Atoms

Contents

Periodic Table

		The Periodic Table in Figure 13-01a is the modern version; while Figure 12-19 depicts the simpler one. The atomic number is the number of positive charges in the atomic nucleus. Atomic masses refer to the masses of neutral atoms, including the masses of the electrons and the mass equivalent of their binding energies. It is expressed in mass units such that the mass of the most abundant type of carbon is exactly
Figure 13-01a Periodic Table, Modern [view large image]	Figure 13-01b Periodic Table, Unconventional [view large image]	12.00 u (1 u = 1.66x10-24 gm).

the boiling point, melting point, density, acidity, basicity, crystal structure, and electronegativity (tendency to keep electrons) of the elements. The s, p, d, and f letters in the electronic configuration designate the orbital quantum number l = 0, 1, 2, 3, ... respectively for the outer shell electrons. A new designation of the groups has a number ranged from 1 to 18. Figure 13-01b is an unconventional Periodic Table. It specifies the phase (solid, liquid, or gas) of the element at room temperature, whether the element is radioactive or man-made, as well as its usage (in daily life) or occurrence (in nature). Only a few of the elements are edible as shown in Figure 13-01c. Other kinds of Periodic Table may incorporate properties such as atomic radius, covalent radius, ionization potential, specific heat, heat of vaporization, heat of fusion, electrical conductivity, and thermal conductivity etc.

Figure 13-01c Periodic Table, Comical [view large image]

The regular pattern in the periodic table is related to the state of the electron in an atom. It is specified by four quantum numbers. The principal quantum number n determines the energy level; its value runs from 1, 2, 3, ... For each n, the orbital quantum number l = 0, 1, 2, ... (n-1); it is related to the magnitude of angular momentum. Then for each l, the magnetic quantum number m can be -l, -l+1, ...l-1, l; it is related to the z component of the angular momentum. The spin quantum number s is either +1/2 or -1/2. For n = 1, l = 0, m = 0, there is only 2 possible quantum states for the electron, with s = +1/2 and -1/2 respectively. For n = 2, l = 0, m = 0 and l =1, m = -1, 0, +1; there is a total of 2 + 6 = 8 possible quantum states. Therefore, it requires 2 electrons to complete the shell for n = 1, and 8 electrons to complete the shell for n = 2, ...and so on. The orbital quantum number l is often designated by a letter, s for l = 0, p for l = 1, d for l = 2, and f for l = 3 ... The quantum number l is non-additive while m is additive and relates to an Abelian group (e.g., the two dimensional rotation about the z-axis). Particles having the same non-additive quantum numbers but differing from each other by their additive quantum numbers are said to belong to the same multiplet. The number of members of a multiplet is called its multiplicity. For a given multiplet l the multiplictiy is equal to 2l+1. The atom tends to lost the outer electrons if the number is far from a complete shell or sub-shell such as the elements in the beginning of a series. It gradually develops a preference for accepting more electrons to complete the outer shell as the progression moves toward the end of a series. This property is responsible for all the chemical reactions. A stable atomic configuration is achieved by completing a shell or sub-shell as illustrated in Table 13-01 below by the inert elements (the rule becomes more complicated in the advanced series as the electrons with high l tend to intermingle with each others), which do not react chemically:

Table 13-01 Electron Configuration of the Inert Elements

The elements in the periodic table has been classified broadly into metals, non-metals and metalloids as discussed in Topic-12. The more refined classification is to categorize according to groups (in new designation from 1 to 18):


• Group 1 elements (known as alkali metals) are silvery colored, soft, low density metals, which react readily with halogens to form ionic salts, and with water to form strongly alkaline (basic) hydroxides. These elements all have one electron in their outermost shell, so the energetically preferred state of achieving a filled electron shell is to lose one electron to form a singly charged positive ion. Their softness and ractivity increase down the group. Lithium is a soft metal (see Figure 13-02a) with such low density that it floats in water. It reacts vigorously with water to produce hydrogen gas, and with oxygen to form Li₂O, such that it is never found in ints free state in nature. Hydrogen (water

creator in Greek), with a solitary electron, nominally belongs in the alkali metals group. However, removal of that single electron requires considerably more energy than for the other alkali metals. Like the halogens, only one additional electron is required to fill in the outermost shell of the hydrogen atom, so hydrogen can be regarded in some respects as behaving like a halogen; its elemental form is a diatomic gas, and it can even form salts (called hydrides) with the alkali metals, where the metal has donated an electron to the hydrogen, almost as if hydrogen were actually a halogen.

