Has a very significant role in the analysis of microworld phenomena. Pauli principle. Pauli put forward this assumption even before the advent of quantum mechanics. Pauli formulated it regarding electrons:
There cannot be two electrons in an atom that would be characterized by the same quadruples of quantum numbers $(n,l,m_l,\ m_s)$, that is, more than one electron cannot be in the same state.
Thus, if two electrons have the same principal quantum numbers $(n)$ and the orbital numbers coincide, then their spins should be oriented oppositely (that is, their quantum numbers $m_s\ are equal\ \frac(1)(2)\ and-\ frac(1)(2)$).
Mathematical notation of the Pauli principle
Consider a system of two electrons. If the interaction of electrons is not taken into account, then the wave function of the electron’s motion in space can be considered:
where the indices $a\ and\ b$ denote the states of electrons conditionally numbered $1$ and $2$. Full function for $2$ electrons is the product of the spin wave function and the wave function of their motion in space. We write the spin wave functions as:
Figure 1.
The multiplications result in eight different complete wave functions that have symmetry. In this case we have: the product of two symmetric and two antisymmetric functions gives a symmetric function. Multiplying a symmetric function by an antisymmetric function is an antisymmetric function. As a result, we find that out of eight complete wave functions, $50\%$ are symmetrical:
Antisymmetric functions include:
Figure 2.
Symmetrical functions:
Figure 3.
Not all of the \Psi -- functions written above are possible if you follow the Pauli principle. If the quantum numbers of the two electrons are equal, then the wave function becomes zero. Let's say the electrons make the same motion in their orbits ($a=b$). In this case (according to the Pauli principle), only the opposite orientation of the electron spins is possible. the wave functions that relate to the description of the spin orientation in one direction (8-10) become equal to zero, since the first factor is zero. Wave function (7) is not zero; it describes opposite spins. It turns out that for $a=b$ the antisymmetric wave functions are consistent with the Pauli principle.
Let's consider the second group of wave functions (11-14). When $a=b$, symmetric functions with the same orientation of spins do not become equal to zero. Therefore, they are not acceptable. Function (14) describes the behavior of electrons with spins that are oriented in the opposite direction, which means that it could not be equal to zero. However, when $a=b$ the first factor of the function under consideration is equal to zero, it turns out that the \Psi-function in such cases is always equal to zero, which is incompatible with the Pauli principle, which in this case allows states with different spins. We conclude that symmetric functions are unacceptable.
Based on the above reasoning, we formulate the Pauli principle:
The total wave function of two electrons must be an antisymmetric function with respect to the permutation of electrons. Since formulas (7) - (14) were written without taking into account the interaction of electrons, but in the reasoning we used exclusively the symmetry properties of $\Psi$ - functions that are associated with the identity of electrons and do not depend on their interaction (If we take into account the interaction of electrons, then there is no exchange degeneracy, but the symmetry properties of the wave functions remain, since the identity of the particles is preserved during their interaction.), then all conclusions will apply to interacting electrons.
In case one has to deal with more than $2$ number of electrons, the above statement can be generalized and formulated as:
The wave function of a collection of electrons must be an antisymmetric function with respect to the permutation of any pair of electrons:
Application of the Pauli principle
This principle was used to substantiate Mendeleev’s periodic system and part of the patterns in the spectra.
Thus, the structure of the electronic shells of an atom is based on two principles:
Pauli's principle. It takes into account the quantum properties of possible states of an atom.
The principle of minimum energy: for a given total number of electrons in an atom, a state with minimum energy is realized. This requirement is natural regarding the stability of the atom.
When analyzing the structure of an atom as a first approximation, the interaction energy of electrons is neglected. It is believed that the sum of the energy of an atom is equal to the sum of the energies of the electrons in the field of the nucleus, which is known. This means that it is not difficult to determine the distribution of electrons over different states, taking into account the Pauli principle. The result is a scheme for filling the shells, which, it should be noted, still differs from the real one, but is useful.
Depending on the value of the orbital quantum number $l\ $, the state of an electron in an atom is designated by different letters. Values $l=0,1,2,3,4,5\dots $ are assigned letters $s,p,d,f,g,h$ and in alphabetical order.
The distribution of electrons by state in an atom is written using spectroscopic symbols (Table 1):
Figure 4.
