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2.1.1: Overview of Atomic Structure - Biology

2.1.1: Overview of  Atomic Structure - Biology


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Atoms are made up of particles called protons, neutrons, and electrons, which are responsible for the mass and charge of atoms.

Learning Objectives

  • Discuss the electronic and structural properties of an atom

Key Points

  • An atom is composed of two regions: the nucleus, which is in the center of the atom and contains protons and neutrons, and the outer region of the atom, which holds its electrons in orbit around the nucleus.
  • Protons and neutrons have approximately the same mass, about 1.67 × 10-24 grams, which scientists define as one atomic mass unit (amu) or one Dalton.
  • Each electron has a negative charge (-1) equal to the positive charge of a proton (+1).
  • Neutrons are uncharged particles found within the nucleus.

Key Terms

  • atom: The smallest possible amount of matter which still retains its identity as a chemical element, consisting of a nucleus surrounded by electrons.
  • proton: Positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element. It weighs 1 amu.
  • neutron: A subatomic particle forming part of the nucleus of an atom. It has no charge. It is equal in mass to a proton or it weighs 1 amu.

An atom is the smallest unit of matter that retains all of the chemical properties of an element. Atoms combine to form molecules, which then interact to form solids, gases, or liquids. For example, water is composed of hydrogen and oxygen atoms that have combined to form water molecules. Many biological processes are devoted to breaking down molecules into their component atoms so they can be reassembled into a more useful molecule.

Atomic Particles

Atoms consist of three basic particles: protons, electrons, and neutrons. The nucleus (center) of the atom contains the protons (positively charged) and the neutrons (no charge). The outermost regions of the atom are called electron shells and contain the electrons (negatively charged). Atoms have different properties based on the arrangement and number of their basic particles.

The hydrogen atom (H) contains only one proton, one electron, and no neutrons. This can be determined using the atomic number and the mass number of the element (see the concept on atomic numbers and mass numbers).

Atomic Mass

Protons and neutrons have approximately the same mass, about 1.67 × 10-24 grams. Scientists define this amount of mass as one atomic mass unit (amu) or one Dalton. Although similar in mass, protons are positively charged, while neutrons have no charge. Therefore, the number of neutrons in an atom contributes significantly to its mass, but not to its charge.

Electrons are much smaller in mass than protons, weighing only 9.11 × 10-28grams, or about 1/1800 of an atomic mass unit. Therefore, they do not contribute much to an element’s overall atomic mass. When considering atomic mass, it is customary to ignore the mass of any electrons and calculate the atom’s mass based on the number of protons and neutrons alone.

Electrons contribute greatly to the atom’s charge, as each electron has a negative charge equal to the positive charge of a proton. Scientists define these charges as “+1” and “-1. ” In an uncharged, neutral atom, the number of electrons orbiting the nucleus is equal to the number of protons inside the nucleus. In these atoms, the positive and negative charges cancel each other out, leading to an atom with no net charge.

Exploring Electron Properties: Compare the behavior of electrons to that of other charged particles to discover properties of electrons such as charge and mass.

Volume of Atoms

Accounting for the sizes of protons, neutrons, and electrons, most of the volume of an atom—greater than 99 percent—is, in fact, empty space. Despite all this empty space, solid objects do not just pass through one another. The electrons that surround all atoms are negatively charged and cause atoms to repel one another, preventing atoms from occupying the same space. These intermolecular forces prevent you from falling through an object like your chair.

Interactive: Build an Atom: Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!


2.1.1: Overview of Atomic Structure - Biology

Atoms are made up of particles called protons, neutrons, and electrons, which are responsible for the mass and charge of atoms.

Learning Objectives

Discuss the electronic and structural properties of an atom

Key Takeaways

Key Points

  • An atom is composed of two regions: the nucleus, which is in the center of the atom and contains protons and neutrons, and the outer region of the atom, which holds its electrons in orbit around the nucleus.
  • Protons and neutrons have approximately the same mass, about 1.67 × 10 −24 grams, which scientists define as one atomic mass unit (amu) or one Dalton.
  • Each electron has a negative charge (−1) equal to the positive charge of a proton (+1).
  • Neutrons are uncharged particles found within the nucleus.

Key Terms

  • atom: The smallest possible amount of matter which still retains its identity as a chemical element, consisting of a nucleus surrounded by electrons.
  • proton: Positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element. It weighs 1 amu.
  • neutron: A subatomic particle forming part of the nucleus of an atom. It has no charge. It is equal in mass to a proton or it weighs 1 amu.

An atom is the smallest unit of matter that retains all of the chemical properties of an element. Atoms combine to form molecules, which then interact to form solids, gases, or liquids. For example, water is composed of hydrogen and oxygen atoms that have combined to form water molecules. Many biological processes are devoted to breaking down molecules into their component atoms so they can be reassembled into a more useful molecule.

Atomic Particles

Atoms consist of three basic particles: protons, electrons, and neutrons. The nucleus (center) of the atom contains the protons (positively charged) and the neutrons (no charge). The outermost regions of the atom are called electron shells and contain the electrons (negatively charged). Atoms have different properties based on the arrangement and number of their basic particles.

The hydrogen atom (H) contains only one proton, one electron, and no neutrons. This can be determined using the atomic number and the mass number of the element (see the concept on atomic numbers and mass numbers).

Structure of an atom: Elements, such as helium, depicted here, are made up of atoms. Atoms are made up of protons and neutrons located within the nucleus, with electrons in orbitals surrounding the nucleus.

Atomic Mass

Protons and neutrons have approximately the same mass, about 1.67 × 10 −24 grams. Scientists define this amount of mass as one atomic mass unit (amu) or one Dalton. Although similar in mass, protons are positively charged, while neutrons have no charge. Therefore, the number of neutrons in an atom contributes significantly to its mass, but not to its charge.

Electrons are much smaller in mass than protons, weighing only 9.11 × 10 −28 grams, or about 1/1800 of an atomic mass unit. Therefore, they do not contribute much to an element’s overall atomic mass. When considering atomic mass, it is customary to ignore the mass of any electrons and calculate the atom’s mass based on the number of protons and neutrons alone.

Electrons contribute greatly to the atom’s charge, as each electron has a negative charge equal to the positive charge of a proton. Scientists define these charges as “+1” and “−1.” In an uncharged, neutral atom, the number of electrons orbiting the nucleus is equal to the number of protons inside the nucleus. In these atoms, the positive and negative charges cancel each other out, leading to an atom with no net charge.

Protons, Neutrons, and Electrons
Charge Mass (amu) Location
Proton +1 1 nucleus
Neutron 0 1 nucleus
Electron −1 0 orbitals

Both protons and neutrons have a mass of 1 amu and are found in the nucleus. However, protons have a charge of +1, and neutrons are uncharged. Electrons have a mass of approximately 0 amu, orbit the nucleus, and have a charge of−1.

