Spectroscopic measurements of the helium atom are performed to high precision using an atomic beam apparatus and electro-optic laser techniques. These measurements, in addition to serving as a test of helium theory, also provide a new determination of the fine structure constant α. An apparatus was designed and built to overcome limitations encountered in a previous experiment. Not only did this allow an improved level of precision but also enabled new consistency checks, including an extremely useful measurement in 3He. I discuss the details of the experimental setup along with the major changes and improvements. A new value for the J = 0 to 2 fine structure interval in the 23P state of 4He is measured to be 31 908 131.25(30) kHz. The 300 Hz precision of this result represents an improvement over previous results by more than a factor of three. Combined with the latest theoretical calculations, this yields a new determination of α with better than 5 ppb uncertainty, α-1 = 137.035 999 55(64).

WesDyne TRC är ett helägt dotterbolag till kärnkraftsteknologiföretaget Westinghouse Nuclear och arbetar främst med oförstörade provning och visuella inspektioner av reaktortankar. Inom kärnkraftsindustrin finns ett starkt intresse och behov av att kunna utföra noggranna geometriska uppmätningar inuti en reaktortank. I detta arbete utfördes först en förstudie där lämpliga tekniker undersöktes för att finna den mest lämpade för ändamålet. Den teknik som ansågs ha störst potential var aktiv triangulering med laser. Efter vidare fördjupning inom området utvecklades och konstruerades en prototyp med tillhörande programmering, som därefter testades och verifierades. Detta i både labbmiljö som i en anpassad tilltänkt applikationsmiljö, vilket i detta fall innebar mätning under vatten i mörker. Mätningar av objekt som EDM gropar, svetsfogar och liknande nyttjades för att utvärdera tekniken. Vid mätningar under vatten presterade tekniken en hög upplösning, ner mot några tiotals mikrometer, och en avvikelse på ungefär 4-6 % från det nominella måttet i djupled. I planet var avvikelsen mycket lägre, kring 1 %. Mätmetoden anses ha mycket stor potential och repeterbarhetstester antyder att en repeterbarhet ner mot upplösningsnivå råder. Vid de tester som har utförts har en upplösning på ca: 5, 38 och 10   μm i de tre riktningarna x, y och z erhållits, där z är djupled. Potential till ytterligare förbättring finns i y-riktningen, vilken vid testerna begränsats av laserns fokusavstånd. Med prototypen har även mycket små avvikelser och tendenser i ytan på mätobjekten registrerats. Detta har varit bl.a. stansade märken, slagmärken, små sprickor, cirkulära fräsmönster samt repor som ej kan uppfattas vid beröring. Vid mätning av en plan, slät, och matt yta var mätbruset mindre än 50 μm.Avvikelsen i djupled kan liknas med en icke linjär expansion som tycks övergå till en svag kompression vid ett ökat avstånd till mätobjektet i bilden. Under arbetet har orsaken till denna expansion/kompression ej lyckats fastställas. Bättre och noggrannare kalibreringsmetoder samt föreslagna förändringar i konstruktionen anses kunna förbättra noggrannheten så till vida att den understiger 1 % i alla riktningar. / WesDyne TRC is wholly owned subsidiary company of the nuclear technology company Westinghouse Nuclear and primarily works with non destructive testing (NDT) and visual inspections of reactor tanks. Within the nuclear industry there exists a strong interest in making accurate geometrical measurements inside a reactor tank. In this work a feasibility study was done where suitable technologies were examined to find the most appropriate for this purpose. The technique that was considered most likely to be successful was active triangulation with laser. After further elaboration in the area of interest a prototype was developed and constructed along with programming, which then was tested and verified. These tests were executed in both a lab environment as in an environment appropriate to the intended application, which in this case involves measurement under water in absence of light. Measurements of items such as EDM pits, welds and the like were utilized to evaluate the technology. In measurements performed under water the technology performs a high resolution, down to a few tens of microns, and a deviation of about 4-6 % from the nominal dimension in depth. The deviation in the plane was much lower at around 1 %. Measurement procedures are considered to have great potential and repeatable tests suggest that repeatability down to the level of resolution is achievable. In the performed tests a resolution of approximately 5, 38 and 10 microns in the three directions x, y and z are obtained, where z is depth. Potential for further improvement in the y direction exists. In these tests the limitation has been the focusing distance of the laser beam which prevents a closer scanning distance. With the prototype measurement and detection of very small deviations in the surface of measured object was possible. This has been such punch labels, dents, small cracks, circular milling patterns and scratches so small that they cannot be perceived by touch. When a flat, smooth, dull surface was measured the noise was less than 50 microns. The discrepancy in depth can be compared with a non-linear expansion that seems change in to a slight compression when the distance to the object in the picture increases. In this thesis the reasons for the expansion/compression has not been determined. Better and more accurate calibration methods, and proposed changes to the design, is most likely to improve the accuracy in the sense that it is less than 1 % in all directions.

