The Makeup of Matter PDF

1 The Makeup of Matter 1.1 Elements, a Concept with a Long History Material objects are one of the main parts of the physical world we live in. Ranging from natu- rally occurring rocks, oceans or trees to an endless number of artifacts craft...

1 The Makeup of Matter 1.1 Elements, a Concept with a Long History Material objects are one of the main parts of the physical world we live in. Ranging from natu- rally occurring rocks, oceans or trees to an endless number of artifacts crafted by humans, ancient thinkers already tried to reduce this enormous diversity to different combinations of a small set of elements. Aristotle summarized this line of natural philosophy by distinguishing four earthly elements, fire, earth, water and air, and the aether as the invariable element stars were made of. As illustrated in Figure 1.1a, these four elements were characterized as either hot or cold and dry or wet, and substances differed in the relative proportions of each element they contained. Obtaining copper or iron by heating an ore, was then described in terms of adding fire. Similar descriptions were found across the world and persisted in Europe through the middle ages. Even in the renaissance, Aristotle’s scheme still inspired emerging chemists, such as Paracelsus, who described medicines as different combinations of three principles, mercury, sulfur and salt. An alternative view on the material world was expressed by the atomists, mostly Leucippus and Democritus. The theory of Democritus held that everything is composed of indivisible buil- ing blocks, so-called atoms, which fill empty space. These atoms are indestructible, but are in permanent motion and can be rearranged all time. Moreover, Democritus expressed different ideas as to how atoms could have different sizes and shapes, and could be linked together to a b Figure 1.1: (a) Illustration of Aristotle’s description of the material world in terms of the four classical elements and their connecting properties. (b) The 100 Greek drachmas banknote, featuring Democritus as a pioneer of the atomic theory. 2 Chapter 1. The Makeup of Matter Figure 1.2: Excerpt of the Table of Nomenclature published by Antoine Lavoisier showing the first six undecomposable substances he distinguished. Note the listing of light and caloric, next to the actual elements oxygen, hydrogen, nitrogen and carbon. Note Lavoisier’s proposing new scientific names, and his identification of, for example, combustion products such as water, nitrous oxide or carbon dioxide and the corresponding acids these gases yield when dissolved in water. form larger objects. While his atomistic theory was not based on any experimental observation and was linked to Aristotle’s notion of elements, the framework of matter and void proposed by Democritus is part of the modern description of mechanics, as introduced by Isaac Newton. In recognition of his pioneering work in natural philosophy, Democritus was represented on the 100 Greek drachma bank note, see Figure 1.1b. From our current perspective, the ancient theories have many flaws. No convincing method was ever put forward to determine the fractions of the four elements in a given substance, let alone obtaining these supposed elements in pure form and combining them on purpose to form new substances. In fact, after the experimental approach to natural philosophy took hold in Europe in the 17th century, it was quickly realized that these ancient ideas had little practical value and they were replaced by modern descriptions, rooted in experiment and verified by their predictive capabilities. Even so, the modern theories of matter remained close to the ancient ideas that material can be understood as made up of indivisible building blocks, and that each building block is one of a limited set of elements. 1.2 The Atomic Theory and the First Chemical Revolution 1.2.1 Pure Elemental Substances and Conservation of Mass In 1787, Antoine Lavoisier published the Méthode de nomenclature chimique together with five co-authors. In this treatise, the classical system of four elements was discarded and replaced by a list of substances that could, according to the authors, not be further decomposed in other known substances. As shown in Figure 1.2, this list was still a highly diverse set, containing sub- stances that were effectively pure elemental substances, such as hydrogen, nitrogen, oxygen and various metals, but also metal oxides, compounds such as ammonia, and immaterial phenomena such as light and caloric, the supposed materialization of heat. Probably the real importance of this work is not so much this – often erroneous – list of non-decomposable substances, nor the introduction of new names for some of these substances. Rather, it was the notion, rooted in experiments, that such pure elemental substances existed, and that most substances are pure compounds or mixtures that can be separated into or prepared out of these pure elemental sub- stances. In this way, Lavoisier summarized in a consistent way a range of experimental findings made by contemporary chemists such as Henry Cavendish and Joseph Priestley. A finding that greatly supported Lavoisier’s interpretation was that of the conservation of mass, which he and the Russian scientist Mikhail Lomonosov formulated independently. By 1.2 The Atomic Theory and the First Chemical Revolution 3 Table 1.1: Illustration of the law of multiple proportions using the oxides of sulfur. Compound Sulfur (g) Oxygen (g) Ratio Current molecular formula light oxide 1.0 1.0 2 SO2 heavy oxide 1.0 1.5 3 SO3 carefully weighing the initial products, and keeping track of the type and weight of final prod- ucts during the combustion of phosphorus, sulfur or carbon, or during fermentation reactions, Lavoisier thus paraphrased the statement of the Greek philosopher Anaxagoras that Nothing is born or perishes, but already existing things combine, then separate again. While the law of the conservation of mass does not provide insight in the nature of the combining and separating things, it finds a direct explanation in the concept of pure elemental substances, and the separation or formation of compounds out of these pure elemental substances. Note that the conservation of mass implies, in fact, that heat or light – possible results of a chemical reaction – have no connotation with mass. 1.2.2 The Law of Multiple Proportions The notion of pure or undecomposable elemental substances as formulated by Lavoisier was a major step for the development of chemistry, especially since reactions involving pure elemental substances yielded deep insight in the building blocks these substances comprise. An intermedi- ate step in this development was the law of multiple proportions, that was put forward by John Dalton in 1804. In this law, Dalton generalized observations already reported by Pelletier and Proust on reactions between the same two elemental substances that can form different com- pounds. Oxygen and copper, for example, can form a black and a red oxide of copper, which we would now describe as CuO and Cu2 O. What Dalton realized, is that in all such cases, the varying weights of the one pure elemental substance which combine with a fixed weight of the other pure elemental substance must be in a simple, numerical relationship to one another. In the case of the two copper oxides, for example, the weight of oxygen that will react with a given weight of copper to form the black or the red oxide will have a ratio 1:2; a point that seems trivial from the way we now express compounds and reactions, but that was far from obvious at the end of the 18th century. Example 1.1: The oxides of sulfur An illustration of the law of multiple proportions follows from the reaction of sulfur and oxygen, two pure elemental substances put forward by Lavoisier. These can react to form two dierent oxides of sulfur, where a rst requires less oxygen for a given amount of sulfur. Expressing these amounts in grams, see Table 1.1, one nds that the amounts of oxide that react in either case have a ratio 2:3; a result in line with the notion of a simple, numerical relationship. 1.2.3 The Atomic Theory of Dalton On the face of it, the law of multiple proportions is nothing more than a generalization of ex- perimental observations on chemical reactions. While building on the idea of pure elemental substances, it is in principle independent of an atomistic description of the composition of such 4 Chapter 1. The Makeup of Matter substances. Still, Dalton realized that his observation of multiple proportions would be a straight- forward result of a description of matter as made up of elemental building blocks. In the same year 1804, Dalton therefore postulated six principles that form a first, experiment-based atomic theory: 1. All matter consists of indivisible particles called atoms. 2. Atoms of the same element are similar in shape and mass, but differ from the atoms of other elements. 3. Atoms cannot be created or destroyed. 4. Atoms of different elements may combine with each other in a fixed, simple, whole number ratios to form compound atoms. 5. Atoms of same element can combine in more than one ratio to form two or more com- pounds. 6. The atom is the smallest unit of matter that can take part in a chemical reaction. While several aspects of Dalton’s atomic theory were later proven wrong – atoms can be divided in sub-units, elements can have different isotopes, atoms of different elements can have the same mass, etc. – this first atomic theory was hugely consequential. Dalton’s postulates not only provided a first set of concrete ideas about the structure of matter, including the point that weight is a central characteristic of atoms, but also linked this atomistic view on matter with chemical reactions between substances. While one could argue that such ideas were already formulated by ancient philosophers, the essential difference is that Dalton’s postulates generalize Figure 1.3: Examples of Dalton’s scheme to represent atoms of different elements, and examples of some compound atoms or molecules. 