The electronic structure is written as follows: the number on the left is the principal quantum number $(n)$, the spectroscopic symbol itself corresponds to the value of the orbital quantum number $(l)$.
Example 1
Apply the Pauli principle, answer the question: what is the maximum number of electrons $N_(max)$ in an atom that can have the same quantum numbers 1) $n,l,m_l,m_s$; 2) $n$?
Solution:
The state of an electron in an atom is uniquely determined by a set of four quantum numbers:
- main $n\ (n=1,2,3...),$
- orbital$\ l\ (l=0,1,2,...,n-1)$,
- magnetic $m_l$ ($m_l=-l,\dots ,\ -1,0,1,\dots ,l$),
- magnetic spin $m_s$($m_s=\pm \frac(1)(2)$).
1) According to the Pauli principle, one electron in an atom can have a certain set of quantum numbers $n,l,m_l,m_s.$
2) For a given principal quantum number ($n$), the orbital quantum number ($l$) can take values from $0$ to $n-1$, with each value $l$ corresponding to $2l+1$ different values $m_l $, in this case the quantity different states, which correspond to the known principal quantum number is equal to:
\[\sum\limits^(n-1)_(i=0)(\left(2l+1\right)=n^2).\]
The quantum number $m_s$ can have only two values, which means the maximum number of electrons that have the same principal quantum numbers can be equal to:
Answer: 1) $N_(max)=1$, 2)$\ N_(max)=2n^2.$
Example 2
The electronic layer, characterized by a principal quantum number equal to $n=3$, is completely filled. How many electrons have the same magnetic quantum numbers equal to $m_l=2$?
Solution:
According to answer $2$ of example $1$ we can say that when $n=3$ there can be $18$ electrons in an atom. In this case, $l=0,1,2;;$ $m_l=0,\pm 1,\pm 2;\ m_s=\pm \frac(1)(2)$. It is convenient to summarize the electron distribution in a table (Table 2):
Figure 5.
The table shows that for a pair of quantum numbers $n=3$,$\ m_l=2$ there are two electrons.
Answer: Two electrons.
An exact solution to the Schrödinger equation can be found only in rare cases, for example, for the hydrogen atom and hypothetical one-electron ions, such as He +, Li 2+, Be 3+. The atom of the element next to hydrogen, helium, consists of a nucleus and two electrons, each of which is attracted to both nuclei and repelled by the other electron. Even in this case, the wave equation does not have an exact solution.
Therefore, various approximate methods are of great importance. Using these methods it was possible to establish electronic structure atoms of all known elements. These calculations show that the orbitals in many-electron atoms are not very different from the orbitals of the hydrogen atom (these orbitals are called hydrogen-like orbitals). The main difference is some compression of the orbitals due to the higher charge of the nucleus. In addition, for multielectron atoms it was found that for each energy level(for a given value of the principal quantum number n) splitting into sublevels. The energy of an electron no longer depends only on n, but also on the orbital quantum number l. It increases in series s-, p-, d-, f-orbitals (Fig. 7).
Rice. 7
For high energy levels, the differences in the energies of the sublevels are large enough that one level can penetrate another, e.g.
6s d4 f p.
The occupation of atomic orbitals for a multielectron atom in the ground (that is, the most energetically favorable) state occurs in accordance with certain rules.
Minimum energy principle
Principle minimum energy determines the order of occupation of atomic orbitals having different energies. According to the minimum energy principle, electrons occupy orbitals with the lowest energy first. The energy of sublevels grows in the series:
1s s p s p s d p s d p s f5 d p s f6 d...
A hydrogen atom has one electron, which can be in any orbital. However, in the ground state it should occupy 1 s-orbital having the lowest energy.
In a potassium atom, the last nineteenth electron can occupy either 3 d-, or 4 s-orbital. According to the principle of minimum energy, an electron occupies 4 s-orbital, which is confirmed by experiment.
Note the uncertainty of entry 4 f 5d and 5 f 6d. It turned out that some elements have lower energy 4 f-sublevel, while others have 5 d-sublevel. The same is observed for 5 f- and 6 d-sublevels.
Introduction
In 1925, Pauli established the quantum mechanical principle (Pauli exclusion principle).
In any atom there cannot be two electrons that are in the same stationary states, determined by a set of four quantum numbers: n, m, ms.