We perform high precision measurements of an isotope shift between the two stable isotopes of helium. We use laser excitation of the 2^3 S_1-2^3 P_0 transition at 1083 nm in a metastable beam of 3He and 4He atoms. A newly developed tunable laser frequency selector along with our previous electro-optic frequency modulation technique provides extremely reliable, adaptable, and precise frequency and intensity control. The intensity control contributes negligibly to overall experimental uncertainty by stabilizing the intensity of the required sideband and eliminating the unwanted frequencies generated during the modulation of 1083 nm laser carrier frequency. The selection technique uses a MEMS based fiber switch and several temperature stabilized narrow band (~3 GHz) fiber gratings. A fiber based optical circulator and an inline fiber amplifier provide the desired isolation and the net gain for the selected frequency. Also rapid (~2 sec.) alternating measurements of the 2^3 S_1-2^3 P_0 interval for both species of helium is achieved with a custom fiber laser for simultaneous optical pumping. A servo-controlled retro-reflected laser beam eliminates residual Doppler effects during the isotope shift measurement. An improved detection design and software control makes negligible subtle potential biases in the data collection. With these advances, combined with new internal and external consistency checks, we are able to obtain results consistent with the best previous measurements, but with substantially improved precision. Our measurement of the 2^3 S_1-2^3 P_0 isotope shift between 3He and 4He is 31 097 535.2 (5) kHz. The most recent theoretic calculation combined with this measurement yields a new determination for nuclear size differences between 3He and 4He: ∆r_c=0.292 6 (1)_exp (8)_th (52)_exp fm, with a precision of less than a part in 〖10〗^4 coming from the experimental uncertainty (first parenthesis), and a part in 〖10〗^3 coming from theory. This value is consistent with electron scattering measurement, but a factor of 10 more precise. It is inconsistent (4 sigma) with a recent isotope shift measurement on another helium transition (2^1 S_0-2^3 S_1). Comparisons with ongoing muonic helium measurements may provide clues to the origin of what is currently called the proton puzzle: electronic and muonic measurements of the proton size do not agree. In the future, the experimental improvements described here can be used for higher precision tests of atomic theory and quantum electrodynamics, as well as an important atomic physics source of the fine structure constant.

As a part of our program to test the unitarity of the Cabibbo-Kobayashi-Maskawa matrix, the decay of the superallowed 0⁺ --> 0⁺ beta emitters ³⁸Ca and ⁴⁶V has been studied in this dissertation. For ³⁸Ca, the half-life, 443.88(36) ms, and superallowed branching ratio, 0.7738(41), have been measured. In our half-life experiment, pure sources of ³⁸Ca were produced and the decay positrons detected in a high-efficiency 4[pi] proportional gas counter. Since the beta⁺ decay of ³⁸Ca feeds ³⁸K^m, which is itself a superallowed beta⁺ emitter, the data were analyzed as a linked parent-daughter decay. Our result for the half-life of ³⁸Ca, with a precision of 0.08%, is a factor of five improvement on the best previous result. The branching-ratio of ³⁸Ca depended on beta-delayed gamma-ray intensities being measured with a high-purity germanium detector calibrated for absolute efficiency to 0.2% precision. This branching-ratio result represents our first step in bringing the ft value for the superallowed ³⁸Ca transition into the desired range of 0.1%. With our half-life and superallowed branching ratio results for ³⁸Ca, we obtain the Ft to be 3072(17) s, in good agreement with the conserved vector current expectation. The half-life of ⁴⁶V has been measured to be 422.66(6) ms, a factor of two more precise than the best previous measurement. Our present result determines the corresponding Ft value to be 3074.5(26) s, which is consistent with the average $\overline{\mathcal{F}t}$ value of 3072.08(79) s established from the 13 best-known superallowed transitions. This demonstrates that previously accepted half-lives of ⁴⁶V were correct in their contribution to a precision test of the conserved vector current hypothesis.