1.3 Organizing the Elements 5 a set of experimental data, and provide a starting point for further experimental verification. In first instance, such studies involved the determination of the weight of the elements, and the composition of so-called compound atoms, which we nowadays describe as molecules. Example 1.2: Dalton's pictorial description of atoms and molecules Next to his postulates, Dalton provided a list of 36 elements whose atoms he represented through pictograms, some of which are shown in Figure 1.3. While Dalton's system was not widely adapted, what is still of interest is his way to de- pict compound atoms or molecules with the composition he thought was correct. As can be seen in Figure 1.3, his representation of water is incorrect, implying a composition we would now write as HO. His view on the composition of carbon monoxide and carbon dioxide, on the other hand, was correct. Knowing that a xed amount of carbon can react with dierent amounts of oxygen, with a weight ratio of 1:2, a description of these two oxides as CO and CO2 , as Dalton pre- sumed, is the most simple way to account for these multiple proportions through atomic theory. For water or ammonia, where Dalton made a similar error, such multiple compounds were absent and Dalton merely assumed a 1:1 ratio. 1.3 Organizing the Elements 1.3.1 Avogadro's Hypothesis, Molecular Formulas and Relative Atomic Weights While Dalton’s postulates were a major step forward in the development of the atomistic un- derstanding of chemistry, they were by no means an endpoint. Two immediate issues that were apparent from Dalton’s own publications were the determination of atomic weights and the sto- ichiometry of molecules. As can be seen in Figure 1.3, for example, Dalton described water as composed of oxygen and hydrogen, a point already made by 18th century chemists such as Cavendish and Priestley, but imagined a single water molecule as consisting of 1 oxygen and 1 hydrogen atom. The problem here is that, while Dalton knew that 1 gram of hydrogen would react with 8 gram of oxygen to form 9 gram of water, these ratios could not be turned into a molecu- lar stoichiometry unless the relative atomic weights of hydrogen and oxygen were known. But determining these weights required knowledge of the stoichiometry of compounds in the first place. Dalton could break this circle for elements that formed multiple compounds according to his law of multiple proportions, but not for other compounds. Many relative atomic weights and molecular stoichiometries he proposed were therefore proven wrong, see Figure 1.4. In 1811, Amadeo Avogadro hypothesized that equal volumes of gaseous substances, whether simple or compound, contain an equal number of molecules. What is meant here by molecules Figure 1.4: Examples of binary compounds proposed by Dalton, including (left to right) water, ammonia, nitrous gas (nitrogen monoxide), ethylene and carbon monoxide. Three out of five proposed compositions later proved erroneous. 6 Chapter 1. The Makeup of Matter Figure 1.5: Representation of the equivalent volumes of hydrogen and oxygen needed to form 1 equiv of water. According to Avogadro’s principle, similar gas volumes hold a similar number of elemental units, i.e.. single atoms or molecules. Combining this principle with the known masses of reagents and products, molecular stoichiometries can be determined through experiments. is individual, separate building blocks, which could be individual atoms or molecules consist- ing of multiple atoms. While Avogadro already used this principle to determine relative atomic weights, his ideas were discarded by Dalton and left dormant, mostly because chemists strug- gled to make the difference between atoms and molecules, in particular when describing pure elemental substances. In 1858, Stanislao Cannizzaro outlined this distinction in his paper Sunto di un Corso di Filosofia Chimica, in which he reformulated Avogadro’s principle and used this principle to propose molecular stoichiometries and relative atomic weights. Applied to water, Cannizzaro’s argumentation runs as follows. As shown in Figure 1.5, it is known that 2 gram of hydrogen and 16 gram of oxygen will react to form 18 gram of water. Looking at the volumes of these gaseous compounds, however, one finds that for a given volume of hydrogen – which we will call 1 equivalent – only half the equivalent volume of oxygen is needed, while an equivalent volume of water is produced. Assuming the units in hydrogen and oxygen gas are single molecules, the equivalent volumes would translate into the following chemical equation: 1 H + O → HO (1.1) 2 Although yielding Dalton’s proposed composition of water, already Lavoisier would dismiss this expression since the amount of oxygen in the reaction is not conserved. To do better, we should interpret the different equivalent volumes of the reagents as reflecting the final stoichiometry of the product. Assuming that the building blocks of pure gaseous substances are single atoms, a first attempt to do so would read: 2H + O → H2 O (1.2) This reaction, however, predicts that the volume of water formed, is only half the equivalent volume of hydrogen. This is not what is seen in experiments. This problem can be solved by the assumption that the building blocks of both gaseous hydrogen and oxygen are not the individual atoms, but the diatomic molecules H2 and O2. Based on that idea, the formation of water can be written down as: 1 H2 + O2 → H2 O (1.3) 2 One readily sees that this relation conserves the number of atoms, and correctly represents the equivalent volumes of hydrogen, oxygen and water used and produced in the reaction. As a final 1.3 Organizing the Elements 7 Table 1.2: Relation between the number of atoms of hydrogen involved in a reaction, the equiva- lence of hydrogen to form one equivalent of product and the supposed stoichiometry of the hydrogen molecule. Combinations in grey have not been observed in reality. Compound Number of Atoms Equiv Compound Equiv Number of Atoms H2 1 0.5 H4 1 0.25 2 1 2 0.5 3 1.5 3 0.75 4 2 4 1 point, one should realize that other stoichiometries than expressed by Eq 1.3 are in line with these two principles. Insight 1.1: The problem of redundant stoichiometries The stoichiometry expressed by Eq 1.3 complies with conservation of mass and Avogadro's principle. One could, however, write an equally valid equation in terms of molecules like H4 , O4 and H4 O2. Cannizarro settled on compositions such as H2 , N2 or O2 based on experimental data that covered a wide set of gas reactions. Let us, for example, compare the formation of H2 O and NH3. To form 1 equiv of water, it takes 1 equiv of hydrogen and half an equiv of oxygen. On the other hand, forming 1 equiv of ammonia requires 1.5 equiv of hydrogen and half an equiv of nitrogen. Note that we use here equiv in the sense of equivalent volumes. These equivalences lead to a range of possible stoichiometries for water, such as H2 O, H4 O2 , H6 O3 ,.... Similarly, for ammonia, one has NH3 , N2 H6 , N3 H9 ,.... What is important here, is that the 0.5 equiv dierence of hydrogen needed to form water or ammonia, stands in these dierent stoichiometries for 1, 2 or 3 hydrogen atoms more in the nal compound, see Table 1.2. Hence, as outlined in Table 1.2, if hydrogen were correctly represented by H4 , reactions should possibly be found where 0.25 or 0.75 equiv of hydrogen forms 1 equiv of product. No such reactions were known to Cannizzaro, and neither have they been discovered since. Hence the deductive step to propose, amongst others, H2 , N2 and O2 as respective stoichiometries for hydrogen, nitrogen or oxygen. Using Avogadro’s principle, Cannizzaro proposed a slate of molecular stoichiometries, and relative atomic and molecular weights for gaseous compounds. As shown in Figure 1.6, he did so using atomic hydrogen as a weight reference and expressed molecular stoichiometries through a combinations of letters and numbers. Moreover, he applied the same approach to determine the atomic weight of metallic elements. Here, he did not use Avogadro’s principle but relied on the so-called law of Dulong and Petit, which stated that all metals have the same specific heat capacity when counted per atom, rather than per weight or per volume. Hence, by 1860, chemists had formulated the notion of pure elemental substances, had identified many elements that actually were pure elemental substances and had a reliable list of the relative weight of these different elements. What was missing, however, was a systematic link between this one physical characteristic of the elements, and the chemistry of these elements. 8 Chapter 1. The Makeup of Matter Figure 1.6: Example of a table with molecular stoichiometries and weights published by Stanisloa Cannizzaro. Note the close-to-modern notation of letters and numbers used to express composition and stoichiometry, and the erroneous prediction of the stoichiometry and weight of ozone. 