion beam sputtering, electrospraying, etc. 3. Chemical—electrodeposition, polymerization, evaporation of droplets, etc. 4. Implanted target. The details of preparing the target using different techniques will be discussed in the second part of the book.

2.5 Detection System Detection system is an integral part of the nuclear physics experiments and hence the key to unfold the nuclear properties. Like five sensors in our physical body, a detector is a device to sense the nuclear radiations, such as those produced by nuclear decay, cosmic radiation, or nuclear reactions. It is used to track and identify those nuclear radiations (gamma, neutron, proton, alpha, heavy ions) and to measure various kinematic properties, such as energy, momentum, position, etc. But, the main question is what kind of materials should be used in the detector. Since the early days of radiation testing by Roentgen [25] and Becquerel [26], nuclear scientists have sought ways to measure and observe the radiation by using some materials. Earlier, a photographic plate was used in the path/vicinity of radiation to capture any sort of data from radioactivity. Another common early detector was the electroscope—a pair of gold leaves that would become charged by ionization caused by radiation and repel with each other. Electroscope could measure the alpha or beta particles both

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depending on the arrangement of device. It was much better than the photographic plate in terms of sensitivity and was an excellent tool for early experiments involving radioactivity. An interesting early device used to detect the radiation was zinc sulfide (ZnS) screen with a microscope [23, 24]. If the radiation incident on ZnS screen, it will produce a tiny flash of light which was further recorded by a microscope. It was one of the first approaches of counting a rate of radioactive source, albeit a very tedious one, as scientists had to work in shifts watching and literally counting the flashes of light. This tendency of certain materials to give off light when exposed to radiation has drawn a great attention to the people to look for the similar kind of materials to be used in the future development of detection technologies. These early devices and many others, such as cloud chamber, GM tubes, and ion chambers, were valuable to understand the basic principles of radiation [27]. With the development of technology, there are many sophisticated detectors available at present for each kind of nuclear radiation. An important part of knowing what type of detector to use is to have an idea of what kind of nuclear radiation we have. Depending on the types of nuclear radiation, the present detectors are classified as follows: 1. 2. 3. 4.

Gamma detector. Neutron detector. Charge particle detector (light and heavy charged particles). Mixed field of gamma and neutron detectors.

2.5.1

Gamma Detector

Detection of gamma radiation is one of the most important research tools in nuclear physics and it provides the information on various properties (excitation energies, angular moments, decay properties, etc.) of nuclei [28]. Detection of gamma radiation is based on the interaction between the radiation and the detector material. The γ rays mainly interact via pair production, Compton scattering, and photoelectric effect while passing through a medium, and produce an electromagnetic shower in the entire volume of the detector. In order to confine this shower, a large interaction volume with high γ -detection efficiency is required. Inorganic scintillators [NaI(Tl), BaF2 , BGO, LaBr3 , etc.] are commonly used in the detection of the gamma rays because of their higher atomic number and high density [27, 28]. Semiconductor detector such as high-purity germanium (HpGe) is also used to detect the gamma rays. The HpGe detector has very good energy resolution (0.3% at 662 keV). However, the highresolution Ge crystals have very small γ -detection efficiency due to small size and low effective atomic number, and very poor timing response. The efficiency, energy resolution, timing resolution, gain stability, and linearity are the main properties of a detector to be used in experiment.

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Fig. 6 A schematic diagram of inorganic scintillator

2.5.2

Inorganic Scintillator

An inorganic scintillation detector is a luminescent material [NaI(Tl), BaF2 , BGO, LaBr3, etc.], which is optically coupled with a photomultiplier tube (PMT) that detects the gamma-ray-induced light emissions (scintillation photons). A schematic diagram of inorganic scintillator is shown in Fig. 6. When a gamma ray is incident on scintillating material, it will produce high-energy electrons (via pair production and photoelectric effect) and low-energy photons (via Compton scattering). Those lowenergy photons and high-energy electrons excite the atoms and molecules present in the scintillating medium and produce very low-energy photons (in the energy range of eV), known as scintillation photons. This scintillation photon could be visible and UV, depending on band structure of scintillating material. If it is visible, a normal glass is used as entrance window of PMT. But, in case of scintillation photon in UV region, either a quartz window PMT is used or small amounts of impurities (called activators) are added to all scintillators to enhance the emission of visible photons. PMT is a device which converts the light into a measurable electrical current. It consists of cathode (made of photosensitive material) followed by an electron-multiplier system (known as dynode) and finally an anode from which the final signal can be taken. All the parts are usually housed in an evacuated glass tube. A high voltage is applied to the cathode, dynode, and anode. When the scintillation photons are incident on photocathode, an electron is emitted via photoelectric process, and then multiplied by the dynode via successive secondary emission of electron and then finally collected by anode. This is how a current signal is produced, which carries all the information of incident gamma radiation.

2.5.3

Semiconductor Detector

A semiconductor detector [27] is a device that uses a semiconductor (usually germanium) to measure the effect of incident gamma radiation. When a gamma radiation is incident on the semiconducting material, it creates electron–hole pairs. The number

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Fig. 7 A schematic diagram of inorganic scintillator

of e-h pair is proportional to the energy of the incident radiation. The basic configuration used for operating a p-n junction diode as radiation detector is shown in figure. Two electrodes must be fitted onto the two sides of p-n junction in order to collect the e-h pairs created by radiation. The electron and holes travel toward the electrodes under the influence of external electric field (reverse bias) and produce a current signal which carries all the information of incident radiation. A schematic diagram of semiconductor detector is shown in Fig. 7.

2.5.4

Neutron Detector

As neutron has mass but no electrical charge, it cannot directly ionize the medium and thereby cannot be directly detected. This means that neutron detectors must rely upon a conversion process where an incident neutron interacts with a nucleus to produce a secondary charged particle or gamma. These charged particles and gammas are then directly detected and from them the presence of neutrons is deduced. When a neutron strikes a piece of matter, it does not interact with the atomic electrons; instead, neutrons interact with the nuclei. Neutron interacts with the nuclei via elastic and inelastic scattering, nuclear reaction, capture process, fission, etc. [28]. The neutron is quite sensitive to light mass nuclei like hydrogen, helium, lithium, oxygen, etc. which have much higher interaction probability with neutrons. Therefore, low-Z materials are generally used for neutron detection. It should be mentioned that thermal neutron (∼0.025 eV) is detected via absorption or capture processes, while slow (< 100 keV) and fast (> 100 keV) neutrons are detected via elastic, inelastic, and nuclear reaction processes. There are different kinds of neutron detectors as follows: 1. 2. 3. 4.

3

He-gas-filled detector; BF3 proportional counter; 6 LiF converter; Organic scintillator (BC501A, NE213, etc.);

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Fig. 8 A schematic diagram of 6 LiF-Si neutron detector

5. Plastic scintillator; 6. Semiconductor detector with HDPE, etc. 3 He-gas-filled detector: It is mainly used for thermal neutron detection [29]. 3 He is an effective neutron detector material because of its large thermal neutron capture cross section. A typical counter consists of a gas-filled tube with a high voltage applied across the anode and cathode. When a thermal neutron is incident on 3 He gas medium, it will produce 1 H and 3 H ion.

n(0.025eV ) +3 H e →3 H + p + 0.765 MeV [∼5400 barn]. The produced 3 H and 1 H ionize the medium and create a signal which carries all the information of incident neutrons. The main disadvantage of this neutron detector is the limited production [30] of 3 He as a by-product from the decay of tritium (t1/2 ∼12.3 year). BF3 proportional counter: It is one of the most famous types of boron-based neutron detector, and generally used for slow neutron detection [29, 31]. In this detector, BF3 gas acts as both a proportional gas and a neutron detection material. Incident neutron interacts with boron nucleus to produce 7 Li and an α particle through 10 B(n,α)7 Li reaction [29, 31]. These produced charge particles further ionize the gas medium and create signal which carries all the information of incident neutron. 6 LiF converter: It is semiconductor-based (Si) charged particle detector (as shown in Fig. 8) in combination with a neutron reactive film (6 Li) which converts neutrons into charge particle radiation [32]. 6 Li acts as converter due to its large cross section (∼940 barn) to thermal neutrons. As said, 6 LiF is used as neutron converter layer and it is coupled with a Si detector. Incident neutron interacts with 6 Li and produced charge particles as shown in the following reaction: n(0.025eV ) +6 Li →3 H (2.73MeV) +4 H e(2.05 MeV).

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Fig. 9 A schematic diagram of semiconductor detector with HDPE [36]

Produced charge particles (3 H and 4 He) incident on Si detector and produce signal which carries all the information of incident neutron. Organic scintillator: This detector is mainly used to detect fast neutrons [27, 33]. The working principle of organic scintillator for neutron detection is exactly same as the inorganic scintillator for gamma detection. The difference is that in case of organic scintillator, liquid organic materials are used as scintillating material and the scintillation photons are produced due to molecular transition. While in case of inorganic scintillator, the scintillation photons are produced due to electronic transition. NE213, BC501A, BC519 [33–35], etc. are some examples of organic scintillator detector. Semiconductor detector: Semiconductor detectors can be used to detect neutrons if they are covered by a conversion layer as discussed earlier, as shown in Fig. 9. Incident neutrons transfer their kinetic energy to hydrogen via elastic nuclear scattering in the conversion layer, and protons are produced as recoils. High-density polyethylene (HDPE) can be used as an excellent conversion medium [36–39], where incident neutrons transfer their kinetic energy to the hydrogen present in HDPE. The recoiled protons are incident on the depletion layer of semiconductor and produce e-h pairs, which are further collected by applying a voltage across the electrode.

2.5.5

Charge Particle Detector

Charged particle primarily interacts with electrons while passing through the medium. Electrons in the atoms are either ejected (ionization) or could jump to higher energy level (excitation) and thus deposit its energy in the medium. The charge particle

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could also interact with the electrons via multiple scattering and also with the nuclei producing bremsstrahlung radiation. Energy loss for the charged particles is strongly energy dependent and it is well described by the Bethe–Bloch formula in the energy range of MeV-GeV including the atomic shell correction at lower energy and radiative loss correction at higher energy. The Bethe–Bloch formula [27, 40] is given as dE = − dx



e2 4π 0

2

    2m e c2 β 2 4π z 2 Z abs ρ N A 2 2 ln − ln(1 − β ) − β , (3) m e c2 β 2 Aabs I

where β = v/c; z is atomic number of the incident particle; ρ, Z abs , and Aabs are density, atomic, and mass number of absorber or medium, respectively; and I is excitation potential. There are different kinds of charge particle detectors such as gas-filled detector, surface barrier detector, CsI scintillator, etc.

2.5.6

Gas-Filled Detector

It consists of a cylindrical tube which contains gas medium, cathode, and anode [27, 40]. A wire is kept along the symmetry axis, which acts as anode kept at high voltage. While the wall of the cylinder acts as a cathode which is kept at zero (grounded) potential. Generally, inert gas is preferred to avoid the contamination (mostly chemical) within the gas chamber. A schematic diagram of gas-filled detector is shown in Fig. 10. The working principle of gas-filled detector is based on the ionization of gas molecules by incident radiation and creation of electron–ion pairs. To collect electron–ion pairs produced in a gas-filled detector, an external voltage must be applied. Due to the application of external voltage, electrons and ions travel toward the anode and cathode, respectively, and thereby create an electric field inside the detector volume. Current is generated due to the collection of electrons and ions

Fig. 10 A schematic diagram of gas-filled detector

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by the electrodes. This current pulse contains all the information of incident charge particle radiation. There are different working regions of gas-filled detector such as ionization region, proportional region, Geiger–Muller region, and continuous discharge region. [27, 40]. Ionization region: If the applied voltage is strong enough that recombination becomes negligible, all the charges are efficiently collected without loss. The steadystate current measured at this condition is called ionization current. Proportional region: Produced electrons and ions make many collisions with neutral gas molecules until collected. When the applied voltage is strong enough, electrons get high kinetic energies which further can ionize neutral molecules. While ions have low mobility and they attain very little energy. The initial and secondary electrons can then be accelerated and further ionizations can be caused, and thus cascades of secondary electrons are produced. This is known as avalanche phenomena. The total number of electrons can therefore be multiplied by a factor of thousands and this amplified signal at the anode is much easier to detect. In this avalanche condition, the current, i.e., collected charge depends on multiplication factor and is proportional to the applied electrical voltage. This is known as proportional region. In this region, multiplication factor is linear. Limited proportional region: If we further increase the applied voltage, the multiplication becomes non-linear. This non-linear effect has occurred because the electrons are collected quickly due to high mobility while positive ions are slowly moving, which takes a long time. Therefore, a (almost) motionless cloud of positive ion is created because of which electric field is distorted, which leads to the distortion in gas multiplication. Geiger–Muller region: In Geiger–Muller region [24], the electric field generated due to the application of high voltage is so strong that secondary and higher order avalanches (e-ion pairs) can occur. The generation and propagation of those avalanches are triggered by the photons emitted by the excitation of atoms in primary avalanches. As these photons are not affected by electric field, they may travel too far (laterally along the anode wire). Hence, entire detector (gas volume) participates in the process.

2.5.7

Surface Barrier Detector

It is a semiconductor detector and its working principle is same as the working principle of semiconductor detector (HpGe) for gamma detection [27, 41]. It is a p-n-type silicon diode wafer characterized by a thin depletion layer as shown in Fig. 11. It is made of p-n-type silicon on which one surface is etched prior to coating with a thin layer of gold (typically 40 µg/cm2 ). The other surface is coated with a thin layer of aluminum (typically 40 µg/cm2 ) to provide an electrical contact. A reverse bias voltage is applied to get the depletion layer which acts as an active volume of the detector. Depending on the applied voltage, the detector can be partially depleted or totally depleted. When a charge particle is incident on depletion layer, it will create e-h pairs. The number of e-h pairs depends on the bandgap of used semicon-

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Fig. 11 A schematic diagram of Si surface barrier detector

ductor and incident energy of charge particle. The produced e-h pair will move to the corresponding electrodes and produces an electrical signal, which contains the information of incident radiation. The thicknesses of such detectors are set by the range of the penetrating particle to be measured. As the particle energy increases, the required thickness of such detector increases. It should be remembered that Si surface barrier detector itself cannot identify the different types of charge particles. Particle identification can be done by using E- E telescope [27], which consists of pairs of thin and thick surface barrier detector. As an example of such telescope, an assemble of E-silicon detector (∼10–30 µm thick) and an E-silicon detector (∼1500 µm thick) may be used for high-energy charge particle detection and particle identification. Thickness of E and E detector depends on types of particle and their kinetic energy.

2.5.8

CsI Scintillator Detector

CsI(Tl) is less expensive, easily machinable, and almost non-hygroscopic (mildly hygroscopic in air) scintillator for charge particle detection [42]. While silicon detectors are much more expensive and require sophisticated cooling to reduce leakage currents (noise source). CsI(Tl) crystals have excellent energy resolution and produce light at wavelengths that match well with the response of silicon-based photodiodes. The energetic light charged particles, such as protons or alphas with E/A > 20 MeV, can penetrate through the thickest commercially available Si detectors. Therefore, Si detectors are generally backed by CsI(Tl) scintillators with thickness between 1 and 10 cm in order to measure those energetic light charged particles.

2.6 Mixed Field of Gamma and Neutron Detection Sometimes we need to measure neutron and gamma both simultaneously. Gamma and neutron detector separately cannot measure the gamma and neutron simultaneously. Therefore, it would be better to have some kind of material which can efficiently and

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effectively measure these two particles simultaneously. Cs2 LiYCl6 (CLYC) [43– 45] is an exciting new material for scintillator detectors because of its very good energy resolution and neutron capture capabilities, which will be very useful for studying the mixed field of γ and neutrons. The presence of 6 Li in CLYC allows us to detect the thermal neutron through [6 Li(n,α)3 H], which produce a high light output (∼3 MeV). The presence of 35 Cl in CLYC enables as a fast neutron spectrometer through the reaction 35 Cl(n, p)35 S. As a γ -ray detector, CLYC features a good energy resolution (typically 5% FWHM at 662 keV for 1 “x1” detector), much better than the commonly used gamma detector such as NaI(Tl) and BaF2 , and comparable to LaBr3 (Ce) scintillator. It has an ultrafast scintillation decay component (∼2 ns) only for γ -ray excitation, while it has slow decay component (∼1 µs) under neutron excitation, which enables excellent pulse shape discrimination (PSD) to separate the neutron and γ -events. All the properties make CLYC a very attractive candidate for different applications in the field of experimental nuclear physics in addition to the simultaneous measurement of neutron and gamma. This detector could also be used for cosmic-induced neutron measurements in connection with radiation background studies for rare decays.

3 Nuclear Electronics and Pulse Processing Most of the radiation detectors require pulse (or signal) processing electronics [46] so that energy or time information involved with radiation interactions can be properly extracted. In nuclear physics experiment, nuclear radiation interacts with detectors and produces a pulse (voltage pulse or current pulse as shown in Fig. 12), which carries all the information of incident radiation. This pulse may contain electronic noise and backgrounds, and therefore needs to be processed in order to get actual pulse. There are two types of signal pulses, namely, linear and logic pulses. In pulse processing system, it is important to distinguish the difference between linear and logic pulse. A linear pulse is carrying information through its amplitude and shape. A logic pulse is a signal pulse of standard size and shape that carries information only by its presence or absence. Generally, linear pulses are produced by incident radiation and then converted to logic pulses in order to get the timing information of incident radiation. Energy information can be obtained either by analyzing the pulse height or by integrating the area under the pulse. There are mainly two pulse processing techniques for processing of various shapes of pulses, which are analog and digital pulse processing. In analog pulse processing techniques, several electronic modules such as amplifier (pre-amplifier, shaping amplifier, fast amplifier, etc.), discriminator (leading edge discriminator, constant fraction discriminator, etc.), gate and delay generator (GDG), logic unit (AND), linear and logic FAN-IN-FAN-OUT, analog-to-digital converter (ADC), charge-to-digital converter (QDC), timing modules (TAC, TDC, TFA), single channel analyzer (SCA), multichannel analyzer (MCA), etc. are used to process the detector pulse. Each electronic module is used for some specific purpose such

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Fig. 12 Electrical pulse generated by a detector [46]

as pre-amplifier (as shown in Fig. 13) which is a very low noise analog circuit that receives the weak signal generated by the detector and produces an output signal with an signal-to-noise ratio suitable for the transmission through a cable up to the readout electronics; shaping amplifier is used to shape the linear pulse; GDGs are used to produce a gate window to integrate the linear pulse; ADC and QDC are used to obtain the energy information by using several mechanisms; timing modules are used to obtain the timing information of incident radiation; SCA and MCA analyze a stream of processed pulses and sort them into a histogram or “spectrum” of number of events; etc. An electronic pulse generator (or pulser) is always required in the initial set-up and calibration of electronic systems. It should be mentioned that rise time and decay time of the detector pulse are two important parameters before we process the pulse to obtain energy and time information. Rise and decay time of detector pulse can be directly obtained by using an oscilloscope. Details of the several electronic modules are discussed in Ref. [46]. It needs to be mentioned that each electronic module has a dead time, which is time after each event during which the electronic module is not able to record another event. The total dead time of a detection system using is due to the contributions of the intrinsic dead time of the detector, of the analog electronic modules, and of the data acquisition. It is found that the total dead time by using analog pulse processing technique is few tens of µs. During analog pulse processing, the detector signal is first shaped and amplified with the help of shaping amplifier (which consists of a combination of integrator and differentiator circuit) and forwarded toward the other analog components either for counting purpose or for further processing. One can amplify the pulse from the mV range into the 0.1–10 V range by using shaping amplifier. In addition, shaping the pulses is required to optimize the energy resolution, and to minimize the risk of overlap between successive pulses. In order to count the pulses reliably, the shaped linear pulses must be converted into logic pulses. Discriminator is such an electronic module that does this operation and consists of a device that produces a logic output pulse only when the linear input pulse height exceeds a threshold, i.e., discriminator level. By setting a discriminator threshold level one can also reject the noise present

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Fig. 13 Signals in traditional analog chain [46]

in the signal. Finally, the logic pulses must be accumulated and their number recorded over a period of time. A counter is used for this purpose and increases one count each time a logic pulse is presented to its input. This is how one can count the number of incident particle in the detector. The height of linear pulse is proportional to the energy deposited by the incident radiation, and thereby is used to obtain the energy information of the incident radiation. A peak sensing ADC measures the height of an analog pulse and converts that value to a digital number. The digital outputs from the ADC are fed into a dedicated memory, and sorted into a histogram. The typical dynamic range of the ADC operation is consistent with the range of the shaping amplifier output, i.e., 0 ∼10 V. The combination of the peak sensing ADC, the histogramming memory, and a display of the histogram form a multichannel analyzer (MCA). Finally, a computer is used to display the spectrum. The peak sensing ADC is mainly used to obtain energy information, but it can also be used for timing information. A time-to-amplitude converter (TAC) has to be connected in between the shaping amplifier and the ADC in order get the time spectrum. When the output of a TAC is connected to the ADC input, the histogram represents the time spectrum measured by the TAC. A block diagram of analog pulse processing to obtain the energy and time information is shown in Figs. 14 and 15. An another way to obtain the energy information of the incident radiation is to integrate the detector pulse [i(t) current pulse] within a time window. The area under the current pulse, t i.e., charge contained in the pulse q(t) = t12 i(t)dt is proportional to the energy of incident radiation. A discriminator or GDG unit is used to make a gate (time window) within which the pulse is integrated. In digital pulse processing technique, the detector signals are directly digitized with a sampling (or digitizing) ADC immediately after the pre-amplifier and processed to extract quantities of interest. The digitized signal pulse is then shaped digitally and the pulse height is extracted. After extraction of the pulse height, one count is added to the memory address corresponding to

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Fig. 14 Block diagram of analog pulse processing using peak sensing ADC [46]

Fig. 15 Block diagram of analog pulse processing using QDC [46]

the pulse height as in analog pulse processing. The digital processor is the key element to perform this operation and either a field programmable gate array (FPGA) or a digital signal processor (DSP) can be employed. This technique has several significant advantages compared to traditional analog pulse processing technique. By using digital signal processing technique, smoothening-, filtering-, correlation, and convolution-like operation are done very rapidly on the digital signals and more precise information are obtained which was very difficult with the help of analog processing technique. Dead times involved in analog electronic modules are greatly reduced, from tens of µs to hundreds of ns. A block diagram of digital pulse processing is shown in Fig. 16. The principle of operation of a waveform digitizer is the same as the digital oscilloscope, i.e., when the trigger occurs; a certain number of samples (within acquisition window) are saved into one memory buffer. Signal

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Fig. 16 Block diagram of digital pulse processing [46]

Fig. 17 Signal digitization and acquisition window defined by the trigger [46]

digitization and acquisition window defined by the trigger is shown in Fig. 17. An algorithm (known as DPP algorithms) is required to obtain the time and energy information of the incident radiation. Timing information of the incident radiation can be obtained precisely by using high-frequency sample digitizer (500 MS/s or more), while 12–14-bit digitizers are well suited for acquisitions where high-energy resolution is a pre-requisite. There are different acquisition modes in a digitizer, such as oscilloscope mode, list mode, mixed mode, and histogram mode. In oscilloscope mode, for each trigger the digitizer saves the sequence of samples (waveform) into one local memory buffer. This mode is normally used to monitor the signals, set the parameters, and see their effect on the signal. Moreover, one can also readout the raw data from the digitizer and apply the digital algorithms by using C-programming or MATLAB software and can make the energy spectrum or histogram. In list mode, the relevant energy (or other quantities) is calculated and written in the memory buffers. In this case, the DPP algorithms are applied runtime by the FPGA that operates on a continuous data stream. In mixed mode, both the waveforms and the energy values are saved into the memory buffers. The histogram mode is same as the list mode, with the difference that the memory buffer is used to accumulate the energy values in a histogram that grows continuously until the user decides to stop it. The histogram can be readout at any time without stopping the acquisition. Details of the digital pulse processing and data acquisition are discussed in Ref. [46]. It should be mentioned that setting up the system with DPP requires a deep knowledge of the digital algorithms and the relevant parameters. It takes more time for the beginners. Rather, beginners should start with analog electronics and then should go for digital pule processing.