Im Jahr 1991 wurde erstmals die Interferenz von Atomen experimentell nachgewiesen. Seitdem wird dieses Phänomen in vielen Bereichen der Grundlagenforschung angewendet, unter anderem zur Bestimmung von Naturkonstanten mit bisher unerreichter Genauigkeit oder für Tests des Äquivalenzprinzips. Grundsätzlich können auch geophysikalische Vermessungen des Schwerefeldes der Erde von dieser neuen Technik profitieren, allerdings waren Atominterferometrie-Experimente aufgrund ihrer Komplexität bisher nur in Laboren möglich. Erst kürzlich wurde mit der Entwicklung mobiler Atominterferometer begonnen, die nun die hochpräzise Messung von Rotationen, Gravitationsgradienten sowie der absoluten Schwerebeschleunigung außerhalb von Laboren ermöglichen. Im Rahmen dieser Arbeit wurde ein absolutes Gravimeter entwickelt, konstruiert und getestet. Es basiert auf Rb87-Atomen, die in einer Vakuumumgebung gefangen, gekühlt und senkrecht entgegen der Erdanziehung beschleunigt werden. Während des anschließenden freien Falls werden die atomaren Ensembles durch drei Raman Lichtpulse aufgespalten und rekombiniert. Die lokale Schwerebeschleunigung kann aus den resultierenden Interferenzmustern bestimmt werden, die abhängig von der Bewegung der Atome in einem Gravitationspotential sind. Wir haben den Wert der lokalen Schwerebeschleunigung, g, mit einer Auflösung von 1 : 10^10 bei einer Integrationszeit von 12 Stunden vermessen. Dies entspricht 2,2 * 10^-7 m/s^2/Sqrt(Hz). Mit dieser Genauigkeit konnten bereits zeitliche Veränderungen des lokalen Schwerefeldes registriert werden, hervorgerufen durch eine Vielzahl an Effekten wie Erd- und Ozeangezeiten oder atmosphärischen Variationen. In einem Vergleich unter ähnlichen Messbedingungen konnte unser Instrument die lokale Schwerebeschleunigung mit einer um fast eine Größenordnung höheren Genauigkeit bestimmen als ein herkömmliches Gravimeter. / Since 1991, matter wave interferometry has been used in many laboratories for a variety of fundamental physics experiments, e.g. measurement of the fine-structure and gravity constants or equivalence principle tests. This new technique is also ideally suited for high-accuracy geophysical gravity measurements. However, due to the complexity of these experiments they were so far confined to laboratory environments. Only in recent years efforts have been undertaken to develop mobile atom interferometers. These new sensors now open up the possibility to perform on-site high-precision measurements of rotations, gravity gradients as well as absolute accelerations. This work reports on the design, construction and first tests of an absolute gravimeter. It is based on interfering ensembles of laser cooled Rb87 atoms in a one meter high atomic fountain configuration. Local gravity is measured by applying three Raman light pulses while the atoms are in free fall, thereby splitting and recombining the atomic wave packets. The resulting interference fringes are sensitive to the movement of the atoms within a gravitational potential. We have measured the value of local gravity g at a resolution of one part in 10^10 at an integration time of 12 hours, or 2.2 * 10^-7 m/s^2/Sqrt(Hz). This was high enough to be sensitive to a number of time varying gravity effects like tides, ocean loading or changes in gravity caused by air pressure. In a comparison under similar measurement conditions, the instrument has surpassed the performance of conventional mobile gravimeters by almost one order of magnitude.