Exploring Electron Properties: Compare the behavior of electrons to that of other charged particles to discover properties of electrons such as charge and mass.

Volume of Atoms

Accounting for the sizes of protons, neutrons, and electrons, most of the volume of an atom—greater than 99 percent—is, in fact, empty space. Despite all this empty space, solid objects do not just pass through one another. The electrons that surround all atoms are negatively charged and cause atoms to repel one another, preventing atoms from occupying the same space. These intermolecular forces prevent you from falling through an object like your chair.

Interactive: Build an Atom: Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!


Radiation Effects in Graphite☆

Anne A. Campbell , Timothy D. Burchell , in Comprehensive Nuclear Materials (Second Edition) , 2020

3.11.5.1 Macroscopic Structure

The atomic structure of graphite is understood, as shown in Fig. 3 , but when discussing the bulk property changes it becomes necessary to understand the larger meso and macroscopic features. The atomic structure of graphite is anisotropic, and this anisotropy is results in the graphite filler particles (i.e., grains) also being anisotropic. The graphite filler particles ( Fig. 13(a) ) are made up of 100’s-10000’s of coherent domains (i.e., crystallites) that have a perfect graphite structure. Within a filler particle these coherent domains have an overall <c>-axis orientation, but there is rotational and tilt misorientation from one crystallite to another. März 76 recently proposed the “Crazy Paver” structure to describe the crystallite orientation in filler particles. The other critical feature in the filler particles is the Mrozowski (a.k.a. thermal cracks) cracks that are stress relief cracks associated with the stresses due to the CTE mismatch between the <a>-axis and <c>-axis directions. 77 Upon heating in TEM these cracks are observed to accommodate the <c>-direction expansion of the crystallites. 78 Bulk graphite is nearly isotropic, which is due to the randomized orientation of filler particles within the bulk material ( Fig. 13(b) ). The other features at the grain-sized scale include the binder regions and the intergranular porosity. The microcracks and intergranular porosity in graphite result in the bulk material having a density that is 15%–20% less than theoretical.

Fig. 13 . (a) Example of the graphite “grain” showing the overall &ltc&gt-axis direction, sub-grain crystallite regions and Mrozowski cracks. (b)

Reproduced from Fig. 1 in Campbell, A.A., Was, G.S., 2014. Proton irradiation-induced creep of ultra-fine grain graphite. Carbon 77, 993–1010.

The processing and forming method of graphite is tailored to produce a bulk material that is nearly isotropic even though the crystal structure is anisotropic. There is still some residual preferential orientation that arises from the orientation of filler particles during forming. In extruded material the basal planes are more likely to orientate parallel to the extrusion direction, in pressed material the <c>-axis is more likely to orientate parallel to the pressing direction, and isostatically pressed materials have been thought to not have any apparent preferential orientation. In pressed and extruded graphite, the orientation is defined relative to the preferred <c>-axis orientation (referred to as Against Grain – AG) and the two orthogonal directions that are parallel to the basal planes (With Grain – WG). In pressed graphite the preferred <c>-axis direction is parallel to the pressing direction, while in extruded graphite the preferred <c>-axis direction is perpendicular to the pressing direction. In isostatically pressed graphite there is minimal preferred orientation and no single pressing direction, so the newer nomenclature is to reference the billet direction that is parallel to the gravitational force during forming as the axial (AX) direction and the orthogonal directions as transverse (TR). Schematics of the preferred orientation in the different forming methods is shown in Fig. 14 .

Fig. 14 . Orientation schematics of pressed graphite (left), extruded graphite (center) and isostatically pressed graphite (right).


2.1 Early Ideas in Atomic Theory

The earliest recorded discussion of the basic structure of matter comes from ancient Greek philosophers, the scientists of their day. In the fifth century BC, Leucippus and Democritus argued that all matter was composed of small, finite particles that they called atomos, a term derived from the Greek word for “indivisible.” They thought of atoms as moving particles that differed in shape and size, and which could join together. Later, Aristotle and others came to the conclusion that matter consisted of various combinations of the four “elements”—fire, earth, air, and water—and could be infinitely divided. Interestingly, these philosophers thought about atoms and “elements” as philosophical concepts, but apparently never considered performing experiments to test their ideas.

The Aristotelian view of the composition of matter held sway for over two thousand years, until English schoolteacher John Dalton helped to revolutionize chemistry with his hypothesis that the behavior of matter could be explained using an atomic theory. First published in 1807, many of Dalton’s hypotheses about the microscopic features of matter are still valid in modern atomic theory. Here are the postulates of Dalton’s atomic theory .

  1. Matter is composed of exceedingly small particles called atoms. An atom is the smallest unit of an element that can participate in a chemical change.
  2. An element consists of only one type of atom, which has a mass that is characteristic of the element and is the same for all atoms of that element (Figure 2.2). A macroscopic sample of an element contains an incredibly large number of atoms, all of which have identical chemical properties.

Dalton’s atomic theory provides a microscopic explanation of the many macroscopic properties of matter that you’ve learned about. For example, if an element such as copper consists of only one kind of atom, then it cannot be broken down into simpler substances, that is, into substances composed of fewer types of atoms. And if atoms are neither created nor destroyed during a chemical change, then the total mass of matter present when matter changes from one type to another will remain constant (the law of conservation of matter).

Example 2.1

Testing Dalton’s Atomic Theory

Solution

Check Your Learning

Answer:

The starting materials consist of four green spheres and two purple spheres. The products consist of four green spheres and two purple spheres. This does not violate any of Dalton’s postulates: Atoms are neither created nor destroyed, but are redistributed in small, whole-number ratios.

Dalton knew of the experiments of French chemist Joseph Proust, who demonstrated that all samples of a pure compound contain the same elements in the same proportion by mass. This statement is known as the law of definite proportions or the law of constant composition . The suggestion that the numbers of atoms of the elements in a given compound always exist in the same ratio is consistent with these observations. For example, when different samples of isooctane (a component of gasoline and one of the standards used in the octane rating system) are analyzed, they are found to have a carbon-to-hydrogen mass ratio of 5.33:1, as shown in Table 2.1.

SampleCarbonHydrogenMass Ratio
A14.82 g2.78 g 14.82 g carbon 2.78 g hydrogen = 5.33 g carbon 1.00 g hydrogen 14.82 g carbon 2.78 g hydrogen = 5.33 g carbon 1.00 g hydrogen
B22.33 g4.19 g 22.33 g carbon 4.19 g hydrogen = 5.33 g carbon 1.00 g hydrogen 22.33 g carbon 4.19 g hydrogen = 5.33 g carbon 1.00 g hydrogen
C19.40 g3.64 g 19.40 g carbon 3.63 g hydrogen = 5.33 g carbon 1.00 g hydrogen 19.40 g carbon 3.63 g hydrogen = 5.33 g carbon 1.00 g hydrogen

It is worth noting that although all samples of a particular compound have the same mass ratio, the converse is not true in general. That is, samples that have the same mass ratio are not necessarily the same substance. For example, there are many compounds other than isooctane that also have a carbon-to-hydrogen mass ratio of 5.33:1.00.