Figure 13-02a Lithium

Group 2 elements have one more electron in their outer shells than group 1 elements. Although they are reactive, the extra electron makes them less reactive than group 1 elements. The reactivity of these elements increases down the group. They are found in compounds, which form important rocks in the Earth's crust. This gives rise to the group's alternative name, the "alkaline earth" metals. The emerald (Be3(Al,Cr)2Si6O18) is the green variety of beryl (see Figure 13-02b), the mineral that is the principal source of beryllium.

Figure 13-02b Beryl

Group 3 -12 are the transition elements, which are usually hard, tough and shiny. They are less reactive and have greater densities than group 1 or 2 elements. Group 11 includes the traditional coinage metals such as copper, silver, and gold (see Figure 13-02c). They are also known as the "noble metals". They are all relatively inert, hard-to-corrode metals which have been used for minting coins, hence their name. Many metals form coloured compounds, (e.g., iron pyrite), and many metals have a coloured sheen. These metals, especially silver, have unusual properties that make them essential for industrial applications outside of their monetary or decorative value. They are all excellent conductors of electricity. The most conductive of all metals are silver, copper and gold in that order. Silver is the most thermally conductive element, and is also the most light

Figure 13-02c Gold

reflecting element. Silver also has the unusual property that the tarnish that forms on silver is still highly electrically conductive.

• Group 13 elements are called light metals. Boron is a hard, brittle, non-metallic element. It is found in nature usually in combination with oxygen. The compound boric oxide is important in making heat-resistant glass. Boric acid is commonly used as eyewash and antiseptic. The compound borax (boron combined with oxygen, water, and sodium) is useful as a cleaning agent and water softener. Aluminum, which is right beneath boron, is by its position a metalloid. But the properties of aluminum are usually those of metals. Aluminum is the most abundant metal and the third most

abundant element in the earth's crust. Aluminum is found as aluminum oxide in the ore called bauxite. Aluminum is extremely valuable in industry (see Figure 13-02d). It is light, strong, and does not tarnish in air. It is a good conductor and is used in wiring, airplane parts, household items. The other members of the boron family ...gallium, indium, and thallium, are metals. Gallium shares many chemical properties with aluminum. It is an extremely soft metal that can be cut with a knife. It also has an extremely low melting point of 29.8oC. Gallium arsenide (GaAs) can function as a "laser diode" and convert electricity directly into a beam of laser light.

Figure 13-02d Aluminum

• Group 14 is the carbon family. It includes the elements carbon, silicon, germanium, tin, and lead. Carbon can combine with other elements in a great variety of ways - in millions of carbon compounds called "organic compounds", which form the basis for life. Carbon exists in its free state in several different forms. The most spectacular of these forms is diamond, the hardest of materials (see Figure 13-02e1). In the form of graphite, the carbon atoms arrange themselves in layer, like the pages of a book. Because the forces between them are not very strong, the layers slide easily over one another, making graphite soft and slippery. It is useful as a lubricant and as a writing material in pencils. Silicon is the second most abundant element in the earth's crust. Silicon is used in glass and in cement. Glass is produced by

		mixing silica (sand - SiO2), soda ash (sodium carbonate - Na2CO3) and limestone (calcium carbonate - CaCO3) in a ratio of 60:80:5 and heated to the melting point of 2500oF. The final product is a supercooled liquid called glass which can be made to be colourful or to have special property by adding impurities (see Figure 13-02e2). Silicon is also used in solar cells, which convert the energy of sunlight into electric energy. Germanium is a metalloid used in transistors, which are devices found in many electronic instruments, such as radios and
Figure 13-02e1 Carbon [view large image]	Figure 13-02e2 Glass	televisions. Tin is a metal which resists rusting and corrosion. The most dense element in the carbon family is the metal lead (Pb), which can form poisonous compounds.