Atom Interferometrie ist eine sehr genaue und sensitive Methode mit einer Vielzahl von Anwendungsmöglichkeiten, zu der auch die Messung der Erdbeschleunigung zählt. Während die meisten Atom Interferometer aus großen, ortsfesten Aufbauten bestehen, werden auf diesem Gebiet häufig mobile Messgeräte benötigt. Das Gravimetric Atom Interferometer (GAIN) Projekt wurde ins Leben gerufen, um dieser zusätzlichen Anforderung bei bestmöglicher Messgenauigkeit gerecht zu werden. Es soll eine Alternative zu anderen modernsten Gravimetertypen geschaffen werden, die wichtige funktionale Eigenschaften wie eine hohe Auflösung und absolute Genauigkeit in einem Gerät vereint. Der GAIN Sensor verwendet lasergekühlte Rb87 Atome in einer 1 m hohen Fontäne. Mit Hilfe von stimulierten Raman Übergängen wird ein beschleunigungssensitives Interferometer realisiert. In dieser Arbeit wurde der Sensor mit Blick auf mobile und driftfreie Langzeitmessungen weiterentwickelt. Dafür wurden einzelne Subsysteme des Laseraufbaus auf die daraus resultierenden Anforderungen hin angepasst oder neu entwickelt. Mit derselben Zielstellung wurden weiterhin systematische Effekte in dem Messaufbau untersucht und Maßnahmen für ihre Reduzierung realisiert. Der Aufbau wurde transportiert und in relevanten Umgebungen getestet. Dabei konnte gezeigt werden, dass die Leistungsfäigkeit dieses Aufbaus mit denen der wichtigsten und modernsten Gravimeter konkurieren kann, sie teilweise übertrifft und dass dieser Sensor zur präzisen Kalibrierung der relativen Gravimeter verwendet werden kann. In den Messungen wurde eine Sensitivität von 138 nm/s^2/Sqrt(Hz) sowie eine Langzeitstabilität von 5 x 10^−11 g über 10^5 s erreicht. / Atom interferometry offers a very precise and sensitive measurement tool for various areas of application whereof one is the registration of the gravity acceleration. While the vast majority of atom interferometers include large and stationary setups, this field very often implies the additional request for a mobile apparatus. The Gravimetric Atom Interferometer (GAIN) project has been started to meet this requirement and to provide best possible accuracy at the same time. It aims to realize an alternative to other types of gravimeters and to combine important qualities such as high sensitivity and absolute accuracy in one instrument. The GAIN sensor is based on laser-cooled Rb87 atoms in a 1 m atomic fountain. Stimulated Raman transitions form a Mach-Zehnder type interferometer which is sensitive to accelerations. In this work it has been advanced to meet all requirements for mobile and drift-free long-term operation. Therefore, selected parts of the laser system have been improved or redeveloped. A second focus has been on systematic effects for the same objective. They have been analyzed and measures for their suppression have been undertaken. The apparatus has been transported, tested in relevant environments, and compared to the most important state-of-the-art gravimeter types where a competitive performance has been achieved. It is demonstrated, that the gravity signal of this sensor allows for a precise calibration of the relative gravimeter types. During the measurements a best sensitivity of 138 nm/s^2/Sqrt(Hz) and a stability of 5 x 10^−11 g after 10^5 s has been reached.

Das gravimetrische Atominterferometer (GAIN) ist ein transportables Atominterferometer welches spezifisch für hochpräzise Schweremessungen in der Geodäsie und Geophysik entwickelt wurde. Es basiert auf einer Rubidium Atomfontäne, stimulierten Ramanübergängen und einer 3-Puls Mach-Zehnder Interferometriesequenz. Die vorliegende Arbeit beschäftigt sich mit der Optimierung und Anwendung von GAIN als transportables Gravimeter für Absolutschweremessungen an geodätischen Observatorien welche über den aktuellen Stand der Technik hinaus gehen. Dabei wurden eine Absolutgenauigkiet von 29 nm/s^2, eine Langzeitstabilität von 0.4 nm/s^2 sowie eine Sensitivität von 82 nm/s^2 in einer Sekunde erreicht. Die gemessene Genauigkeit und Langzeitstabilität stellen, nach dem Wissen des Authors, die bis heute besten publizierten Werte für ein transportablen Atominterferometer dar und repräsentieren einen bedeutenden Fortschritt im Bereich der Gravimetrie. Um dies zu erreichen wurden umfangreiche Verbesserungen am Gerät umgesetzt und eine ausführliche Analyse der systematischen Messabweichungen durchgeführt. Unter anderem wurden ein System zur Kompensation von Corioliskräften und Ausrichtungsfehlern, ein verbessertes Schwingungsisolationssystem zur nachträglichen Korrektur von Umgebungsvibrationen und eine magnetische Abschirmung instrumenteller Streufelder implementiert. Darüber hinaus wurden insgesamt vier Messkampagnen in Berlin, sowie an den geodätischen Observatorien in Wettzell, Deutschland und Onsala, Schweden