1.3.2 The Avogadro Constant The formulation of Avogadro’s principle immediately leads to the question as to how many atoms or molecules a given volume of gas contains. However, macroscopic materials comprise such an immense number of atoms that counting atoms one-by-one is not realistic. Chemists therefore settled on the mole as a unit to count atoms. The mole has been defined in multiple ways, such as the number of atoms in 16 gram of oxygen-16, or in 12 gram of carbon-12. In 2017, however, this quantity, which is known as the Avogadro constant and often indicated as NA , was simply given a fixed numerical value called Avogadro’s number: NA = 6.02214076 1023 As a result, the number of atoms in 12 gram of carbon-12 is no longer exactly equal to Avogadro’s number, but still close enough to use that description. Insight 1.2: Estimating the atomic size Avogadro's number leads to a straightforward estimate of atomic sizes. Take, for example, diamond, which is a crystalline form of pure carbon. Diamond has a known density of 3.52 g/cm3. Given carbon's molar mass of 12 g/mol, this density corresponds to a molar volume Vm of 3.41 cm3 /mol. Hence, the volume vat a single atom occupies within the diamond lattice is given by: Vm vat = = 5.85 10−30 m NA Such a volume corresponds to a cube with an edge of 0.18 nm, a gure that provides a reasonable estimate for typical atomic dimensions. 1.3 Organizing the Elements 9 c secnatsid dne fo noitcarF 0.3 0.2 0.1 0.0 0 5 10 15 20 2 µm End distance ( µm) Figure 1.7: (a) Examples of trajectories of pigment particles as observed by Perrin. (b) Overview the end points of all trajectories recorded by Perrin, where the center of the circle was always chosen as the starting point. (c) Distribution of end points from Perrin’s observation, grouped in bins with a width of 2 µm, together with (red line) the distribution predicted by Einstein. (a) and (b) and the data plotted in (c) are taken from Perrin’s 1909 paper Mouvement Brownienne et Réalité Moléculaire. The possibility of counting atoms is inherent to accepting atomic theory. The first determi- nation of Avogadro’s number by Jean Perrin in 1909 was therefore a true milestone that ushered in the general acceptance of atomic theory by the scientific community. Perrin, who defined Avogadro’s constant as the number of atoms in 16 gram of oxygen-16, made his estimate of the Avogadro constant using a description developed by Einstein of Brownian motion. Brownian motion is the random movement small particles – such as micron-sized pigments – exhibit when dispersed in a liquid. This motion is driven by the continuous collisions between liquid molecules and the particle. What Perrin did, is monitor the trajectories of such particles and determine the statistical distribution of the distance r between the start and end point of a trajectory. Figure 1.7a shows an example of a few trajectories, while Figure 1.7b represents all trajectories analyzed by Perrin by means of their end point plotted relative to the center of a circle. The random nature of a single trajectory is clearly visible, and so is the randomness in the end points. In 1906, Albert Einstein had developed a theory that described Brownian motion as a random walk, that resulted in an expression for the distribution of end points that was characterized by a mean square displacement ⟨r2 ⟩ as given by: RT t ⟨r2 ⟩ = NA 3πηd Here, R is the gas constant, T temperature, η the viscosity of the liquid, d the diameter of the particles and t the observation time. As can be seen in Figure 1.7c, Perrin’s observations were in very good agreement with the distribution of distances predicted by Einstein. Moreover, as the only quantity unknown to Perrin was NA , this correspondence between experiment and theory enabled him to obtain a numerical value for Avogradro’s constant. His best estimate amounted to: NA = 7.05 1023 This outcome is about 15% larger than the current definition. 1.3.3 The Periodic System of the Elements The knowledge available to chemists at around 1860 involved a list of 63 elements, including the relative atomic weight for most of these, and an extensive set of chemical reactions that were 10 Chapter 1. The Makeup of Matter translated in reaction equations with the appropriate stoichiometry. While scientists such as John Newlands noticed the periodicity of chemical properties throughout the set of elements, a first consistent system to organize the elements was first provided by Dmitri Mendeleev in 1869 and Julius Lothar Meyer in 1870. In fact, Meyer was the first to draft this periodic table in 1866, but did not publish his findings before 1870. The central insight of Mendeleev was to prioritize chemical properties over weight to or- ganize the elements. To understand what Mendeleev did, one can consider the metal hydrides as an example. Starting from lithium, it was known that reaction with hydrogen would form a compound, lithium hydride, with the molecular formular LiH: 1 Li(s) + H2 (g) → LiH(s) (1.4) 2 Taking beryllium, the next element by weight, the same reaction yields a dihydride, rather than a monohydride: Be(s) + H2 (g) → BeH2 (s) (1.5) When running down the list of elements, this sequence of stoichiometries repeats itself with sodium and magnesium, and with potassium and calcium. Therefore, Mendeleev proposed that elements such as lithium, sodium and potassium, or beryllium, magnesium and calcium, belong to the same group. Organizing each group as a vertical column, this led him to propose the pe- riodic system represented in Figure 1.8, which is characterized by successive horizontal periods, i.e., sequences of elements exhibiting similar changes in chemical properties and ordered by the weight of the elements. The order of many of the elements in Mendeleev’s periodic table, such as the second (Li to F) and the third (Na to Cl) period, are retained in the modern version of the periodic system. While attesting to the impressive step Mendeleev took in organizing the elements, this agreement at Figure 1.8: Representation of the periodic system of the elements as proposed by Dmitri Mendeleev, listing each elements symbol and relative atomic weight. Note the organisation of the elements with comparable chemical properties in vertical groups, while the periodic appearance of these properties gives rise to different periods organized by the weight of the elements within a period. 1.4 The Internal Structure of Atoms 11 large with our current periodic systems also makes the deviations more notable. For one thing, by sticking to an 8-fold period, Mendeleev struggled with the position of what we currently know as the transition metals. He put, for example, vanadium in the same group III as aluminium and introduced a group VIII as a basket for multiple elements. Moreover, completely absent from his periodic system are the noble gases, which would only be discovered at the end of the 19th century by William Ramsay, while Mendeleev also left gaps in his table. These correspond to elements, such as scandium, gallium and germanium, unknown at the time. Mendeleev, however, predicted their existence based on the underlying periodicity of the elements, and even proposed approximate atomic weights of 44, 68 and 72 for these three examples. Obviously, the subsequent discovery of, in particular, gallium (1875), scandium (1879) and germanium (1886), with atomic weights close to the expected, was a major validation of the idea behind the periodic system. Even so, what Mendeleev’s periodic system did not provide, was an explanation of the periodicity of the elements. This step was only made after scientists obtained a detailed insight in the internal structure of atoms. Before we go there, however, let us see how the scientific community as a whole finally accepted the ideas of the atomic theory that chemists had been developing since Dalton. 1.4 The Internal Structure of Atoms 1.4.1 The First Step  the Discovery of the Electron Dalton’s atomic theory mentions weight as the only quantifiable property in which atoms of dis- tinct elements will differ. As outlined in §1.3, relative atomic weights were used to determine the stoichiometry of molecules, and initiate the periodic ordering of the elements. However, to understand rather than describe the chemistry of the elements, the atomic weight proofed less interesting as a property and scientists looked for different ways to characterize atoms and un- derstand their inner structure. For this, they would mostly harness new experimental approaches involving electricity, magnetism and radiation. Joseph J. Thomson made the first step in the analysis of the internal structure of atoms in 1897. As outlined in Figure 1.9, Thomson used a setup consisting of a set of vacuum tubes. A first chamber contained a metal electrode and a set of plugs with aligned slits, while a second chamber held two parallel electrodes and ended with a phosphorescent screen. What Thomson Figure 1.9: Reproduction of the drawing published by Joseph Thomson of the set of tubes used to generate, deflect and analyze so-called cathode rays, depicted here in blue. Thomson rightly inter- preted cathode rays as a stream of electrons emitted by the cathode plate C as a result of an applied voltage difference between the cathode and the plug A. The slits in the sequence of plugs A and B serve to create a directional ray that can be deflected by applying a voltage difference between the electrodes D and E. The ray is detected by the spot it leaves on the phosphorescent screen at the end of the tube. 12 Chapter 1. The Makeup of Matter Table 1.3: Currently proposed masses and charges for the electron, the proton and the neutron and, where relevant, the mass-over-charge ratio. Mass m (10−27 kg) Charge q (10−19 C) m/q (10−12 kg · C−1 ) electron (e− ) 9.1093837 10−4 −1.60217663 −5.68563 proton (p+ ) 1.67262192 1.60217663 10439.7 neutron (n) 1.67493 0 – showed is that by applying a negative voltage to the metal electrode relative to the first plug in the first chamber, so-called cathode rays could be generated that would leave an imprint on the phos- phorescent screen. Moreover, by applying a voltage difference