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4 Vacuum System The beam of particles travels inside a vacuum in the metal beam pipe and makes reactions with the target present in target chamber. The vacuum is crucial to maintaining an air- and dust-free environment for the beam of particles to travel unobstructed. The vacuum is also required to avoid colliding with gas molecules inside the beam pipe so that the beam of particles can reach to the target without having energy loss. The most common way of creating a vacuum is to pump the gas out of a vessel that is initially at atmospheric pressure (1 atm = 1.01325 bar) [47, 48]. There are many different ways of pumping on a vessel, but all of them have a limiting pressure below which they are ineffective. The ultimate pressure of the pump is the lowest possible pressure achieved by a particular pump. The use of those pumps must be preceded by another pump, called a “roughing pump”, which brings the pressure in the vessel within their working range. Usually pumps which require a “roughing pump” require a backing pump as well. The backing pump is used to extract residual gases from the main pump to keep it at low enough pressure to operate. The pressure of the backing pump is called the backing pressure. Rotary pump, roots pump, scrawl pump, etc. are some examples of backing pumps, by which one can reach backing pressure up to 10−3 torr, not more than that. However, we need high vacuum at least 10−3 –10−7 torr to perform the nuclear physics experiment. Such high vacuum can be reached by using diffusion pump, turbo molecular pump, etc. First we need to make rough vacuum by using backing pumps, and then diffusion pump or turbo molecular pump is used to reach to the vacuum level at ∼10−3 –10−7 torr. It should be mentioned that we need to separate accelerator hall from the beam hall while making the rough vacuum to the target chamber and associated beam pipe. Details of the vacuum pumps will be discussed in the second part of the book.

5 Faraday Cup, Beam Dump, and Its Shielding 5.1 Faraday Cup The particle beam current is an important parameter for doing the nuclear physics experiment. The detection system used in the experiment has a tolerable value of beam current beyond which the detectors may not work properly. In addition, accurate value of beam current is important for precise measurement of cross section. The measurement of beam current is done using a device called “Faraday cup”. The device is a metallic cup and electrically insulated from the surroundings including the beam pipe. This collects beam particles and determines the number of particles (ions or electrons) hitting the Faraday cup. The beam of charged particles penetrates the cup and thereafter neutralizes after scattering with the electrons and nuclei of the metal, whereas the metal gains a net charge albeit small. The metal then discharges. Since the cup is connected to a current meter through an electric circuit, the leakage

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Fig. 18 Geometry of the Faraday cup

current from the cup can be measured. From the equation, Nt = eI , the number of particles impinged can be calculated. Here, N is number of observed ions in a time t and I is measured current in ampere, and e is the electronic charge. However, it is important that number of ions impinged and observed should be same, otherwise there will be mistake in calculation. There is a possibility that some ions would be lost from the cup. Mainly the electrons have probability of backscattering and thus may escape. For this reason, to prevent the backscattering of particles sometimes a bias voltage is applied in front of the cup. A Faraday cup is shown in Fig. 18. C is the charge collecting cup, F is the window for the entrance of charged particles, and P is the insulator. The interior is kept in vacuum to eliminate spurious charges. To prevent backscattered electrons, electric field is applied through G (guard ring) or magnetic field is applied through windings (W). The dimension of the Faraday cup is decided from the energy of the accelerator beams. It is important so that the cup can contain all the secondary or tertiary particles as the charged particles hit the cup.

5.2 Beam Dump and Its Shielding An energized particle or photon beam, after passing through the target, is stopped in a controlled manner in a beam dump that is positioned at the extreme end of the experimental beam line. The condition is that the beam dump must absorb the energy of the beam completely. For efficient absorption of beam, the beam dump material should be very good conductor of heat and must have high melting and

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yield stress point. Moreover, the material must show low radioactivity for safety purpose. A beam dump material may be solid, liquid, or gas, depending on the type of beam. Generally, beryllium, carbon, aluminum, copper, etc. are selected for beam dump fabrication. Besides, water and gas beam dumps are also sometimes used for high power beam containment. As, for example, the Korean proton accelerator uses graphite and oxygen-free high conductivity copper as beam dump material. Graphite has the properties of high melting point, durability, and low radioactivity in nature. At CERN Super Proton Synchrotron, the beam dump is made of tungsten, molybdenum, and graphite. In case of sensitive nuclear physics experiments, background produced from the beam dump may contaminate the actual result. Moreover, the background radiation could affect adversely by increasing the dead time of the data acquisition system. Besides, there will be radiation damage of the detector and electronics. Therefore, shielding of the beam dump is highly important, which would otherwise damage all the experimental findings. However, planning for beam dump shielding requires the knowledge of experiment, detectors, data acquisition, and electronics. Lead (Pb) brick, concrete (contains carbon, oxygen, silicon, hydrogen, calcium, magnesium, etc.), etc. are generally used as passive shielding for preventing the unwanted gamma rays. Since lead has high atomic number and density, it is a very good attenuator of photons. Again, for neutron shielding, barite concrete slab (boron mixed concrete), high-density polyethylene (HDPE), etc. are used. Boron has very high neutron capture cross section and hence used in concrete slab for neutron absorption. Beside this, sometimes active beam dump shielding is also done using detectors along with passive shielding. The signals coming from the main events at the detectors used for experiments are taken in anti-coincidence with the signals of active beam dump shielding detectors to remove the background events. Beam dump and its shielding plays a crucial role in nuclear physics experiment.

6 Laboratory Standard Radioactive Sources Laboratory standard radioactive source is an important additional requirement for nuclear physics experiment [27]. It is required either for detector calibration and testing of detectors, or as object of measurements itself. It is also important to know the natural background radiation. There are mainly three kind of sources required for detector calibration, which are charge particle (α, β, and heavily charged particles), gamma, and neutron radiation sources as discussed below: Alpha Source: Heavy nuclei are energetically unstable and decay spontaneously through the emission of alpha (4 He2 ) particle. The probability of alpha is governed by quantum tunneling process. The decay process is written schematically as A

X Z → A−4 Y Z −2 +4 H e2 + Q.

The Q-value is shared between alpha particle (4 He2 ) and daughter nucleus ( Y Z −2 ). As mass of the alpha particle is much smaller than the daughter nucleus, A−4

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most of the Q-value is taken by alpha particle as kinetic energy. Each alpha particle appeared with the same kinetic energy given by Ek = Q[MY /(M H e +MY )] = Q(A4)/A. This alpha particle with the kinetic energy Ek is used to calibrate the charge particle detector. Heavy nuclei could also emit more than one alpha particle with different energies if energetically possible. Most of the alpha particles are limited to between 4 and 6 MeV. There is a strong correlation between alpha particle kinetic energy and half-life of the nuclei: higher kinetic energy means shortest half-life and lower kinetic energy means longer half-life. Examples of some alpha sources used in laboratory are 232 Th (Ek ∼ 4 MeV), 230 Th (Ek ∼ 4.6 MeV), 239,240 Pu (Ek ∼ 5.1 MeV), 241 Am (Ek ∼ 5.5 MeV), 243 Am (Ek ∼ 5.2 MeV), etc. [27]. As alpha particle loses its energy rapidly in material, alpha sources are prepared in very thin layers and covered with a metallic foil or other material. Heavily charged particle source: The spontaneous fission process is only the source of energetic heavy charged particle with mass greater than that of the alpha particle [27]. Fission fragments in spontaneous fission process are therefore widely used in the calibration and testing of detectors for heavy ion measurement. The most widely used example is 252 Cf, which undergoes spontaneous fission as well as alpha decay. The probability of alpha emission is considerably higher than that of spontaneous fission. A sample of 1 μg of 252 Cf emits 1.92 × 107 alpha particles and undergoes 6.1 × 105 spontaneous fission per second. The actual half-life of 252 Cf source is t1/2 ∼ 2.65 yrs [27]. β particle source: β-particles (β + or β − ) are emitted due to the decay of a neutron (or proton) in nuclei which contain excess of respective nucleon. Energy spectrum of β − particle is continuous in nature. Nuclides that directly decay to the ground states via β emission are known as “pure beta emitters”. Examples of such “pure β-emitters” are 90 Sr and 90 Y, which emit a continuous spectrum of β − particle up to the maximum energies 0.546 and 2.27 MeV, respectively [27]. Some of the β-active nuclei (such as 22 Na, 60 Co, 137 Cs) could also decay to the excited states and then go to the ground state by emitting a gamma rays. Gamma-ray source: Gamma ray is emitted due to the transition of a nucleon from higher energy level to its lower energy level. Nucleons (proton and neutron) inside an atomic nucleus are constantly moving due to quantum fluctuation. Thus, it is possible to write a charge and current distribution for the nucleus. Changes to these distributions would cause electromagnetic emission known as gamma radiation. When a nucleus is at excitation energy below particle threshold, it emits gamma radiation as it tries to reach the ground states by changing its charge and current distributions. Energy of the gamma ray is the difference between the two energy levels in which transition occurs. Excited nucleus can directly go to the ground states by emitting a gamma rays or it can make gamma transition followed by β − decay or electron capture (EC). Examples of such gamma-ray sources used in laboratory are 22 Na (0.511 and 1.274 MeV), 60 Co (1.173 and 1.332 MeV), 137 Cs (0.662 MeV), etc. [27]. Level scheme of the above gamma-ray sources is shown in Fig. 19. The gammaray sources are generally encapsulated by sufficiently thick material which stops the emitted β − particles. In case of parent nuclei (such as 22 Na) which undergo β + decay, an additional gamma is emitted due to the annihilation of positron with the electron

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Fig. 19 Level scheme of some gamma-ray sources

present in the encapsulating material. In general, the gamma-ray sources based on β decay and EC are limited to low energies up to 2.5 MeV except 56 Co which emits 3.55 MeV gamma following both β + and EC. Though 56 Co has short half-life (∼77 d), it can be used as a potential source for high-energy calibration point. However, we need to calibrate the nuclear detector to energy more than 4 MeV (sometimes up to 15 MeV), which can be achieved by nuclear reaction. Examples of such high-energy gamma-ray sources are 241 Am-9 Be and 238,239 Pu-13 C. Alpha particles are emitted from Am and Pu, and make the reaction with 9 Be and 13 C, and produce 4.44 and 6.13 MeV gamma rays from the excited states of 12 C∗ and 16 O∗ nuclei, respectively [27]. The nuclear reactions are given by 4 2He 4 2He

∗ 1 12 1 +94 Be →12 6 C +0 n →6 C +0 n + γ (4.44MeV); 16 ∗ 1 16 1 +13 6 C →8 O +0 n →8 O +0 n + γ (6.13MeV).

In case of the gamma-ray energy more than 6 MeV, in-beam experiments are generally performed. An another important gamma-ray source widely used in nuclear physics experiment is 152 Eu, which emits a large number of gamma rays such as 122, 245, 344, 779, 964, 1112, and 1408 keV. Neutron source: An excited nucleus can decay via neutron emission, if the excitation energy is greater than that of neutron binding energy. Such highly excited states of nuclei cannot be produced as a result of any convenient radioactive decay process. 87 Br is an exceptional radioactive nucleus which decays via βdecays that leads to an excited state of 87 Kr which de-excites via neutron emission: 87 Br →87 K r ∗ + β − →86 K r + n . However, 87 Br source is not used as laboratory neutron source as half-life of this source is 55 sec. Therefore, the possible way to find the neutron source is based on nuclear reaction or spontaneous fission source. (α, n) reactions as discussed earlier are generally used to produce neutron source. The Q-value of 9 Be(α, n)12 C is 5.71 Mev, and therefore the emitted neutron could have energy up to ∼ 5 MeV. Examples of such neutron source based on nuclear reactions are 241 Am-9 Be and 238,239 Pu-9 Be, 210 Pu-9 Be, etc. [27]. In addition, spontaneous fission source can also be used as neutron source. The most common spontaneous

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Fig. 20 A schematic experimental set-up for nuclear physics experiment

source is 252 Cf (t1/2 ∼2.65 years) and its neutron yield is 0.116 neutron/sec per Bq. On an average, 2.3 × 106 n/s are produced per μg of the sample. The neutrons are emitted from the excited states of fission fragments and are in the energy range of 1–10 MeV [27]. It is very important point to keep in mind that users need to follow the radiation safety rules while working with any kind of radioactive sources.

7 A Schematic Experimental Set-Up A schematic experimental set-up for nuclear physics experiment with all required components is shown in Fig. 20.

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Offline Measurements and Extraction of Fusion Cross Section Pankaj K. Giri, Rudra N. Sahoo, and P. K. Rath

Abstract This chapter is mainly focused to explain the fusion cross-section measurements around the barrier and is organized in the following manner. First, a brief introduction is given on nuclear reactions, useful kinematics to plan an experiment, the importance of cross sections, the fusion cross-section expressions, details techniques of offline measurements, details of derivation of experimental fusion cross section, astrophysical applications of offline measurements, and end up with the summary and conclusion.

1 Introduction to Heavy Ion Reaction The nuclear reactions are classified into two categories in terms of reaction times. The fast “direct reaction” process, where the time interval between the incident particle and particle emission is close to the time required for a nucleon to cross the nucleus, i.e., 10−22 s. On the other hand, the slow “compound nuclear reaction” timescale is ∼10−19 to 10−16 s. The two types of reaction can be distinguished by various experimental features, such as the shape of the excitation function, emitted particle spectra, and angular distribution. Although these times of interactions are very difficult to measure experimentally, there was indirect experimental evidence for different types of reactions. For example, a familiar bell shape of the excitation function reveals a compound nuclear mechanism. The cross section for a particular channel first increases with increasing energy and then decreases with a further increase in energy due to competition from other reaction channels, which becomes P. K. Giri Department of Physics, Central University of Jharkhand, Ranchi 830222, India India and UGC-DAE Consortium for Scientific Research, Kolkata Centre, Kolkata 700098, India R. N. Sahoo (B) Racah Institute of Physics, Hebrew University of Jerusalem, 9190401 Jerusalem, Israel e-mail: [email protected] P. K. Rath Centurion University of Technology and Management, Odisha, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Deshmukh and N. Joshi (eds.), Understanding Nuclear Physics, https://doi.org/10.1007/978-981-19-8437-2_4

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Fig. 1 Energy spectra of the emitted particles from different reaction processes [1]

energetically possible. Indirect reaction, the excitation function depends sensitively upon the level structure of the residual nucleus. The outgoing particles from compound nuclear reaction show a continuous Maxwellian distribution in their energies whereas those emitted in direction reaction have discrete energies characteristics of the residual nucleus as shown in Fig. 1. In direct reaction, the emitted particles are partially polarized whereas they are completely unpolarized in a compound nuclear reaction. An important distinction lies in the angular distribution of the emitted particles of compound nucleus reaction is characterized by fore and after symmetry (symmetry around 90◦ ) whereas it is predominantly forward peaked in direct reaction. In low energy studies, the bulk of the particle emission can be attributed to one of these processes. As both reactions are iso-chronous, the main feature of the heavy ion (HI) induced reaction includes (i) large excitation energy transfer, (ii) large angular momentum transfer, (iii) exchange of large number of nucleons, and (iv) special type of interactions, such as multi Coulomb excitation and short range interactions. Due to these special features, it is possible to study the properties of the nucleus under unusual conditions, which are not normally meet with light ion studies. Thus, nuclear matter with unusual high density (super dense nuclei) rotating at extremely high speed (due to high angular momentum) and decaying with extremely short radioactive half-lives (due to being highly proton-rich and hence far away from the stability line) can be studied in these types of experiments. In addition, super heavy (in the far transuranic region) and super charged (nuclear molecules) nuclei can also be studied. The exchange of mass, charge, energy, and angular momentum between two interacting nuclei is termed as nuclear reaction. After the interaction of the projectile and target, two or more particles are produced at outgoing channels which are different from interacting nuclei. The interaction between two nuclei initiates when they come

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close approximately, within an order of a few femtometers. To bring two subatomic particles down to such a small distance is only possible by particle accelerators, which is also discussed in the previous chapters. To summarize, a particle accelerator is a machine that is used to propel charged particles (heavy ions) to very high energy and to focus them in well-defined beams. There are many particle accelerators all over the world. However, In India, major accelerator related programs are being pursued at Inter-University Accelerator Centre (IUAC) [2], Bhaba Atomic Research Centre (BARC)/Tata Institute of Fundamental Research (TIFR)-Mumbai [3], Variable Energy Cyclotron Centre (VEECC) Calcutta [4], Facilities for Research in Experimental Nuclear Astrophysics (FRENA) at Saha Institute of Nuclear Physics (SINP) [5] and Institute of Physics (IOP) Bhubaneswar [6]. Varieties of accelerator-based research are being pursued with the 14 MV Pelletron accelerator at and BARC/TIFR Pelletron accelerator and the 15 MV Pelletron accelerator at IUAC NEW Delhi. In general experimentally cross-section is measured. Practically in the laboratory two nuclei will strike with each other and fuse together as a result it creates a new nucleus. In the collision of two heavy-ions, different process may take place as they are many body quantum mechanical particles. A typical nuclear reaction may be written as a + X → Y + b + .... + Q (1) where “a” is projectile and ‘X’ is target nuclei. ‘Y’ is residual nuclei and ‘b+’...” are considered ejectiles. “Q” is called the energy kinematics of the reaction, which is the mass difference between the incident and exit channels multiplied by the square of the light velocity, which is based on the Einstein mass energy relation. Q = [M(a + X ) − M(Y + b + ....)]c2 .

(2)

For short notation, this reaction may be written, X(a, b)Y. Isotopes are indicated by the use of their mass number as a superscript on the left of the chemical symbol. Special symbols are used to designate elementary particles, and some of the light nuclei; for example, e for electron, p for proton, n for neutron, d or 2 H for deuteron, t or 3 H for triton, α or 4 He for alpha particle, γ for photon or gamma-ray, π for pion, μ for muon, etc. Sometimes, b or Y may be produced in an excited state. This is indicated by the use of an asterisk, Y∗ , etc. The symbol Q in Eq. (1) is the energy released in a reaction; if both b and Y are left in the ground state, this is denoted by Q = 0 (elastic scattering). If Q = 0, it means that a part of kinetic energy has gone into excitation energy and/or a new type of nuclei. If E f and Ei are the total kinetic energy in the final and initial state, then Q = E f − Ei

(3)

If Q is positive, the reaction is said to be exoergic (or exothermic as in chemical reactions) and when it is negative (Q = −ve) signifies that the reaction is endoergic or endothermic. In this case, a definite minimum kinetic energy, called the threshold

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energy is required for the projectile to initiate the reaction. The threshold energy needed is equal to -Q in the center of mass system. If Q = 0, then it represents elastic scattering in which case total kinetic energy is conserved. Given enough energy for the bombarding particle, a collision may result in more than two particles in the final state. To initiate a nuclear reaction in the lab, the beam of high energy incident particles is bombarded on a thin film, i.e., called target. Targets of different thicknesses are prepared based on the physics motivations. For low energy nuclear reactions, the thickness of the target is much thinner to avoid the energy loss and straggling of the incident ions as well as reaction residues. The enriched isotopic materials are used to prepare the targets. A brief account about target preparation and its thickness measurement is given in the following sections. Before projectile slams/interacts with the target, it experiences three types of potential, such as Coulomb potential (V B ), nuclear potential (V N ), and Centrifugal potential (VC ). Among these three potentials, the Coulomb and centrifugal potential oppose the projectile to fuse with the target due to its repulsive nature, but nuclear potential supports as it is attractive. The addition of these three potentials results in an effective potential between two interacting nuclei. These three potentials depend on the masses and charges of their interacting nuclei, and the analytic form of the total potential is shown in Fig. 2. To initiate a reaction, the bombarding energy of the projectile must be greater than or equal to the effective barrier between the interacting partners. After the interaction of two nuclei, it is very difficult to predict the properties of outgoing events as the nuclei are many-body quantum mechanical systems. Nuclear reactions occur directly through a single step or many more intermediate steps. A brief account regarding the classification of nuclear reactions is given. The heavy-ion induced reactions are classified by manifesting outgoing events based on mass, charge, and energy.

Fig. 2 Potential energy plot for the system 16 O + 148 Nd at energy 6.3 MeV/u

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Considering the structure of the nucleus and the way of interaction between the interacting partners, the nuclear reactions are classified as direct nuclear reactions and compound nuclear reactions, which are briefly introduced as follows [7–9].

2 Direct Nuclear Reactions (DNR) Those reactions in which the projectile cannot completely fuse with the target but mass, charge, and energy exchanges between them belong to direct reactions. Based on the properties of the ejectiles, these are classified as elastic scattering, inelastic scattering, Deep elastic collision (DIC), and Quasi-inelastic collision (QIC). In elastic scattering, only the direction of the projectile changes due to Coulomb interaction without changing its energy, which is the simplest process among all reaction mechanisms. However, in inelastic scattering, both interacting nuclei are excited due to the exchange of energy between them. There is no sharp boundary between QIC and DIC. Within the regimes of QIC and DIC, there are so many reactions, like—transfer, quasi-fission, incomplete fusion, and charge exchange is dominated, where the partial mass, energy, and orbital angular momentum transfers between the projectile and target. These reactions come into play when the fermi surface of both nuclei overlap, or inter-mixing takes place between valence shells of both nuclei, where the transfer reactions probability gets dominated. Transfer reactions are mainly two types: 1. pick up reactions—nucleons are transferred from target to projectile, and 2. stripping reactions—nucleons are transferred from the projectile to target. When more than two nucleons transfer between the interacting nuclei, that particular reaction is termed as the multi-nucleon transfer. Since, all the aforesaid reactions predominantly occur within very small regions, from QIC to DIC, it is very difficult to distinguish each individual outgoing channel. Direct reactions have more importance in many aspects because these are the interface between the compound nuclear reactions and elastic scattering. These reactions are not only helping us to understand the exchange dynamics of a few nucleons but also a few static and dynamic properties of the nucleus, like—collective excitations, the idea about the smoothness of nucleus, nucleonic co-relations in nuclear medium, spectroscopic factors and shell structure of the nucleus. The time scale of direct reaction is less than or equal to the transit time of 1 MeV nucleon to cross the nuclear dimension, and its numerical value is nearly equal to 10−22 s, which is much shorter as compared to the compound nuclear reaction time scale [10, 11]. There are other verities of reactions also, some of them have been given below.