durchgeführt, um GAIN mit anderen hochmodernen Absolut- und Relativgravimetern zu vergleichen. Der direkte Vergleich zwischen GAIN und anderen Gravimetern stellt den prinzipbedingten Vorteil der Atominterferometrie durch die Kombination aus Absolutgenauigkeit, Stabilität und Langzeitbetrieb klar hervor. Dies wurde in der Arbeit durch die um einen Faktor 2-5 verbesserte Kalibrierung des Skalenfaktor von zwei supraleitenden Gravimetern demonstriert. / The gravimetric atom interferometer (GAIN) is a transportable setup which was specifically designed to perform high-precision gravity measurements at sites of interest for geodesy or geophysics. It is based on a Rb atomic fountain, stimulated Raman transitions and a three-pulse Mach-Zehnder atom interferometry sequence. The presented work is concerned with the optimization and application of GAIN as a transportable gravimeter in order to perform gravity measurements beyond the state-of-the-art. An absolute accuracy of 29 nm/s^2, long-term stability of 0.4 nm/s^2 and short-term noise level as low as 82 nm/s^2 in one second was achieved. The obtained long-term stability and accuracy values are, to the knowledge of the author, the best published performance of any transportable atom interferometer to date and represent a significant advancement in the field of gravimetry. A comprehensive analysis of the systematic error budget was performed to improve the accuracy and stability of the measured gravity value. Several setup improvements were implemented to this end, including Coriolis force and alignment control systems, an improved vibration isolator with post-correction and magnetic shielding which reduces spurious coupling due to stray fields. Measurement campaigns were conducted in Berlin and at geodetic observatories in Wettzell, Germany, and Onsala, Sweden, in order to compare GAIN to other state-of-the-art absolute and relative gravimeters. The direct comparison of GAIN to other absolute and relative gravimeters shows the general advantage of atom interferometers due to their unique combination of absolute accuracy, stability and robust architecture enabling continuous measurements. This was demonstrated during the presented campaigns by the improvement of the scale factor calibration of two superconducting gravimeters by a factor 2 to 5 using GAIN data.

Atomic parity violation measurements provide a way to probe physics beyond the Standard Model. They can provide constraints on conjectures of a massive Z′ bosonor a light boson, or searches of dark energy. Using the two-pathway coherent control technique, our group plans to make a new measurement of the weak interaction induced parity non-conservation (PNC) transition moment (<i>E_PNC</i>) on the cesium 6S→7S transition. We will coherently interfere a 2-photon transition with the Stark and PNC transitions to amplify and extract the PNC amplitude. Previously, our lab has measured the magnetic dipole transition moment on the same 6S→7S transition to about 0.4% uncertainty using this technique. In this dissertation, I discuss improvements made to the system, and review what future upgrades are needed for a new<i> E</i>_{<i>PNC </i>}measurement. Key systematics are also described. For an accurate determination of <i>E_PNC</i>, properties of cesium such as the scalar (<i>α</i>) and vector (<i>β</i>) transition polarizabilities are needed. I present improved determinations of keyelectric dipole matrix elements, and calculate new high precision determinations of <i>α</i> and <i>β</i>. Finally, using <i>β</i> and the previously measured value of <i>E_PNC/β</i>, I calculate new values for the weak charge of the cesium nucleus Q_w.<br>

In dieser Dissertation werden drei Messungen des Produktionswirkungsquerschnitts von Top-Quark-Antiquark-Paaren in Proton-Proton-Kollisionen bei einer Schwerpunktsenergie von 7 TeV vorgestellt. Die Daten wurden mit dem ATLAS-Experiment am Large Hadron Collider in den Jahren 2010 und 2011 aufgezeichnet. Für die Analyse werden Endzustände mit genau einem Myon oder Elektron, mindestens drei Jets sowie großem fehlenden Tranversalimpuls selektiert. Während eine Analyse ausschließlich kinematische Informationen für die Trennung von Signal- und Untergrundprozessen verwendet, nutzen die anderen beiden zusätzlich Informationen zur Identifizierung von Bottom-Quark-Jets. Mit Hilfe von multivariaten Methoden werden die präzisesten Messungen in dieser Ereignistopologie erreicht. Dies ist für zwei der Analysen insbesondere dank der Profile-Likelihood-Methode möglich, welche sorgfältig untersucht wird. Desweiteren wird zum ersten Mal ein sogenannter sichtbarer Wirkungsquerschnitt in Top-Quark-Ereignissen gemessen. Alle Ergebnisse sind in Übereinstimmung mit den theoretischen Vorhersagen in angenäherter nächstnächstführender