Dalton also used data from Proust, as well as results from his own experiments, to formulate another interesting law. The law of multiple proportions states that when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small, whole numbers. For example, copper and chlorine can form a green, crystalline solid with a mass ratio of 0.558 g chlorine to 1 g copper, as well as a brown crystalline solid with a mass ratio of 1.116 g chlorine to 1 g copper. These ratios by themselves may not seem particularly interesting or informative however, if we take a ratio of these ratios, we obtain a useful and possibly surprising result: a small, whole-number ratio.

This 2-to-1 ratio means that the brown compound has twice the amount of chlorine per amount of copper as the green compound.

This can be explained by atomic theory if the copper-to-chlorine ratio in the brown compound is 1 copper atom to 2 chlorine atoms, and the ratio in the green compound is 1 copper atom to 1 chlorine atom. The ratio of chlorine atoms (and thus the ratio of their masses) is therefore 2 to 1 (Figure 2.5).

Example 2.2

Laws of Definite and Multiple Proportions

Solution

In compound B, the mass ratio of oxygen to carbon is:

The ratio of these ratios is:

This supports the law of multiple proportions. This means that A and B are different compounds, with A having one-half as much oxygen per amount of carbon (or twice as much carbon per amount of oxygen) as B. A possible pair of compounds that would fit this relationship would be A = CO and B = CO2.

Check Your Learning

Answer:

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    Atomic Number and Mass Number

    The atomic number is the number of protons in an element, while the mass number is the number of protons plus the number of neutrons.

    Learning Objectives

    Determine the relationship between the mass number of an atom, its atomic number, its atomic mass, and its number of subatomic particles

    Key Takeaways

    Key Points

    • Neutral atoms of each element contain an equal number of protons and electrons.
    • The number of protons determines an element’s atomic number and is used to distinguish one element from another.
    • The number of neutrons is variable, resulting in isotopes, which are different forms of the same atom that vary only in the number of neutrons they possess.
    • Together, the number of protons and the number of neutrons determine an element’s mass number.
    • Since an element’s isotopes have slightly different mass numbers, the atomic mass is calculated by obtaining the mean of the mass numbers for its isotopes.

    Key Terms

    • mass number: The sum of the number of protons and the number of neutrons in an atom.
    • atomic number: The number of protons in an atom.
    • atomic mass: The average mass of an atom, taking into account all its naturally occurring isotopes.

    Atomic Number

    Neutral atoms of an element contain an equal number of protons and electrons. The number of protons determines an element’s atomic number (Z) and distinguishes one element from another. For example, carbon’s atomic number (Z) is 6 because it has 6 protons. The number of neutrons can vary to produce isotopes, which are atoms of the same element that have different numbers of neutrons. The number of electrons can also be different in atoms of the same element, thus producing ions (charged atoms). For instance, iron, Fe, can exist in its neutral state, or in the +2 and +3 ionic states.

    Mass Number

    An element’s mass number (A) is the sum of the number of protons and the number of neutrons. The small contribution of mass from electrons is disregarded in calculating the mass number. This approximation of mass can be used to easily calculate how many neutrons an element has by simply subtracting the number of protons from the mass number. Protons and neutrons both weigh about one atomic mass unit or amu. Isotopes of the same element will have the same atomic number but different mass numbers.

    Atomic number, chemical symbol, and mass number: Carbon has an atomic number of six, and two stable isotopes with mass numbers of twelve and thirteen, respectively. Its average atomic mass is 12.11.

    Scientists determine the atomic mass by calculating the mean of the mass numbers for its naturally-occurring isotopes. Often, the resulting number contains a decimal. For example, the atomic mass of chlorine (Cl) is 35.45 amu because chlorine is composed of several isotopes, some (the majority) with an atomic mass of 35 amu (17 protons and 18 neutrons) and some with an atomic mass of 37 amu (17 protons and 20 neutrons).

    Given an atomic number (Z) and mass number (A), you can find the number of protons, neutrons, and electrons in a neutral atom. For example, a lithium atom (Z=3, A=7 amu) contains three protons (found from Z), three electrons (as the number of protons is equal to the number of electrons in an atom), and four neutrons (7 – 3 = 4).


    2.1 Early Ideas in Atomic Theory

    The earliest recorded discussion of the basic structure of matter comes from ancient Greek philosophers, the scientists of their day. In the fifth century BC, Leucippus and Democritus argued that all matter was composed of small, finite particles that they called atomos, a term derived from the Greek word for “indivisible.” They thought of atoms as moving particles that differed in shape and size, and which could join together. Later, Aristotle and others came to the conclusion that matter consisted of various combinations of the four “elements”—fire, earth, air, and water—and could be infinitely divided. Interestingly, these philosophers thought about atoms and “elements” as philosophical concepts, but apparently never considered performing experiments to test their ideas.

    The Aristotelian view of the composition of matter held sway for over two thousand years, until English schoolteacher John Dalton helped to revolutionize chemistry with his hypothesis that the behavior of matter could be explained using an atomic theory. First published in 1807, many of Dalton’s hypotheses about the microscopic features of matter are still valid in modern atomic theory. Here are the postulates of Dalton’s atomic theory .

    1. Matter is composed of exceedingly small particles called atoms. An atom is the smallest unit of an element that can participate in a chemical change.
    2. An element consists of only one type of atom, which has a mass that is characteristic of the element and is the same for all atoms of that element (Figure 2.2). A macroscopic sample of an element contains an incredibly large number of atoms, all of which have identical chemical properties.

    Dalton’s atomic theory provides a microscopic explanation of the many macroscopic properties of matter that you’ve learned about. For example, if an element such as copper consists of only one kind of atom, then it cannot be broken down into simpler substances, that is, into substances composed of fewer types of atoms. And if atoms are neither created nor destroyed during a chemical change, then the total mass of matter present when matter changes from one type to another will remain constant (the law of conservation of matter).

    Example 2.1

    Testing Dalton’s Atomic Theory

    Solution

    Check Your Learning

    Answer:

    The starting materials consist of four green spheres and two purple spheres. The products consist of four green spheres and two purple spheres. This does not violate any of Dalton’s postulates: Atoms are neither created nor destroyed, but are redistributed in small, whole-number ratios.

    Dalton knew of the experiments of French chemist Joseph Proust, who demonstrated that all samples of a pure compound contain the same elements in the same proportion by mass. This statement is known as the law of definite proportions or the law of constant composition . The suggestion that the numbers of atoms of the elements in a given compound always exist in the same ratio is consistent with these observations. For example, when different samples of isooctane (a component of gasoline and one of the standards used in the octane rating system) are analyzed, they are found to have a carbon-to-hydrogen mass ratio of 5.33:1, as shown in Table 2.1.