For example, an energy level can contain no more than two electrons, but with opposite spin directions.
The Pauli principle made it possible to theoretically substantiate Mendeleev's periodic system of elements, create quantum statistics, modern theory solids etc.
Pauli principle
The state of each electron in an atom is characterized by four quantum numbers:
1. Principal quantum number n (n = 1, 2 ...).
2. Orbital (azimuthal) quantum number l (l = 0, 1, 2, ... n-1).
3. Magnetic quantum number m (m = 0, +/-1, +/-2, +/-... +/-l).
4. Spin quantum number ms (ms = +/-1/2).
For one fixed value of the principal quantum number n, there are 2n2 different quantum states of the electron.
One of the laws of quantum mechanics, called the Pauli principle, states:
In the same atom there cannot be two electrons that have the same set of quantum numbers (that is, there cannot be two electrons in the same state).
The Pauli principle provides an explanation for the periodic repetition of the properties of the atom, i.e. Mendeleev's periodic system of elements.
Periodic table elements of D. I. Mendeleev
In 1869, Mendeleev discovered the periodic law of changes in the chemical and physical properties of elements. He introduced the concept of the serial number of an element and obtained complete periodicity in changes in the chemical properties of elements.
At the same time, some of the cells of the periodic system remained unfilled, because their corresponding elements were unknown at that time. In 1998, the isotope of element 114 was synthesized in Russia.
Mendeleev predicted a number of new elements (scandium, germanium, etc.) and described their chemical properties. Later, these elements were discovered, which completely confirmed the validity of his theory. It was even possible to clarify the values of atomic masses and some properties of elements.
The chemical properties of atoms and a number of their physical properties are explained by the behavior of external (valence) electrons.
Stationary quantum states of an electron in an atom (molecule) are characterized by a set of 4 quantum numbers: principal (n), orbital (l), magnetic (m) and magnetic spin (ms). Each of them characterizes the quantization of: energy (n), angular momentum (l), projection of angular momentum onto the direction of the external magnetic field(m) and spin projections (ms).
According to the theory, the serial number chemical element Z is equal to the total number of electrons in an atom.
If Z is the number of electrons in an atom that are in a state that is specified by a set of 4 quantum numbers n, l, m, ms, then Z(n, l, m, ms) = 0 or 1.
If Z is the number of electrons in an atom that are in states determined by a set of 3 quantum numbers n, l, m, then Z(n, l, m)=2. Such electrons differ in spin orientation.
If Z is the number of electrons in an atom that are in states determined by 2 quantum numbers n, l, then Z(n, l)=2(2l+1).
If Z is the number of electrons in an atom that are in states determined by the value of the principal quantum number n, then Z(n)=2n2.
Electrons in an atom, occupying a set of states with the same values of the principal quantum number n, form an electronic layer: at n=1 K - layer; at n=2 L - layer; at n=3 M - layer; at n=4 N - layer; at n=5 O - layer, etc.
In each electron layer of an atom, all electrons are distributed among shells. The shell corresponds to a certain value of the orbital quantum number (Table 1 and Fig. 1).
| n | Electronic layer | Number of electrons in shells | Total number of electrons | ||||
| s(l=0) | p(l=1) | d(l=2) | f(l=3) | g(l=4) | |||
| 1 | K | 2 | - | - | - | - | 2 |
| 1 | L | 2 | 6 | - | - | - | 8 |
| 3 | M | 2 | 6 | 10 | - | - | 18 |
| 4 | N | 2 | 6 | 10 | 14 | - | 32 |
| 5 | O | 2 | 6 | 10 | 14 | 18 | 50 |
For a given l, the magnetic quantum number m takes 2l+1 values, and ms takes two values. Therefore, the number of possible states in the electron shell with a given l is equal to 2(2l+1). So the l=0 shell (s - shell) is filled with two electrons; shell l=1 (p - shell) - six electrons; shell l=2 (d - shell) - ten electrons; shell l=3 (f - shell) - fourteen electrons.