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(i) Elastic Scattering: Here b = a and Y = X. The internal states are unchanged so that Q = 0 and the kinetic energy of the particles in the center of mass system (CMS) is unchanged before and after the scattering. In general a+X →a+X

(4)

for example : p + 7 Li → p + 7 Li or 7 Li(p, p)7 Li (ii) Inelastic Scattering: Here b = a, but ‘X’ is raised to an excited state, Y = X∗ , so that Q = −E x , where E x is the excitation of the state. Since “a” is emitted  with reduced energy, it is usually written as a so the equation becomes 

a + X → a + X∗ 

(5)



for example : 10 B + α →10 B∗ + α or 10 B(α, α )10 B∗ . If ‘a’ is itself a complex nucleus, it may get excited instead of the target, or both may be excited. An example of the latter is 12 C + 16 O →12 C∗ + 16 O∗ (iii) Nuclear Reactions: Here b = a and Y = X so that there is a rearrangement of the constituent nucleons between the colliding pair, known as transmutation. A number of possibilities are open; X + a → Y1 + b1 + Q 1

(6)

X + a → Y2 + b2 + Q 2

(7)

Or

Some examples are Mg + 14 N →27 Mg + 13 N 7 Li + p →7 Be + n (iv) Capture Reactions: This is a special case of class (iii); the pair X + a coalesce, forming a compound system in an excited state which decays via one or more γ -rays, (8) X + a → C∗ → C + γ + Q 26

For example 197 Au(p, γ )198 Hg (v) Fission: A neutron absorbed by a heavy nucleus like 235 U causes the nucleus to split it into two and sometimes three large fragments with the emission of a few neutrons. An example is n +235 U →141 Ba +92 K r + 3n + 189MeV

(9)

The energy released is much greater than in other types of reactions. Fission can also be caused by other projectiles’ like p, α or pions.

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(vi) Other reactions: If sufficient energy is available then in the final state there can be more than two particles. In general, X +a →Y +b+c+ Q

(10)

For example   α + 40 Ca → p + α + 39 K or 40 Ca(α, α ,p)39 K The bombarding particle and the target nucleus a + X constitute the entrance channel, while the products such as b + Y form the exit channel. Open channels are those which are energetically available. All the above explain the different types of nuclear reaction out of which the compound nuclear fusion has been explained below.

3 Compound Nucleus Fusion Reaction In this case, two interacting nuclei (projectile and target) completely amalgamate and lead to an equilibrated composite system by repetitive interaction of the internucleonic degrees of freedom. The mass of the equilibrated compound nucleus is equal to the sum of masses of interacting partners. Fusion is possible if the kinetic energy of the projectile must be greater than that of the effective barrier (Coulomb + Nuclear + Centrifugal), and collision should be central in nature to impart all driving input orbital angular momentum and energy of projectile with a target nucleus. The compound nuclear reaction is a two-step process. In the first stage, the projectile entirely fuses with the target nucleus within a very short time, which eventually leads to a thermally equilibrated excited compound nucleus. In the second stage, the excited compound nucleus de-excites via the emission of light nuclear particles and characteristic γ -rays based on the availability of excitation energy. The formation and decay of the compound nucleus are two independent processes, theoretically first proposed by Bohr in 1936, and Ghosal was verified experimentally in 1950 [12]. The compound nuclear reaction lasts for a long time and achieves thermal equilibrium is a matter of conjecture as it goes through many intermediate processes. The compound nucleus must live for at least several times longer than the transit time of an 1 MeV incident particle. The compound nucleus loses its memory of formation due to the repetitive interaction of nucleons in the composite system. The de-excitation depends on the energy, angular momentum, and parity of the quantum state of the compound nucleus, and the final product (after the de-excitation) is termed as the evaporation residues (ERs) of the nucleus. The time scale of compound nucleus reactions is of the order of 10−19 s–10−16 s, and the lifetime is around 10−14 s have been observed [11]. In the compound nuclear reactions, the incident particle is captured by the target nucleus, and its energy (kinetic+ binding energy (∼8 MeV)) is shared among the nucleons of the compound nucleus until it attains a state of statistical equilibrium. After a time of the order of 10−14 s–10−15 s at low incident energies and 10–100

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times greater at high energies, a nucleon or a group of nucleons near the surface may undergoes a statistical fluctuation, receive enough energy to escape, in the manner of evaporation of a molecule from a heated drop of liquid. This statistical process favors the emission of low energy particles which form the Maxwellian distribution. If the excitation energy of the compound nucleus is high enough, several particles may be emitted in succession until the energy of the nucleus has dropped below the threshold for particle emission. Then the nucleus emits γ -rays until the ground state is reached. The nucleus may decay in a variety of other ways such as fission into two large fragments, if the compound nucleus is very heavy or through the production of radioisotopes. The type of information that the study of compound nucleus yields includes the properties of excited energy levels the mechanism of nuclear de-excitation, the density of high energy states, the role of angular momentum and nuclear deformation in affecting the evaporation process. The measurement of γ ray energies and intensities and their angular correlations find important applications for the structure studies of low energy levels. The direct reactions take place in the time the incident particle takes to traverse the target nucleus which is typically of the order of 10−22 s. Here the incident particle may interact with a nucleon or a group of nucleons or the entire nucleus and emission takes place immediately. The simplest direct process is the elastic scattering in which the target nucleus is left in the ground state. In non-elastic processes, the states of the residual nuclei which are excited bear a simple structural relationship to the ground state of the target nucleus. Inelastic scattering predominantly excites collective states, one nucleon transfer excites single-particle states and multi-nucleon transfer excites cluster states. Measurements of cross sections of these states, the angular distribution of the emitted particles, and their state of polarization permit the study of these states. Much of our knowledge of nuclear structure has originated from the study of direct reactions. It is possible that after the interaction the particle may not be emitted immediately as in the direct reactions nor after a long time in the statistical way as from the compound state. The particle may be emitted before reaching the statistical equilibrium, such processes are termed as pre-compound or pre-equilibrium reactions and constitute the third category. Heavy ion collisions at energies above the Coulomb barrier have a high probability to proceed through the complete fusion of the projectile and the target, with the formation of a compound nucleus (CN). In the fusion process, the kinetic energy available in the center of mass system is completely dissipated through a series of nucleon-nucleon interactions inside the di nuclear complex. The CN is produced in an excited state and decays subsequently with times ≥10−19−22 s through two main mechanisms: particle evaporation and fission The main features of this decay rely on the thermo dynamical equilibrium reached by the system which is characterized by an excitation energy U and an angular momentum J. In agreement with the observation of the long lifetimes that characterize the CN, in 1936 Bohr suggested the independence between the two processes: formation and decay of the system. This allows to factorize the cross section in two terms: the fusion cross section of the colliding ions σ f us in the entrance channel a(x; A) and the decay probability G(b) of the compound

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Fig. 3 The formation of a same compound nucleus (64 Zn) through different entrance channel. The asterisk (*) shows that the compound nucleus is produced in an excited state

nucleus in the exit channel b (y; B): σa→b = σ f us G(b); When an isolated state is populated, the reaction cross section has a typical resonant behavior expressed by the Breit-Wigner formula [Hog78] Such situation concerns states at low excitation energy. Here we will discuss mostly the compound nucleus formation process. In the early 1930s a large number of nuclear reactions were studied. Detailed studies showed sharp peaks in cross sections at selective bombarding energies, (resonance reaction). This led Bohr and independently Breit and Wigner to postulate that these reactions proceed through two stages, first the formation of an intermediate state called the compound nucleus state, and second its break up in a relatively long time into the observed products. According to Bohr: (i) The same compound nucleus can be formed in a variety of ways. For example, the compound nucleus in a particular excited state in the compound state designated as 64 Zn∗ can be produced by many ways as pictorially shown in Fig. 3. (ii) A particle that hits the nucleus is captured and the energy released consists of its binding energy plus the kinetic energy. In its strong interaction with one or more nucleons, the available excitation energy is dissipated and shared by several nucleons in the collision processes. The excitation energy (Eexc ) is given by adding the binding energy B = Q (Q value of the reaction) and the available kinetic energy in the c-system. Need discussion B + Ea [mx /(ma + mx )], where Ea is the bombarding energy in the Lab system. The new nucleus is thus formed in the intermediate stage (compound nucleus) Sooner or later lot of energy may be deposited on a single or a group of nucleons of the compound nucleus, resulting in the emission of a particle (s). The compound nucleus is long lived (∼ 10−16 s) compared to the natural nuclear time which may be taken as the time taken to cross the nuclear diameter. (iii) The final break-up of the compound nucleus is independent of the mode of formation. The time involved in break-up is so long that ‘memory’ is lost. The formation and break-up can be regarded as independent events. For example, the compound nucleus 64 Zn∗ can decay into a variety of ways as in Fig. 4. (iv) The exit channel with neutron emission is much more favored than the charged particles like proton and α as the charge particles have to overcome the Coulomb

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Fig. 4 The decay of the same compound nucleus (64 Zn) through different exit channels

Fig. 5 The experimentally measured cross section for the exit channel where the entrance channel is different for the same compound nucleus

barrier. One can define a Reaction Channels as: various ways in which a compound state is formed are called entrance channels, and the various ways in which a compound nucleus breaks up are called exit channels. The entrance and the exit channel are independent i.e., the formation mode may be different for a compound nucleus but the decay mode can be same. This has been visualized in below Fig. 5 where the same compound nucleus has been formed through different ways and the same exit channel cross section has been measured experimentally and found correct. Experimentally what measured is the cross section which has explained below. The density of nuclear levels depends strongly on the excitation energy and on the mass number. In the light nuclei, near the ground state the levels are about 1 MeV apart, while in

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heavy nuclei about 50 keV apart, except in magic number nuclei. When the excitation energy is in the vicinity or the neutron binding energy (∼8 MeV) slow neutron resonances in medium-heavy elements are in a few electron volts apart. Using Fermi’s gas model one can obtain the level density ρ(E) for the total system energy E by an approximate formula where ρ(0) and ’a’ are empirical constants 1/2

ρ(E) = ρ(0)e2(a E)

(11)

3.1 Decay of Compound Nucleus The compound nucleus formed by the absorption of neutron is de-exited by evaporating one or more neutrons or other particles. If the excitation energy is sufficient then the evaporation of one particle may leave enough energy to enable the second particle to leave and so on. We may then have (p,Xn ), for example, with X = 1, 2, 3, . . . up to 6 or 7. The probability of decay through a larger number of particles increases with greater excitation. When little excitation is left particle emission ceases and only gamma emission is possible. The energy distribution of the evaporated neutrons is described by the Maxwellian distribution at the temperature T of the residual nucleus. Number or neutrons emitted between En and En + dEn being nd E n ≈ const × E n e−En/kt d E n

(12)

In the case of charged particles, the Coulomb barrier inhibits the emission of low energy particles. The energy distribution is modified by multiplying the Maxwell distribution by the Coulomb barrier penetration factor. There are many complicated quantum mechanical approaches to find the cross section theoretically and one of the methods is the partial wave analysis which is beyond our scope of the derivation. The most empirical formula for the cross section is described below.

3.2 Cross Sections Formula In order to measure the probability quantitatively that a given nuclear reaction will take place, we introduce the concept of the cross section. Consider a reaction of the type X(a,b)γ . If I0 is the incident flux of particles ’a’ incident per unit area on a target consisting of N nuclei of type X, then the number of particles b emitted per unit time (I) will be proportional to both I0 and N. The constant of proportionality (σ ) is called the cross section which has the dimensions of the area. A schematic representation of the same is shown in Fig. 6. Thus, experimentally measured cross section for the exit channel where the entrance channel is different for the same compound nucleus is I = I0 × N × σ ,

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Fig. 6 The schematic representation of the incident beam and the scattered beam over a solid angle dω

σ = I / I0 × N. In words cross section = number of particles b emitted/s/(number of particles ‘a’ incidentunit areas) (number of a target nuclei within the beam). In nuclear physics the unit of cross section is a Barn (b) 1 b = 10−24 cm2 = 10−28 m2 ; the sub-multiples are millibarn, 1 mb = 10−3 b, microbarn, 1 μb = 10−6 b, nanobarn, 1 nb = 10−9 b, etc. The number of particles b emitted per unit time within an element of solid angle d in the direction with polar angles (θ ,φ) with respect to the incident beam will be proportional to d as well as I0 and N. The constant of proportionality is known as the differential cross section dσ (θ ,φ)/d, also written as (dσ /d) or σ (θ ,φ) so that dσ /d = I/I0 × N × d. The unit of dσ /d is barn/steradian. If the particles are unpolarised then the scattered particles will not depend on the azimuth angle φ, and the scattering will be symmetrical about the beam axis. In that case the differential cross section will depend only on the polar angle θ and will be written as dσ (θ )/d or σ (θ ). The total elastic cross sections σ and dσ /d are related by 

4π

σ = 0



 dσ .d d

(13)

The classical picture of a cross section has been shown in Fig. 7. For the same entrance channel, a number of exit channels will be open corresponding to different reaction products at a given energy. As the exit channels are independent, there will not be any quantum interference and the cross section of different reaction channels may be added. The sum of all these non-elastic channels cross sections is called the reaction or absorption cross sections and is denoted by σr . When the elastic cross section is also added we speak of the total cross section σtotal = σr + σel . Strictly

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Fig. 7 The classical picture of the cross section formation due to spherical nucleus of radius R1 and R2

speaking, the finite dimensions of the projectile must also be taken into account in the calculation of cross section. Let a sphere 1 of radius R1 be at rest and sphere 2 proceed toward it with impact parameter b, see Fig. 7. The two spheres will collide only if the impact parameter b ≤ (R1 + R2 ). The effect is the same as for the collision of a point particle with a disc of radius R1 + R2 , the disc being perpendicular to the axis joining the centers of the spheres. The area of the disc which is the projected area of the two spheres touching each other is equal to π (R1 + R2 )2 . This will be the maximum cross-section of the collision. This means that if the radius of the target nucleus is to be determined the radius of the bombarding particle must be taken into account.

4 Evaporation Residue Channel The complete fusion cross section (σ f us ) can be obtained by measuring the fusionevaporation σ F E and fusion-fission σ F F cross sections: σ f us = σ F E +σ F F . The prevailing of one of the two terms depends essentially upon the mass of the compound nucleus. In general, in the case of light nuclei (A < 100 amu) the term σ F E dominates while, with the increase of the mass of the compound nucleus, σ F F becomes comparable to σ f us , the former becoming dominant for heavy nuclei. One can understand the Evaporation Residue (ER) and the particle emission (evaporated particle) using the statistical mechanics’ formalism. As we discussed earlier the excitation energy of the compound nucleus and the angular momentum decides the emission process, if one will plot the U-J it looks like Fig. 8. U-J plane represents the decay of a compound nucleus. The Yrast-line represents the rotational energy of the nucleus. The initial angular momentum distribution (upper part) and the level density (left) are also shown in Fig. 8. Evaporation residue channel owing to the angular momentum transferred to the compound nucleus by the reaction, the excitation energy U is given by U = Eth + Er ot , where Er ot is the collective rotational energy and Eth is the thermal energy associated to the excitation of intrinsic degrees of freedom. The classical relation between the rotational energy Er ot and the total angular momentum J is Er ot = J2 /(2I), where I is the moment of inertia of the nucleus. A schematic description of the CN decay in the evaporation channel, to the final states of the evaporation

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Fig. 8 U-J plane representation of the decay of a compound nucleus

residue, can be obtained using the U-J plane shown in Fig. 8. The values of the rotational energy in this plane identify the Yrast-line, below which no states are allowed to the CN. In the figure, one Yrast-line approximates all the decays of the nuclei involved in the evaporative chain. The upper part of the figure is shown the triangular distribution of the angular momentum in the entrance channel, while on the left is the level density. The CN starts to decay from the initial excitation energy Ui in the continuum region and with an angular momentum Ji at high excitation energy. Light particle emission dominate with respect to the electromagnetic radiation emission and the nucleus decays initially with the emission of n, p and α particle losing part of excitation energy and angular momentum. The process continues with further emissions till the excitation energy of the residual nucleus is less than the minimum needed for the emission of a particle. The final residue is called evaporation residue (ER), will be still in an excited state in the continuum region, called region of Entry States as shown in Fig. 8. The decay in this region occurs only by emission of γ rays till the ground state is reached. The γ ray emission from the entry states starts in the continuum region reaching the region of the discrete level density in the proximity of the Yrast-line. The emission of a light particle is the result of a competitive process, the neutron emission is favored with respect to the emission of charge particles because of the

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Fig. 9 Dominant decay modes (partial widths >50%) predicted for the nucleus 44 Sc. In the righthand part of the figure, the most likely decay chains in the decay are indicated for different angular momenta of compound nucleus 46 Ti [13]

Coulomb barrier. For heavy nuclei the neutron emission becomes dominant, while for medium-light nuclei the charged particles compete more effectively with the neutrons. The angular momentum plays an important role in particle emission, especially for light nuclei. To show these effects, in Fig. 9 the decay of nuclei 44 Sc and 46 Ti are presented in the U-J plane. From the figure it is evident that the plane can be divided qualitatively in many regions, each of them being characterized by the prevalence (>50%) of the decay of a kind of particle. The most probable evaporative cascades of the nucleus 46 Ti at E L AB = 76 MeV is presented in the right part of Fig. 9. One can see that the increase of the angular momentum of the compound nucleus enhances the probability of alpha particles emission, these particles being more effective to take away angular momentum. Once created, the compound nucleus proceeds along the beam direction and the recoil produced by the particles emission determines the angular distribution of evaporation residues inside a cone with an angular opening of few degrees around the beam direction. Particle evaporation is governed by two main quantities: the transmission coefficients and the level density. The observation of ER’s with indirect methods can be done through the measurement of the discrete characteristics gamma-rays and is more practicable in the case of low excitation energy, which involves a limited variety of nuclei. The direct observation of residues implies the use of mass spectrometers, electrostatic detectors, or techniques based on the measure of time of flight (TOF).The observed energetic spectra of the emitted particles in the CN decay showed an evaporative behavior. Their angular distributions of the evaporated particles, in the center of mass system, show a symmetry around 90◦ with respect 0◦ the direction of the beam reacting with the evaporative nature of the emission. The

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fission process consists in the scission of the compound nucleus in two fragments of comparable mass. The excitation energy of the compound nucleus is transferred not only to the relative motion of the fragments, but also to their intrinsic degrees of freedom and angular momentum.

5 Basic Formula of Nuclear Reaction Kinematics For nuclear reactions’ experiments we have the initial information like mass and charge of projectile and target, and the incident energy of the projectile. From these values, many other pieces of information can be extracted to plan and execute an experiment. Other information about nuclear reactions can be derived using reaction kinematics. Here a few kinematics formulas are given to plan fusion experiments as well as for analysis of data. Let the mass and charge of the projectile be M P and Z P , and the target is MT and ZT , respectively. E P is the incident energy of the projectile in the lab frame (The energy is coming out from the particle accelerator). The schematic representation of a nuclear reaction is given by, M P + MT → M R + M E + QV alue . Generally, in a low energy nuclear reaction, the target is at rest, and the target is bombarded by the projectile beam for their interactions. The center of mass-energy available for the reaction will be E C M = [M P /(M P + MT )]E P

(14)

If the projectile energy is around the barrier then the probability of compound nucleus formation gets dominated. Then after collision, the recoil energy of the compound nucleus will be E Recoil = E P − E C M

(15)

Then if one adds the recoil energy and the center of mass energy then we can get the original energy of the projectile. The excitation energy of the compound nucleus is calculated by E E X = E C M − Q V alue

(16)

QV alue is the energy kinematics of the reaction. QV alue can be estimated from the mass difference of incident and exit channels. Since QV alue is the dimension of energy then the mass can be converted to energy through Einstein’s mass-energy equivalence principle (E = Mc2 ). Q V alue = [(M P + MT ) − (M R + M E )] c2

(17)

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where M R and M E are the mass of residual and the ejectile nuclei after the reaction. When projectile and target come close approximate then a Coulomb potential arises and that depends upon the interacting lie as distance, and charge of both projectile and target. The Coulomb potential can be calculated by  VB = 1.22

Z P .Z T R

 ,

(18)

where R is the distance between the interacting nuclei which is 1/3

1/3

R = r0 [A P + A T ]

(19)

where r0 is the radius parameter whose numerical value is around 1.2 fm and A P and AT are the mass of projectile and target. The exact form of nuclear potential is not yet explored as nuclei individually behave like many-body dynamical systems. Therefore the nuclear potential is calculated empirically or by fitting the experimental results with predicted theoretical models. The clear understanding of the nuclear potential has probability to explore many more properties of the nucleus. Furthermore, when a projectile comes toward the target with a certain trajectory, it is involved with some orbital angular momentum (l). This orbital angular momentum is just the product of the linear momentum and the impact parameter l = b × p = b. p.sinθ

(20)

The impact parameter is the distance between the central axis of the reaction to the initial trajectory of the projectile. As the direction of both p and b are perpendicular to each other, i.e., θ = 90◦ which corresponds to sin θ = 1, therefore l = bp. The input angular momentum changes with the impact parameter and the input linear momentum. During the experiment, the input linear momentum of the projectiles in a beam is the same, but the impact parameter changes with the position of the projectile, consequently, the input angular momentum. The input angular momentum also influences the interaction between projectile and target. The interaction between projectile and target due to input angular momentum is called the centrifugal potential. It is denoted as VCent =

l(l + 1)2 2μR 2

(21)

where μ is called reduced mass. The equations of motion of two mutually interacting bodies can be reduced to a single equation by using the reduced mass. The reduced mass of the two interacting particle can be written as, μ=

(M P .MT ) (M P + MT )

(22)

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We can write the kinetic energy of projectile E P =(1/2)M P . V2P , where V P is the velocity of the projectile. We can also express the kinetic energy of the projectile in terms of the reduced mass (μ). The beam energy, number particles in the incoming beam and number of particles in the target nuclei are the important parameters to predict the rough cross section before the experiment. In nuclear physics, nA current is used and target thickness is around 100–1000 µg/cm2 . The number of particles in a target is calculated in the following way. 200 µg/cm2 of 12 C target contains how many numbers of the atoms/cm2 ? We know the Avogadro’s (N) = 6.023× 1023 , 1 gm = 6.023 × 1023 no. of particles, 1 µg = 6.023 × 1023 ×10−6 = 6.023 × 1017 no of particles. 200 µg/cm2 gm of 12 C = (200/12) × 6.023 × 1017 no. of particles. The number of particles varies inside the target according to the thickness. Along with that, the isotopic abundance factor is multiplied as it varies from the isotopes to isotopes. As per the definition, the current is defined as the charge per sec (current = charge/s). 1e = 1.63 × 10−19 c, then 1c = 1/1.63 × 10−19 no of “e” = 6.25 ×1018 “e”. 1A = 1 C/1 s (1A = 1c/1s), 1A = 6.25 × 1018 p/s, 1nA = 6.25 × 1018 × 10−9 p/s = 1 nA = 6.25 × 109 p/s. Particle nA is defined as pnA, which is current in nano ampere divided by the charge state of the incoming beam. After that the beam energy also plays an important role for the nuclear reactions as the nuclear reaction cross section depends on the incoming energy. Beam energy (23) (E beam ) = E 0 + (q + 1)VT where E0 is the energy of the incident ion after removal/addition of 2–3 electrons. q = charge state of the incoming ions and VT is the terminal potential of the accelerator.