    SampleCarbonHydrogenMass Ratio
    A14.82 g2.78 g 14.82 g carbon 2.78 g hydrogen = 5.33 g carbon 1.00 g hydrogen 14.82 g carbon 2.78 g hydrogen = 5.33 g carbon 1.00 g hydrogen
    B22.33 g4.19 g 22.33 g carbon 4.19 g hydrogen = 5.33 g carbon 1.00 g hydrogen 22.33 g carbon 4.19 g hydrogen = 5.33 g carbon 1.00 g hydrogen
    C19.40 g3.64 g 19.40 g carbon 3.63 g hydrogen = 5.33 g carbon 1.00 g hydrogen 19.40 g carbon 3.63 g hydrogen = 5.33 g carbon 1.00 g hydrogen

    It is worth noting that although all samples of a particular compound have the same mass ratio, the converse is not true in general. That is, samples that have the same mass ratio are not necessarily the same substance. For example, there are many compounds other than isooctane that also have a carbon-to-hydrogen mass ratio of 5.33:1.00.

    Dalton also used data from Proust, as well as results from his own experiments, to formulate another interesting law. The law of multiple proportions states that when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small, whole numbers. For example, copper and chlorine can form a green, crystalline solid with a mass ratio of 0.558 g chlorine to 1 g copper, as well as a brown crystalline solid with a mass ratio of 1.116 g chlorine to 1 g copper. These ratios by themselves may not seem particularly interesting or informative however, if we take a ratio of these ratios, we obtain a useful and possibly surprising result: a small, whole-number ratio.

    This 2-to-1 ratio means that the brown compound has twice the amount of chlorine per amount of copper as the green compound.

    This can be explained by atomic theory if the copper-to-chlorine ratio in the brown compound is 1 copper atom to 2 chlorine atoms, and the ratio in the green compound is 1 copper atom to 1 chlorine atom. The ratio of chlorine atoms (and thus the ratio of their masses) is therefore 2 to 1 (Figure 2.5).

    Example 2.2

    Laws of Definite and Multiple Proportions

    Solution

    In compound B, the mass ratio of oxygen to carbon is:

    The ratio of these ratios is:

    This supports the law of multiple proportions. This means that A and B are different compounds, with A having one-half as much oxygen per amount of carbon (or twice as much carbon per amount of oxygen) as B. A possible pair of compounds that would fit this relationship would be A = CO and B = CO2.

    Check Your Learning

    Answer:

    As an Amazon Associate we earn from qualifying purchases.

    Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

      If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    • Use the information below to generate a citation. We recommend using a citation tool such as this one.
      • Authors: Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson, PhD
      • Publisher/website: OpenStax
      • Book title: Chemistry 2e
      • Publication date: Feb 14, 2019
      • Location: Houston, Texas
      • Book URL: https://openstax.org/books/chemistry-2e/pages/1-introduction
      • Section URL: https://openstax.org/books/chemistry-2e/pages/2-1-early-ideas-in-atomic-theory

      © Jan 22, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.


      S Orbitals

      Three things happen to s orbitals as n increases (Figure (PageIndex<2>)):

      1. They become larger, extending farther from the nucleus.
      2. They contain more nodes. This is similar to a standing wave that has regions of significant amplitude separated by nodes, points with zero amplitude.
      3. For a given atom, the s orbitals also become higher in energy as n increases because of their increased distance from the nucleus.

      Orbitals are generally drawn as three-dimensional surfaces that enclose 90% of the electron density , as was shown for the hydrogen 1s, 2s, and 3s orbitals in part (b) in Figure (PageIndex<2>). Although such drawings show the relative sizes of the orbitals, they do not normally show the spherical nodes in the 2s and 3s orbitals because the spherical nodes lie inside the 90% surface. Fortunately, the positions of the spherical nodes are not important for chemical bonding.


      Contents

      Consider a three-dimensional harmonic oscillator. This would give, for example, in the first three levels ("" is the angular momentum quantum number)

      We can imagine ourselves building a nucleus by adding protons and neutrons. These will always fill the lowest available level. Thus the first two protons fill level zero, the next six protons fill level one, and so on. As with electrons in the periodic table, protons in the outermost shell will be relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore, nuclei which have a full outer proton shell will have a higher binding energy than other nuclei with a similar total number of protons. All this is true for neutrons as well.

      This means that the magic numbers are expected to be those in which all occupied shells are full. We see that for the first two numbers we get 2 (level 0 full) and 8 (levels 0 and 1 full), in accord with experiment. However the full set of magic numbers does not turn out correctly. These can be computed as follows:

      In particular, the first six shells are:

      • level 0: 2 states ( = 0) = 2.
      • level 1: 6 states ( = 1) = 6.
      • level 2: 2 states ( = 0) + 10 states ( = 2) = 12.
      • level 3: 6 states ( = 1) + 14 states ( = 3) = 20.
      • level 4: 2 states ( = 0) + 10 states ( = 2) + 18 states ( = 4) = 30.
      • level 5: 6 states ( = 1) + 14 states ( = 3) + 22 states ( = 5) = 42.

      where for every there are 2+1 different values of ml and 2 values of ms, giving a total of 4+2 states for every specific level.

      These numbers are twice the values of triangular numbers from the Pascal Triangle: 1, 3, 6, 10, 15, 21, .

      Including a spin–orbit interaction Edit

      We next include a spin–orbit interaction. First we have to describe the system by the quantum numbers j, mj and parity instead of , ml and ms, as in the hydrogen–like atom. Since every even level includes only even values of , it includes only states of even (positive) parity. Similarly, every odd level includes only states of odd (negative) parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are

      • level 0 (n = 0): 2 states (j =
      • 1 ⁄ 2 ). Even parity.
      • level 1 (n = 1): 2 states (j =
      • 1 ⁄ 2 ) + 4 states (j =
      • 3 ⁄ 2 ) = 6. Odd parity.
      • level 2 (n = 2): 2 states (j =
      • 1 ⁄ 2 ) + 4 states (j =
      • 3 ⁄ 2 ) + 6 states (j =
      • 5 ⁄ 2 ) = 12. Even parity.
      • level 3 (n = 3): 2 states (j =
      • 1 ⁄ 2 ) + 4 states (j =
      • 3 ⁄ 2 ) + 6 states (j =
      • 5 ⁄ 2 ) + 8 states (j =
      • 7 ⁄ 2 ) = 20. Odd parity.
      • level 4 (n = 4): 2 states (j =
      • 1 ⁄ 2 ) + 4 states (j =
      • 3 ⁄ 2 ) + 6 states (j =
      • 5 ⁄ 2 ) + 8 states (j =
      • 7 ⁄ 2 ) + 10 states (j =
      • 9 ⁄ 2 ) = 30. Even parity.
      • level 5 (n = 5): 2 states (j =
      • 1 ⁄ 2 ) + 4 states (j =
      • 3 ⁄ 2 ) + 6 states (j =
      • 5 ⁄ 2 ) + 8 states (j =
      • 7 ⁄ 2 ) + 10 states (j =
      • 9 ⁄ 2 ) + 12 states (j =
      • 11 ⁄ 2 ) = 42. Odd parity.

      where for every j there are 2j + 1 different states from different values of mj.