• Group 15 is the nitrogen family. This group is sometimes referred to as pnictogens (choke generator in Greek as nitrogen was discovered in 1772 by demonstrating that it can asphyxiate a mouse). It consists of nitrogen, phosphorus, arsenic, antimony, and bismuth. Although a relatively unreactive gas, nitrogen forms hundreds of thousands of compounds that are of crucial importance for agriculture and industry. Considering the stability of nitrogen itself, some of these compounds are, ironically, extremely unstable and explosive. Despite the lack of reactivity of nitrogen, nature has developed several mechanisms (such as lightning) for converting nitrogen for use by living cells. The process of converting nitrogen from the atmosphere into usable nitrogen compounds is called nitrogen "fixing". Phosphorus (bearer of light in Greek) is a non-metal that glows in dark and bursts into flames in warm air (see Figure 13-02f). It is used in

the military to make incendiary bomb. Arsenic is a brittle, crystalline solid at room temperature. It is often thought of as a semi-metal, or metalloid. For example, it is a poor conductor of electricity, yet it has a steel-gray color. In the form of arsenious oxide, it is a well-known poison. It is used as a weed killer and insecticide. Arsenic has become a material of great importance in the world of solid-state electronics. Small amounts of arsenic are added to such semi-conductors as germanium and silicon to transform them into transistors.

Figure 13-02f Phosphorus

• Group 16 consists of the elements Oxygen (O), Sulfur (S), Selenium (Se), Tellurium (Te), and the radioactive Polonium (Po). Their compounds, particularly the sulfides, selenides and tellurides are collectively known as chalcogenides. The

name is generally considered to mean "ore former" (from the Greek chalcos "ore" and -gen "formation"). Oxygen and sulfur (see Figure 13-02g) are nonmetals, polonium is a true metal, and selenium and tellurium are metalloid semiconductors (i.e., their electrical properties are between those of a metal and an insulator). Nevertheless, tellurium, as well as selenium, is often referred to as a metal when in elemental form. Chalcogenides are quite common as minerals, e.g., FeS2 pyrite is an iron ore and AuTe2 gave its name to the gold rush town of Telluride, Colorado in the USA. The oxidation number of the chalcogen is generally -2 in a chalcogenide but other values (e.g. -1 in pyrite) can be attained. The highest oxidation number +6 is found in sulfates, selenates and tellurates, e.g., in Na2SeO4 - sodium selenate.

Figure 13-02g Sulfur

• Group 17 elements are known as the halogens (salt creator). These elements are diatomic molecules in their natural form. They require one more electron to fill their outer electron shells, and so have a tendency to form a singly-charged negative ion. This negative ion is referred to as a halide ion, which tend to form ionic compounds with metals. Salts containing these ions are known as halides. Halogens form covalent compounds with non-metals. Examples include

  most carbon-containing, or organohalogen, molecules. Organohalogen chemicals (such as DDT, PCB, CFC, ...) are often believed to be solely industrial compounds, but many living organisms and geological phenomena also produce them naturally. In the case of plants and animals (such as sponges, corals, seaweeds, evergreen trees, some arthropods, and some frogs), the substance is produced for defence against predators and parasites. The halogens become less reactive, and their melting points increase further down the group. For example, fluorine is a yellow gas at room temperature, whereas bromine is a liquid, and iodine is a black solid (see Figure 13-02h).

  Figure 13-

  02h Iodine

  Group 18 contains the noble or inert gases. Their outer electron shells are full, which makes it difficult for them to gain or lose electrons. They are therefore almost totally unreactive and non-flammable. The neon signs contain neon gas that glows in orange-red color (see Figure 13-02i) when it is energized by an electrical discharge. Every gas, when excited, radiates its own characteristic color. Argon, for example, produces a purple light, while xenon produces a blue-green light.

  Figure 13-02i Neon

Band Theory, Metal

		About 80% of the free elements at room temperature exists in the form of metal. The conditions to form metal are vacant valence orbitals and low ionization energies. Similar to the splitting of energy levels when two or more atoms come close to each other; (See Figure 12-15.) energy levels broadened to a band (many closely spaced energy levels)
Figure 13-03 Energy Bands [view large image]	Figure 13-04 Band Theory [view large image]	for an aggregate of many atoms as shown in

Specific Heats of Solids and Phonons

		These waves, like electromagnetic waves in a cavity, have quantized energy contents. A quantum of vibrational energy in a solid is called a "phonon", and it travels with the speed of sound. The concept of phonones is quite general and has applications in connection with the thermal conductivity of some solids, and the way electrons in the crystal structure interact with sound waves.
Figure 13-05 Crystal Types [view large image]	Figure 13-06 Specific Heats [view large image]