6 Radioactivity The knowledge in the offline technique to measure the cross section is one of the most important parts. In these offline measurements, first, a target is irradiated with beams for a few hours. After irradiation the desired populated particles, in our case the evaporation residues, are measured by detecting the gamma-ray from the decay of that nuclei. A brief about radioactivity is given based on offline measurements. Radioactivity is one of the most important properties of exotic nuclei in which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. The most common types of decay are alpha decay (α-decay), beta decay (β-decay), and gamma decay (γ -decay). In these processes, the active nucleus comes to the ground state by emitting one or more particles. For a significant number of identical atoms, the overall decay rate can be expressed by a parameter called a decay constant or half-life. Half-life and decay constant are inversely proportional to each other with the multiplication of a factor, ln2. The half-lives of radioactive nuclei range from nearly instantaneous to far longer than the age of the universe. Except for gamma decay or internal conversion from a nuclear excited state, the decay is a nuclear transmutation resulting in a daughter

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containing a different number of protons or neutrons (or both). The International System of Units (SI) unit of radioactive activity is the Becquerel (Bq), named in honor of the scientist Henri Becquerel. One “Bq” is defined as one transformation (or decay or disintegration) per second. An older unit of radioactivity is the curie, Ci, which was originally defined as “the quantity or mass of radium emanation in equilibrium with one gram of radium (element)”. Today, the curie is defined as 3.7×1010 disintegrations per second. A = A0 e−λt

(24)

where λ is the decay constant, and “t” is the lapse time. The activity is related to the number of nuclei by A = λN (25) “N” is the initial number of nuclei. For a particular time period of the measurement, a certain number of gamma-rays will be detected corresponding to a particular nucleus, which is nothing but the area under a peak. The ratio of the counts (the net area inside the peak) to the time in a sec is called the counts per second (cps). The counts per second divided by the intensity of the gamma-ray is called the gamma per sec (gps) and the gps divided by the efficiency is called the decay per second, which is termed as the activity of the nuclei (A). Then the activity multiplied by the decay constant will give the number of nuclei at the initial time of the measurements. This information is important and helps to calculate the fusion cross section.

7 Dose Information Safety is one of the most important issues during the nuclear reaction’s experiments. Nuclear reactions are performed always with proper precaution. Lifes have evolved in a world containing significant levels of ionizing radiation. In other words, radiation has always been present all around us, and these come from space, the ground, and even within our own bodies. When ionizing radiation penetrates to the human body/living cells or an object, it deposits energy. The energy absorbed from exposure to radiation is called a dose. The absorbed dose is measured in a unit called the gray (Gy). A dose of 1 gray is equivalent to 1 J of energy deposited in a kilogram of substance. Denser tissues tend to absorb a greater dose as compared to normal tissue. For example, the bone marrow has much more absorptive capacity to radiation than the normal tissue of the body. As we know the radioactive decay is a nuclear process, the energy is measured in terms of Mega Electron Volt (MeV) per Becquerel per second (J/Bq*sec). Then the dose is calculated based on how much of this radiation is absorbed in 1 kg of material. The absorbed dose for the different radiations is the same but its effect on the human tissue is different. For an example, the alpha particles are more biologically damaging to live tissue than other types of radiation although the absorbed dose are same for both. Therefore, a factor is multiplied with

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the absorbed dose (rad or gray) to obtain a quantity that expresses, on a common scale for all ionizing radiation, the biological damage (rem or sievert) to the exposed tissue. As the science of radiation dosimetry developed, it was realized that the ionizing effect, and hence tissue damage, was linked to the energy absorbed, not just radiation exposure. 1 gray = 1 J/kg and equals 100 rad. One sievert equals 100 rem. One sievert is equal to one gray multiplied by a relative biological effective factor. Absorbed dose (DT) x radiation weighting factor (WR) = equivalent dose (HT) The International Commission on Radiological Protection (ICRP) has published the latest set of numerical values of radiation weighting factors as below [7]: • • • •

WR WR WR WR

= 1: photons, electrons = 2: protons = 20: alpha particles, fission fragments, heavy ions = variable: neutrons (determined by complex equations).

8 Cross-Sectional Determination The physical observable quantity of a nuclear reaction is termed as “cross section”, which corresponds to the interaction area between two interacting partners at a certain energy. It connects the abstract world of theory and the real world of the experiment. Although complexity is increasing in experimental techniques and theoretical calculations compared to the early days, the concept of the cross section has not been changed. To measure the interactions the term “cross section” is used, although alternatives like the area, probability, and reaction rate are existing to quantify the interaction as it is independent of the intensity and size of the incoming beams, and the number of nuclei available in the target per unit area. Therefore, it is easy to compare the experimental cross sections of two different labs, which correspond to two different accelerators despite how powerful the accelerators are. This is the beauty of the cross section measurement in nuclear reactions [8]. As the cross section represents the interaction area between two interacting nuclei at a particular energy, the total cross section of a nuclear reaction cannot exceed the geometrical cross section between them. The geometrical cross section is the sum of the area of both projectile and target, σ = π(R 2p + Rt2 )

(26)

where R p and Rt are the radius of the projectile and target nucleus, respectively. The cross section contains detailed information about the interacting partners, such as the bombarding energy, the incoming input angular momentum, the wavelength of the incoming projectile, the phase shift of incoming particles after interaction with the target, and scattering length. The way to calculate cross section is quite different in theory as well as in experiments. Theoretically, the cross section of the different reaction mechanisms is anticipated by comparing the phase of incoming and out-

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going particles after the interaction with the target. The phase shift is calculated by taking the incident channel parameters, like energy, angular momentum, spin, and interaction potential [9]. However, experimentally the cross section depends on the ratio of outgoing particles to all possible interactions between the incoming beam and target nuclei and the corresponding experimental setup. The comparison of both cross sections provides a way to understand more about the concerning interaction between the two nuclei if the theoretical result does not agree with the experimental observations. However, to detect the particles at outgoing channels different experimental set-ups and detectors are designed; those insist to insert different parameters in the basic formula of cross section calculation. Experimentally, the cross section of different reaction mechanisms can be calculated differently based on the user facility and detector set-ups, which change the structure of the cross section calculation formula without changing its original meaning. Simply, the cross section (σ ) of a particular reaction is defined as the ratio of the number of specific particles produced at outgoing channels corresponding to that reaction (N) to the number of incoming beam particles on a target (Nb ) multiplied by the number of target particles per unit area (NT ), i.e., σ = N/(Nb NT ). However, in this chapter, the focus is given only to the offline measurement techniques for fusion cross-section calculations [14]. In principle, according to the classical picture, the fusion cross section should vanish suddenly when the bombarding energy becomes less than the interaction barrier, which is termed as the sharp cut-off model. The expression of fusion cross section according to the sharp cut-off model is given by   Vb σ f us = π Rb2 1 − E

(27)

where Rb is the barrier radius, Vb is the interaction barrier and E is the bombarding energy. Here, if the barrier between two nuclei is greater than the bombarding energy ( E  Vb ), then their ratio will be greater than one (Vb /E 1). This makes the crosssectional negative, which is unphysical as the cross section is the interaction area between two interacting nuclei. This signifies that the fusion of two nuclei is only possible provided the incident energy is greater than that of the interaction barrier between the colliding partners. However, many experimental observations have been obtained where the fusion is also possible at the below barrier and after that the sharp cut-off model becomes invalid. To explain the measured experimental cross sections at sub-barrier energies, Hill and Wheeler [15] in 1953 introduced quantum mechanical tunneling phenomenon in heavy ions interaction by approximating the shape of the interaction barrier as inverted parabola using the WKB approximation, and provided an analytic expression for the tunneling probability, which quite well described the experimental cross section at below barrier for fusion with a light particle with heavy nuclei, except for the excitation of resonant states [16]. This model is termed as a one-dimensional potential barrier penetration model (1-d BPM). This was mainly constructed as the distribution of potential is uniform for spherical nuclei. This was the first approxi-

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mation to calculate the fusion cross section below the barrier. The expression of 1-d BPM is given as  σ f us =

ω0 .Rb 2E







2 .ln 1 + ex p ω0 (E − Vb )

 (28)

where E is the incident energy of the projectile, .ω0 is the barrier curvature, and Vb is the barrier height due to Coulomb and nuclear potential for s-wave approximation. If, E Vb , the formulation is reduced to the sharp cut-off model, and when E  Vb , the above formula will be reduced to  σ f us =

ω0 .Rb 2E

   . ex p

2 ω0 (E − Vb )

 (29)

The calculation of fusion cross section by theory as well as from the experiment is quite different. The theoretical formula mainly depends on the intrinsic properties as well as the energy of the nuclei; however, experimentally the cross section is calculated taking the parameters of the experimental setup and properties of the detector. As this chapter is mainly focused on the offline measurements techniques, the expression of the cross section is defined as σ f us (E) =

Aλex p(λt2 ) N0 φεG θ K [1 − ex p(−λt2 )][1 − ex p(−λta )]

(30)

Details description is provided in Sect. 12.4.

9 Pre-irradiation Preparation for Off-Line Cross-Sectional Measurement The experimental fusion cross sections measurement techniques can be broadly categorized into two types, online measurement and another is offline measurement. The online measurements technique is the one, in which the data recording either for the particle and/or gamma/X-rays are done during the beam irradiation on target with an appropriate detector [14, 17, 18]. On the other hand, in the case of offline measurements, the data recording was done after the beam irradiation on target. Here we will discuss the offline measurement technique to measure the experimental fusion cross section of evaporation residues populated [19–22]. Before experiment and measurement of cross section one of the important things to consider is the targets and detector setup for such an experiment, in the following subsection brief information for the target fabrication and characterization, detector arrangement, energy and efficiency calibration are discussed for the designing of offline cross-sectional measurement experiments.

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10 Target Fabrication and Characterization Target fabrication takes an important role for the low energy nuclear reactions. The isotopic enrichment and isotropic distribution are the two main factors during the target fabrication. The thickness of the target mainly depends on the type of nuclear experiment one is going to perform. Usually, in low energy nuclear reactions, very thin targets are used to avoid the energy loss and straggling of the reaction’s residues, however, in case of gamma measurements, the thickness of the target is slightly more to achieve statistics, around 1–5 mg/cm2 , as the energy of the gamma does not affect much by the target thickness. For a representative case, the fabricated 130 Te targets with thickness around 150–200 µg/cm2 can be found in Ref. [23]. A photograph of the prepared target is in Fig. 10. There are many methods to fabricate thin and uniform targets for offline nuclear physics experiments, some of those techniques are vacuum evaporation (using thermal resistive heating or evaporation using e-gun [24]), rolling methods which were used very often. The typical vacuum evaporation using e-gun setup for the target fabrication can be seen in Fig. 11. The order of 10−5 to 10−6 mbar vacuum pressure typically needed inside the chamber during the evaporation of target materials for optimum evaporation and deposition rate. The vacuum evaporation and deposition technique is used for the very thin selfsupporting target materials like holmium, gold, carbon, aluminum, etc. However, if the target material is not good for self-supporting then low Z materials like—carbon (C) or aluminum (Al) are used for the fabrication. On the other hand, the rolling method is only employed to prepare self-supporting targets, where a desired metallic isotopic sample rolled very slowly and made it thinner as per the requirement. After

Fig. 10 The fabricated 130 Te target at IUAC [23] using vacuum evaporation method

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Fig. 11 a–b Inside picture of the resistive heating chamber with tantalum boat and the quartz crystal monitor [23–25] Fig. 12 Measurements of thickness by alpha-transmission method. The black peak represents the energy of the 241 Am source and other peaks are the energy of same alpha particles after passing through the different target [23]

the fabrication, a precise target thickness measurement is one of the very important factors in case of offline cross-sectional measurement. To do this, one can use either the Rutherford backscattering (RBS) or the alpha-transmission method. In the case of the alpha-transmission method, the thickness is measured by estimating the energy loss of the alpha particles after passing through the target. The energy loss is measured by comparing the energy of alpha particles of used source (i.e., 241 Am) in a blank sample holder and energy loss through target samples. The centroid shifting with respect to blank in energy/channel will provide the thickness of target sample, a plot of centroid shifting with respect to the blank using 241 Am source can be seen for reference in Fig. 12 and details can be found in Ref. [23].

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Fig. 13 The Rutherford Backscattering Spectrometry (RBS) spectrum of 148 Nd enriched sandwiched targets with Al-backing and capping. The red line represents the simulation to the experimental RBS data [19, 25]

Fig. 14 EDS spectrum for the 148 Nd sandwiched targets, having Al-as capping and Al-as backing material [19, 25]

Using the RBS, one can do thickness measurement and as well as check the impurity in the target during fabrication. Another method that can be adopted for characterization is Scanning Electron Microscope (SEM). By this one can see more clearly and in a precise manner the impurities in the fabricated target. For a representative case RBS and EDS spectrum for the 148 Nd sandwiched targets, having Al-as capping and Al-as backing material is shown in Figs. 13 and 14 The details can be found in Ref. [25].

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11 Irradiation of Sample In the activation analysis the decay properties such as the half-life and energy of radioactive decay of a nuclide can be identified uniquely for individually produced nuclide after fusion and/or fission. The analysis is achieved by the formation of radioactivity through irradiation of the sample either by neutrons or charged particles. Irradiation of target samples can be done inside a scattering chamber having sample mounting and removing arrangement. For a representative case a typical photograph of the General-Purpose Scattering Chamber (GPSC) at Inter University Accelerator Centre (IUAC), New Delhi is shown in Fig. 15.

11.1 Stacked Foil Activation Technique The stacked foil activation technique is a quite powerful method, which can be used for the offline measurements like the excitation functions (EFs) [26, 27], forward recoil range distributions (FRRDs) and angular distributions (ADs) of evaporation residues [28], fission fragment (FF) or fission like fragment (FLF) [29]. This technique has been widely used to measure the cross sections of ERs and used to obtain useful information about the mechanism of different reaction dynamics involved in nuclear reactions. In this method, target and suitable catcher/degrader foils were arranged in the form of a stack. The stack is then irradiated by an energetic beam of particles in a fixed geometry. A schematic of target-catcher assembly and, (b) photo of target-catcher stack mounted on the target ladder along with the quartz crystal for beam tuning and focusing used for the measurement of excitation functions is shown in Fig. 16.

Fig. 15 A typical photograph of the General-Purpose Scattering Chamber (GPSC) at Inter University Accelerator Centre (IUAC), New Delhi

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Fig. 16 a A schematic target-catcher assembly used for the measurement of excitation functions measurement [19]

The activities produced in the stacks are recorded offline after the irradiation. The recording of induced activities in the target-catcher assembly depends upon the lifetimes of desired nuclear reaction products. The advantages of this technique are— (i) the activation method can effectively reduce beam-time requirements by providing the possibility of cross sections measurements for large numbers of populated ERs at different beam energies in a single irradiation, (ii) measurement of activities produced in the irradiated samples is done after the irradiation of the stack with the ion beam, hence, the possibility of contribution from the background of the beam is substantially rejected, (iii) the separation of the contributions of different ERs decaying by γ -rays of nearly the same energy, is now possible using high-resolution γ -ray detectors, and (iv) this technique provides more accurate results as each ER can be identified through its characteristic γ -ray and half-life both by measuring the activities with different times and also the background in γ -ray spectra is much smaller [19]. This technique is very simple and able to give accurate cross section. However, in some cases, it may be more complicated because of the common γ -rays for multiple residues. The contribution can be separated out by their half-life curve analysis in case of common γ -rays from each ER. The unique decay mode of each radionuclide provides a specific path for its identification and measurement. This technique is however limited only for the ERs with measurable half-lives range, i.e., ∼3 min to a few days. After the irradiation, how fast the sample is taken out from the scattering chamber and placed in front of the detector determines the lower limit of the measurable half-life. On the other hand the upper limit is dependent on the activity build-up or the irradiation time of the sample.

12 Post Irradiation Setup and Analysis for Off-Line Cross-Sectional Measurement Post irradiation is one of the most important parts of the offline cross-section measurements and calculations. The irradiated samples are placed in front of the detector, which is now free from the particles and prompt gamma-rays (those are produced

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Fig. 17 A typical photograph of offline γ -ray spectra recording set-up using HPGe detector with lead brick shielding [19]

at the time of irradiation). Before placing the sample in front of the detector, first the detector is calibrated with a known source, and the efficiency has been evaluated at different distances. The efficiency is the intrinsic properties of the detector. The detector can’t detect all the γ -rays coming from the source as it has certain efficiency toward the energy and geometry of target placement in front of it. The detector efficiency is decreased with the distances as the solid angle is gradually decreased. The signal processing from the detector and identification of characteristic γ -rays and their analysis have been discussed in the following subsections.

12.1 Efficiency, and Energy Calibration of HPGe Detector In heavy-ion interactions, large numbers of ERs are produced which may emit a large number of consecutive γ -rays. For identification of the inherent characteristic γ -rays of different ERs in the complex γ -ray spectra, a detector of high resolution and proper calibration is needed (Fig. 17).

12.2 Energy Calibration and Efficiency The calibration of γ -ray spectrometers includes the energy calibration, and then the detection efficiency. The energy calibration is done to identify the unknown γ rays as well as to measure the efficiency. The relationship between peak position of known γ -ray energy and the channel number is called calibration, which is normally performed before the data recording with the detector. Efficiency is one of the most

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important factors which is considered at the time of the cross section calculations because γ -rays are measured only for a small solid angle by the HPGe detector. The absolute efficiency of the detector is first calculated using a standard source with known activity. The current activity of the source can be calculated by the S = S0 e−λt expression, where “S” is the present activity after time (t) has passed after manufacturing, S0 is the initial activity at the time of manufacture, λ = 0.693/T1/2 is the decay constant. For calibration and determination of efficiency of the detector the source is placed in front of the detector with a known distance (i.e., 1 cm, 2 cm..etc.), depending up on the activity of the source. The γ -spectra for preferably known strength sources, are measured for a few hours, and then analyzed. The net area under each γ peak is measured. The efficiency can be calculated according to the following equation: A (31) ε= Sθt where A is a net peak area, S is the activity (Bq), θ is the absolute γ -ray emission probability or branching ratio, “t” is the measured time. However, to calculate the efficiency at any other desired energy points, all the data points are fitted with a suitable function. From that fitting function, the efficiency can be easily predicted. For example, 152 Eu is considered as a source. It has many gamma lines. Consider the number of counts of 121.78 keV peak is 87654. The measurement time is 1 h (3600 s). The intensity is 28.53 (%) = 0.2853. Consider the present activity is 9.16E+3 Bq. The efficiency can be calculated as follows: ε = 87654/(0.2853 × 3600 × 9.6E+3) = 0.00888. Similarly, we can calculate the efficiency at all energy points. With increase in energy the efficiency decreases. Then all the efficiencies data points are fitted with a suitable function, ε = a.E−b , where “a” and “b” are two constants, “E” is the energy of the corresponding energy of gamma-ray. Using this equation, we can calculate the efficiency at any other energy. To be clearer, the gamma spectra of 152 Eu and its efficiency curve are given below. The characteristic γ -rays of 152g Eu [14], which were used in the calibration process and their branching ratios are listed in Table 5.1. A typical γ -ray spectrum was recorded by a 100 cc HPGe detector for 152g Eu γ -ray source at 2 cm distance as shown in Fig. 18 for a representative case. During the data collection from the irradiated samples and source should be taken in the same geometry and distance from the detector. The efficiency plots recorded 2 cm distances between 152g Eu γ -source to HPGe detector are shown in Fig. 19. The 5th-order polynomial provides the best fit for the γ -ray energy dependence of efficiency as expressed in the following form: ε=

5  i=0

ni E i

(32)

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Fig. 18 Experimental characteristic γ -rays spectra of 152g Eu source at a distance of 2 cm recorded by HPGe detector[19] Fig. 19 Experimental efficiency plot for 2 cm separations between the point source 152g Eu to the HPGe detector. The solid line is the best fit of the measured data

where n1 , n2 , n3 , n4 , and n5 are the constants, which can be determined by the leastsquare fitting. The above constants have different values at various source to detector distances. In Eq. (4.32), E is the energy of the emitted γ -rays from the source. A typical block diagram of electronics of the offline γ -ray spectrometry using HPGe detectors for the measurement of EFs for ERs is shown in Fig. 20. The irradiated samples were placed in front of the HPGe detector, emitting delayed γ -rays. These γ -rays interact with the HPGe detector and generate pre-amplified signals called energy (E) and timing (T) signals. The E and T signals are identical in nature. The E-signal from the pre-amplifier was fed into the amplifier for further amplify the signal and shaping from the detector, then for data recording to the Multichannel Analyzer (MCA)-based PC.