      For example, consider the states at level 4:

      • The 10 states with j =
      • 9 ⁄ 2 come from = 4 and s parallel to . Thus they have a positive spin–orbit interaction energy.
      • The 8 states with j =
      • 7 ⁄ 2 came from = 4 and s anti-parallel to . Thus they have a negative spin–orbit interaction energy.
      • The 6 states with j =
      • 5 ⁄ 2 came from = 2 and s parallel to . Thus they have a positive spin–orbit interaction energy. However its magnitude is half compared to the states with j =
      • 9 ⁄ 2 .
      • The 4 states with j =
      • 3 ⁄ 2 came from = 2 and s anti-parallel to . Thus they have a negative spin–orbit interaction energy. However its magnitude is half compared to the states with j =
      • 7 ⁄ 2 .
      • The 2 states with j =
      • 1 ⁄ 2 came from = 0 and thus have zero spin–orbit interaction energy.

      Changing the profile of the potential Edit

      The harmonic oscillator potential V ( r ) = μ ω 2 r 2 / 2 r^<2>/2> grows infinitely as the distance from the center r goes to infinity. A more realistic potential, such as Woods–Saxon potential, would approach a constant at this limit. One main consequence is that the average radius of nucleons' orbits would be larger in a realistic potential This leads to a reduced term ℏ 2 l ( l + 1 ) / 2 m r 2 l(l+1)/2mr^<2>> in the Laplace operator of the Hamiltonian. Another main difference is that orbits with high average radii, such as those with high n or high , will have a lower energy than in a harmonic oscillator potential. Both effects lead to a reduction in the energy levels of high orbits.

      Predicted magic numbers Edit

      Together with the spin–orbit interaction, and for appropriate magnitudes of both effects, one is led to the following qualitative picture: At all levels, the highest j states have their energies shifted downwards, especially for high n (where the highest j is high). This is both due to the negative spin–orbit interaction energy and to the reduction in energy resulting from deforming the potential to a more realistic one. The second-to-highest j states, on the contrary, have their energy shifted up by the first effect and down by the second effect, leading to a small overall shift. The shifts in the energy of the highest j states can thus bring the energy of states of one level to be closer to the energy of states of a lower level. The "shells" of the shell model are then no longer identical to the levels denoted by n, and the magic numbers are changed.

      We may then suppose that the highest j states for n = 3 have an intermediate energy between the average energies of n = 2 and n = 3, and suppose that the highest j states for larger n (at least up to n = 7) have an energy closer to the average energy of n − 1 . Then we get the following shells (see the figure)

      • 1st shell: 2 states (n = 0, j =
      • 1 ⁄ 2 ).
      • 2nd shell: 6 states (n = 1, j =
      • 1 ⁄ 2 or
      • 3 ⁄ 2 ).
      • 3rd shell: 12 states (n = 2, j =
      • 1 ⁄ 2 ,
      • 3 ⁄ 2 or
      • 5 ⁄ 2 ).
      • 4th shell: 8 states (n = 3, j =
      • 7 ⁄ 2 ).
      • 5th shell: 22 states (n = 3, j =
      • 1 ⁄ 2 ,
      • 3 ⁄ 2 or
      • 5 ⁄ 2 n = 4, j =
      • 9 ⁄ 2 ).
      • 6th shell: 32 states (n = 4, j =
      • 1 ⁄ 2 ,
      • 3 ⁄ 2 ,
      • 5 ⁄ 2 or
      • 7 ⁄ 2 n = 5, j =
      • 11 ⁄ 2 ).
      • 7th shell: 44 states (n = 5, j =
      • 1 ⁄ 2 ,
      • 3 ⁄ 2 ,
      • 5 ⁄ 2 ,
      • 7 ⁄ 2 or
      • 9 ⁄ 2 n = 6, j =
      • 13 ⁄ 2 ).
      • 8th shell: 58 states (n = 6, j =
      • 1 ⁄ 2 ,
      • 3 ⁄ 2 ,
      • 5 ⁄ 2 ,
      • 7 ⁄ 2 ,
      • 9 ⁄ 2 or
      • 11 ⁄ 2 n = 7, j =
      • 15 ⁄ 2 ).

      Note that the numbers of states after the 4th shell are doubled triangular numbers plus two. Spin–orbit coupling causes so-called 'intruder levels' to drop down from the next higher shell into the structure of the previous shell. The sizes of the intruders are such that the resulting shell sizes are themselves increased to the very next higher doubled triangular numbers from those of the harmonic oscillator. For example, 1f2p has 20 nucleons, and spin–orbit coupling adds 1g9/2 (10 nucleons) leading to a new shell with 30 nucleons. 1g2d3s has 30 nucleons, and addition of intruder 1h11/2 (12 nucleons) yields a new shell size of 42, and so on.

      The magic numbers are then

      • 2
      • 8 = 2 + 6
      • 20 = 2 + 6 + 12
      • 28 = 2 + 6 + 12 + 8
      • 50 = 2 + 6 + 12 + 8 + 22
      • 82 = 2 + 6 + 12 + 8 + 22 + 32
      • 126 = 2 + 6 + 12 + 8 + 22 + 32 + 44
      • 184 = 2 + 6 + 12 + 8 + 22 + 32 + 44 + 58

      and so on. This gives all the observed magic numbers, and also predicts a new one (the so-called island of stability) at the value of 184 (for protons, the magic number 126 has not been observed yet, and more complicated theoretical considerations predict the magic number to be 114 instead).

      Another way to predict magic (and semi-magic) numbers is by laying out the idealized filling order (with spin–orbit splitting but energy levels not overlapping). For consistency s is split into j = 1⁄2 and j = -1⁄2 components with 2 and 0 members respectively. Taking leftmost and rightmost total counts within sequences marked bounded by / here gives the magic and semi-magic numbers.

      • s(2,0)/p(4,2) > 2,2/6,8, so (semi)magic numbers 2,2/6,8
      • d(6,4):s(2,0)/f(8,6):p(4,2) > 14,18:20,20/28,34:38,40, so 14,20/28,40
      • g(10,8):d(6,4):s(2,0)/h(12,10):f(8,6):p(4,2) > 50,58,64,68,70,70/82,92,100,106,110,112, so 50,70/82,112
      • i(14,12):g(10,8):d(6,4):s(2,0)/j(16,14):h(12,10):f(8,6):p(4,2) > 126,138,148,156,162,166,168,168/184,198,210,220,228,234,238,240, so 126,168/184,240

      The rightmost predicted magic numbers of each pair within the quartets bisected by / are double tetrahedral numbers from the Pascal Triangle: 2, 8, 20, 40, 70, 112, 168, 240 are 2x 1, 4, 10, 20, 35, 56, 84, 120, . and the leftmost members of the pairs differ from the rightmost by double triangular numbers: 2 − 2 = 0, 8 − 6 = 2, 20 − 14 = 6, 40 − 28 = 12, 70 − 50 = 20, 112 − 82 = 30, 168 − 126 = 42, 240 − 184 = 56, where 0, 2, 6, 12, 20, 30, 42, 56, . are 2 × 0, 1, 3, 6, 10, 15, 21, 28, . .