Superductivity

If mercury is cooled below 4.1 K, it loses all electric resistance. This discovery of superconductivity by H. Kammerlingh Onnes in 1911 was followed by the observation of other metals and intermetallic compounds (made of two or more metallic elements) which exhibit zero resistivity below a certain critical temperature Tc. The fact that the resistance is zero has been demonstrated by sustaining currents in superconducting lead rings for many years with no measurable reduction. Table 13-02 shows the elements, which can become superconducting at the indicated critical temperature. [view large image]

Table 13-02 Super-conducting Elements

Ceramic materials are expected to be insulators -- certainly not superconductors, but that is just what Georg Bednorz and Alex Muller found when they studied the conductivity of a lanthanum-strontium-copper oxide ceramic in 1986. Its critical temperature of 30 K was the highest, which had been measured to date. Their discovery started a surge of activity which discovered superconducting behavior as high as 125 K. However, these compounds are hard to make and difficult to shape. They pose a multitude of physical challenges to researchers and engineers.

Figure 13-07 High Temperature Superconductors [view large image]

Figure 13-07 lists the high temperature (above 4o K) superconductors discovered during the last one hundred years.

		The effect of superconducting is often demonstrated by cooling a disk made of superconducting material with liquid nitrogen to below the critical temperature Tc. A magnet placed above the disk is repelled, i.e., it is levitated above the superconductor. (See Figure 13-08a.) This phenomenon is caused by the Meissner effect which is related to the fact that a superconductor will exclude magnetic fields within the superconducting material
Figure 13-08a Superconductivity	Figure 13-08b Meissner Effect	(see Figure 13-08b).

Laser

		Finding substances in which a population inversion can be set up is central to the development of new kinds of laser. The first material used was synthetic ruby. Ruby is crystalline aluminum (Al2O3) in which a small fraction of the Al3+ ions have been replaced by chromium ions, Cr3+. It is the chromium ions that give rise to the characteristic pink or red colour of ruby and it is in these ions that a population inversion is set up in a ruby laser.
Figure 13-09 Ruby Laser [view large image]	Figure 13-10 Laser, Energy Levels [view large image]

Laser cooling utilizes the collective momentum of many photons to reduce the thermal motion of an atom. Since the approaching and recessing speed of the atoms differs slightly due to the Doppler effect and the atoms can only absorb a certain frequency, the laser beam can be tuned such that it slows down only the approaching atoms. The six crossed laser beams shown in Figure 13-10a create a space in which atoms moving in this region (the bright area in the center of the picture) are trapped and cooled by absorption of photons from the crossed beams. With this technique, researchers have already reached temperatures lower than a millionth of a degree Kelvin. That's an average atomic speed on the order of a few cm/sec.

Figure 13-10a Laser Cooling

Plasma

		In the solid state, the atoms are firmly imprisoned inside a rigid network (like ice for example). When we raise the temperature, they go into a liquid state (the ice melts), where the atoms may slide around in relation to the others, thus enabling a liquid to adapt to the shape of a container. If we heat it up even more, we arrive at the gas state: atoms then move around freely and independently of each other (water turns into steam). (See Figure 13-11.) Finally, when we get to very high temperatures (typically several million degrees), the ingredients of the atom separate, nuclei and electrons move around independently and form a globally neutral mixture: this is the plasma state (See Figure 13-12).
Figure 13-11 The Three Phases [view large image]	Figure 13-12 The Fourth Phase [view large image]

		The vast power radiated by the Sun is generated by the fusion process wherein light atoms combine with an accompanying release of energy. In nature, proper conditions for fusion occur only in the interior of stars. Researchers are attempting to produce the conditions that will permit fusion to take place on earth. Since nuclei carry positive charges, they normally repel one another. The higher the temperature, the faster the atoms or nuclei move. When they collide at these high speeds, they overcome the force of
Figure 13-13 Plasma Occurrence [view large image]	Figure 13-14a Plasma Con- finement [view large image]	repulsion of the positive charges, and the nuclei fuse. In such collisions, energy is released. The