The sequence of filling electronic layers and shells in Mendeleev’s periodic system of elements is explained by quantum mechanics and is based on 4 provisions:
1. The total number of electrons in an atom of a given chemical element is equal to the atomic number Z.
2. The state of an electron in an atom is determined by a set of 4 quantum numbers: n, l, m, ms.
3. The distribution of electrons in an atom over energy states must satisfy the minimum energy.
4. The filling of energy states in an atom with electrons should occur in accordance with the Pauli principle.
When considering atoms with large Z, due to an increase in the charge of the nucleus, the electron layer is drawn towards the nucleus and the layer with n=2 begins to fill, etc. For a given n, the state of s-electrons (l=0) is first filled, then p-electrons (l=1), d-electrons (l=2), etc. This leads to periodicity in the chemical and physical properties of elements. For elements of the first period, the shell 1s is filled first; for electrons of the second and third periods - shells 2s, 2p and 3s and 3p.
However, starting from fourth period(potassium element, Z=19), the sequence of shell filling is disrupted due to competition between electrons that are close in binding energy. Electrons with larger n but smaller l may turn out to be more tightly (energetically more favorable) bound (for example, 4s electrons are more tightly bound than 3d).
The distribution of electrons in an atom across shells determines its electronic configuration. To indicate the electronic configuration of an atom, the symbols for filling the electronic states of the shells nl are written in a row, starting with the one closest to the nucleus. The index at the top right indicates the number of electrons in the shell that are in these states. For example, the sodium atom has 2311Na, where Z=11 is the ordinal number of the element in the periodic table; number of electrons in an atom; number of protons in the nucleus; A=23 - mass number (number of protons and neutrons in the nucleus). The electronic configuration has the form: 1s2 2s2 2p6 3s1, i.e. in the layer with n=1 and l=0 - two s-electrons; in the layer with n=2 and l=0 - two s-electrons; in the layer with n=2 and l=1 - six p-electrons; in the layer with n=3 and l=0 - one s-electron.
Along with the normal electronic configuration of the atom, which corresponds to the strongest binding energy of all electrons, excited electronic configurations arise when one or more electrons are excited.
For example, in helium, all energy levels are divided into two level systems: the orthohelium level system, corresponding to the parallel orientation of electron spins, and the parahelium level system, corresponding to the antiparallel spin orientation. The normal configuration of helium 1s2 due to the Pauli principle is possible only with an antiparallel orientation of electron spins, corresponding to parahelium.
Conclusion
So, the Pauli exclusion principle explains, long considered mysterious, the periodic structure of elements discovered by D.I. Mendeleev.
References
1. Detlaf A.A., Yavorsky B.N. Physics course. - M., 1989.
2. Kompaneets A.S. What is quantum mechanics? - M., 1977.
3. Orir J. Popular physics. - M., 1964.
4. Trofimova T.I. Physics course. - M., 1990.
The history of atomic physics has many ups and downs. But thanks technical progress any assumption that arose in the minds of theorists could be tested in laboratory conditions. Since many aspects of the behavior of elementary particles still defy the laws of logic, the scientists who discovered the microworld agreed to accept them “as is,” without explaining the reasons. The Pauli principle refers to the results of those experiments that have not yet found their only explanation.
Controversies in atomic theory
One of the most common successful misconceptions in atomic physics was the planetary atomic model proposed by the English scientist Ernest Rutherford. In the end, it turned out to be not entirely reliable, but it made it possible to draw so many correct conclusions that its benefits were undoubted.
One of the main contradictions of the Rutherford atom was the ability of electrons to radiate. As a result of the loss of energy, any electron would eventually stop moving and fall onto the nucleus. But any atom (except radioactive) is essentially stable, can exist for an indefinitely long time and does not show any signs of self-destruction. To solve this problem, it took the talent of the brilliant Danish physicist Niels Bohr.
Bohr's theory
In 1913, a young unknown physicist from Denmark proposed two changes to be included in classical physics, with the help of which it was possible to explain the facts of observations and make many useful discoveries. Bohr could not explain the reason for the behavior of the electron in orbit, so he based his rules on the “as is” principle. These rules served well in the future and paved the way for new discoveries.