Ordnung der Störungstheorie (approx. NNLO). / This doctoral thesis presents three measurements of the top quark-antiquark pair production cross section in proton-proton collisions at a centre-of-mass energy of 7TeV recorded in 2010 and 2011 with the ATLAS Experiment at the Large Hadron Collider. Events are selected in the single lepton topology by requiring an electron or muon, large missing transverse momentum and at least three jets. While one analysis relies on kinematic information only to discriminate the top quark-antiquark pair signal from the background processes, the other two also make use of b-tagging information. With the help of multivariate methods the most precise measurements in this topology are obtained. This is for two of the measurements in particular possible due to the use of a profile likelihood method which is studied in detail. For the first time a fiducial inclusive cross section measurement for top quark events is performed allowing a measurement almost independent of theoretical uncertainties. All measurements are in agreement with theory predictions performed in perturbation theory at approximate NNLO.

The subject of this thesis is the study of coherent interaction of light with matter (atoms) to improve the precision measurements and techniques. Special attention is drawn to get the narrow subnatural electromagnetically induced transmission (EIT), electromagnetically induced absorption (EIA) and nonlinear magneto-optic rotation (NMOR) caused by alkali atoms contained in a vapor cell. Subnatural polarization rotation introduces by a strong circularly polarized light in the absence of any external magnetic field was also studied. A detailed theoretical treatment, given in this dissertation, allows to associate each of the features of the spectra with a special physical mechanism. Many quantum phenomena related to interferences, coherences, optical pumping etc. experiments are studied using home-built diode lasers. This thesis also describes laser cooling and trapping of rubidium atoms using two techniques. Deflection of cold atoms horizontally from MOT using pushing beams are discussed in close consideration for the improvements in the precision measurements. This thesis is organized as follows. In Chapter 1, an introduction to the importance of sub-natural narrow resonance and simplified technique in the precision measurement will be discussed. In Chapter 2, an introduction to EIT, EIA and NMOR resonance are discussed. This chapter will provide a basic theoretical background of atom-field interactions, especially for Λ-type and N -type systems and its steady state solution using density matrix analysis and experimental tools. The most important notion of laser cooling of atoms, ions or molecule i.e., exchange of momentum between light and atoms combining with the Doppler effect will be discussed. In Chapter 3, the observation of subnatural EIT and subnatural EIA in closed and open degenerate two-level system using room temperature vapor cell filled with Rb will be presented. Physical mechanisms that contribute to EIT and EIA, and the contrast of our results from the coherent population trapping (CPT)–type resonances are discussed in detail in appendix A.2 according to our experimental results. In Chapter 4, the narrowing of subnatural EIT and subnatural EIA linewidth in closed and open systems again in degenerate two level transition, using the “Laguerre-Gaussian” control beam instead of generally studied Gaussian beam, will be discussed in detail. In Chapter 5, the conversion between subnatural EIT to subnatural EIA in a degenerate Λ system will be discussed. The physical mechanism created by the introduction of a coherent counter propagating control laser to the co-propagating probe and the control laser are studied. The effect of polarization and axial velocity Doppler averaging will also be studied. In Chapter 6, we will discuss the sensitive technique for precise measurement of small magnetic fields using the NMOR, by chopping the resonant laser beam. We will study the sensitivity and the potential application of this technique in the measurement of an atomic electric-dipole moment. In Chapter 7, we will be studying about the induced optical rotation by a circularly polarized control laser on the linearly polarized probe laser. The effect of the intensity of the control laser beam on the higher order optical rotation will also be studied. In Chapter 8, we will be studying about the cooling and trapping of 87Rb in magneto-optic trap. We will be studying two techniques of trapping of atoms in MOT. The cold cloud of atoms from the MOT are deflected horizontally by using different configuration of pushing beam are studied. A brief summary and outlook of my thesis work will be discussed at Chapter 9.