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Fig. 20 The typical block diagram of data acquisition electronics of the offline γ -ray spectrometry using HPGe detector

12.3 Identification of Evaporation Residues and Conformation in Offline Measurement The ERs populated in the interaction of projectile with target, decay into their ground state by the emission of characteristic γ -rays. The cross sections of produced ERs are determined by identifying their characteristics γ -rays. As discussed earlier in this chapter, the γ -activities induced in the irradiated stack can be recorded using a HPGe detector associated with a personal computer through a suitable data acquisition system using any standard gamma spectrum recording software after the irradiation. For the representative case study, we recorded the data using 100 cc HPGe-n-type detectors coupled with CAMAC crate data acquisition setup. The activity of individual target/catcher samples was recorded for increasing time intervals. Hence, the recorded activity would be useful for the decay curve analysis. The data collection was done in the live-time mode for incorporating the dead time loss. The ERs were preliminarily identified by their characteristic γ -lines in γ -ray spectra. The identified ERs were further confirmed by the analysis of their decay curves. This technique provides a proper way for the identification of ERs formed in HI collisions due to the unique decay mode of individual ER. Some ERs emit common γ -rays of similar energy, for which the simple identification of γ -ray energy may not be enough. In the case of these ERs, the photo-peak intensity can be plotted as a function of decay time to get the decay curve of the ERs. Therefore, the ERs were identified through their characteristic γ -ray energies along with analysis of their decay curves. Typical γ -ray spectra obtained from the irradiation of 148 Nd targets by 16 O at the same projectile energies ≈ 94.7 MeV is shown in Fig. 21 The γ -ray peaks have been assigned to corresponding ERs populated through different modes. Typical decay curves of ERs 159 Er (T1/2 = 36.0 min) detected at beam energy ≈99.90 MeV for the system 16 O + 148 Nd are shown in Fig. 22.

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Fig. 21 Typical γ -ray spectra obtained from the irradiation of projectile energies ≈94 MeV [19, 20]

148 Nd

targets by

16 O

at the same

Fig. 22 Typical decay curves of ERs 159 Er (T1/2 = 36.0 min) detected at beam energy ≈99 MeV for the system 16 O + 148 Nd [19, 20]

12.4 Determination of Measured Cross Section An energetic charged particle beam incident on the target nucleus may initiate different nuclear reactions. In this interaction process, many ERs are produced by light particle emission. The ERs formed in different nuclear reactions are left in the excited states. These ERs can decay through characteristic γ -emissions. If ’N0 ’ is the number of nuclei initially present in the target, ’’ is the beam flux, and ’σn ’ is the nuclear reaction cross section for a channel, the production rate of the particular reaction residue may be expressed as [19], N = N0 σn

(33)

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91

The rate of disintegration of the activity builds up in an irradiated sample after time ‘t’ from the irradiation stop can be given as 

dN dt

 =N t

[1 − ex p(−λt1 )] ex p(λt)

(34)

where ‘t1 ’ is the irradiation time of the target and ‘λ’ is the decay constant of induced activity of the evaporation residues. The decay constant (λ) is related to the half-life (T1/2 ) of the reaction product given by the expression, λ=

ln2 T1/2

(35)

The term [1-exp(-λ t1 )] is known as the saturation correction. It should also be considered that the radioactive nuclei formed in nuclear reactions might also decay during the irradiation of the target. The number of radioactive nuclei decays in a small-time interval ‘dt’ can be given as d N = N [1 − ex p(−λt1 )].ex p(−λt).dt

(36)

If the induced activity of the irradiated target is recorded after a lapse time ‘t2 ’ for a time duration ‘t3 ’ then the total number of nuclides decayed during the time ‘t2 ’ and ‘(t2 + t3 )’ will be given by 

t2 +t3

Ct =

dN

(37)

t2

Ct = N

[1 − ex p(−λt1 )].[1 − ex p(−λt3 )] λex p(λt)

(38)

If the activity induced in the irradiated target is recorded by a suitable γ -ray detector of geometry dependent efficiency (εG ), then absolute counting rate ‘C’ and the observed counting rate ‘A’ may be related as C=

A (εG ).θ.K sac

(39)

where ‘θ ’ is the branching ratio of the characteristic γ -ray, ‘Ksac ’ is the selfabsorption correction factor for the γ -ray in the target. This correction factor is given by the following expression as K sac =

[1 − ex p(μd)] μd

(40)

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where ‘μ’ is the γ -ray absorption coefficient for the target and ’d’ is the thickness of the target. Therefore, the reaction cross section ’σr e (E)’ at a particular projectile energy ‘E’ can be written as [8]; σr e (E) =

Aλex p(λt2 ) N0 .φ.(εG ).θ.K sac [1 − ex p(−λt1 )].[1 − ex p(−λt3 )]

(41)

where ‘A’ is the area under the photo-peak of the characteristic γ -ray, a factor [1exp(-λt1 )] takes care of the decay of evaporation residue during irradiation time ‘t1 ’. The factor [exp(λt2 )] takes care of correction due to the decay of irradiated target during time-lapse ‘t2 ’. The correction factor due to the decay of the irradiated target during counting time ‘t3 ’ is taken as [1-exp(-λt3 )].

12.5 Determination of the Weighted Average of the Measured Cross Sections The ER formed in a nuclear reaction may emit more than one γ -rays. In such cases, the cross section for the same ER was determined individually from the observed intensities of γ -ray of different energies. The weighted average of the cross sections was taken as the final measured value of the cross section. The formulation used for the determination of the weighted average cross sections for a nuclear reaction is given in [30]. Let us consider that σ1 , σ2 , σ3 ,.......,σn , are the measured cross sections and σ1 , σ2 , σ3 ,......σn , are experimental errors, respectively, for the same ER due to different γ -rays. Then σ1 ± σ1 , σ2 ± σ2 , σ3 ± σ3 ,.......,σn ± σn are the measured reaction cross sections for a particular ER due to different γ -rays. Thus, the weighted average cross section is determined as  Wi σi σ =  Wi

(42)

where Wi =1/(σi )2 . The internal error (I. E.) is given by I.E. = [Wi ]−1/2

(43)

Therefore, I.E. entirely depends on the individual observations. However, the external error (E.E.) is given by [10],  E.E. =

Wi (σ − σi )2  n(n − 1) Wi

1/2 (44)

The E.E. depends on the difference between observed and the mean value. As such, the I.E. depends on the internal consistency, while the E.E. is a function of the

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external consistency of the observations. This error calculation has been incorporated in the computation of cross sections at different energies.

12.6 Error Incorporation in the Measured Cross Section Several factors are responsible for the uncertainties in the measured cross sections. The main factors are (i) The uncertainty due to the non-uniformity of the target, and thickness measurement, (ii) The uncertainty in the efficiency calibration of the HPGe detector (iii) The error arising from the fluctuations in beam current, (iv) To minimize the error, the counting is done for the dead time below 10%. (v) Uncertainty due to the straggling effect of the projectile passing through the stack. In general, the overall uncertainties from various factors including statistical errors in the photo-peak area are estimated to be 0 and r ± √ r 2 + s > 0. For the elastic case (Q = 0), if the projectile is lighter than the target (m a < m A ), the ejectile energy E b will have a single value for each scattering angle θb . This condition is known as direct kinematics. When m a > m A (inverse kinematics) there is a maximum scattering angle θbmax and two different energies E b1 and E b2 for all angles smaller than θbmax .

G.S. 23.8 keV 89.5 keV

787.0 keV

1089 keV

1354 keV

920.5 & 921.4 keV

103

Elastic

1173.3 keV

2400.3 keV

count

718.4 keV

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E (MeV) Fig. 4 Energy spectrum for 10 B projectiles (E a = 39.7 MeV) scattered at θb = 170◦ on a 120 Sn target. The main peak (black arrow) corresponds to the elastically scattered 10 B ions, whereas the two inelastic peaks (red arrows) correspond to excited states of the 120 Sn target: 2+ at E x = 1171 keV and 3− at E x = 2400 keV. The smaller peak located close to 27.8 MeV (blue arrow) corresponds to the first excited state of the 10 B projectile: 1+ at E x = 718.4 keV. Additionally, peaks corresponding to 11 B, in its excited or ground state (g.s.), stemming from the neutron pickup reaction, are also shown (green arrows). [7]

Peaks formed by inelastically scattered ions lie at lower energies than the elastic peak, roughly E binel ≈ E belast − E x , where E x is the energy of the internal excitation. Nevertheless, an exact calculation should use Q = −E x in Eq. 3. Typically, the location of the elastic peak in the spectrum allows an effective energy calibration of the detection system, and the identification of inelastic peaks, together with transfer peaks, unequivocally identify the interacting particles (see Figs. 4 and 5). In some cases, target or projectile nuclei having low-energy excited states produce inelastic peaks too close to the elastic peak. When the energy difference is close or less to 150–200 keV they can not be resolved1 and the whole peak is regarded as quasielastic. The cross section for quasielastic scattering can be calculated by models like Distorted Wave Born Approximation (DWBA) [8] and Coupled Channels Calculations (CCC) [9] and compared with experimental results.

1

The energy width of a peak do not depends solely on the detector resolution, but also on the target thickness, the kinematic widening due to the angular acceptances of the detector among other factors, as it will be discussed in Sect. 2.3.

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Fig. 5 Energy of the elastic peak (solid line), inelastic peaks (dashed and dotted), and neutron pickup leading to 11 B (dashed-dotted) for the 10 B + 120 Sn system as a function of the scattering angle. The elastic scattering on 27 Al (a lighter element that could be present in the target holder) is shown for comparison. At very forward angles it will represent a contamination for the scattering on 120 Sn

2 Experimental Setup A scattering experiment involves a heavy ion accelerator, a beam transport system, collimators, a target, a detector array, and the data acquisition system. In this section, we will comment on these issues.

2.1 Heavy Ion Accelerator Studying the nuclear interaction potential through elastic scattering involves measuring the angular distribution of ejectiles at several energies below, close and above the Coulomb barrier. Hence, the accelerator should allow varying the beam energy at arbitrary steps. Cyclotrons can achieve higher energy than electrostatic accelerators, but they generally have few fixed beam energies and the desired beam energy must be achieved by the use of degradation foils. These foils worsen the energy distribution of the beam and its angular emittance. Therefore, electrostatic accelerators are more commonly used in research on nuclear reactions. Single-ended Van de Graaff may be used for light systems (projectile and target), while tandems are mainly used for heavier systems, which require higher energies.

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Table 1 Minimum terminal voltages of a tandem accelerator required to study the elastic scattering of different projectiles with 27 Al and 197 Au targets (20% above the respective coulomb barrier). Voltages values in italics are above the possibilities of most tandem facilities 27 Al target 197 Au target Projectile

1H

Energy (MeV)

3.7 9.9 9 Be 13.8 9 Be (16 O) 13.8 16 O 31.8 27 Al 61.4 27 Al (16 O) 61.4 6 Li

Charge state

yield (%)

1 3 4 3 6 8 7

100 43 61 35 9.6 7.6 14

Terminal voltage (MV) 1.8 2.5 3.2 4.1 4.5 6.8 8.0

Energy (MeV)

Charge state

yield (%)

14.3 36.8 48.6 48.6 97 184 184

1 3 4 4 8 10 9

100 100 85 21 1 7.7 1

Terminal voltage (MV) 7.1 9.2 9.7 11.1 10.8 16.7 17.3

Tandem accelerators use a negative-ion source and in a first stage, these ions (either atomic or molecular) are accelerated from ground potential to a terminal at a high positive voltage. There, ions cross a thin foil or a gas region in which they are stripped off from their extra electron and a number q of their own electrons. In this process, molecules are dissociated. Positively-charged ions are then accelerated back to ground potential, achieving a kinetic energy E a = (V P + VT )

ma + VT q, mm

(4)

where V P is the pre-acceleration voltage (including the ion source voltages) which is typically in the order of V P ∼ 50−400 kV, VT is the terminal voltage (typically between 3 and 16 MV), m m is the mass of the molecule accelerated in the low-energy side of the tandem, m a is the mass of the atom selected at the analyzing magnet after the high-energy side and q its charge state. Selecting the full-stripped ion (q = Z a ) at the analyzing magnet is always possible but, depending on the terminal voltage, it has not always an acceptable yield. In Table 1, we give examples of the minimum terminal voltage needed to accelerate different projectile up to an energy 20% above their Coulomb barrier2 with 27 Al and 197 Au targets. In these estimates, we require a (arbitrarily chosen) minimum charge-state yields of 1%, estimated by the prescription of [11] for solid (carbon) strippers. It must be recalled than ions heavier than 127 I tend to burn out solid strippers, and suffer larger both energy and angular straggling, with the consequent loss of intensity. For these cases, gas strippers are used, with the drawback of having the charge-state distribution shifted 1 or 2 units to lower values. This results in a lower final energy.

2

Estimated following [10].

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Molecular beams such as 9 Be16 O− (m m = 25 u) or 27 Al16 O− (m m = 43 u) are regularly extracted from negative-ion sources since their production overcome that of atomic negative ions, 9 Be− and 27 Al− , by more than an order of magnitude. However, the Coulomb explosion of the molecule at the stripper causes a significant loss of transmission, especially for 9 Be, which is lighter than its companion 16 O. A possible solution for this problem is to use both gas and foil stripper [12]. If the gas stripper is located upstream from the foil one, it breaks molecules into ions with lower charge states, hence suffering less defocusing due to a weaker Coulomb explosion. Then, the foil strips the ions to a higher charge states. On the other hand, for achieving the maximum possible energy, the atomic ion, even with its much lower yield at the ion source, should be used.

2.2 Beam Collimation Obtaining a precise angular distribution of scattered particles with respect to the beam incidence direction, requires establishing this direction as well as possible. Magnetic quadrupoles are employed to focus the beam, while steerers (magnetic dipoles) are used to center it on the axis. However, due to collisions with acceleration tubes or with residual gas molecules, the beam usually has a halo of particles around it which should be blocked with slits. By scattering on its borders, slits may also produce a halo, so a second set of limiting slits, around 1 m away, are frequently used, as shown in Fig. 6. As depicted in Fig. 7, a deviation x in the position of the incident beam within the reaction plane produces a shift in the scattering angle. The effective scattering angle of the ejectile θbeff is related with the nominal scattering angle θb by

Fig. 6 Scheme of slits for beam collimation (not to scale). The scattered particles produced in the borders of the first slit are suppressed by the second one, about 1 m away. The current induced in the first set of slits, conveniently amplified, can be used to monitor online the centering of the beam (see Sect. 3)

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Fig. 7 A deviation x in the position of the incident beam within the reaction plane produces that the effective scattering angle θbeff differs from the detector angular position θb . The distance between the target and the detector is d, whereas θT is the angle of the target

tan θbeff =

sin θb − x/d x − x = , z + z cos θb + x/d tan θT

(5)

where d is the distance from the detector to the target and θT is the angle between the target normal and the beam direction. For a distance d = 300 mm, a target angle θT = 0◦ , a deviation of x = 3 mm introduces an angular shift of 0.5◦ at θb = 30◦ . The angular shift can be experimentally determined by comparing the events in two monitor detectors at both sides of the beam (see Sect. 6.2). On the other hand, deviations in the direction perpendicular to the reaction plane (y axis in Fig. 7) do not produce shifts in the scattering angle at first order in x/d.

2.3 Target Targets should be thin enough in order to have a well determined reaction energy but also to avoid the energy spread of the ejectiles (see bellow). On the other hand, the target orientation should guarantee that scattered ions heading towards the detector emerge at angles not far from the normal to its surface (green arrows in Fig. 8). Large emergence angles (>60◦ ) have too long tracks within the target (red arrows), producing a large energy loss, energy, and angular straggling. Moreover, a fraction of these ions may hit the target holder, making the cross section measurement unreliable.

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Fig. 8 Positioning of the target for the detection of particles scattered at forward angles and at backward angles Fig. 9 Positioning of the target for the detection of particles scattered at forward angles and at backward angles

To avoid this, two orientations of the target can be use, for e.g. θT = +40◦ for the detectors placed at forward scattering angles (θb < 95◦ ) and θT = −40◦ for backward scattering angles (θb > 85◦ ). θT is the angle of the target normal and θb the angle of the ejectile, both with respect to the beam incidence direction. As an alternative, the target may remain at θT = +40◦ and the detectors be placed at different sides of beam, as shown in Fig. 9. An important issue to assure a good energy resolution, which will allow to distinguish between elastic and the several inelastic peaks, is the difference in track length within the target material between projectiles which are scattered on the front layer of the target and those scattered on the back one. Given the stopping power of projectiles a and ejectiles b in the target, d E a /d x and d E b /d x, this difference implies an energy spread of the particles to be detected. For the measurements of forward angles of detection θb (Fig. 10a), in which all scattered ions cross through the entire target thickness t, this energy spread is

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Fig. 10 Track lengths within the target for the scattering on the first layer (blue line) and on the last layer (red line) a) for forward and b) backward angles. θT is the angle of the target normal and θb the angle of the ejectile, both with respect to the beam incidence direction

E =

t dE a t dE b , − d x cos θT d x cos (θT − θb )

(6)

which is rather small or can be even close to zero for θb ∼ 2θT and E a ∼ E b . However, for backward detection angles (see Fig. 10b), the difference in track length is much larger: ions scattered on the front layer have no penetration while those scattered on the back layer must cross the target twice. In this case, the energy spread is t dE a t dE b . (7) E = + d x cos θT d x | cos (θT − θb )|

Fig. 11 Energy spectra for 9 Be projectiles elastically scattered on a 197 Au target foil, with a thickness t = 250 µg/cm2 oriented at a target angle θT = 40◦ : a at a forward angle (θb = 32◦ , E b = 36 MeV) and b at a backward angle (θlab = 105◦ , Elab = 39 MeV) [13]. In the former case, the difference in the track length is 2.53 t and the energy spread, given by Eq. 7, is 348 keV, very close to the experimental width

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Fig. 12 Difference in effective thickness, teff (normalized by the nominal thickness t) as a function of the ejectile angle θb for target angles θT = 30◦ (red line) and θT = 40◦ (blue) for forward (solid) and backward (dashed) configurations. It can be seen that for angles θb < 110◦ , the forward configuration has much smaller differences and, hence, smaller energy spread of the elastic peak

In Fig. 11 elastic peaks at a forward and a backward angle are compared. At forward angles the peak has a width which main contribution is the energy resolution of the detector, whereas at backward angles the energy loss at the target is dominant. In Fig. 12 the difference in track length is plotted as a function of the detector angle θb for target angles θT = 30◦ and 40◦ . There, it can be seen that the forward configuration yields much smaller track length differences up to θb = 110◦ . Hence, a better energy resolution can be expected in this way as long as it can be guaranteed that scattered ions will not be partially blocked by the target holder, as suggested in Fig. 9.

2.4 Detection System In this section, different detection methods with an increasing degree of complexity will be briefly discussed. The elastic scattering is the most intense channel, at least at energies below and close to the barrier, and/or at forward angles. Hence, an array of single silicon detectors, which only register the total energy (E tot ) of ejectiles, should be enough to obtain the elastic and inelastic peaks (see item a). Otherwise, it is advisable to use a telescope configuration (E − Er es ) to identify the atomic and/or the mass number (see item b) or even a magnetic spectrometer or a timeof-flight to distinguish the mass number of heavy ions (item c). With these tools, the elastic events can be discriminated from the background and separately counted.

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Fig. 13 Detection setup at the Institute of Physics of the University of São Paulo, Brazil: a compact array of nine single detectors (5◦ between detectors) with collimating slits is placed on a rotatable arm to cover forward angles. At backward angles, an array of nine telescopes (E − Er es , which can also provide particle identification) is placed on a fixed position (10◦ between detectors). Courtesy from L. Gasques

These three methods consist in the detection of the ejectile b in single mode. The last one (item d), also involves the detection in coincidence of the recoil B. (a) Detection of energy E b and angle θb of the ejectile For the measurement of the angular distribution of the elastic scattering products, an array of several silicon surface-barrier detectors [14] is usually implemented. To define the measurement angle, each detector must have a collimating slit. It is also possible to have two collimators, joined by a tube so as to guarantee that only particles stemming from the target direction can enter the detector, as shown in Fig. 13. The detector array should be moved from forward to backward angles to cover the whole angular range. Since the elastic differential cross section has a very strong angular dependence, a measurement run at forward angles can be completed in few minutes, while at backward angles can last several hours (recall Fig. 3). Thus, it is advisable to dispose the detectors of the array as close to each other as possible, for e.g. with an angular separation of 5◦ . If not, forward detectors may have too high counting rate, thus blocking the acquisition system or even damaging the detector, and backward detectors may turn to be useless, due to their low amount of events. Moreover, in order to assure comparable counting rates in all detectors of the array, the solid angle of detectors may increase as their angular position in the array. Although there is not an optimal combination for all angular ranges and energies, solid angles may range from ∼0.05 msr for the first detector up to ∼1 msr for the back one [15]. Another possibility is to place additional detectors fixed at the most backward angles. While the main detector array varies its angle in each measurement run, the backward detectors sum up the

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Fig. 14 a) Three silicon strip detectors with rectangular shape covering forward, intermediate and backward angles at the Institute of Physics of the University of São Paulo, Brazil. The detectors are segmented in 16 vertical strips, each of them corresponding to a scattering angle θb . Courtesy from L. Gasques b) An annular shaped double-side strip detector placed at 180◦ with respect to the beam incidence direction at the TANDAR Accelerator in Argentina. The first panel shows the 1.5 µm mylar plus 300 Å Al foil to suppress δ electrons. The rear side is segmented into 16 angular sectors while the front side has 48 rings. Each ring correspond to a scattering angle θb

events from all these different runs, thus achieving a peak with statistical significance [16]. Instead of using an array of several single detectors, more modern silicon strip detectors provide an individual signal from each of its sections [17, 18]. Thanks to their large effective area, they can provide an almost complete angular distribution even remaining at a fixed angular position. For intermediate angles, detectors with rectangular geometry are the more suitable (see Fig. 14a). For very forward or very backward angles, those with annular shape are the best choice, since they cover all azimuthal angles φ for a given scattering angle θb . In this former case, the beam passes through its central hole (see Fig. 14b). Alternatively, position sensitive detectors (PSD) can be employed. These detector have three outputs. The one in the middle yields the energy deposited E. One of the outputs must be short-circuited3 and then the third output will yield a signal proportional to the energy times the position of incidence along its sensitive path, E× P. The position of incidence of each event can then be obtained by an electronic module or by software processing as the ratio between its E× P and E signals. In this way, a continuous angular distribution can be measured along its path. Combining several PSD’s, a significant angular range can be covered, with some dead zones between detectors, which can be completed in a complementary run. An advantage of this method, is that the angular range of each histogram bin can be adjusted in the offline analysis in terms of the desired granularity and the available amount of events for the statistical significance. 3

Switching the shortcut to the other side, the position is measured from the other end.