Bohr's rules
The first rule stated that Rutherford's planetary model of the atom was still correct. But the electrons in it move in their orbits without radiation. Bohr's second rule states that electrons can only move in certain “allowed” orbits. For an electron moving along a permitted orbit, the product of momentum and the radius of this orbit is always a multiple of Planck's constant. Thus, electron orbits can only be at those energy levels for which the following rule holds:
(electron momentum * orbital circumference) = n * h,
where h is the bar constant and n is natural number. Thus, at the smallest allowed orbit, n = 1. The third rule says that the electrons of atoms can be moved (for example, by bombarding them with heavy particles) into a free outer orbit. After this, the electron is able to return to the free inner orbit. In this case, the atom emits excess energy in the form of a quantum of light.
Quantum Limits
Bohr's quantum rule suggests that electrons that are closest to the nucleus have the smallest allowed orbit. At this level, the electron has minimal energy. One would expect that all the electrons in an atom would occupy this orbit and remain in this level. However, this does not happen. The Pauli principle helped explain this contradiction.
Wolfgang Pauli
This famous Austrian physicist was born in Vienna in 1869. At the University of Munich he received an excellent comprehensive education, but devoted all his scientific works to quantum physics. At the age of twenty, Pauli wrote a review article for the Physical Encyclopedia, many pages of which are still relevant today. His scientific works rarely published, Pauli voiced his most important thoughts and hypotheses in correspondence with his colleagues scientific activity. The most active correspondence was with N. Bohr and W. Heisenberg. It was the joint work of these three scientists that laid the foundations for modern quantum physics. Based on the experimental data of these three prominent scientists, Pauli formed his principle. For him in 1945, the Austrian scientist received the Nobel Prize.
Electron movement
While studying the motion of the electron, W. Pauli came across many strange aspects in the behavior of this elementary particle. For example, electrons, when moving, behave as if they were rotating around their axis. The electron's own angular momentum is called spin. Two electrons can fit in one place in the orbit, and their spins must be opposite to each other, as the Pauli principle states. The physics of this limitation applies not only to electrons, but also to other particles with a half-integer spin value.
Periodic table and Pauli principle
Chemistry has used the uncertainty principle to explain the internal structure of substances. Now it becomes quite understandable why there are only two elements in the first row of the periodic table. Both hydrogen and helium have at their disposal a single lower orbit, in which there is only one twin place for electrons having opposite spins. The next orbit already contains eight such places. Therefore, eight elements could occupy the second row of the periodic table. This pattern extends to all rows of the periodic table.
Physics of stars
Oddly enough, the laws of behavior of elementary particles extend far beyond the microcosm. For example, inner world aging stars are dealt with by stellar physics. The Pauli principle also works here, but it is understood a little differently. Now this rule says that in a certain spatial volume it is possible to accommodate only two elementary particles with opposite backs. This law is especially clear when observing aging stars. As is known, after an explosion, a supernova rapidly collapses, but not all stars turn into black holes. With an increase in the threshold density limit (and for an aging star this value is about 10 7 kg/m 3), the internal pressure cosmic body begins to grow rapidly. This process has a special scientific term - pressure of degenerate electron gas. Thus, the star stops losing its volume and turns into a small celestial body the size of our Earth. In astrophysics, such stars are called white dwarfs.
Results
The uncertainty principle is one of the first laws of a new type, which differs from all known ideas about the world around us. The new laws are fundamentally different from the rules of classical physics known to us from childhood. If the old rules told us what could happen when carrying out certain actions, then the new type of laws tell us what should not happen.
Algorithms for solving many problems should be built using a slightly modified Pauli principle. By cutting off impossible options for solving problems at the very beginning, there is a chance to find the only correct answer. The practical use of the uncertainty principle significantly reduces the time required for computer processing of information. Previously known only among theoretical physicists, the Pauli principle has long gone beyond the boundaries of quantum physics, thereby identifying new methods for studying the laws of nature.
In an atom by states
If identical particles have the same quantum numbers, then their wave function is symmetric with respect to the permutation of particles. It follows that two identical fermions included in the same system cannot be in the same states, since for fermions the wave function must be antisymmetric. Summarizing the experimental data, W. Pauli formulated the principle according to which systems of fermions occur in nature only in states described by antisymmetric wave functions (quantum-mechanical formulation of the Pauli principle).
From this position follows a simpler formulation of the Pauli principle, which he introduced into quantum theory (1925) even before the construction of quantum mechanics: in a system of identical fermions, any two of them cannot simultaneously be in the same state. Note that the number of bosons of the same type that are in the same state is not limited.