across the pair of electrodes in the second chamber, Thomson found that these cathode rays were attracted to a positive electrode as indicated in Figure 1.9. He therefore concluded rightly that what was actually emitted was a stream of negatively charged particles. While measuring the mass or the charge on individual par- ticles was beyond the capabilities of his setup, Thomson could determine the mass-over-charge ratio m/q of the particles from the deflection of the ray as caused by electric or magnetic forces. Since he obtained the same ratio m/q, regardless of the material the electrode was made off, or the trace gas used in the vacuum tube, Thomson hypothesized that the particles making up the cathode rays were a fundamental constituent of any atom, which were later called electrons. The idea that atoms of different elements comprise the same sub-atomic particles had been formulated already before Thomson’s study on cathode rays. However, Thomson’s results en- abled him the relate this hypothesis for the first time to a particle that was effectively observed as being emitted by different pure elemental substances alike. Knowing that atoms had no net electric charge, Thomson therefore tried to describe atoms as consisting of negatively charged electrons, embedded in a body of positive charge. This so-called plum-pudding model could, however, predict little atomic properties and was given up completely after Ernest Rutherford first probed the internal structure of the atom in 1911. Example 1.3: The mass-over-charge ratio of the electron From his experiments, Thomson's best estimate of the mass-over-charge ratio of the electron was, in absolute values, ≈ 10−11 kg · C−1. This gure was about 1000 times smaller than the mass-over-charge ratio of the hydrogen ion, the smallest known at the time. Although Thomson had no knowledge of the elementary charge, let alone the charge on the electron or the hydrogen ion, he argued that this 1000-fold dierence was largely due to the very low mass of the electron, a conclusion later studies proofed correct, see Table 1.3. Note that current values for the electron's mass-over-charge ratio of amount to −5.68563 10−12 kg · C−1. 1.4.2 The Atomic Structure According to Rutherford By the beginning of the 20th century, several materials had been discovered that emitted different types of radiation, conveniently called α, β and γ rays. The phenomenon of radioactive decay raised deep questions about the make up of matter; α rays were found to consist of positively charged particles, later identified as helium nuclei, β rays were shown to be a stream of electrons identical to Thomson’s cathode rays while γ rays appeared immaterial, carrying no mass nor 1.4 The Internal Structure of Atoms 13 c 5 10 π ) 2( 3 10 θN N/) ( 1 10 -1 10 0 π/4 π/2 3 π/4 π Deflection angle Figure 1.10: (a) Scheme outlining Rutherford’s scattering experiment, depicting the incoming beam of α particles, i.e., helium nuclei, the gold foil, the deflected α ray and the deflection angle θ. (b) De- tailed depiction of the deflection as a result of the Coulomb repulsion between an incoming α particles and a gold nucleus. (c) Comparison of (full line) the predicted and (markers) the expermentally mea- sured variation of the number of deflected particles as a function of the deflection angle, normalized relative to full back scattering. Note the logarithmic scale in the vertical axis. The data represented were published by Geiger and Marsden in 1913. The inset shows the well-known pictorial representa- tion of an atom, which is clearly inspired by Rutherford’s planetary model. charge. On the other hand, the rays emitted by radioactive elements also provided researchers with a new set of tools to probe the structure of the atom. This approach was pioneered by Ernest Rutherford, who used the deflection of α rays upon passing through thin metal foils as depicted in Figure 1.10a to develop a new understanding of the atomic structure. Rutherford’s discovery of the atomic structure resulted from a combination of experimen- tal data – mostly acquired by Hans Geiger and Ernest Marsden in 1909 – and predictions of the experimental outcome starting from an atomistic model. What he imagined, unlike Thom- son’s plum-pudding description, is that the positive countercharge needed to balance the negative charge of the electrons was all centered in a tiny, massive atomic nucleus. Taking that atomic structure and the small mass of the electrons as Thomson proposed for granted, the deflection of an incoming, positively charged α particle would be mostly caused by the repulsive Coulomb interaction between the α particle and the atomic nucleus, see Figure 1.10b. By considering the Coulomb repulsion between an incoming particle with mass m and charge Zα e and an infinitely heavy nucleus with charge Znuc e, with e the elementary charge, Rutherford calculated the number of particles N(θ ) deflected by an angle θ as: 2 Zα Znuc e2  Jα N(θ ) = (1.6) 8πε0 mv2α sin4 θ2 In this expression, the so-called Rutherford cross-section, ε0 is the permittivity of the vacuum, a known constant from Coulomb’s law, while vα is the incoming speed of the α particle and Jal pha represents the number of α particles hitting the metal foil per unit area. This formula contains multiple interesting points: Angular dependence. If deflection occurs because of a point-like interaction, the number of deflected α particles should drop as a function of the deflection angle as 1/ sin4 θ2. As shown in Figure 1.10c, such a relation implies a dramatic drop of deflection events with increasing angle, which can be readily compared with experimental data. 14 Chapter 1. The Makeup of Matter Charge dependence. Having the nuclear charge Znuc e in the numerator of Eq 1.6, deflection will be more pronounced for more heavy nuclei. Moreover, when the experiment is truly quantitative, the nuclear charged can be obtained from the number of deflected particles. Backscattering. The deflection at full backscattering, i.e., θ = π, is not zero. Hence, according to Rutherford, some α particles may simply bounce back from the metal foil, rather than passing through. The observation of such backscattering events was later de- scribed by Rutherford as quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. Figure 1.10c compares the angular dependence of deflected α particles as predicted by Rutherford’s formula, with experimental data obtained by Hans Geiger and Ernest Marsden. Given the variation of N(θ ) by four orders of magnitude across the range of angles considered, the agreement between experiment and theory remains amazing, even today. This result strongly supported Rutherford’s starting idea of an atom as consisting of a massive, point-like, positively charged nucleus at its center that is somehow surrounded by a set of electrons, not unlike the sun is surrounded by a number of planets. Insight 1.3: Rutherford's upper limit for the size of the nucleus Since the backscattering at high angles was still determined by the Coulomb force, Rutherford argued that even backscattered α particles did not touch the actual nucleus. Otherwise, he expected dierent, nuclear forces would be at play, which would yield a dierent distribution of backscattered particles over the deection angle. Hence, when an incoming α particle comes to a standstill in a head on collision with the nucleus, all the initial kinetic energy will be converted into potential energy linked to the Coulomb interaction. This relation can be expressed as: 1 2 1 Zα Znuc e2 mval pha = · 2 4πε0 r0 Here, r0 is the distance of closest approach of the α particle from the center of the nucleus. Knowing all other quantities, Rutherford obtained for r0 a distance of 2.7 10−14 m. He put forward this number as an upper limit to the size of the gold nucleus, which is indeed far smaller than the typical estimate of atomic dimensions, see Ÿ1.3.2. In principle, a quantitative analysis of experimental deflection data using Eq 1.6, should yield the charge Qnuc of the atomic nucleus. For gold foils, Rutherford thus estimated a charge of Znuc ≈ 100 times the elementary charge e. This figure somewhat overestimates the actual number of Znuc = 79, highlighting that the method was not the best at the time for accurately determining nuclear charges and, in our current wording, atomic numbers. However, at the time Rutherford published his results, better methods were already under development to accomplish this. 1.4.3 The Ordering of the Elements Mendeleev ordered the elements in his periodic table through a combination of chemical prop- erties and increasing relative atomic weight, which mostly led to a gradually increasing weight 1.4 The Internal Structure of Atoms 15 along the listed elements. However, a few notable exceptions existed, such as nickel and cobalt, and the relative atomic weight did not increase in a very systematic manner throughout the peri- odic system. This made that atomic weight was not a reliable physical characteristic that could account for the atomic number, a figure at that time simply seen as the place of an element in the periodic table. The work of Henry Moseley on X-ray emission by the elements changed this state-of-affairs. What Moseley observed, is that any element that was irradiated by the newly discovered X-rays, would re-emit a series of X-rays with fixed wavelengths, where he used the labels K and L to indicate those with the two shortest wavelengths, see Figure 1.11. This phe- nomenon is nowadays called X-ray fluorescence, and analyzing X-ray emission lines has become a widely used method for elemental analysis. The relevant aspect of Moseley’s work in view of atomic theory, was that the wavelength or frequency of the X-ray lines elements emitted, changed in a systematic manner as a function of the atomic number. As shown in Figure 1.11b, Moseley found that, within a given line, the atomic number Z could be seen as a linear function of the square root of the frequency: √ Z = a+b ν (1.7) This relation, known as Moseley’s law, led to a listing of all elements in line with the periodic tabel, and provided a means to determine the atomic number for, as yet, undiscovered elements. Even more, using Bohr’s newly developed theory for the hydrogen atom, see §2.3, Moseley argued that the atomic number Z should be interpreted as the charge of the atomic nucleus, measured in units of the elementary charge e. Hence, what Mendeleev’s classification system represented, was a periodicity of chemical properties as a function of the nuclear charge and thus, since atoms are charge neutral, as a function of the number of electrons. 