      Other properties of nuclei Edit

      For nuclei farther from the magic numbers one must add the assumption that due to the relation between the strong nuclear force and angular momentum, protons or neutrons with the same n tend to form pairs of opposite angular momenta. Therefore, a nucleus with an even number of protons and an even number of neutrons has 0 spin and positive parity. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the parity of the last neutron (or proton), and the spin equal to the total angular momentum of this neutron (or proton). By "last" we mean the properties coming from the highest energy level.

      In the case of a nucleus with an odd number of protons and an odd number of neutrons, one must consider the total angular momentum and parity of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus spin will be one of the possible results of the sum of their angular momenta (with other possible results being excited states of the nucleus).

      The ordering of angular momentum levels within each shell is according to the principles described above – due to spin–orbit interaction, with high angular momentum states having their energies shifted downwards due to the deformation of the potential (i.e. moving from a harmonic oscillator potential to a more realistic one). For nucleon pairs, however, it is often energetically favorable to be at high angular momentum, even if its energy level for a single nucleon would be higher. This is due to the relation between angular momentum and the strong nuclear force.

      Nuclear magnetic moment is partly predicted by this simple version of the shell model. The magnetic moment is calculated through j, and s of the "last" nucleon, but nuclei are not in states of well defined and s. Furthermore, for odd-odd nuclei, one has to consider the two "last" nucleons, as in deuterium. Therefore, one gets several possible answers for the nuclear magnetic moment, one for each possible combined and s state, and the real state of the nucleus is a superposition of them. Thus the real (measured) nuclear magnetic moment is somewhere in between the possible answers.

      The electric dipole of a nucleus is always zero, because its ground state has a definite parity, so its matter density ( ψ 2 > , where ψ is the wavefunction) is always invariant under parity. This is usually the situations with the atomic electric dipole as well.

      Higher electric and magnetic multipole moments cannot be predicted by this simple version of the shell model, for the reasons similar to those in the case of deuterium.

      For nuclei having two or more valence nucleons (i.e. nucleons outside a closed shell) a residual two-body interaction must be added. This residual term comes from the part of the inter-nucleon interaction not included in the approximative average potential. Through this inclusion different shell configurations are mixed and the energy degeneracy of states corresponding to the same configuration is broken. [4] [5]

      These residual interactions are incorporated through shell model calculations in a truncated model space (or valence space). This space is spanned by a basis of many-particle states where only single-particle states in the model space are active. The Schrödinger equation is solved in this basis, using an effective Hamiltonian specifically suited for the model space. This Hamiltonian is different from the one of free nucleons as it among other things has to compensate for excluded configurations. [5]

      One can do away with the average potential approximation entirely by extending the model space to the previously inert core and treat all single-particle states up to the model space truncation as active. This forms the basis of the no-core shell model, which is an ab initio method. It is necessary to include a three-body interaction in such calculations to achieve agreement with experiments. [6]

      In 1953, the first experimental examples were found of rotational bands in nuclei, with their energy levels following the same J(J+1) pattern of energies as in rotating molecules. Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was nonspherical. In principle, these rotational states could have been described as coherent superpositions of particle-hole excitations in the basis consisting of single-particle states of the spherical potential. But in reality, the description of these states in this manner is intractable, due to the large number of valence particles—and this intractability was even greater in the 1950s, when computing power was extremely rudimentary. For these reasons, Aage Bohr, Ben Mottelson, and Sven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape. The first successful model of this type is the one now known as the Nilsson model. It is essentially the harmonic oscillator model described in this article, but with anisotropy added, so that the oscillator frequencies along the three Cartesian axes are not all the same. Typically the shape is a prolate ellipsoid, with the axis of symmetry taken to be z. Because the potential is not spherically symmetric, the single-particle states are not states of good angular momentum J. However, a Lagrange multiplier − ω ⋅ J , known as a "cranking" term, can be added to the Hamiltonian. Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tilted-axis cranking can also be considered. Filling the single-particle states up to the Fermi level then produces states whose expected angular momentum along the cranking axis ⟨ J x ⟩ angle > is the desired value.

      Igal Talmi developed a method to obtain the information from experimental data and use it to calculate and predict energies which have not been measured. This method has been successfully used by many nuclear physicists and has led to deeper understanding of nuclear structure. The theory which gives a good description of these properties was developed. This description turned out to furnish the shell model basis of the elegant and successful interacting boson model.

      A model derived from the nuclear shell model is the alpha particle model developed by Henry Margenau, Edward Teller, J. K. Pering, T. H. Skyrme, also sometimes called the Skyrme model. [7] [8] Note, however, that the Skyrme model is usually taken to be a model of the nucleon itself, as a "cloud" of mesons (pions), rather than as a model of the nucleus as a "cloud" of alpha particles.


      Contents

      For light atoms, the spin–orbit interaction (or coupling) is small so that the total orbital angular momentum L and total spin S are good quantum numbers. The interaction between L and S is known as LS coupling, Russell–Saunders coupling (named after Henry Norris Russell and Frederick Albert Saunders, who described this in 1925. [2] ) or spin-orbit coupling. Atomic states are then well described by term symbols of the form

      S is the total spin quantum number. 2S + 1 is the spin multiplicity, which represents the number of possible states of J for a given L and S, provided that LS. (If L < S, the maximum number of possible J is 2L + 1). [3] This is easily proven by using Jmax = L + S and Jmin = |LS|, so that the number of possible J with given L and S is simply JmaxJmin + 1 as J varies in unit steps. J is the total angular momentum quantum number. L is the total orbital quantum number in spectroscopic notation. The first 17 symbols of L are:

      L = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .
      S P D F G H I K L M N O Q R T U V (continued alphabetically) [note 1]

      The nomenclature (S, P, D, F) is derived from the characteristics of the spectroscopic lines corresponding to (s, p, d, f) orbitals: sharp, principal, diffuse, and fundamental the rest being named in alphabetical order from G onwards, except that J is omitted. When used to describe electron states in an atom, the term symbol usually follows the electron configuration. For example, one low-lying energy level of the carbon atom state is written as 1s 2 2s 2 2p 2 3 P2. The superscript 3 indicates that the spin state is a triplet, and therefore S = 1 (2S + 1 = 3), the P is spectroscopic notation for L = 1, and the subscript 2 is the value of J. Using the same notation, the ground state of carbon is 1s 2 2s 2 2p 2 3 P0. [1]

      Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or their quantum numbers.