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Fig. 15 a Scheme of telescope mounting: a thin detector registers the energy loss E of the particles, while a thicker one completely stops the ions, yielding the residual energy E res . b Two dimensional plot E − E res for 7 Li projectiles (E a = 30 MeV) on a 144 Sm target. Each dot represents a detected event. The curves are calculated relations for the E and E res signals for each nuclide

A much cheaper alternative is to use commercial PIN photodiodes instead of surface-barrier detectors [19]. For the detection of particles, the transparent plastic cover must be manually removed, leaving the silicon wafer exposed. Most of them have a very small sensitive area (∼1 mm2 ) but models with area up to 10 mm × 10 mm, and a depletion layer with a depth of 100 µm can be found (Hamamatsu S3590-09). An array of 50 or even 100 units can be placed directly welded on circuit boards, covering the whole angular range. In this case, the amount of data acquisition modules and vacuum feed-through is the only limit to their number. To further reduce the whole setup cost, low-cost preamplifier and shaper-driver can be home made as describe in [20, 21]. (b) Detection of energy E b , angle θb and atomic number Z b of the ejectile In the case of some weakly-bound nuclei such as 6 Li and 7 Li, the exothermic (Q > 0) transfer of deuterons and/or tritons prolifically produces α particles with energies around the elastic peak. This produces a background which hampers the counting of elastic events, particularly at backward angles and/or at high energies, where the elastic cross section turns very small. To discriminate the elastically scattered ions from the fragments and/or their transfer products, a telescope configuration can be used. This consist in at least two aligned detectors, as shown in Fig. 15a. In the first one, the incoming ions lose a fraction (1/4 − 1/2) of their total energy, noted as E, while in the second detector the residual energy Er es is deposited. Plotting each event in a E − E res two-dimensional plot (see Fig. 15b), the atomic number of incoming ions can be identified [14]. In case of light ions, also isotope (i.e. mass) discrimination can be achieved. For very light ejectiles, telescopes may consist in two (or more) silicon surfacebarrier detectors, with appropriate thicknesses, e.g. 100 μm for the E and 1000 μm for Er es . The energy loss of a projectile with mass Mb , atomic number Z b and kinetic

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Fig. 16 Ionization chamber with three silicon position-sensitive detectors (PSD) at the TANDAR Laboratory in Argentina. The anode is split in two to yield E 1 and E 2 signals, whereas the residual energy Er es is given by a PSD. This signals allow the identification of the atomic number of heavy particles (M  16 u). Additionally, the position given by the PSD allows the determination of the incidence angle in a 30◦ angular range [22, 23]

energy E b can be calculated by the Bethe-Bloch formula [14], but very roughly it scales as E ∝ Mb Z b2 /E b . Hence, for ejectiles such as 16 O, a E as thin as ∼10 μm is necessary.4 For heavier ions, a gas detector should be used as E. Although they have more requirements, such as gas recirculation and pressure stabilization, their thickness (gas pressure) can be easily adjusted to fit the E ∼ (1/4 − 1/2)E tot condition. Additionally, the anode can be segmented to yield several E signals (see Fig. 16). (c) Detection of energy E b , angle θb and mass number Mb of the ejectile While the dependence of the energy loss E is quadratic in Z b , it is linear in Mb . Hence, isotope discrimination turns more difficult as the ion mass increases. In these cases, the use of a magnetic spectrometer or a time-of-flight (ToF) can very significantly improve the mass and energy resolution of ejectiles. A magnetic spectrometer imposes a selection in magnetic rigidity, Bρ = (2Mb E b )1/2/q, where B is the magnetic field and ρ the particle bending radius (see Fig. 17a). The particles accepted by the slits (scattering angles within θb ± θb /2) are focused on the detector plane. The position of incidence along this plane, related to the bending radius ρ, allows to separate particles with different masses and/or different energies. Another complementary determination of the mass can be achieved by a ToF measurement. This can be done either with a pulsed beam and one stop detector or by two consecutive 4

Here, we assume an energy of 40 MeV, which is close to the Coulomb barrier between 16 O and a medium-mass target such as 58 Ni.

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Fig. 17 a) Scheme of a magnetic spectrometer (Q2D). Ejectiles scattered within θb ± 1/2θb are accepted by the slit and focused on the focal plane detector. Particles 1 (blue lines) have less magnetic rigidity than particles 2 (red lines), i.e. ρ1 < ρ2 . Hence, they impinge on different positions on the focal plane. b) Scheme of a ToF system. The time between the start and stop detectors, together with the energy, determines the mass

detectors separated by a length l (e.g. a micro-channel plate as a start and a silicon detector as a stop, as shown in Fig. 17b). The simultaneous measurement of energy and ToF (i.e. velocity), determines the mass of the particle. One example of the TOF technique used in conjunction with a magnetic spectrometer (Q3D) and E − Er es technique is [24] where the elastic scattering 16 O(16 O,16 O)16 O was measured at 350 MeV at angles in which there is a ratio to Rutherford (actually Mott scattering) of 10−4 . (d) Overdetermination of kinematics The elastic or inelastic scattering leads to two particles with kinetic energy: the ejectile and the recoil. However, in most cases the target is much heavier than the projectile and hence, on one side the transferred kinetic energy is very low, and, on the other side, the stopping power of the recoil within the target foil is very high. In such cases, the recoil can not exit the foil or its energy is below the detection threshold. For systems in which the mass of projectile and target are comparable, both the ejectile and the recoil can be detected in kinematic coincidence (KC) at angles determined by linear momentum conservation, on opposite sides of the beam. Detection in KC is very effective reducing background and this technique has been successfully applied to the measurement of very low cross sections. One early example of this technique is [25], in which the reaction 9 Be(14 N,14 N)9 Be was measured at E a = 28 MeV. At that time, the energy resolution of the scintillation detectors was not enough to discriminate against the contaminants in the target, like minor amounts of oxygen. By measuring energy and angle for the ejectile and for the recoil (E b , θb , E B , and θ B , respectively), they were able to clean those contaminants as well as residues of 8 Be breakup (after neutron transfer). An extreme application of the KC technique is given in [26] in which the 12 C(12 C,12 C)12 C reaction was measured at 25.8 MeV in an attempt to measure direct gamma transitions arising from the quasi-molecular states in the continuum of 24 Mg. The KC technique allowed discrimination against all background sources of about one part in 106 .

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Fig. 18 Conceptual and simplified layout of a data acquisition system for a measurements in single (OR condition) and b measurements in kinematic coincidence (AND condition). The charges of each detector are collected by preamplifier (Pre), the generated voltage is increased by an amplifier (Amp) and this signal is delivered to the analogue-to-digital converter (ADC). When the amplifier signal is over a conveniently set lower limit, the single-channel analyzer (SCA) generates a logical pulse which triggers the ADC acquisition

2.5 Data Acquisition In case that the KC is being applied, the data acquisition should be triggered by the simultaneous arrival (AND condition) of an ejectile and a recoil to each corresponding detector (see Fig. 18). However, in most cases only ejectiles signals are sought and the data acquisition system is operated in single mode, i.e., any detector can trigger the acquisition (OR condition). For triggering the acquisition, signals should surpass a conveniently set threshold, in order to reject electronic noise and maintain the rate below few kHz. In case that recoils have enough energy to trigger the detector, they may form a low-energy peak in the spectrum.5 Usual analogue-to-digital converters (ADC) have conversion times τc of several microseconds. Given an acquisition rate R, dead-time fractions close to Rτc are produced.6 Hence, rates R well above 1 kHz implies dead-time fractions in the order of few percent and this effect needs to be accounted and corrected for in the calculation of cross sections. An effective way of determining the correction factor is to add a pulser or a function generator to the detector signal, through the test input of the preamplifier. The pulser amplitude is usually set so as to have the pulser peak at the highest part of the spectrum. From the number of counts of the pulser peak N p , the acquisition time tacq , and the pulser frequency f , the dead-time correction factor for the detector events can be derived as Cdt = tacq f /N p . It is advisable, though, not to have dead-times over 10%, since the final cross section value turns too sensitive to this correction factor. More modern digitizers, which shorter conversion times, have much less dead times and circumvent this problem to a large extent.

5

These recoil nuclei have their corresponding ejectiles on the other side of the beam and, hence, are uncorrelated to ejectiles recorded in the same detector. 6 See [14], Chap. 4, Sect. VII for a more precise treatment.

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3 Alignment and Beam Focusing In order to obtain a precise angular distribution, the beam line, collimating slits, the target and the detectors should be aligned as precisely as possible. The beam direction is determined by the center of the last bending element of the accelerator (analyzing or switching magnet) and the target position. This direction can be established by using a theodolite or with a laser. Collimating slits must be then adjusted to the magnettarget line, leaving the desired clearance according to the distance from the slit to the target. If allowed by the detector cables and rotation possibilities of the array, each detector should be placed at 180◦ and observed with the theodolite (Fig. 19a) or, alternatively, placed at 0◦ and aligned with a laser device (Fig. 19b). Once the beam is on, several instruments such as beam profile monitors, inductive coils, secondary emission monitors, luminescent screens and Faraday cups are generally available to characterize the beam transversal profile, intensity, position and focus [27]. Also, the collimating slits can have a current amplifier to obtain a readout of the current induced by tail of the beam on them (see Fig. 6). The intensity of these signals can be shown in a panel to easily visualize the deviation of the beam (Fig. 20a). With these readouts, magnetic quadrupoles and steerers can be online adjusted to focus and center the beam, respectively. In a typical procedure before a measurement run, a target-like luminescent screen with a 2 mm hole is placed instead of the target (Fig. 20b). The screen can be spotted with a small webcam inside the scattering chamber. The beam is first focused on the screen adjusting the quadrupoles. Then it can be centered in the hole adjusting the steerers, maximizing the current in the Faraday cup at the end of the line, while minimizing the tail of the beam impinging on the screen and on the collimating slits.

Fig. 19 Aligning of detectors with use of a) a theodolite at the end of the beam line, placing each detector a 180◦ or b) a laser placed behind the analyzing or switching magnet, placing each detector at 0◦

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Fig. 20 Elements to diagnose the beam focus and centering: a) LED panel indicating the current induced by the incidence of the beam tail in the first set of collimating slits, upstream from the scattering chamber. b) Target holder including an alumina luminescent screen for focusing the beam prior to a measurement run (pink color, in the central position)

4 Irradiation Times The irradiation time must be decided as a trade-off between the amount of events needed in the elastic (or inelastic) peak for achieving a determined statistical significance, the number of different energies and angular positions of the detection array necessary to study the elastic scattering process, and the total beam time available for the experiment. Due to the strong angular dependence of the elastic scattering (see Fig. 3), which has a fall with the scattering angle even stepper than the Rutherford cross section (∝ sin−4 θ ) due to the nuclear potential, measurements at forward angles are much shorter (may be few minutes) than at backward ones (up to several hours). On one hand, measuring times lower than a few minutes are not an actual spare of time, since comparable times are required to move the detectors to another angle and/or to save acquired spectra. On the other hand, it also does not help to acquire much more than ∼104 events in the elastic peak, which implies a 1% statistical uncertainty. Systematical and uncontrolled uncertainties such as those produced by the beam displacement, solid angle estimation or peak integration may have larger contributions. Hence, it is more profitable to repeat the measurement either placing a different detector of the array at the same angle or after checking the focus of the beam (see Sect. 3). In this way, the reproducibility of the results are ascertained and systematic errors may be averaged out. In particular, the measurement of cross sections for the elastic scattering at backward angles and sub-barrier energies require the best accuracy, since they are quite close to Rutherford and assessing their dif-

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ference is very relevant to determine the depth and shape of the optical potential between the interacting nuclei. Above the barrier and at backward angles, where the cross section is very small, about two orders of magnitude below Rutherford, ∼102 events or even less can be enough and very worthy to establish the optical potential parameters. Another key region to take special care are the intermediate angles in which forward and backward target configuration overlap (recall Fig. 8). The cross section at these angles should be measured in both configurations to assure that no bias is introduced by this change. These angles are of particular interest because the Fresnel peak, produced by the interference of Coulomb and nuclear scattering, takes place at them. It should be recalled, however, that other processes as transfer which peaks could be clearly identified even in a one-dimensional spectrum, most probably need more irradiation time than the elastic peak to achieve statistical significance.

5 Peak Integration Although the elastic scattering is the most intense channel at most angles and energies and its peak dominates the spectrum, its integration interval must be correctly delimited and its background subtracted. The background can be either projectiles scattered from the collimating slits or the target frame, or it may consist on breakup or transfer fragments. The width of the integration interval for the elastic peak depends on the existence of inelastic or transfer peaks close to it. In some cases, these inelastic peaks can not be resolved at all from the elastic one and both contributions are counted together. This is called quasielastic scattering. When the separation is possible but not trivial, a simultaneous fit to the elastic and inelastic peaks yields the more reliable result (see an example in Fig. 21). When the elastic peak is well separated from inelastic contributions or these are negligible, a direct integration of the peak area can be done. For setting the integration limits, a Gaussian fit can be performed. These limits can be then adopted as E 0 − 3σ and E 0 + 3σ (E 0 is the peak centroid and σ its standard deviation), which includes the 99.7% of the events. If the cross section is normalized by the elastic events in a monitor detector (Sect. 6.2) or by events from a heavier target (Sect. 6.3), the final results is actually independent of this choice. For estimating the background average, a region wider than the elastic peak and far away from it should be considered. Then, the contribution of the background to the peak is simply corrected by the number of channels of each region. In this way, the possibility of introducing a bias due to the inclusion of inelastic or transfer peaks in the background region are minimized (see Fig. 22). Hence, the net number of detected events NDet can be estimated from the total number of events in the peak, Ntot , and the corresponding to the background, Nbkg as (8) NDet = Ntot − Nbkg .

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Fig. 21 Energy spectrum for the 9 Be + 197 Au system at θb = 75◦ , E a = 40 MeV. The elastic and two inelastic peaks are fitted by gaussian functions with the same width, while a linear function was used to estimate the background. Actually, the elastic peak includes inelastic events corresponding to a state with an excitation energy of 77 keV which can not be resolved. In the same way, the first inelastic peak may include events from states at 269 keV (3/2+ ) and 279 keV (5/2+ ), which are not resolved from each other. The second inelastic peak corresponds to the 7/2+ state at 547 keV (taken from [13])

The corresponding statistical uncertainty have then both contributions: 2 2 1/2 + Nbkg ) = (Ntot + Nbkg )1/2 . NDet = (Ntot

(9)

6 Calculation of the Differential Cross Section Experimentally, the differential cross section in the laboratory system for a certain reaction channel, dσlab /d, can be determined by a) the number NDet of detected particles corresponding to this channel which are scattered in a direction with an angle θb with respect to the beam direction and within a solid angle  (i.e. particles that impinge within the detector entrance slit, see Fig. 23), b) the number of incoming beam particles that crossed the target, Ninc , and c) the number of target atoms per surface unit ω: NDet dσb (θb ) = . d ωNinc

(10)

The usual units for differential cross sections are millibarn/steradian (mb/sr), where 1 b = 10−28 m2 and one steradian is the angle subtended by a unit area on a

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Fig. 22 Integration of an elastic peak acquired by a monitor detector placed at θ = 16◦ , for the 9 Be + 197 Au system (E = 40 MeV). The peak was fitted with a Gaussian function, and the intea gration limits were set to include ±3σ (99.7% of the events). The inset shows this same peak and its fit in linear scale. The background was estimated averaging a wide region far from the elastic peak ([13])

Fig. 23 Parameters involved in the experimental determination of the differential cross section dσ/d: θb is the angle of the ejectile with respect to the beam direction, in the laboratory system; d is the differential solid angle centered at the angle θb and dσ is the differential cross section

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sphere of radius of one unit. The number of target atoms per surface unit is usually expressed as δNA 1 , (11) ω= m A cos θT where δ is the mass surface density of the target (usual units: μg/cm2 ), m A is the atomic mass of the target and N A the Avogadro’s number. The factor 1/ cos θT takes into account the effective thickness of the target due to the angle θT of its normal and the beam direction (recall Fig. 10). To convert the ejectile scattering angle in the laboratory system θb to the c.m. angle θcm the expression sin θcm , (12) tan θb = cos θcm + τb can be used, being  τb =

E cm ma mb mb + m B m A m B m a + m A E cm + Q

1/2 .

(13)

For the elastic case, this parameter reduces to simply τb = m a /m A . To convert the experimental cross section from the laboratory system (Eq. 10) to the c.m. system (in which the Rutherford cross section of Eq. 1 is expressed) the Jacobian factor J has to be used:     dσ dσ =J (14) d cm d lab where (1 − τb2 sin2 θb )1/2 d(cos θb ) 1 + τb cos θcm = = . 2 d(cos θcm ) (1 + τb + 2τb cos θcm )3/2 [τb cos θb + (1 − τb2 sin2 θb )1/2 ]2 (15) To determine the number of projectiles Ninc which have impinged on the target, and in this way normalize a cross section measurement, there are at least three methods. J=

6.1 Normalization with a Faraday Cup If the target is not very thick (e.g. more than few mg/cm2 ) the reaction probability of projectiles is less than 10−6 and the intensity of the beam emerging from the target will be indistinguishable from the incident one. If, in addition, the angular straggling is not too high, a Faraday cup placed downstream from the target can online register

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the beam intensity during the whole measurement run.7 By means of a conveniently calibrated charge integrator, which converts the electric charge to a number of pulses n p (k = 10−10 C/pulse is the usual range) and a pulse counter, the accumulated charge is determined. For converting this to number of particles, it has to be recalled that the target also acts as another stripper and the projectiles will emerge with a charge state distribution corresponding to their energy. The average charge state q¯ can be calculated following the prescription of [11]. Normally, q¯ is higher than the charge state of the incoming beam, q, and it is close to the atomic number of the projectile, Za . Hence, the number of incoming beam particles can be estimated as Ninc =

k np eq¯

(16)

and the differential cross section as dσcm e q¯ m A cos θT Cdt NDet (θcm ) = J (θcm ), d k np N Aδ 

(17)

where Cdt is the correction factor for the dead time of the acquisition system discussed in Sect. 2.5. Although the charge collection of the Faraday cup is very reliable, note that differential cross section normalized in this way remains dependent on four parameters which can not be determined with high precision: • The mass surface density of the target δ. The production procedure of thin targets (usually by evaporation) does not allow to determine the target thickness better than a 10–20% of uncertainty. If the target is not very thin (e.g. more than 200 μg/cm2 ), its thickness could be checked by the energy loss of α particles provided by a radioactive source. Another value for its thickness is given by the width of the elastic peak in backward configuration (see Fig. 11b). • The target angle θT . This angle is usually not determined with the same accuracy as the detector angle. On addition, wrinkles on the target foil may make that the effective target angle varies randomly along the beam spot. • The solid angle of the detector . This can be geometrically determined from the detector active area A and its distance to the target d as  ∼ A/d 2 . But the measurement of the width of the detector slit, usually 1 or 2 mm, introduces errors in the order of 10%. • The average charge state q. ¯ Although it is not far from the projectile atomic number Z a , this value depends on empirical formulas and, hence, it has a considerable uncertainty.

7

A Faraday cup upstream from the target can register the beam intensity only before and after each run. The average of those values, multiplied by the run duration and divided by the charge state q tuned by the accelerator, could be used just as a consistency check for the number of incoming projectiles Ninc .

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Of course, the product of these four factors can be renormalized imposing dσ elastic/dσ Ruth = 1 at very forward angle and/or low energy. However, the following normalization methods circumvent these uncertainties.

6.2 Normalization with a Monitor Detector If an extra detector is placed at a forward angle (e.g. 10◦ − 25◦ ) for which the elastic cross section can be reliably assumed as coincident with Rutherford for all energies, the events on this detector (hereafter called monitor), NMon , can be used to deduce the number of incoming beam particles Ninc . In this way, Eq. 10 can be written also for the monitor, with the differential cross section being the one given by the Rutherford formula of Eq. 1. Since this calculation for the monitor uses the same number of incoming particles, Ninc , and the same the number of target atoms per surface unit, ω, as the detectors of the array, these uncertain factors simply cancel out. Thus, for the detector i we obtain i i dσ i J (θcm ) dσ Ruth Mon Mon Cdti NDet (θcm ) = (θcm ) i . Mon Mon d d Det Cdt NMon J (θcm )

(18)

As it can be seen, this expression depends on the ratio of solid angles of the monitor Mon and of the detector i, iDet . To obtain these ratios without depending on geometrical measurements, the same detector array and monitor should be used to measure the angular distribution of a system well below the barrier, for which the cross section coincides with Rutherford for all angles. This could be the same system under study at low energy (e.g. half of the barrier), or a heavier projectile and/or target can be chosen (e.g., 16 O + 197 Au). Hence, the solid angle ratios are obtained as  Mon CdtMon NMon = i iDet Cdt i NDet

dσ Ruth i (θcm ) d dσ Ruth Mon ) (θcm d

Mon J (θcm ) ,  i J (θcm )

(19)

where the primed variables indicates values corresponding to this calibration run. Note that even if the monitor may remain at the same laboratory angle, its c.m. angle, Mon , changes if another system is used. Although a single measurement run seems θcm to be enough to obtain the solid angle ratios, it is strongly advisable to make several and redundant measurements at different angles, both in forward and in backward configurations, to check the consistency of the measuring system. The deviations of the different values obtained for the Mon /iDet ratios are a measure of the systematic uncertainty of the experiment. In other words, the standard deviation from the Rutherford cross section when measuring the angular distribution of a system well below its Coulomb barrier should be considered in the uncertainty estimation.

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The uncertainty for differential cross section dσ/d(θb ) when normalizing with a monitor can be estimated taking into account the numbers of events in the ith i , in the monitor, NMon , and the uncertainty in the ith solid angle ratio, detector, NDet (Mon /iDet ),



 dσ d

i (θcm )



2 1/2  dσ i 1 1 (Mon /iDet ) (θ ) = + + . i d cm NDet NMon Mon /iDet

(20)

If two monitors are used, one of each side of the beam, the deviations of the beam in the reaction plane can be controlled. Since the Rutherford cross section is very sensitive to the angle at forward angles, a small shift x in the incidence position will result in an increment of the events in one monitor and a decrease in the other (recall Eq. 5). Hence, the ratio between the events of the two monitors is a good indication of the centering of the beam. Moreover, for forward angles the cross section should be normalized using the monitor on the same side as the detector array. In case of a beam deviation, the angular shifts for this monitor and for the detectors, although they will not exactly cancel, they will have the same sign and will compensate to a large extent. For backward angles, the monitor on the other side than the detector array should be used for the normalization.