	ITER's six international partners could be in place by the winter of 2005, allowing construction to begin in 2006, and operation in 2016. ITER is designed to heat hydrogen to hundreds of millions of degrees centigrade, and then squeeze energy from the resulting plasma, while holding it stable for minutes at a time. It is based on the tokamak model, which up until today has only one machine that has begun to approach the "break-even point". It is believed that by building a tokamak with bigger size, it will allow the high-temperature high-pressure plasma to remain stable longer (~ 7-10 minutes) producing 500 megawatts of energy within the interval.
Figure 13-14b ITER [view large image]

Quantum Computing

Physics and Computer science have combined to create a new field: quantum computing and quantum information. The spark that ignited world wide interest in this new field sprang forth in 1994 with Peter Shor's discovery of a theoretical way to use quantum mechanical resources to unravel a mathematical problem at the heart of electronic commerce and cryptography. Basic steps towards the creation of a quantum computer have been taken, with the demonstrations of elementary data storage and manipulation using photons and atoms or trapped ions as the quantum bits, or "qubits". Recently, it has been shown that it is possible to build solid-state qubits made from tiny samples of superconducting material. Figure 13-15 shows some of the subjects, which are currently being investigated in the field of quantum computing. [view large image]

Figure 13-15 Qunatum Computing

Qunatum computing exploits two resources offered by the laws of quantum mechanics: the principle of superposition of states and the concept of entanglement. Superposition is a "one-particle" property; while entanglement is a characteristic of two or more particles. Consider a particle with spin such as the electron. With reference to a given axis (say along the z axis), the spin of the particle can point in two opposite directions, say "up" and "down", and the spin states can be denoted as |1 > and |0 >. But by the laws of quantum mechanics, the particle can exist in a superposition of these two states, corresponding to arbitrary orientation as shown in Figure 13-16.

Figure 13-16 Qubit [view large image]

		Now, consider a two-particle state: there are four "basis states", |1 >1|1 >2, |0 >1|0 >2, |1 >1|0 >2 and |0 >1|1 >2, where the subscript indicates particle 1 and 2. Again, superpositions can be made of these states, including in particular, the four "maximally entangled Bell states": |1 >1|1 >2 + |0 >1|0 >2 ------ (2) |1 >1|1 >2 - |0 >1|0 >2   ------ (3)
Figure 13-17 Entanglement [view large image]	Figure 13-18 Entanglement Implementation [view large image]	|1 >1|0 >2 + |0 >1|1 >2  ------ (4) |1 >1|0 >2 - |0 >1|1 >2   ------ (5)

Suppose particle 1 which Alice wants to teleport is in the initial state: |f >1 = a |1 >1 + b |0 >1 ------ (6) and the entangled pair of particles 2 and 3 shared by Alice and Bob is in the state: |f >23 = (|1 >2|0 >3 - |0 >2|1 >3)/21/2 ------ (7) which is produced by an Einstein-Podolsky-Rosen (EPR) source2. The teleportation scheme works as follows. Alice has the particle 1 in the initial state |f >1 and particle 2. Particle 2 is entangled with particle 3 in the hands of Bob. The essential point is to perform a joint Bell-state measurement (BSM)3 on particles 1 and 2 which projects them onto the entangled state:

Figure 13-19 Teleportation [view large image]

|f >12 = (|1 >1|0 >2 - |0 >1|1 >2)/21/2 ------ (8)

polarizing beam splitters (PBS) for Bell-state measurement (BSM). The logic electronics identify the Bell state and convey the result through the microwave channel (RF unit) to Bob's electro-optic modulator (EOM). Depending on the message, it either leaves the photon state unaltered or changes it to the opposite state. Note that because of the reduced velocity of light within the fibre-based quantum channel, the classical signal arrives about 1.5 microseconds before photon 3. Thus, there is enough time to set the EOM correctly before photon 3 arrives. Polarization rotation (which introduces errors) in the fibres is corrected by polarization controllers (PC) before each run of measurements.

Figure 13-20 Teleportation over River Danube [view large image]

Polarization stability proved to be better than 10o on the fibre between Alice and Bob, corresponding to an ideal teleportation fidelity of 0.97.