      The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or molecules (see molecular term symbol). For molecules, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.

      For a given electron configuration

      • The combination of an S value and an L value is called a term, and has a statistical weight (i.e., number of possible microstates) equal to (2S+1)(2L+1)
      • A combination of S, L and J is called a level. A given level has a statistical weight of (2J+1), which is the number of possible microstates associated with this level in the corresponding term
      • A combination of S, L, J and MJ determines a single state.

      The parity of a term symbol is calculated as

      When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:

      2 P o
      ½ has odd parity, but 3 P0 has even parity.

      Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for gerade (German for "even") or ungerade ("odd"):

      2 P½,u for odd parity, and 3 P0,g for even.

      It is relatively easy to calculate the term symbol for the ground state of an atom using Hund's rules. It corresponds with a state with maximum S and L.

      1. Start with the most stable electron configuration. Full shells and subshells do not contribute to the overall angular momentum, so they are discarded.
        • If all shells and subshells are full then the term symbol is 1 S0.
      2. Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, fill the orbitals with highest m ℓ > value with one electron each, and assign a maximal ms to them (i.e. +½). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning ms = −½ to them.
      3. The overall S is calculated by adding the ms values for each electron. According to Hund's first rule, the ground state has all unpaired electron spins parallel with the same value of ms, conventionally chosen as +½. The overall S is then ½ times the number of unpaired electrons. The overall L is calculated by adding the m ℓ > values for each electron (so if there are two electrons in the same orbital, add twice that orbital's m ℓ > ).
      4. Calculate J as
        • if less than half of the subshell is occupied, take the minimum value J = |LS|
        • if more than half-filled, take the maximum value J = L + S
        • if the subshell is half-filled, then L will be 0, so J = S .

      As an example, in the case of fluorine, the electronic configuration is 1s 2 2s 2 2p 5 .

      1. Discard the full subshells and keep the 2p 5 part. So there are five electrons to place in subshell p ( ℓ = 1 ).
      2. There are three orbitals ( m ℓ = 1 , 0 , − 1 =1,0,-1> ) that can hold up to 2 ( 2 ℓ + 1 ) = 6 electrons . The first three electrons can take ms = ½ (↑) but the Pauli exclusion principle forces the next two to have ms = −½ (↓) because they go to already occupied orbitals.
        m ℓ >
        +10−1
        m s >↑↓↑↓
      3. S = ½ + ½ + ½ − ½ − ½ = ½ and L = 1 + 0 − 1 + 1 + 0 = 1 , which is "P" in spectroscopic notation.
      4. As fluorine 2p subshell is more than half filled, J = L + S = 3 ⁄ 2 . Its ground state term symbol is then 2S+1 LJ = 2 P
      5. 3 ⁄ 2 .

      Atomic term symbols of the chemical elements Edit

      Term symbols for the ground states of most chemical elements [4] are given in the collapsed table below (with citations for the heaviest elements here). In the d-block and f-block, the term symbols are not always the same for elements in the same column of the periodic table, because open shells of several d or f electrons have several closely spaced terms whose energy ordering is often perturbed by the addition of an extra complete shell to form the next element in the column.

      For example, the table shows that the first pair of vertically adjacent atoms with different ground-state term symbols are V and Nb. The 6 D1/2 ground state of Nb corresponds to an excited state of V 2112 cm −1 above the 4 F3/2 ground state of V, which in turn corresponds to an excited state of Nb 1143 cm −1 above the Nb ground state. [1] These energy differences are small compared to the 15158 cm −1 difference between the ground and first excited state of Ca, [1] which is the last element before V with no d electrons.

      Background color shows category:

      The process to calculate all possible term symbols for a given electron configuration is somewhat longer.

      Case of three equivalent electrons Edit

      Alternative method using group theory Edit

      For configurations with at most two electrons (or holes) per subshell, an alternative and much quicker method of arriving at the same result can be obtained from group theory. The configuration 2p 2 has the symmetry of the following direct product in the full rotation group:

      which, using the familiar labels Γ (0) = S , Γ (1) = P and Γ (2) = D , can be written as

      The square brackets enclose the anti-symmetric square. Hence the 2p 2 configuration has components with the following symmetries:

      S + D (from the symmetric square and hence having symmetric spatial wavefunctions) P (from the anti-symmetric square and hence having an anti-symmetric spatial wavefunction).

      The Pauli principle and the requirement for electrons to be described by anti-symmetric wavefunctions imply that only the following combinations of spatial and spin symmetry are allowed:

      1 S + 1 D (spatially symmetric, spin anti-symmetric) 3 P (spatially anti-symmetric, spin symmetric).

      Then one can move to step five in the procedure above, applying Hund's rules.

      The group theory method can be carried out for other such configurations, like 3d 2 , using the general formula

      Γ (j) × Γ (j) = Γ (2j) + Γ (2j−2) + ⋯ + Γ (0) + [Γ (2j−1) + ⋯ + Γ (1) ].

      The symmetric square will give rise to singlets (such as 1 S, 1 D, & 1 G), while the anti-symmetric square gives rise to triplets (such as 3 P & 3 F).

      More generally, one can use

      Γ (j) × Γ (k) = Γ (j+k) + Γ (j+k−1) + ⋯ + Γ (|jk|)

      where, since the product is not a square, it is not split into symmetric and anti-symmetric parts. Where two electrons come from inequivalent orbitals, both a singlet and a triplet are allowed in each case. [6]

      Basic concepts for all coupling schemes:

      • l → >> : individual orbital angular momentum vector for an electron, s → >> : individual spin vector for an electron, j → >> : individual total angular momentum vector for an electron, j → = l → + s → >=>+>> .
      • L → >> : Total orbital angular momentum vector for all electrons in an atom ( L → = ∑ i l i → >=sum _>>> ).
      • S → >> : total spin vector for all electrons ( S → = ∑ i s i → >=sum _>>> ).
      • J → >> : total angular momentum vector for all electrons. The way the angular momenta are combined to form J → >> depends on the coupling scheme: J → = L → + S → >=>+>> for LS coupling, J → = ∑ i j i → >=sum _>>> for jj coupling, etc.
      • A quantum number corresponding to the magnitude of a vector is a letter without an arrow (ex: l is the orbital angular momentum quantum number for l → >> and l ^ 2 | l , m , … ⟩ = ℏ 2 l ( l + 1 ) | l , m , … ⟩ >^<2>>left|l,m,ldots ight angle =<^<2>>lleft(l+1 ight)left|l,m,ldots ight angle > )
      • The parameter called multiplicity represents the number of possible values of the total angular momentum quantum number J for certain conditions.
      • For a single electron, the term symbol is not written as S is always 1/2, and L is obvious from the orbital type.
      • For two electron groups A and B with their own terms, each term may represent S, L and J which are quantum numbers corresponding to the S → >> , L → >> and J → >> vectors for each group. "Coupling" of terms A and B to form a new term C means finding quantum numbers for new vectors S → = S A → + S B → >=>+>>> , L → = L A → + L B → >=>+>>> and J → = L → + S → >=>+>> . This example is for LS coupling and which vectors are summed in a coupling is depending on which scheme of coupling is taken. Of course, the angular momentum addition rule is that X = X A + X B , X A + X B − 1 , . . . , | X A − X B | ,X_+X_-1. |X_-X_|> where X can be s, l, j, S, L, J or any other angular momentum-magnitude-related quantum number.