6.3 Normalization with an Additional Target Layer Another very reliable way of normalizing the measurement of the cross section of an elastic scattering is to have an extra layer of a heavy element evaporated onto the target foil. A very thin layer (∼10 − 20 µg/cm2 ) of gold is definitively the best choice. If the projectile bombarding energy E a is well below the Coulomb barrier between the projectile and 197 Au, it can be assumed that the scattering over it will correspond to Rutherford for all angles. If the target nuclide under study is light enough (A  100), the elastic peaks produced by the projectile scattering on these nuclei and on 197 Au will have discernible energies on the spectrum (see Eq. 3). The expression for the cross section (Eq. 10) can be also written for this 197 Au layer, but assuming the Rutherford differential cross section (Eq. 1). In analogous way as before, this layer has the same number of incoming particles Ninc and, since it is registered by the same detector as the scattering under study, the same solid angle  and dead-time correction factor Cdt apply. Hence, these factors cancel out, obtaining Ruth dσAu dσ Au ωAu NDet J (θcm ) (θcm ) = (θcm , ) Au ) d d ω NAu JAu (θcm

(21)

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where NDet and N Au are the number of events in the elastic peaks of the system under study and of the extra 197 Au layer. Note that, although the laboratory scattering angle Au , and the Jacobians, J and θb is the same for both systems, the c.m. angles θcm and θcm J Au , differ from each other. The ratio of the numbers of target atoms per surface unit of both layers, ω Au /ω, can be obtained by calibration runs done at a lower energy, at very forward angles and/or with a heavier projectile for which the Rutherford cross section is granted for both target layers: ω Au = ω

dσ Ruth  (θcm ) d Ruth dσ Au Au ) (θcm d

 Au NAu Jb (θcm ) .   ) NDet J (θcm

(22)

In this case, the uncertainty of the cross section is

 1/2   dσ 1 dσ 1 (ωAu /ω) 2  (θcm ) = (θcm ) + + . d d NDet NAu ωAu /ω

(23)

7 Plotting the Results Once the experimental results and their uncertainties are established, a convenient plot is mandatory to get a proper insight of the underlying physics of the scattering phenomenon. The most usual way to present elastic scattering data is to plot the ratio of the experimental cross section to the Rutherford cross section for each bombarding energy (such as in Fig. 3). In this way, several features are evident. At energies well bellow the Coulomb barrier, the elastic cross section coincides with Rutherford, i.e. the plotted ratio, σ elastic /σ Ruth ∼ 1. At energies close or even above this barrier, this is valid only for forward angles which correspond to large impact parameters. At backward angles, which correspond to small impact parameter, the ratio σ elastic /σ Ruth is well below unity (Fig. 25a). The grazing angle θgr , defined as the angle for which σ elastic /σ Ruth = 1/4 (Fig. 25b), corresponds to a distance of closest approach D which approximately coincide with the sum of the nuclear radii of the projectile and the target, Ra + A . Since this distance is D=

D0 2



 1 +1 sin θ/2

(24)

Z a Z A e2 , E cm

(25)

where D0 =

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Fig. 24 Projectiles with two different impact parameter, bC and bC+n , may be scattered to the same angle θ F and interfere with each other. One of them, bC , interacts solely with the Coulomb field of the target, while the other, bC+n interacts with the sum of the Coulomb and the nuclear field

Fig. 25 a) Angular distributions of the elastic scattering (normalized to Rutherford) for the 7 Li + 27 Al system at several bombarding energies around the Coulomb barrier (V = 8.3 MeV). b) Plot C for E b = 11 MeV in linear scale, where the the Fresnel peak at θ F and the grazing angle θgr are indicated. Data taken from [28]

the c.m. grazing angle can be estimated as  1 . = 2 arcsin 2Ra + A /D0 − 1 

θgr

(26)

Another remarkable feature of angular distributions is the quantum interference. Ions with impact parameter bC are scattered solely by the Coulomb potential, while those with a smaller impact parameter, labeled as bC+n in Fig. 24, are scattered by the sum of Coulomb and nuclear potentials. Both may end scattered at the same angle θ F and, thus, they can interfere giving rise to Fresnel oscillations. The main peak of these oscillations, located at θ F , can be clearly seen in the linear plot of Fig. 25b. Another complementary way to display the elastic scattering data is to plot them versus the distance of closest approach of Eq. 24. This allows to see several bom-

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Fig. 26 Experimental elastic scattering differential cross sections (normalized to Rutherford) for the 16 O+ 64 Zn (circles, data from [30]) and 7 Li+ 27 Al (squares, data from [28]) systems as a function of a) the distance of closest approach (Eq. 24), and b) the reduced distance (Eq. 27). The energies are expressed in the laboratory frame. The fit follows the prescription of [29] (Eq. 28). The critical interaction and strong-absorption distances, d I and d S , are indicated. While the strong-absorption distance is similar for both systems (d SO+Zn = 1.59 fm and dsLi+Al = 1.54 fm), the critical interaction distance is clearly larger for the weakly bound projectile 7 Li (d IO+Zn = 1.84 fm and d ILi+Al = 2.65 fm). Courtesy from J. Gómez

barding energies in a single curve,8 which is a strong consistency check for the data (Fig. 26a). Moreover, to compare different systems the reduced distance d=

D 1/3 Aa

1/3

+ AA

(27)

can be used (see Fig. 26b). Following the prescription of [29], this data can be fitted by p1 dσ fit = , (28) dσ Ruth 1 + e− p2 (d− p3 ) where p1 , p2 and p3 are adjustable parameters. In particular, p1 is the asymptotic value for large distances and it is close to unity. In Fig. 26b the critical interaction and strong-absorption distances, d I and d S , are indicated. They are defined as the reduced distances for which dσ fit /dσ Ruth = 0.98 and 0.25, respectively. These distances qualitatively indicates where each process is expected to dominate. At reduced distances larger than the critical interaction d I , the elastic scattering is dominant. Between it and the strong-absorption distance d S , inelastic and other direct reactions are most important. For reduced distances shorter than d S , it can be expected that fusion is the most relevant channel. The distances d I and d S can be related to the binding energy and other structure details of the involved nuclei [29].

8

The Fresnel peak, which occurs for energies well above the barrier, lies somehow out of this curve.

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Proton Induced Spallation Reactions S. Sharma , U. Singh , and B. Kamys

Abstract In this chapter, an overview is given of the results obtained over more than eighty years in the study of proton-induced spallation reactions. The main facts discussed here concern the qualitative behavior of the spallation cross sections as a function of the target nuclei involved, the proton beam energy, and the reaction products detected. Their phenomenological parameterization as well as microscopic theoretical models used to explain the observed facts are discussed. The problems that have not yet been satisfactorily solved are mentioned.

1 Introduction The term spallation was introduced by Seaborg in 1937 [1] for reactions in which the incident proton knocks out several nucleons in a series of two-body collisions, leaving behind a highly excited nucleus. This nucleus decays further by evaporation of charged particles and neutrons, forming a continuous distribution of nuclei descending in mass number A from the target mass number. This definition has been generalized by other scientists, and in fact, according to the Encyclopedia Britannica, the spallation process is understood as “a high-energy nuclear reaction in which a target nucleus struck by an incident (bombarding) particle with energy greater than about 50 million electron volts (50 MeV) ejects numerous lighter particles and becomes a product nucleus correspondingly lighter than the original nucleus.” Knowledge of the cross sections for the various products of spallation reactions is essential for a variety of applications in science and technology. They affect such diverse topics as the design and operation of neutron spallation sources, e.g., in the United States [2], Europe [3, 4], and Japan [5], the optimization of isotope production [6], the accelerator-driven sub-critical reactor systems, being considered for nuclear waste transmutation [7], the interpretation of the reaction products of cosmic ray interaction with terrestrial and extraterrestrial matter [8], etc. S. Sharma (B) · U. Singh · B. Kamys The Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Deshmukh and N. Joshi (eds.), Understanding Nuclear Physics, https://doi.org/10.1007/978-981-19-8437-2_6

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2 Phenomenology Experiments discussed in ref. [9] and later published have shown that at high proton beam energies (several GeV) the total and differential cross sections of spallation reactions provide strong evidence for the applicability of two very important hypotheses: • the hypothesis of limiting fragmentation, • the factorization hypothesis. The first of these hypotheses states that at high proton beam energies (several GeV or more) both the total and differential cross sections of the spallation reactions cease to vary. The energy dependence of the total cross sections is well illustrated in Fig. 1. The symbols represent experimental cross sections, while the lines are the result of the parameterization proposed by Wellisch and Axen [10]. The mass number of the target nuclei varies from A = 12 to A = 238. The following characteristic properties of the energy dependence are evident: • The cross-sectional values vary at low beam energies but cease to vary at GeV beam energies—this simply illustrates the hypothesis of limited fragmentation. • The beam energies at which the character of energy dependence changes are lower for light targets than for heavy targets. • The values of the cross sections on the energy dependence plateau are larger for heavy targets than for light targets. The last property is interpreted as evidence for the factorization hypothesis, which states that the dependence of the spallation cross sections on the mass of the target nucleus A can be explained, at sufficiently high energies, by a factor that depends only on the target mass number A. In proton, helium, and heavy-ion induced reactions, it has been observed that the elemental cross sections σ (Zp) for the production of intermediate-mass fragments (i.e., ejectiles heavier than helium but lighter than fission products) exhibit a power-law dependence on the atomic number of the product, i.e., σ (Zp)≈ Zp−τ . Moreover, the power index τ depends in a very regular way on the energy of the projectile beam Elab as shown in Fig. 2. The power-law parameter τ decreases monotonically with the beam energy up to a value of about τ = 2 at an energy of about Elab = 1 GeV and retains this value at higher energies even up to 10 TeV. This is concrete evidence for the independence of the spallation cross sections from the beam energy at sufficiently high-energy values, i.e., for the validity of the hypothesis of limited fragmentation. Figure 3 illustrates the applicability of the hypothesis of limited fragmentation for the differential cross sections of spallation reactions: The left part of the figure shows the differential cross sections measured at 350 in p+Ni collisions at proton beam energies E = 1.2, 1.9, and 2.5 GeV (open circles, full squares, and open triangles, respectively), while the right part shows the same observables for p+Au collisions. It is evident that the shape of the energy spectra for a given product (4 He, 7 Li, 9 Be, and 11 B) is practically identical at all three energies

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Fig. 1 Total cross section from [10]

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Fig. 2 Energy dependence of the τ power index of the Zp dependence of the cross sections [11]

Fig. 3 Picture (a) from ref. [12] and (b) from ref. [13]

for each of the targets (Ni in the left panel and Au in the right panel), while the values of the cross sections are energy independent for the Ni target, but they increase with beam energy for the Au target. This shows that the beam energies for the Ni target are high enough to enter the asymptotic region where the limited fragmentation hypothesis works well, while these energies are still too low for the Au target.

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2.1 Parameterization of Total Cross Section The regular behavior of the spallation cross sections at sufficiently high beam energies enables one to look for some numerical formulas to parameterize the values of the total and differential cross sections in this energy range. The parameterization of the total inelastic cross section (σi ) of proton-nucleus collisions at high energies (E > 2GeV /nucleon) is presented in the work of Silberberg and Tsao [14] shown by Eq. 1: (1) σi = 45A0.7 [1 + 0.016sin(5.3 − 2.63ln(A))]mb where A denotes mass number of the nucleus. At lower energies E (but larger than 20 MeV/nucleon—below which resonance effects are important) the energy dependent formula of Letaw et al. [15] may be applied:   E σi (E) = σi (Eq. 1)[1 − 0.62 exp − sin(10.9E − E −0.28 )] 200

(2)

A parameterization of the total reaction cross sections for any system of colliding nuclei, valid for the energy range from a few A MeV to a few A GeV, was proposed by Tripathi et al. [16]. Proton-nucleus collisions are treated as a special case of these reactions. Another parameterization of such cross sections, valid for the target mass 12 < A < 238, was given by Wellisch and Axen [10]. Its quality is illustrated in Fig. 1 by comparing their formula predictions (lines) with the data represented by dots. The parameterization of the total reaction and partial cross sections in proton-nucleus (with atomic number ZT of the target less than or equal to 26) and nucleus-nucleus reactions (with atomic number of the projectile Z P and target ZT less than or equal to 26) was published by Sihver et al. [17]. Their formulas apply to energies of incident protons above 15 MeV and incident nuclei above 100 MeV/nucleon. The cross sections of all products whose charge number is smaller than that of the target nucleus (projectile) can be estimated. The above parameterization of partial cross sections was extended by Silberberg et al. to a broader range of target nuclei up to the Au target (Z = 79) in proton-nucleus collisions [18]. In 1990, Summerer et al. [19] proposed an empirical parameterization (called EPAX) of the izotopic cross sections for the fragmentation of nuclei interacting with a hydrogen target in the limited fragmentation energy region. This parameterization was improved by Summerer and Blank [20] to correct for the description of the fragmentation yields near the projectile and the memory effect of neutron-deficient projectiles. The above parameterization was extended by Suemmerer in 2003 [21] to describe izotopic fragmentation cross sections outside the limiting- fragmentation region, i.e., for proton energies in the vicinity of 1 GeV. The quality of the EPAX parameterization is shown in Fig. 4. The histogram shows the values of the partial cross sections evaluated in this parameterization, while the symbols represent data taken at different energies of the Fe beam. It can be observed that the agreement improves with increasing energy, i.e., when approaching the region of limited fragmentation energy.

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Fig. 4 Taken from [22] for spallation residues in the reaction 56 Fe+ p at 0.3A, 0.5A, 0.75A, 1.0A, and 1.5A GeV

A new semi-empirical parameterization SPACS of isotopic spallation cross sections was proposed by Schmitt et al. [23] for the interaction of protons or neutrons with light (C) to heavy non-fissile nuclei (Pb, Bi) from about 50 MeV/nucleon up to several hundred GeV/nucleon. Figure 5 shows the very good agreement obtained with this parameterization. It must be emphasized that the form of the parameterization formula and the values of the parameters were chosen taking into account both the extended data set (compared to previous parameterizations) and the information on the mechanism of the reactions underlying the emission of the different products.

2.2 Parameterization of Differential Cross Sections The following experimental information was used for experiments to parameterize the differential cross sections d 2 σ/d Ed. In many experiments it was observed that the spectra d 2 σ/d Ed(E | ) of the differential cross sections for any fixed emission angle () have the Maxwellian shape—cf. Figure 3. Moreover, the slope of the high-energy tail of the spectrum is less steep, especially for light ejectiles— lighter than Li isotopes—than for energies less than 30–50 MeV. It appears that the energy spectrum contains two Mawellian-like parts—the low-energy part and the high-energy part. Moreover, it was observed that the low-energy Maxwellian part does not change with the scattering angle, while the slope of the high-energy part increases with increasing emission angle. Such specific dependence of the cross sections on the energy and emission angle of the particle was usually interpreted as an indication of izotropic emission of particles from two sources with different velocities and different temperatures moving along the beam direction. The energy spectrum of the slow source has a large slope, i.e., a low temperature, whereas small

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Fig. 5 This figure was taken from ref. [23]. Data for 208 Pb nuclei at 500 MeV/nucleon (circles) and at 1000 MeV/nucleon (squares) impinging onto hydrogen target illustrate the quality of agreement between the SPACS parameterization predictions (dashed blue lines for 500 MeV/nucleon and solid red lines for 1000 MeV/nucleon). The cross sections of the data and the parameterization for higher energy are scaled by a factor of 10 to distinguish them from those for lower energy

slope of the fast source indicates a high temperature, see, e.g., [24–26] and references therein. A typical example of such data can be found in Fig. 6, taken from ref. [12]. The spectra d 2 σ/d Ed(E) measured for 35, 50, and 100 ◦ (in LAB system) are shown in the left, middle, and right columns of the panels, respectively. The rows of panels show the results for 6 Li, 7 Li, 7 Be, 9 Be, and 11 B ejectiles (downward). The dots represent experimental data, whereas the lines show the contributions to the differential cross sections from two moving sources. As can be seen, the red dashed line does practically not change with increasing of the emission angle while the slope of the high-energy tail of the blue dashed line increases significantly as the emission angle increases. The red line describes emission of particles from a slow source, whereas the blue line describes emission from a faster-moving source, both along the beam direction. The sum of the two contributions reproduces very well the experimental differential cross sections. It is interesting to note that such a simple geometrical model is able to reproduce both the variation of the properties of the moving sources with the mass of the target [12] and with the energy of the bombarded protons [27]. However, it is not able to predict the number of moving sources necessary to describe the data well (see, e.g., [28]), and furthermore, such a model does not provide information about the mechanism of the processes leading to the formation and decay of the sources emitting the ejectiles. The most natural explanation is to say that the group of nucleons of the target nucleus that appears

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Fig. 6 The figure presents a quality of differential data d 2 σ/d Ed reproduction by two-source model for p+Ni reactions at proton beam energy of 2.5 GeV [12]

in the straight-line path of the bombarding proton through the target can be treated as the “effective target” (see, e.g., references [24, 29]), whereas all other nucleons behave as spectators during the first stage of the collision. Depending on the impact parameter this effective target may consist of a different number of nucleons. Meng Ta-chung [24] estimated this average number to be of the order of A1/3 , where A denotes the mass number of the target nucleus. The effective target may emit energetic nucleons or heavier H and He isotopes. The rest of the target nucleus is excited during such a process and may emit heavier ejectiles in addition to the light particles mentioned above. Depending on the impact parameters and the energy transferred from the impacting proton to the target nucleus, different scenarios of this emission can occur. For example, at high proton energies (above 10 GeV), the so-called “fast breakup” of the heavy remnant of the proton-nucleus collision can be manifested by the appearance of two excited parts of this remnant emitting different particles. Such a process was proposed by Huefner and Sommermann [30] to explain unusual backward enhancements in the angular distributions of heavy fragments (Sc, Cu) from uranium targets at proton beam energies from 40 to 400 GeV. This picture explains the occurrence of two (or even more) moving sources of the observed particles. The properties of the moving sources are usually parameterized, as in the

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work of Westfall et al. [31], or estimated from quasi-classical considerations, as in the so-called firestreak model of Myers [32], where the collision region is divided into tubes. A tube from the projectile fuses with the corresponding tube in a target nucleus, forming a “firestreak”. Therefore, the main efforts of scientists are focused on the creation of microscopic models of the interaction of the projectile with the target nucleus and models of the emission of the reaction products. In this context, the question arises what is the origin of the emission of particles from moving sources.

3 Theoretical Models of the Reaction Mechanism Historically, the first idea for the reaction mechanism of energetic protons (with energy greater than 100 MeV) interacting with atomic nuclei was proposed by Serber [33]. He emphasized that protons with energies so high that their deBroglie wavelength is shorter than or comparable to the average nucleon-nucleon distance within the nucleus interact with individual target nucleons. Therefore, the first stage of the proton-nucleus collision can be viewed as a sequence of nucleon-nucleon scatterings leading to the emission of multiple nucleons and leaving the remnant of the target in the excited state. Depending on the energy deposited in this remnant, different deexcitation processes with emission of nucleons and their groups can then occur. Such a picture is the background of the modern two-step models describing the spallation reactions (Fig. 7). All these models could be realized exactly only by solving the many-body Schroedinger equation, which is not possible at present.

3.1 Models of the First Step of Spallation Reactions In this chapter, the ideas underlying the three most popular models of the first stage of spallation reactions are briefly presented. These are (1) the Intranuclear Cascade (INC), (2) the Boltzmann-Uehling-Uhlenbeck (BUU) model, and (3) the Quantum Molecular Dynamics (QMD) model.

3.1.1

Intranuclear Cascade—INC

According to this model, the proton impinging onto the atomic nucleus triggers a cascade of nucleon-nucleon and nucleon-pion collisions leading to the emission of nucleons, pions and (in the most involved version of INC—Intranuclear Cascade Li’ege—INCL) light nuclei. Such models were developed by various authors, starting with Metropolis et al. [34], Bertini [35, 36], Yariv and Fraenkel [37, 38], and INCL by Cugnon and collaborators [39–43].

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Fig. 7 A schematic of the two-stage mechanism of the spallation reaction. The upper (colored) part of the figure represents the first, fast step of the reaction, which consists of a cascade of two-body nucleon-nucleon (eventually pion-nucleon), collisions in which only these particles are emitted (possibly with some light composite particles) and at the end the excited, equilibrated residual nucleus remains. This residual nucleus can emit nucleons and composite particles in the second stage of the process, by various mechanisms shown in the lower (colorless) part of the figure

The main assumption of the INCL model are described below: The nucleus is treated as an ensemble of nucleons whose spatial distribution has a known shape and fixed parameters. The momenta of the nucleons are distributed randomly inside a sphere with the radius equal to the Fermi momentum p F . The nucleons are placed in a static mean field in the form of a square well shape, with fixed depth V0 and radius Rp, dependent on the momentum p of given nucleon. The nucleons move inside the nucleus along straight-line trajectories until two of them collide or until a nucleon reaches the surface of the nucleus where it can be transmitted or reflected. The collisions are allowed only between the nucleon from the beam and nucleons of the target (first collision) or between nucleons that participated in previous collisions and other nucleons (sequential collisions). The collisions between other nucleons of the target (“spectator nucleons”) are prohibited. The collision occurs when the distance d between two nucleons is smaller than √ dmin = σtot /π , where σtot is the free space total nucleon-nucleon cross section. Pauli blocking is introduced for the possible collisions, i.e., only those collisions

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are allowed which lead to final states with the momentum of nucleon larger than the Fermi momentum. The nucleons can tunnel through the edge of the assumed square well potential of the target nucleus or be reflected back based on the calculated transmission probability. Both elastic and inelastic collisions (with creation or annihilation of  resonance) are allowed. Delta resonances decay to nucleons and pions either on the basis of their lifetime or they are forced to decay after the stopping time of the cascade. The unique feature of INCL compared to other INC models is the ability to emit light nuclei consisting of multiple nucleons (clusters with mass number A not larger than 8). The clusters of nucleons are formed by a process called coalescence. The following conditions must be met: (1) Each nucleon leaving the cascade of collisions is checked whether it is capable of overcoming the potential barrier at the surface of the nucleus. A nucleon with such a capability is treated as a leading nucleon of the possible cluster. (2) Its position on the surface of the nucleus is traced back along the straight line to a radial distance D = R0 + h, where h is a parameter (h = 1 fm) and 0) = 1/2. It is checked if there R0 is half the density radius of the nucleus, i.e., ρ(R ρ(0) are some nucleons in close vicinity (in r- and p-space) around this point, and in such a case the next conditions are checked. (3) The most stable cluster (starting from largest (Zcl = 5, Acl = 8) to as light as deuteron (Zcl = 1, Acl = 2)) is selected. (4) The emission of such a cluster is allowed only if its kinetic energy is large enough to overcome the Coulomb barrier and if its direction of flight is close enough to the normal to the surface of the nucleus. At the end of the cascade, unstable clusters (with lifetimes of < 1 ms) are forced to decay isotropically. The stopping time of the cascade is determined self-consistently as a function of the target mass A: tstop = f stop t0 (A/208)0.16

(3)

where t0 = 70 fm/c and fstop is a free parameter usually taken to be equal to unity. Conservation of mass, charge, energy, momentum, and angular momentum is respected by the simulation of each event. Further details of the INCL model may be found in Ref. [43].