1.4.4 The Problem of Isotopes and the Discovery of the Neutron The identification between the atomic number and the nuclear charge that resulted from Mose- ley’s work, quickly seemed inconsistent with the known atomic weights of the elements. By a b 100 Sample Au 80 rebmun cimotA W L line L-line Eu λ ,ν L L , 60 Ag Ag Mo Mo K-line 40 Zn Fe λ ,ν K K , Ca K line 20 Al Incoming X-ray Re-emitted X-ray 0.5 1.0 1.5 2.0 2.5 9 1/2 Square root of frequency (10 Hz ) Figure 1.11: (a) Schematic representation of Moseley’s experiment, showing the irradiation of an elemental substances by X-rays and the re-emission of a set of X-ray lines – denoted K and L – characterized by fixed wavelengths λ or frequencies ν for a given element. (b) Representation of data published by Moseley in 1914, showing the atomic number as a function of the square root of the frequency of the re-emitted √ X-rays of these K and L lines. The straight lines are best fits of the data to a linear relation a + b ν, a relation known as Moseley’s law. Some specific elements have been highlighted. Note that molybdenum and silver appear both with their K and L line. 16 Chapter 1. The Makeup of Matter 2 )g( thgiew cimotA 100 4 2 A =2Z r 10 4 A =Z r 2 1 2 4 6 2 4 6 1 10 100 Atomic number Figure 1.12: Representation of (markers) the atomic weight Ar as a function of the atomic number Z. The reference lines indicate the hypothetical cases where the atomic weight is equal to the atomic number, or the double of the atomic number. Note the log-log representation of both quantities. 1920, it was understood that the nuclear charge was related to a single atomic constituent, which Rutherford coined the proton. The nucleus of the hydrogen atom, the lightest known element, consisted of a single proton, while higher nuclear charges reflected the presence of multiple pro- tons in the nucleus. However, if this were the ultimate description of the nucleus, than the atomic weight Ar measured relative to hydrogen, should be identical to the atomic number Z. As can be seen in Figure 1.12, it is not. Even worse, the atomic weight of many elements is not a simple multiple of that of hydrogen. In fact, an increase in atomic number can even result in a reduc- tion of the atomic weight, for example when moving from cobalt to nickel, while on average the atomic weight seems to increases more than linear with the atomic number, see Figure 1.12. By 1912, Thomson had started analyzing beams – or rays as he still called them – of positively charged atoms for their mass-over-charge ratio by again studying the deflection of such beams in magnetic fields. Analyzing neon cations, he observed a double imprint. The difference in mass- over-charge ratio was too small to assign this double imprint to doubly-charged neon atoms, so the inevitable conclusion was that two different types of neon atoms existed, a first with a relative mass of 20, and a second with a relative mass of 22. Francis Aston, who started working with Thomson, confirmed this result, and found that many of the elements were similar mixtures of atoms with different masses. Chlorine, for example, was found to consist of atoms with relative masses of 35 and 37, see Figure 1.13. Importantly, the mass of these different, so-called, isotopes was always a multiple of the hydrogen mass. Hence, elemental substances should be seen as mixtures of different atoms that have the same atomic number and identical chemical properties but a different mass. In that case, the known atomic weights correspond to the average weights of these different isotopes, taking their abundance into account; a point that could explain the erratic variation of the atomic weight as a function of atomic number. Even after the discovery of isotopes, the discrepancy remained between the atomic number – 17 for chlorine for example – and the atomic weight of the isotopes – 35 and 37 for chlorine. To account for this difference, Rutherford hypothesized that the nuclei of such chlorine atoms would contain 35 or 37 protons, which were charge-compensated by 18 or 20 nuclear electrons, i.e., electrons he assumed to be confined to the nucleus. While this idea accounted for the differ- ence between atomic number and atomic mass, it ran into major difficulties by 1930, when more properties of protons and electrons – such as spin – had been discovered. In 1932, James Chad- wick showed that beryllium atoms would release an as-yet undiscovered particle when exposed to α rays. He found that these particles were electrically neutral, and had a mass almost identical to the proton mass. Unlike the nuclear electron idea, the notion that the neutron, as this particle 1.5 Talking Atoms Today 17 Figure 1.13: First mass spectra of neon and chlorine as published by Francis Aston in 1920. The red ellipses highlight the imprints of neon-20 and neon-22, and of chlorine-35 and chlorine-37, re- spectively. Imprints at other mass-to-charge ratios are assigned to molecular fragments consisting of hydrogen, oxygen and carbon, the three main contaminants in Aston’s setup. was named, was a constituent of the atomic nucleus that added mass but no charge, provided a fully consistent picture of the atomic and nuclear properties, and its discovery completed the search for the make up of the atom. 1.5 Talking Atoms Today 1.5.1 Denitions and Symbols For scientific purposes, a set of definitions and symbols have been introduced to identify atoms and specify their characteristics. As often, however, the recommended terminology has changed over time, which leads to an often confusing dual use of terms, or the persistent use of obsolete terms. Nevertheless, it is most useful to see how insight in the atomic structure is currently translated into the definitions according to Compendium of Chemical Terminology, informally known as the Gold Book published by the International Union of Pure and Applied Chemistry (IUPAC). According to the IUPAC, a chemical element is a species of atoms, which all have the same number of protons in the atomic nucleus. In parallel, a chemical element is described as a pure chemical substance, composed of atoms with the same number of protons in the atomic nucleus. Hence, one can use the word element either to refer to a specific set of atoms with the same chemical properties, such as oxygen, chlorine or silver, or to describe the notion of a pure ele- mental substance that was already put forward by Lavoisier. So the periodic system proposed by Mendeleev lists the different elements. On the other hand, a single atom is not an element. A nuclide is a species of atom, characterized by its mass number, atomic number and nuclear energy state. As the latter point is less relevant for chemistry, the difference between an element and a nuclide is the specification of the mass number. Hence, while neon is an element, neon-20 is a nuclide. Note that chemists will often use the wording isotope rather than nuclide, which is more common in nuclear physics. In principle, however, IUPAC only accepts the plural isotopes as referring to a set of nuclides having the same atomic number but different mass numbers. Hence, isotopes is often used so as to stress that a given element is a mixture of multiple nuclides. Statements such as the isotopes of carbon, thus involve a reference to the set of nuclides carbon- 12, carbon-13 and carbon-14. An atom is the smallest particle still characterizing a chemical element, characterized by a nucleus of a positive charge Ze, where Z is the proton number and e the elementary charge, carrying almost all its mass (more than 99.9%) and Z electrons determining its size. Atoms are represented through an atomic symbol, which are one, two or three letters used to represent the atom in chemical formulae. As discussed before, the proton number is also called the atomic 18 Chapter 1. The Makeup of Matter a b Ne 10 22 A Ne X 10 Z neon 20.180 Figure 1.14: (a) Example of the element neon, as specified by the atomic number, the atomic symbol and the atomic weight. (b) Representation of (left) the nuclide neon-22 and (right) a nuclide in general, where X refers to the atomic symbol, while A and Z are the atomic mass number and the atomic number, respectively. number. The atomic mass ma is characterized by the atomic mass number A, which is the total number of heavy particles (protons and neutrons jointly called nucleons) in the atomic nucleus. Atomic masses, which are a property of a given nuclide, are often expressed in terms of the unified atomic mass unit u, a quantity defined as one twelfth of the mass of a carbon-12 atom in its ground state: u = 1.6605402(10) 10−27 kg (1.8) The atomic weight Ar of an atom is then defined as the average mass of the atom, accounting for the abundance pi of the different isotopes: ma,i (X) Ar (X) = ∑ pi (1.9) i u Here, the sum runs over all isotopes i of a given element