      LS coupling (Russell–Saunders coupling) Edit

      • Coupling scheme: L → >> and S → >> are calculated first then J → = L → + S → >=>+>> is obtained. From a practical point of view, it means L, S and J are obtained by using an addition rule of the angular momenta of given electron groups that are to be coupled.
      • Electronic configuration + Term symbol: n ℓ N ( ( 2 S + 1 ) L J ) ^><<(>^<(2S+1)>><_>)> . ( ( 2 S + 1 ) L J ) ^<(2S+1)>><_>)> is a Term which is from coupling of electrons in n ℓ N ^>> group. n , ℓ are principle quantum number, orbital quantum number and n ℓ N ^>> means there are N (equivalent) electrons in n ℓ subshell. For L > S , ( 2 S + 1 ) is equal to multiplicity, a number of possible values in J (final total angular momentum quantum number) from given S and L. For S > L , multiplicity is ( 2 L + 1 ) but ( 2 S + 1 ) is still written in the Term symbol. Strictly speaking, ( ( 2 S + 1 ) L J ) ^<(2S+1)>><_>)> is called Level and ( 2 S + 1 ) L >> is called Term. Sometimes superscript o is attached to the Term, means the parity P = ( − 1 ) ∑ i ℓ i ^<>>,<_>>>> of group is odd ( P = − 1 ).
      • Example:
        1. 3d 7 4 F7/2: 4 F7/2 is Level of 3d 7 group in which are equivalent 7 electrons are in 3d subshell.
        2. 3d 7 ( 4 F)4s4p( 3 P 0 ) 6 F 0
          9/2 : [7] Terms are assigned for each group (with different principal quantum number n) and rightmost Level 6 F o
          9/2 is from coupling of Terms of these groups so 6 F o
          9/2 represents final total spin quantum number S, total orbital angular momentum quantum number L and total angular momentum quantum number J in this atomic energy level. The symbols 4 F and 3 P o refer to seven and two electrons respectively so capital letters are used.
        3. 4f 7 ( 8 S 0 )5d ( 7 D o )6p 8 F13/2: There is a space between 5d and ( 7 D o ). It means ( 8 S 0 ) and 5d are coupled to get ( 7 D o ). Final level 8 F o
          13/2 is from coupling of ( 7 D o ) and 6p.
        4. 4f( 2 F 0 ) 5d 2 ( 1 G) 6s( 2 G) 1 P 0
          1 : There is only one Term 2 F o which is isolated in the left of the leftmost space. It means ( 2 F o ) is coupled lastly ( 1 G) and 6s are coupled to get ( 2 G) then ( 2 G) and ( 2 F o ) are coupled to get final Term 1 P o
          1 .

      Jj Coupling Edit

      J1L2 coupling Edit

      LS1 coupling Edit

        3d 7 ( 4 P)4s4p( 3 P o ) D o 3 [5/2] o
        7/2 : L 1 = 1 , L 2 = 1 , S 1 = 3 2 , S 2 = 1 _<1>>=1,

      Most famous coupling schemes are introduced here but these schemes can be mixed to express the energy state of an atom. This summary is based on [1].

      These are notations for describing states of singly excited atoms, especially noble gas atoms. Racah notation is basically a combination of LS or Russell–Saunders coupling and J1L2 coupling. LS coupling is for a parent ion and J1L2 coupling is for a coupling of the parent ion and the excited electron. The parent ion is an unexcited part of the atom. For example, in Ar atom excited from a ground state . 3p 6 to an excited state . 3p 5 4p in electronic configuration, 3p 5 is for the parent ion while 4p is for the excited electron. [8]

      Paschen notation is a somewhat odd notation it is an old notation made to attempt to fit an emission spectrum of neon to a hydrogen-like theory. It has a rather simple structure to indicate energy levels of an excited atom. The energy levels are denoted as n′l#. l is just an orbital quantum number of the excited electron. n′l is written in a way that 1s for (n = N + 1, l = 0), 2p for (n = N + 1, l = 1), 2s for (n = N + 2, l = 0), 3p for (n = N + 2, l = 1), 3s for (n = N + 3, l = 0), etc. Rules of writing n′l from the lowest electronic configuration of the excited electron are: (1) l is written first, (2) n′ is consecutively written from 1 and the relation of l = n′ − 1, n′ − 2, . , 0 (like a relation between n and l) is kept. n′l is an attempt to describe electronic configuration of the excited electron in a way of describing electronic configuration of hydrogen atom. # is an additional number denoted to each energy level of given n′l (there can be multiple energy levels of given electronic configuration, denoted by the term symbol). # denotes each level in order, for example, # = 10 is for a lower energy level than # = 9 level and # = 1 is for the highest level in a given n′l. An example of Paschen notation is below.


      Grain boundaries (GBs) are one of the most widely existing interfaces in polycrystalline materials. Scientists have devoted great efforts in understanding the structures and behaviors of GBs for more than a century. However, due to the complexity of GBs and the limitations of widely used two-dimensional imaging techniques, the structure of general GBs still remains largely unknown. To this end, by using atomic-resolution electron tomography, we present quantitative study of the three-dimensional atomic structure and crystallography of general GBs in nanometals. Our findings will have significant impact on the fundamental understanding of GB behaviors and properties of polycrystals in general, and this research also shows the importance of developing methods to include the non-planar nature of GBs in order to statistically evaluate their behaviors in modeling studies. The application of atomic-resolution electron tomography could be extended to film and bulk materials with proper sample-preparation techniques.

      Grain boundaries (GBs) determine the properties of polycrystals, and tailoring the GB structure offers a promising method for the discovery and engineering of new materials. However, GB structures are far from well understood because of their structural complexity and limitations of conventional projection imaging methods. Here, we decipher three-dimensional atomic structure and crystallography of GBs in nanometals using atomic-resolution electron tomography. Unlike conventional descriptions, whereby they are either straight or curved planar planes with one-dimensional translational symmetry, we show that the high-angle GBs completely lose translational symmetry due to undulated curvature related to configurations of structural units. Moreover, we directly visualize kinks and jogs at the single-atom scale in dislocation-type GBs and investigate their mobilities. Our findings bring new insights to the conventional wisdom of GBs and show the importance of developing methods to include the non-planar nature of GBs to statistically evaluate the behavior of GBs in modeling studies.


      Watch the video: Γενετικός κώδικας Βιολογία Κατ. ΓΛυκείου (November 2022).