3.1.2

Boltzmann-Uehling-Uhlenbeck Model (BUU)

The transport equation which was proposed by Boltzmann in 1872 to describe the statistical behavior of non-equilibrated classical thermodynamic system was generalized by Uehling and Uhlenbeck [44] to include the Einstein-Bose and the Fermi-Dirac statistics of the particles:  4 dσ ∂f + vi · r f i − r U ·  p f i = − vi j d 3 p j d 3 p j  d 6 ∂t (2π ) d   × f i f j (1 − f i  )(1 − f j  ) − f i  f j  (1 − f i )(1 − f j ) × (2π )3 δ 3 (pi + pj − pi − pj )

(4)

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The above Boltzmann-Uehling-Uhlenbeck (BUU) formula allows to determine single particle phase space distribution f ≡ f 1 (r, p, t) of nucleons moving in the mean field U and interacting one with another according to the differential cross     dσ what leads to their new positions r1 , r2 and momenta p1 , p2 . The usual section d choice of the normalization of the f function is to put f = 1 for the case when all quantum states are occupied and f = 0 when all of them are empty. This normalization allows to interpret the f ≡ f 1 (r, p, t) function as the occupation probability of the d 3 r d 3 p phase space element in the moment t and the factor (1 − f 1 ) as the probability that this phase space element is empty. Therefore, the scattering     1 + 2 → 1 + 2 of two nucleons is possible only when f 1 f 2 (1 − f 1 )(1 − f 2 ) does not vanish. Since the BUU is an integrodifferential equation in seven-dimensional space (r, p, t) it cannot, in general, be solved analytically. The numerical recipe to solve BUU equation was proposed by Bertsch, Kruse, and Das Gupta in their study of heavy-ion collisions [45]. This method—with further modifications, which were necessary when different systems of the particles were considered—is still actually used by most sophisticated, modern BUU models as, e.g., so-called Giessen BUU (GiBUU) model [46]. According to this method each nucleon is represented by group of N so-called test particles. Their spatial positions at beginning of the process have to reproduce the density distribution of the target nucleus [47]:    r − R0 ρ (r ) = ρ0 / 1 + ex p a

(5)

with ρ0 = 0.168 nucleon/fm 3 , R0 = 1.124 A1/3 fm, a = 0.025A1/3 + 0.29 fm. The initial momentum distribution is homogenous in the Fermi sphere of radius p F (r ) dependent on the local spatial density ρ(r):  p F (r ) =

3π 2 ρ(r ) 2

1/3 (6)

The test particles representing the proton impinging on to the target nucleus are randomly distributed in the cylinder whose axis is parallel to the proton beam momentum and goes through the center of the target nucleus. Its radius is equal to the maximal expected impact parameter, and the height is very small. The mean field U(r) in which the target nucleons are moving is constructed by adding the Yukawa term to the Skyrme force, and (for protons) the Coulomb potential: 





4/3



 ex p(−μ|r − r |) 

ρ r + Vcoul  μ|r − r | (7) with A = −141.62 MeV, B = 165.23 MeV, V0 = −378 MeV, μ = 2.175 fm−1 , ρ0 = 0.168 fm−3 . Due to such a parameterization, the mean field is unambiguously

U (r) = A

ρ(r) ρ0

+B

ρ(r) ρ0

+ V0

d 3r



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determined in each moment of time t by density distribution ρ(r) which, in turn, is calculated as the weighted average of phase space function f over N*A(t) test particles present in the nucleus of atomic mass A(t). The test particles move as classical particles in the mean field U between collisions occurring only between test particles representing different √ nucleons when the smallest distance between two test particles is smaller than σtot /π (as in the intranuclear cascade models). Pauli blocking is implemented in the BUU model by requiring that only collisions leading to empty or partially occupied final states are allowed (similar to intranuclear cascade models). In summary, this method of solving the BUU equation is very similar to that of the intranuclear cascade. The main differences are as follows: 1. In the BUU model, the time-dependent mean field U is used instead of the static field of INC. 2. Calculation of the phase space function f , the density ρ and the mean field U proceeds in constant time steps (not only for collisions of nucleons, as it is the case at INC). The first of these two main differences allows BUU models to account for collective effects, which is not possible in the INC model. However, this is done at the expense of following the fate of N times larger number of test particles in BUU than the A + 1 nucleons in INC. This is necessary to determine the time dependence of the nucleon density ρ and the mean field U . The time required to solve the BUU equation is therefore larger than that required for the model INC, in particular because the time steps between collisions in INC may be larger than the time steps required to follow the time dependence of the density and mean field in BUU (second point above).

3.1.3

Quantum Molecular Dynamics (QMD)

Presentation of the QMD (Quantum Molecular Dynamics) model, the assumptions introduced and the method of solution may be found in review article of Aichelin [48]. The extension of the model to take into consideration relativistic effects is discussed by Bass and collaborators in another paper [49]. This last version of the model is called Ultrarelativistic QMD (UrQMD). To explain the main ideas of QMD we follow the description of the older realization of QMD model— so-called Jaeri Quantum Molecular Dynamics (JQMD) by Niita et al. [50]. The group of nucleons participating in the reaction is represented by the product of single particle wave functions φi (r, p, t) taken as Gaussians what leads to the following form of the wave function of the group: φ (r, p, t) =

 i

φi (r, p, t) =

 i

  i 1 (r − Ri (t)2 ) + p.r ex p − (2π L)3/4 4L 

(8)

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where L determines width of the Gaussian (in JQMD L = 2fm2 ) and Ri (t)) corresponds to position of i th nucleon. The next step consists in transforming this wave function to create so-called Wigner density f (r, p, t) (see, e.g., [51]): f (r, p, t) ≡

 i

f i (r, p, t) =

 i



2L(p − Pi (t))2 (r − Ri (t))2 − 8ex p − 4L 2

 (9)

In the above formula the vector Pi (t) denotes the momentum coordinate of the center of wave packet representing i th nucleon in the momentum space. Initial positions of the nucleons in the coordinate space Ri (t = 0) and in the momentum space Pi (t = 0) are sampled in an analogous manner as in the BUU model described above, i.e., the coordinates are sampled to reproduce Woods-Saxon distribution with radius parameter R = 1.124 A1/3 − 0.5 fm and diffuseness a = 0.2 fm. To exclude very large distances between nucleons the tail of Woods-Saxon distribution is cut at Rmax = 1.124 A1/3 whereas to decrease fluctuations of the spatial density the relative distance between nucleons of the same kind is limited to values larger than 1.5 fm and between neutrons and protons to values larger than 1.0 fm. The knowledge of local density of nucleons allows to evaluate local Fermi momentum p F (Ri ) and then to sample randomly the centre of the wave packet Pi (t = 0) in the sphere of the radius equal to this local Fermi momentum. To accept the sampled set of nucleons as representative of the ground state of the nucleus additional conditions have to be fulfilled: The sum of kinetic energy of nucleon and its potential energy (derived from density dependence potential) cannot be positive for bound nucleon. Furthermore, the binding energy per nucleon of the nucleus should not deviate from the experimental binding energy more than 0.5 MeV. The time evolution of positions of nucleons in the coordinate and momentum space is determined by Newtonian equations: ∂H dR(t) = dt ∂Pi ∂H dPi (t) = dt ∂Ri

(10)

however with the additional condition for the possibility of collisions, which are allowed to take √ place when the minimal distance dmin between two nucleons is smaller than σ/π (σ is the total nucleon-nucleon cross section). Elastic and inelastic collisions (with creation of  and/or N ∗ resonances) are taken into consideration. The Pauli blocking of collisions is introduced in the same manner as in BUU model, i.e., by introducing the factor [1 − f (r, p, t) ]. As can be seen the JQMD has many features in common with the BUU model, but it also differs in many details. For example, one attractive feature of JQMD that BUU lacks is the ability to generate nucleon clusters. On the other hand, the computer calculations of this model can be cumbersome, since many conditions must be controlled to ensure correct physical interpretation of the obtained results and to

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eliminate numerous non-physical samples of interacting nucleons. More detailed information on QMD can be found in the publication [50] (for JQMD) and in the reviews [48, 49].

3.2 Models of the Second Step of the Reaction As mentioned earlier, the two-step model of spallation reactions assumes that after a fast cascade of nucleon-nucleon collisions, the heavy remnant of the cascade reaches thermodynamic equilibrium. This excited remnant can emit nucleons and/or nuclei in various processes. In many studies it has been found that the main mechanisms of such emission are: (1) evaporation of particles, (2) fragmentation—sequential or simultaneous, i.e., multifragmentation, and (3) fission. They may occur for the same group of excited nuclei, but separate subgroups of excited remnants (differing in excitation energy, mass, or some other properties) may favor a particular deexcitation mechanism. In the following, we discuss the main mechanisms of emission of particles from the second stage of the spallation process and the models used to describe them.

3.2.1

Evaporation

The evaporation of neutrons and light charged particles, i.e., hydrogen and helium isotopes not heavier than the alpha particle, is the main mechanism for deexcitation of the equilibrated remnant nucleus of the fast cascade of nucleon-nucleon collisions into proton-nucleus collisions. Such a process does not usually involve large angular momenta, especially when confined to light target nuclei. In this case, the formalism of Weisskopf and Ewing [52, 53], which evaluates the cross section for the evaporation of particles using the density of states of the decaying (parent) nucleus and the final (daughter) nucleus, and the cross section for inverse reaction, may be successfully applied. It was shown by Furihata [54] that her Generalized Evaporation Model (GEM2), which uses this formalism with a simple parameterization of appropriate densities of states as well as the cross sections is able to successfully describe the total cross sections for the evaporation of 7 Be from reactions involved by protons with energies of 100 MeV—3 GeV interacting with 16 O, 27 Al, nat Fe, and 93 Nb targets. However, this model was found to be unable to reproduce differential cross sections for the emission of particles with mass number A heavier than 9 in p+Ag collisions at Ep = 480 MeV [55]. The same conclusion emerged from the analysis of the total isotopic cross sections measured for p+Xe collisions at proton energy of 1 GeV [56]. The same Weisskopf-Ewing formalism is used to describe the evaporation of particles from the excited compound nucleus by another commonly used model, ABLA07 [57]. However, it should be emphasized that ABLA07 extends the Weisskopf-Ewing formalism to include a random sampling of the angular momen-

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tum change in the evaporation process from a Gaussian distribution with specifically chosen parameters. It is worth noting that the data from last mentioned reactions, i.e., proton-induced spallation on Xe nuclei at 1 GeV [56], were significantly better reproduced by ABLA07 than by GEM2. Other popular models (GEMINI++ [58] and SMM [59]) use the Hauser-Feshbach formalism to describe particle evaporation, which explicitly accounts for spin effects. The Hauser-Feshbach method [60], which describes the evaporation of particle i from the excited compound nucleus with excitation energy E ∗ and spin JC N , uses the following formula for the partial decay width i :

∗

i E , JC N

1 = 2πρC N (E ∗ , JC N ) ×

J +Si

 d

∞ 

JC N +Jd

Jd =0 J =|JC N −Jd |

Tl ( )ρd (E ∗ − Bi − , Jd )

(11)

l=|J −Si |

where Jd is the spin of the daughter nucleus, Si , l and J are the spin, orbital, and total angular momenta of the evaporated particle; and Bi are its kinetic and separation energy, respectively. Tl is the transmission coefficient of the barrier whereas ρC N and ρd are densities of states of the compound nucleus and the daughter nucleus, respectively. Details concerning the determination of transmission coefficients and densities of states in GEMINI++—a typical model using the above formalism, may be found in Ref. [61] (Fig. 8). It is interesting to note that GEMINI++ and SMM (which also uses the Hauser-Feshach formalism) were able to reproduce the total izotopic cross sections from p+Xe collisions at Ep = 1 GeV [56] quite well, similar to what ABLA07 did, while GEM2 did not achieve the comparable quality of data reproduction. This analysis was performed with the same model used to describe the fast stage of the reactions. This may suggest that spin effects are important in the evaporation of particles from excited remnant nucleus of the first stage of the spallation reactions.

3.2.2

Fission

While nucleon evaporation is very likely for highly excited remnants of the fast cascade of nucleon-nucleon collisions, the probability of this process decreases with decreasing excitation energy of this nucleus and with its mass. This is because the postulated thermodynamic equilibrium requires a uniform sharing of excitation energy among all nucleons, i.e., the energy of each nucleon decreases with decreasing total excitation energy and with the number of nucleons in the excited nucleus. Moreover, equilibrium requires a distribution of excitation energy among all degrees of freedom of the excited group of nucleons, i.e., the collective deformations of the excited remnant nucleus associated with fission participate in the absorption of the available energy (Fig. 9). In such a situation, i.e., for heavy nuclei (with mass num-

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Fig. 8 The figure above (taken from [56]) illustrates quality of reproduction of izotopic cross sections σ (A,Z) from p+136 Xe collisions at proton energy 1 GeV. Black dots represent the data whereas predictions of INCL++ coupled to GEM2, to ABLA07, to SMM, and to GEMINI++ are represented by yellow dots, green dashed line, red solid line, and dash-dotted blue line, respectively

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Fig. 9 The above figure (taken from ref. [62]) illustrates the quality of reproduction of the experimental total production cross section σ (A) as a function of the mass number A of the products from p + Pb reaction at Ep = 1 GeV. The data are shown as black dots, while the histograms show the predictions of the INCL4.5 + GEMINI++ models for different versions of the parameter values. The fission is responsible for the production of the residual nuclei with mass number A in the interval 50 < A < 130

ber A > 100) at low excitation energy, fission becomes a significant competitor of particle evaporation. Therefore, all models of the second stage of spallation reactions induced by protons on heavy target nuclei take into consideration the fission process. All of them start from the Bohr-Wheeler model of the fission process [63] based on the liquid-drop model of the nucleus, where the fission width is described by the formula:    1 (12) d ρ f E ∗ − B(J ) − , J BW = ∗ 2πρn (E , J ) where the symbols ρn and ρ f denote the density of states of the excited nucleus with excitation energy E ∗ and spin J in the ground state and in the saddle point, respectively, and B(J) represents the spin-dependent fission barrier. Individual models of the second stage of the spallation reaction such as ABLA07 [57], GEMINI++ [58], SMM [59], GEM2 [54] use some individual recipes for height of the fission barier and other parameters of the above formula.

3.2.3

Multifragmentation

Both particle evaporation and nuclear fission are based on the picture of a long leaving compound nucleus, i.e., a set of nucleons in equilibrium created during the first stage of the spallation reaction. While this may be true at small energies shared by these nucleons, such an assumption may not be valid for high excitation energies, e.g., above 3 MeV/nucleon. This is because then each nucleon or group of nucleons escaping from a highly excited compound nucleus may violate the equilibrium and

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cause further emission. Such a picture of sequential violation of equilibrium can be treated as a sequential fragmentation, which in fact leads to multifragmentation. The picture of sequential fragmentation is used in the GEMINI++ model of Charity [58, 61]. Let us briefly summarize the formalism used in GEMINI++ for sequential fragmentation. The emission of the lightest particles (with Z not greater than 4) is treated according to the Hauser-Feshbach evaporation formalism. The partial decay widths for the emission of heavier fragments leading to asymmetric division of the compound nucleus were taken from the generalized transition state formalism of Moretto [64], while for heavy nuclei where symmetric division predominates, the Bohr-Wheeler method was used [63]. A different picture of the fragmentation of highly excited compound nucleus is also considered. It takes into account the fact that the excitation energy of the residuum of the first stage of the reaction is so high that equilibrium can be reached only for subsets of the nucleons, and then they are emitted simultaneously and appear as products of simultaneous fragmentation. This last picture may be considered as the equivalent of a phase transition that occurs when the temperature of the nuclear matter reaches the critical value. The simultaneous fragmentation is realized in the SMM model of Botvina et al. [59, 65]. This model assumes that the thermalized residual nucleus of the first stage of the proton-nucleus collision undergoes a statistical breakup. At first, the nucleus expands to a certain volume and then breaks up into nucleons and hot fragments. All possible breakup channels are considered. The probability w j of a given decay channel j of the nucleus excited to energy E ∗ is proportional to the exponential function of entropy S j (E ∗ ), which depends (besides the excitation energy) also on other parameters of the system:   W j ∼ ex p S j (E ∗ )

(13)

The model treats the formation of a compound nucleus as one of the decay channels. This allows the transition from evaporation at low energies to multifragmentation at high excitations based on the available phase space. It is assumed that at the breakup time the nucleus is in thermalized equilibrium characterized by the channel temperature T. The light fragments with mass number A ≤ 4 and atomic number Z ≤ 2 are treated as structureless particles, i.e., they have only translational degrees of freedom. The heavier fragments are considered as heated droplets of a nuclear liquid, so their individual free energies are parameterized according to the liquid-drop model, i.e., they are equal to the sum of the bulk, surface, Coulomb, and symmetry energies. The Coulomb interaction between all fragments is taken into account via the Wigner-Seitz approximation. The breakup channels are simulated using the Monte Carlo method according to their statistical weighting. After the system is broken up, the fragments propagate independently in their mutual Coulomb fields and undergo secondary decays. The deexcitation of large fragments (with a mass number greater

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than 16) is described by the evaporation-fission model, whereas that of smaller fragments is described by the Fermi breakup model. A detailed description of the model and its parameters can be found in Ref. [65].

4 Validation of Spallation Models Validation of the theoretical model consists in checking whether the model is able to reproduce the experimental data. It is obvious that the model works well if the values of the cross sections predicted by the model match the experimental cross sections. However, the experimental data are always biased by statistical errors, which are present even when all systematic inaccuracies are removed. This is because the fact that the experimental cross sections are proportional to the number of events collected in the experiment that satisfy the appropriate conditions, e.g., number of particles detected at a certain angle, number of particles with a given energy, etc. Since the number of events is a random variable with Poisson probability distribution, the experimental cross sections of particular effects are also random variables with appropriate probability distributions. In such a situation, it is obvious that the good model cannot reproduce the experimental data exactly. Then the question arises when it can be said that the value of the experimental cross section is satisfactorily close to the value predicted by the model. Moreover, such a comparison usually concerns not only one experimental cross section of action, but a set of them, e.g., the angular or energy distribution of the cross sections. The solution to this problem, i.e., the validation of the theoretical model, consists in the quantitative estimation of whether the difference between the set of experimental cross sections and the set of corresponding theoretical values is small enough. For this purpose, a function called deviation factor is usually used, which estimates an average value over the distances between individual experimental cross section and the corresponding theoretical cross section, and allows to check if this value is close enough to its expected value. This procedure is possible if the expectation value of the deviation factor and its standard deviation are known under the condition of equality of the expectation value of the experimental and theoretical cross sections. Unfortunately, such a probability distribution is not known for most of the deviation factors used so far. Therefore, it is usually checked whether the observed value of a given deviation factor is close to its so-called reference value, i.e., such a value of the deviation factor that occurs when all model cross sections are equal to the corresponding experimental cross sections. As mentioned above, due to the statistical nature of the experimental cross sections, this can occur only in very special situations. The following effects should be taken into consideration when using the deviation factors:

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1. The contributions of individual cross sections to the deviation factor must not cancel themselves, i.e., the overestimation and the underestimation of data by model cross sections should contribute to the deviation factor in the same way. As an example the deviation factor R EC defined as below (see, e.g., Ref. [66]) does not have this desirable property: R

EC

N ex p 1  σi = N i=1 σicalc

(14)

The equality of R EC to unity (expected for ideal agreement of data and model cross sections) may be achieved accidentally when too large and too small (in respect to appropriate data) theoretical cross sections participate to the above sum. 2. It should be noted that the deviation factor which uses the experimental errors in weights of individual experimental cross sections overestimates contributions of the large cross sections. This is because the experimental cross section is proportional to number of registered experimental events whereas the statistical error to square root of this number. Therefore the contributions of individual cross sections to the deviation factor are scaled proportionally to square root of number of registered events. This may be observed, e.g., for popular H-deviation factor (for its properties see Ref. [67]):  H≡

1/2 N  ex p 1  σi − σicalc ex p N i=1 σi

(15)

The A—deviation factor, proposed in Ref. [68]:  N  ex p 1  σi − σical  A= N i=1 σiex p + σical

(16)

is free of these two above-mentioned negative properties: (1) It is symmetrical in respect to exchange of the theoretical and experimental cross sections therefore the overestimation and underestimation of the data by theoretical cross sections influences the A-deviation factor in the same manner, and (2) The large value of the experimental cross section does not cause that its contribution to the A-factor is larger than that of the small experimental cross section. The same is true for theoretical cross sections. Whereas the probability distribution function of the A-factor is not known, value of this factor has very intuitive interpretation: • It measures the average absolute value of distance between the data and model cross sections in units of the sum of them. • Its values belong to the interval [0, 1].

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• Vanishing of A-factor corresponds to equality of all theoretical cross sections to appropriate experimental data, i.e., the reference value of the A-factor is A = 0. • When A is close to zero the experimental and theoretical cross sections are almost equal and A represents half of the absolute value of the average relative distance of the model and experimental cross sections. • The unit value of the A-factor corresponds to the largest possible distance between theoretical and experimental cross sections which appears when (a) all theoretical cross sections vanish, (b) they become infinite. The following situation occurs frequently: There are several models competing in describing the available data, but none of them is able to reproduce them perfectly. Then, a less ambitious task may be solved, which consists in ranking the models. It is assumed that the model whose predictions give the smallest distance of the given deviation factor from the reference value of this deviation factor is superior to the other models. The A-deviation factor discussed above seems to be very useful in such a ranking procedure (see, e.g., ref. [68]).

5 Summary In this chapter, the main facts about the long history of experimental and theoretical studies of proton-induced spallation reactions have been discussed. Their importance in various fields of science, medicine, and technology was mentioned. The very wide range of applications of these reactions requires a good understanding of their mechanism, both qualitatively and quantitatively. Whereas the general picture of their mechanism seems to be known, there are still unanswered questions concerning the interpretation of the observed facts. They concern, for example, the mechanism of emission of intermediate-mass fragments (IMF), i.e., nuclei heavier than alpha particles but lighter than fission fragments. The observed facts suggest that IMFs are emitted in both the fast and slow phases of the reaction. While the fast process can be understood qualitatively as emission from hot, moving sources, the general, microscopic description of this process has yet to be determined. Another question to which a convincing answer has not yet been found is the competition between the sequential emission of particles in the second stage of the spallation process, as postulated by the GEMINI++ model, and the simultaneous emission of many fragments, as assumed by the SMM model. Since these mechanisms do not exclude themselves, we would like to point out that the more involved, coincidence measurements of the spallation reactions seem to be necessary to solve the problem. We emphasized at the end of the chapter that the assessment of the quality of the data description, and thus the decision on the selection of appropriate models, must be based on some quantitative measures—deviation factors. It is surprising that the selection process often uses factors that are biased by the statistical properties of the experimental data and their errors, as well as factors that can lead to incorrect

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conclusions because the contributions of overestimation and underestimation of the data by the models cancel each other out. We have presented the deviation factor that is not biased by the above shortcomings.

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