X. Note that these mass definitions imply that the atomic mass is a property of a nuclide, while the atomic weight is a property of an element. Elemental properties are often used to make a tag or identity card for each element, for example for listing the elements in a periodic system. Figure 1.14a provides such an elemental ID for the case of Neon, which includes the atomic number, the atomic symbol and the atomic weight. While elements are represented through the corresponding atomic symbols, the representa- tion of nuclides further specifies the mass number and the atomic number as a superscript and a subscript preceding the atomic symbol. Figure 1.14b exemplifies this notation for the specific case of neon-22, and displays the general notation of a nuclide. Note that in this notation, the atomic number is to some extend redundant, since it is implied in the atomic symbol. Therefore, the representation of a nuclide or isotope is often reduced to the mass number superscript. So carbon-13 is often written as 13 C rather than the more lengthy expression 13 6 C. 1.5.2 The Current Version of the Periodic System Figure 1.15 represents the periodic system of the elements as currently recommended by the IU- PAC. As can be seen, the elements are represented by tags similar to the one given in Figure 1.14a and explained in Figure 1.15. In line with Mendeleev’s approach, elements are organized in verti- cal columns – groups – of chemically similar substances, and the periodicity of properties results in different rows or periods. As so often, different parallel systems exist to label the groups (see Table 1.4): IUPAC new. The IUPAC recommends to label the groups successively from 1 to 18. So the 1.5 Talking Atoms Today 19 Figure 1.15: The periodic system of the elements as currently recommended by IUPAC, including for each element the atomic number, the atomic symbol, the name of the element in English and the atomic weight. 20 Chapter 1. The Makeup of Matter Table 1.4: Currently proposed masses and charges for the electron, the proton and the neutron and, where relevant, the mass-over-charge ratio. New IUPAC Old IUPAC Name Trivial Name 1 IA Lithium family Alkali metals 2 IIA Beryllium family Alkaline earth metals 3 IIIA Scandium family – 4 IVA Titanium family – 5 VA Vanadium family – 6 VIA Chromium family – 7 VIIA Manganese family – 8 VIII Iron family – 9 VIII Cobalt family – 10 VIII Nickel family – 11 IB Copper family – 12 IIB Zinc family – 13 IIIB Boron family – 14 IVB Carbon family – 15 VB Nitrogen family Pnictogens 16 VIB Oxygen family Chalcogens 17 VIIB Fluorine family Halogens 18 VIIIB Helium family Noble gases alkali metals, containing Li, Na, K and so on, are group 1, while the chalcogens or oxygen family are 16 and the noble gases 18. IUPAC old. The old IUPAC convention labeled group 1 to 7 using Roman letters as IA to VIIA , then called group 8, 9 and 10 simply group VIII and indicated group 11 to 18 as IB to VIIIB. This leads to the persistent description of the pnictogens or nitrogen family as group 5, or the fluorine family or halides as group 7. In addition, groups 3 to 12 are often referred to as the d-block elements or transition metals, while the separate, so-called f -block elements or inner transition metals, from lanthanum (La) to lutetium (Lu), and actinium (Ac) to lawrencium (Lr) are not labeled by any group numbers but are denoted as lanthanides or rear-earth elements, and actinides, respectively.  2 The Puzzling Properties of Atoms 2.1 Particles and Waves  Macroscopic Templates for Understanding Atoms 2.1.1 Particles The notion of a particle as a small, indivisible constituent of matter was already put forward by the ancient Greek philosophers, yet it is probably the work of Isaac Newton that made the idea of point-like particles a useful scientific concept. Newton showed, for example, that the motion of the planets around the sun could be neatly described by reducing the sun and each planet to a point that (1) coincided in space with the center-of-mass and (2) was assigned the full mass of that planet or the sun. Subjecting this point-like particles to his laws of motion and of gravity, yielded the correct relation between the orbital period and the orbital radius of all planets in the solar system. This idea of a particle was widely used in the further developments of classical mechanics that followed Newton’s work. The essential aspects of the underlying concept are that a point- a b m y sixa dr v= dt y r = (x,y) 0 0 x x axis Figure 2.1: (a) Representation of a point particle, as characterized by a mass, and a precise position r and velocity v in space according to classical mechanics (b) Typical depiction of Rutherford’s planetary atomic model, which underscores the underlying interpretation of the electron, the proton and the neutron as classical point-like particles that are, for example, fully localized in space. 22 Chapter 2. The Puzzling Properties of Atoms like particle has a mass m and takes at every moment in time t a well-defined position x – or r = (x, y, z) in three dimensions in space, see Figure 2.1. This localization in a single point works both ways, in the sense that each point in space can be occupied by one particle at the most. Moreover, a particle’s position and velocity, i.e., the derivative of the position to time, are continuous functions of time. Hence, one could say that a particle can be weighed on a balance, and is always fully localized in space. Given this description of point-like particles, one readily understands that the study of atoms, up until Rutherford’s work, was fully aligned with the idea that atoms and the atomic constituents, such as the electron and the nucleus, should be seen as particles in this classical sense. This is apparent from the typical representation of Rutherford’s atomic model, which depicts electrons, protons and neutrons as dots in combination with lines representing the electron orbits, see Fig- ure 2.1b. Moreover, the notion of atoms – or at least the atomic constituents – as indivisible, fundamental particles that was already expressed by Democritus and Dalton, ushered in a view on particles as discrete objects. One could have a proton, a hydrogen atom or a water molecule, but not half a proton, half a hydrogen atom or half a water molecule. 2.1.2 Forces and Fields According to Isaac Newton, particles change their velocity under the action of a force. By in- troducing a specific expression for the gravitational force, which depends on the inverse square of the distance separating two particles, he then continued to demonstrate the aformentioned re- lation between orbital period and orbital radius of the planets. A similar approach was followed by Coulomb when he described the electric force acting between two point charges as a vector F along the line connecting both charges and oriented such that equal charges repel and opposite charges attract each other, see Figure 2.2a. Writing the two charges as Q and q, the magnitude F of this so-called Coulomb force reads: 1 q×Q F= (2.1) 4πε0 r2 Here, ε0 is the so-called permittivity of the vacuum, a physical constant. Later work showed that the force F exerted by the charge Q on q can be conveniently replaced by an electric field, linked to the charge Q. As shown in Figure 2.2b, this procedure assigns a so-called electric field vector E(r) to every point r in space. This vector is defined such that the Figure 2.2: (a) The Coulomb force depends on the inverse square of the distance between two charges, and is oriented such that two positive charges Q and q repel each other. (b) The force exerted by the charge Q can be replaced by an electric field, which is a vector E assigned to each point r in space. The electric field yields to force on a charge q if it were placed on point r as qE. 2.1 Particles and Waves  Macroscopic Templates for Understanding Atoms 23 a b c dx f ).u.a( ecnatsid laidaR Q q F dW=Fdx r 0 r ¥ W(r) f(r)= q Electric potential (a.u.) Figure 2.3: (a) Displacing a charge q over an infinitesimal distance dx against the Coulomb force F of a charge Q requires an infinitesimal work dW corresponding to the product Fdx. This relation is used to introduce the electric potential φ (r) as the force per unit charge needed to bring a charge from infinity to the position r. (b-c) The Coulomb potential of a central charge is characterized by (bold red) circles of constant electrostatic potential that are perpendicular to (thin red) the direction of the Coulomb field. The potential varies with the inverse of the distance r from the central charge. force on a test charge q – if that charge were positioned in point r – equals the product qE(r). Hence, based on Eq 2.1, it follows that the Coulomb field of a point charge Q is a radial field, for which the magnitude E of the electric field vector changes with the distance r from the central point charge as: 1 Q E= (2.2) 4πε0 r2 Note that the electric fields introduced in this way are a mathematical construct. Scientist started thinking in terms of fields, because that approach facilitates the calculation of forces on charges. This, however, does not necessarily mean that fields exist as a physical reality by themselves. Insight 2.1: The electric potential When a force moves an object over a arbitrarily small distance dx, the corre- sponding innitesimal work dW done by that force is dened as the product Fdx  more precisely the scalar product F · dx  of the force and the displacement, see Figure 2.3a. The work W done in a nite displacement from point r1 to r2 , is then obtained by integrating dW along the path taken: Z r2 W= F · dx r1 In much the same way as done for electric elds, the notion of work enables us to assign a number φ (r)  called to electric potential  to each point r in space, where the product qφ (r) yields to work done when a charge q is moved from an innite distance to that point r. Referring to the above relation and the denition of the electric eld, we thus have: Z r W Q φ (r) = φ (r) = =− Edx = q ∞ 4πε0 r 24 Chapter 2. The Puzzling Properties of Atoms Note that we obtained the electric potential as a function of the distance r between point r and the origin by using a path along a straight line passing through the origin and the point r , see Figure 2.3a. When multiplied with the charge q, this so-called Coulomb potential yields the potential energy V (r) of that charge in the Coulomb eld of the central charge Q, using the potential energy at very large distances as the zero of energy: q×Q V (r) = qφ (r) = 4πε0 r Hence, next to electric elds, the electric potential provides a second way to describe interactions between charges, with a focus on interaction energy rather than force. 