Therefore, quantum error correction is performed differently as shown in the followings:
1. Prepare the primary physical qubit such as |f > = a |1 > + b |0 >, which is to be protected from error.
2. Prepare two auxiliary qubits |0 >|0 >, which are then entangled (encoded) with the primary qubit to form a logical state.
3. Noise is applied to this logical state.
4. The primary qubit is decoded from the auxiliary qubits.
5. Syndrome measurement is performed on the four possibilities for the auxiliary states, e.g.,
   
   1. |1 >|1 > for no error - association of the most often outcome with the most easily distinguishable measurement.
   2. |1 >|0 > for auxiliary state 1 flipped - no correction required.
   3. |0 >|1 > for auxiliary state 2 flipped - no correction required.
   4. |0 >|0 > for primary qubit flipped - the primary qubit before correction is: -a |0 > + b |1 >.
6. For the first and last cases, the original primary state has been altered. Appropriate correction is applied to recover the primary qubit initial state.

Such error-correction protocols have been implemented in 2004 using three beryllium atomic-ion qubits (the qubits comprise the two electronic ground state hyperfine levels, which are equated to the two spin 1/2 states - up and down) confined to a linear, multi-zone trap. It has been demonstrated that fidelity of 0.7 - 0.8 can be achieved in the experiments. However, the method works well only when at most one of the three qubits undergoes a spin-flip error. Figure 13-21 shows the transportation of the ions in the trap during the error-correction protocol as a function of time. Each experiment requires approximately 4 ms to perform. The ions are kept together by careful tuning of the phases of the optical-dipole force. Refocusing operations are required to counteract qubit dephasing caused by fluctuations in the local magnetic field.

Figure 13-21 Quantum Error Correction [view large image]

Data encryption is used to protect messages and files from prying eavesdroppers. In its simplest form, a coded message can be created in which each letter is substituted with the letter that is two down from it in the alphabet. So "A" becomes "C", "B" becomes "D", ... and so on. The recipient was told that the code (key) is: "Shift by 2". He/she can then decodes the message accordingly. Anyone else will only see a garbled message. Modern encryption employs two keys to provide greater security. The sender selects a public-key such as 1525381, which is the product of two prime numbers: 10667 and 143. This key is used to convert a block of text via an algorithm (a formula for combining the key with the text). The recipent must used a private key such as 143 to decode the encrypted text in the reverse process. The security of public-key cryptography depends on factorization - the fact that it is easy to compute the product of two large numbers but extremely hard to

Figure 13-22 Quantum Encryption [view large image]

factor it back into the primes. But the advent of the quantum information era - and, in particular, the capability of quantum computers to rapidly perform monstrously challenging factorizations - may portend the eventual demise of such cryptographic scheme.

Unlike public-key cryptography, quantum cryptography should remain secure when quantum computers arrive on the scene. One way of sending a quantum-cryptographic key between sender and receiver requires that a very low intensity laser transmits single photons that are polarized in one of two modes as shown in Figure 13-22. Followings illustrate the steps in establishing the key (see Figure 13-22):


1. Alice sends a photon through either the 0 or 1 slot of the rectilinear or diagonal polarizing filters, while making a record of the various orientations.
2. For each incoming bit, Bob chooses randomly which filter slot he uses for detection and writes down both the polarization and the bit value.
3. After all the photons have reached Bob, he tells Alice over a public channel, perhaps by telephone or an e-mail, the sequence of filters he used for the incoming photons, but not the bit value of the photons.
4. Alice tells Bob during the same conversation which filters he chose correctly. Those instances constitute the bits that Alice and Bob will use to form the key that they will use to encrypt messages.
5. If Eve the eavesdropper tries to spy on the train of photons, quantum mechanics prohibits her from using filters to detect the orientation of a photon. If she chooses the wrong filter, she will create errors by modifying their polarization.

Footnotes

References:

• Periodic Table -- http://chemserv.bc.edu/web-elements/
• Band Theory -- http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html
• Band Theory, Metal -- http://www.owlnet.rice.edu/~chem152/lecture/Reading/bands.html
• Superconductivity -- http://www.physics.carleton.ca/courses/75.364/mp-2html/node16.html
• Laser -- http://www.hk-phy.org/articles/laser/laser_e.html
• Plasma Physics -- http://www.fys.uio.no/plasma/english/texts/plsmgenintro/plsmgenintro.html
• Quantum Computing, Centre for Quantum Computation -- http://www.qubit.org/

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