2.1.3 Waves After throwing stone in a pond, it is easy to imagine the circular wave propagating from the point of impact to the edges of the water surface as shown in Figure 2.4a. While straightforward, the example highlights the essence of a wave as a propagating disturbance – the elevation of the water surface – of a material system. This makes a wave in many ways the opposite of a particles. As a propagating disturbance, such as the height of the water surface, a wave is not localized and has no mass. Moreover, two waves can pass simultaneously through the same point in space, a phenomenon called interference, by which the local disturbance can be amplified or extinguished. Finally, waves are in essence continuous, not discrete. The elevation of the water surface, for example, can be halved or doubled at will; it simply takes in initial disturbance with less or more energy. So unlike atomic building blocks, one can imagine having a knob by which a wave can be gradually turned on or off. While the mathematical description of waves on a water surface is complex, Figure 2.4a suffices to show that waves often involve a sequence of crests and throughs, in this case elevations and depressions of the water surface. The distance between two such crests determines the wavelength λ of the wave. As a crest propagates along the surface, the water level at each given a c λ t=T t=5T/6 t=2T/3 t=T/2 T b t=T/3 t=T/6 t=0 time 0 λ 2λ 3λ distance Figure 2.4: (a) Ripples on a water surface show a sequence of elevations (crests) and depressions (throughs) of the water surface, where the distance between two crests is the wavelength. (b) Monitor- ing a single point at the water surface in time shows a sequence of crests and depressions, where the time between two successive crests is the period of the wave. (c) A crest of a traveling wave propagates over a wavelength within the time span of a period, a relation that yields the speed of a wave. 2.1 Particles and Waves  Macroscopic Templates for Understanding Atoms 25 point will also show a succession of crests and depressions, see Figure 2.4b. Here, the time between the passage of two crests is called the period T of the wave, while the number of crests passing per second is the frequency ν of the wave. Obviously, period and frequency are inversely related: 1 ν= (2.3) T Finally, Figure 2.4c shows that a given crests will propagate over a distance as given by the wavelength within a single period. Hence, the propagation speed c of the wave can be written as: λ c= = λ ×ν (2.4) T The relation between propagation speed, wavelength and frequency as expressed by Eq 2.4 is a characteristic for a given wave. Sound waves, waves on a water surface or a wave travelling on a string have all a given propagation speeds that makes that the frequency and the wavelength of a wave are not independent quantities. On the other hand, freely propagating waves can have any frequency (or wavelength), depending on the way the wave is generated. In scientific language, one says that the frequency can be varied continuously. 2.1.4 Standing Waves Organ pipes, as depicted in Figure 2.5a, confine an air column to a finite volume of space. Sound waves generated inside that volume will bounce back against the edges of the pipe and the re- sulting interference between the incident and the reflected wave will most often extinguish the wave. The specific conditions that result in constructive interference, can be best understood by thinking about the speed of the gas molecules. A sound wave corresponds to a repetitive change back and forth of the speed of these molecules. However, at the edge of the tube, this speed must a b c t=0 t = T/2 L t=T deepS deepS 0 L/2 L 0 L/2 L Figure 2.5: (a) Sound waves in organ pipes are confined to the inner volume of the pipe. (b) Rep- resentation of the speed of gas molecules in a pipe as they move (blue) right or (red) left. At the edge of the pipe, the speed must be zero, leading to a standing wave inside the pipe with nodes and crests that do not propagate. The example shows the fundamental (n = 1), for which the pipe length L corresponds to half a wavelength. (c) The same for the first overtone (n = 2), for which L is a full wavelength. 26 Chapter 2. The Puzzling Properties of Atoms be zero since the gas molecules cannot escape from the pipe. Hence, the only waves that can be sustained within the pipe, must have a zero or a so-called node at the edge of the pipe. Within the center of the pipe, on the other hand, gas molecules can still change their speeds back and forth at the frequency of the wave, see Figure 2.5b. When looking back at the propagating wave depicted in Figure 2.4c, one sees that crests, troughs and nodes all move together with the same speed. Hence, when the edges of the pipe fix the nodes of a wave, this implies that sound waves inside an organ tube cannot propagate. As can be seen in Figure 2.5b, the gas molecules in the center of the pipe continuously change the direction of their speed, yet next to the nodes, also the crest of the sound wave does not move left or right. So wave motion is reduced to an oscillation of a quantity in time, not in space. Such waves are called standing waves, and they will occur in any situation where waves are formed in a finite space, such as organ pipes, guitar strings or the surface of a liquid in a beaker. The example of the organ pipe helps us to understand an important property of standing waves. As shown in Figure 2.5b, the edges of the pipe determine the nodes of the standing wave. Since the nodes of a wave are separated by half a wavelength, the conclusion is that a given pipe can only sustain those standing waves for which the length L of the pipe is a multiple of half the wavelength of the wave. This condition is most conveniently written as a constrained on the wavelength: 2L λ= with n = 1, 2, 3,... (2.5) n Hence, unlike running waves, the wavelength or frequency of standing waves cannot vary con- tinuously, but will only take discrete values. Figure 2.5b and 2.5c show examples of standing waves, where n = 1 and n = 2, which are also called the fundamental and the first overtone. Also note that according to Eq 2.5, larger pipes can sustain standing waves with larger wavelengths and thus, in the case of sound, with a lower pitch. This, of course, is a relation well-known from musical instruments. 2.1.5 Light as an Electromagnetic Wave Imagine two opposite charges separated by some vertical distance; a configuration resembling an electron and a nucleus in Rutherford’s planetary atomic model. In each point in space, the electric field of this charge configuration is obtained by adding the field vectors of both charges. fixed time λ fixed distance 1/ν dleiF dleiF Distance Time Figure 2.6: (a-c) Representation of two vertically stacked opposite charges and the resulting vertical electric field at points along a horizontal line that bisects the two charges. From (a) to (c), one sees the direction of the field tracking the orientation of the charges. (d) Continuously flipping charges yield a propagating electric field disturbance, characterized by a wavelength λ and a frequency ν; this propagating disturbance is known as an electromagnetic wave. 2.1 Particles and Waves  Macroscopic Templates for Understanding Atoms 27 Figure 2.7: Representation of different electromagnetic waves as part of the electromagnetic spec- trum. Short wavelength radiation, such as gamma rays and X-rays are represented at the left, long wavelength radiation, such as radiowaves, at the right. The small part of the visible spectrum corre- sponding to visible light is highlighted. So think of a point on a horizontal line that intersects the line connecting the charges halfway. As shown in Figure 2.6a, the horizontal field components related to the positive and negative charge cancel. What is left, is a vertical field, pointing in the direction of the negative charge. Hence, if the order of the charges is flipped, the field will flip as well, see Figure 2.6b, and it will flip again if the order of the charges is changed back, see see Figure 2.6c. What James Maxwell showed in 1867, is that electric fields do not adjust instantaneously to such changes in charge configuration. The flipping of the field rather propagates through space at a finite speed, which we call the speed of light c. Now imagine the charge configuration shown in Figure 2.6a permanently switching orienta- tion with a frequency ν. In such a case, Figure 2.6d highlights that the propagating field will show a sequence of crests and troughs in space, in line with the upward and downward orienta- tion of the charges. Moreover, at a fixed position, the field will switch from upward to downward at the same frequency ν at which the charges are rotating. What we are looking at is an electro- magnetic wave, a propagating distortion of the electric field, characterized by a frequency ν and a wavelength λ , whose product yields the speed of light according to Eq 2.4. You will remember that fields were introduced as a mathematical construct that helps keeping track of forces. However, what Maxwell showed is that an electromagnetic wave is more than a way to keep track of how forces change in space and time. Most importantly, electromagnetic waves propagate energy, regardless of us providing a test charge to turn a field into a measurable force. The shift in the meaning one should give to fields, from a mere representation of electro- static forces to something more tangible, capable of transmitting energy, puzzled scientists for a while, and inspired the concept of an aether as the underlying vibrating medium. However, no evidence for such an aether was ever found. Accepting that fields can properly describe the propagation of electromagnetic energy, sci- entists found that Maxwell’s electromagnetic waves could be related to a multitude of natural phenomena, ranging from radiowaves and microwaves to infrared radiation, visible light and ultraviolet radiation and the penetrating X-rays and gamma rays. All these were found to be electromagnetic waves, that all propagate at the same speed of light and only differ in their wavelength. This finding led to the nowadays familiar notion of the electromagnetic spectrum, in which all these electromagnetic waves are represented as a function of wavelength or frequency, 28 Chapter 2. The Puzzling Properties of Atoms see Figure 2.7. Example 2.1: Visible, near infrared and ultraviolet light As can be seen in Figure 2.7, visible light corresponds to a narrow wavelength range in the electromagnetic spectrum extending roughly from 400 to 700 nano- meters, wavelengths that we would see as violet and deep red light. Changing the wavelength from 400 to 700 nm, results in us seeing dierent colors in line with the color ordering in a rainbow or as obtained by a prism. Ultraviolet and infrared light, which are both invisible by the human eye, are found at wavelengths shorter than 400 nm or longer than 700 nm. 2.2 Atoms and Light Have Unexpected Properties 2.2.1 The Rutherford Atom Cannot be Stable Already upon its formulation, Rutherford’s planetary model of the atom had a major issue. If indeed electrons were orbiting the nucleus like planets the sun, Figure 2.6 shows atoms would continuously lose energy by electromagnetic radiation. As this energy can only come from the kinetic energy of the electrons in their orbit, these orbitals should eventually collapse as depicted in Figure 2.8. So electromagnetic theory and the Rutherford atom cannot provide both a correct descriptions of nature, even if the experiments of Geiger and Mardsen clearly point towards the presence of a small, massive, positively charged nucleus in atoms. 2.2.2 Planck and the Quantization of Light The detailed study of electromagnetic radiation led to more apparent inconsistencies in the de- scription of matter. Figure 2.9 represents, for example, the so-called emission spectrum of the sun, a graph yielding the emitted power as a function of the wavelength of the radiation. As can be seen, the emitted power peaks at around 550 nm, a wavelength corresponding to yellow light, and shows a gradual decay in the direction of the infrared and the ultraviolet. However, when Max Planck sought to describe this emission spectrum by combining Maxwell’s description of electromagnetic radiation with established thermodynamic principles, he obtained a spectrum electron emitted radiation nucleus Figure 2.8: Representation of the radiation emitted by an orbiting electron, a process eventually leading to the collapse of the electron orbit that highlights the instability of atom when described using Rutherford’s planetary model. 2.2 Atoms and Light Have Unexpected Properties 29 a b 2.0 )mn· mc(/W( ytisned rewoP classical 2 continuous energy flow prediction 1.5 1.0 Planck's quantum = e hn = e hn = e hn model 0.5 quantized energy flow 0.0 0 1000 2000 3000 Wavelength (nm) Figure 2.9: (a) Depiction of (red) the power density of the electromagnetic spectrum emitted by the sun in comparison with predictions based on (thin red line) a classical combination of Maxwell’s elec- tromagnetic theory and thermodynamics and (filled background) Planck’s theory of energy quantiza- tion of electromagnetic radiation. The dips in the solar spectrum are related to absorption of radiation by, for example, water in the earths atmosphere. (b) Schematic representation of Planck’s proposal to replace the continuous flow of radiative energy by energy transfer in discrete energy units or quanta, referred to as photons. that showed an ever increasing power density with decreasing wavelength. Since the emitted power had to be finite, Planck correctly concluded that this so-called UV catastrophe indicated a major flaw in the underlying understanding of radiation. Interestingly, in 1900, Planck managed to describe the sun’s emission spectrum when he assumed that the energy transmitted by an electromagnetic wave is not a continuous quantity, but must be a multiple of a fixed unit. According to Planck, this unit or quantum of energy ε was proportional to the frequency of the radiation: ε = hν (2.6) Here, the factor h is nowadays known as Planck’s constant, a physical constant amounting to: h = 6.62607015 10−34 J · s (2.7) As shown in Figure 2.9, the assumption that radiative energy is quantized, enabled Planck to per- fectly describe the sun’s emission spectrum, notwithstanding the gaps in the emission spectrum that are related to absorption of light in the earth’s atomosphere. Planck’s theory, which was fully confirmed in later years, ushered in a dramatic change in the way scientists looked at the physical reality. Research in the 19th century had firmly shown that light should be seen as a wave, a point fully confirmed by Maxwell’s theory. What Planck showed, is that such a wave somehow had particle-like properties. Indeed, with the energy carried by an electromagnetic wave coming in discrete multiples of a fundamental unit, an electromagnetic wave could also be seen as a stream of particles or photons, where each photon carries such a single unit of energy. As we will see, this description of photons as the energy carriers of electromagnetic radiation proved extremely useful to understand the interaction between light and atoms, and is currently widely accepted. 30 Chapter 2. The Puzzling Properties of Atoms ∆V =1 V -19 1 eV =1.602 10 J q -19 =-1.602 10 C Figure 2.10: The electron volt is defined as the energy gain by a single electron when it crosses a potential-energy difference of 1 V, which is about the voltage across a small battery. Insight 2.2: The electron volt, a unit for counting energy quanta Let us apply Planck's relation to determine the energy carried by λ = 620 nm photons, which correspond to red light. Using the relation between wavelength, frequency and the speed of light, we obtained: h × c 6.626 × 3 −17 ε620 nm = = 10 J = 3.2 10−19 J λ 620 As compared to the thousands of kJ we are used to encounter in our daily life, one sees that the energy carried by one such a photon is extremely small. No surprise that we do not directly experience the quantization of energy in our daily life. However, this dierence also implies that the Joule is not the most useful quantity when describing photon energies or, as we will see later, atomic energies. A more convenient unit to do this, is the so-called electron volt (eV). As shown in Figure 2.10, the electron volt is dened as the energy gained by a single electron when passing through a electric potential dierence of one volt. Since the electron carries a single elementary charge, one has: 1 eV = 1.602 10−19 J Using the eV as a unit of energy, we have: ε620 nm = 2 eV 2.2.3 The Line Spectra of Atoms The study of light emission and absorption by atoms brought about additional surprises. Opposite from the sun, Johann Balmer discovered, for example, that a gas of hydrogen atoms emits light at only a few different wavelengths, four of which are shown in Figure 2.11. Next to this series of visible lines, which became known as the Balmer series, later studies showed similar series in the ultraviolet and the near infrared, such as the Lyman series and the Paschen series, respectively. Moreover, the wavelength or photon energy of the different lines in the Balmer series could be expressed using a straightforward mathematical relation named after Johannes Rydberg:   1 1 hνn,Ba = Ry − 2 for n = 3, 4, 5,... (2.8) 4 n 2.2 Atoms and Light Have Unexpected Properties 31 Photon energy (eV) 3.2 3 2.8 2.6 2.4 2.2 2 1.8 Continuous spectrum of visible light Hydrogen emission spectrum mn 014 mn 434 mn 684 mn 656 Hydrogen absorption spectrum 400 450 500 550 600 650 700 750 Wavelength (nm) Figure 2.11: Representation of (top) the continuous spectrum of visible light, (middle) the emission spectrum of hydrogen atoms, showing four lines of the Balmer series with their corresponding wave- length, and (bottom) the absorption spectrum of hydrogen atoms, where the same wavelengths are imprinted as black lines. Here, the natural number labels the different lines, while Ry is the so-called Rydberg constant, which reads: Ry = 13.605693 eV (2.9) Example 2.2: The Rydberg equation for all the hydrogen line series Following the description of the Balmer series by Rydberg's formula, a similar expression was found to describe the Lyman and Paschen series of emission lines, provided the term 1/4 in Eq 2.8 was replaced by 1 and 1/9, respectively:   1 hνn,Ly = Ry 1 − 2 for n = 2, 3, 4,... n   1 1 hνn,Pa =