Let us recall that the state of an electron in an atom is uniquely determined by a set of four quantum numbers:
main n(n =1, 2, 3, ...),
orbital l (l= 0, 1, 2, ..., n-1),
magnetic m l(m l = - l, .... - 1, 0, +1, ..., + l),
magnetic spin (ms = + 1/2, - 1/2).
The distribution of electrons in an atom obeys the Pauli principle, which can be used in its simplest formulation: the same atom cannot have more than one electron with the same set of four quantum numbers n , l, m l and m s, t. e.
where Z(n, l, m l, m s) - the number of electrons in a quantum state described by a set of four quantum numbers: n , l, m l, m s. Thus, the Pauli principle states that two electrons bound in the same atom differ in the values of at least one quantum number.
According to formula (223.8), given n corresponds to n 2 different states with different meanings l and m l. Quantum number m , can take only two values (± 1/2).
Therefore, the maximum number of electrons in states determined by a given principal quantum number is equal to
The collection of electrons in a multielectron atom that have the same principal quantum number n , called the electron shell. In each shell, electrons are distributed among subshells corresponding to a given l. Since the orbital quantum number takes values from 0 to n - 1, the number of subshells is equal to the ordinal number of the nshell. The number of electrons in a subshell is determined by the magnetic and magnetic spin quantum numbers: the maximum number of electrons in a subshell with a given l equals 2(2 l+ 1). Shell designations, as well as the distribution of electrons across shells and subshells are presented in Table. 6.
Table 6
Periodic table of elements
Mendeleev
The Pauli principle, which underlies the systematics of filling electronic states in atoms, allows us to explain the Periodic Table of Elements by D. I. Mendeleev (1869) - fundamental law nature, which is the basis modern chemistry, atomic and nuclear physics.
D.I. Mendeleev introduced the concept of the atomic number Z of a chemical element, equal to the number of protons in the nucleus and, accordingly, the total number of electrons in the electron shell of the atom. By arranging the chemical elements in order of increasing serial numbers, he obtained periodicity in the changes in the chemical properties of the elements. However, for the 64 chemical elements known at that time, some cells of the table turned out to be empty, since the corresponding elements (for example, Ga, Se, Ge) were not yet known. D.I. Mendeleev, thus, not only correctly positioned known elements, but also predicted the existence of new, not yet discovered elements and their basic properties. In addition, D.I. Mendeleev managed to clarify the atomic weights of some elements. For example, the atomic weights of Be and U, calculated on the basis of the periodic table, turned out to be correct, but those obtained earlier experimentally were erroneous.
Since chemical and some physical properties elements are explained by external (valence) electrons in atoms, then the periodicity of the properties of chemical elements should be associated with a certain periodicity in the arrangement of electrons in atoms. Therefore, to explain the table, we will assume that each subsequent element is formed from the previous one by adding one proton to the nucleus and, accordingly, adding one electron in the electron shell of the atom. We neglect the interaction of electrons, making appropriate corrections where necessary. Let us consider atoms of chemical elements that are in the ground state.
The only electron of the hydrogen atom is in the 1s state , characterized by quantum numbers n = 1, l= 0, m l= 0 and m s = ± 1/2 (the orientation of its spin is arbitrary). Both electrons of the He atom are in the 1s state , but with an antiparallel spin orientation. Electronic configuration for atom Not written as 1s 2 (two 1s electrons). The filling of the K-shell ends at the He atom, which corresponds to the completion of the first period of Mendeleev’s Periodic Table of Elements (Table 7).
The third electron of the Li atom (Z=3), according to the Pauli principle, can no longer fit into a completely filled A-shell and occupies the lowest energy state with n=2 (L-shell), i.e., the 2s state. Electronic configuration for the Li atom: 1s 2 2s. The Li atom begins the second period of the Periodic Table of elements. The fourth electron, Be (Z=4), completes the filling of the 2s subshell. The next six elements from B (2=5) to Ne (Z=10) complete the filling of the 2p subshell (Table 7). The second period of the Periodic System ends with neon, an inert gas for which the subshell is completely filled.
The eleventh electron Na (Z=11) is located in the M-shell (n=3), occupying the lowest state 3s. The electronic configuration is 1s 2 2s 2 2p 6 3s. The 3s electron (like the 2s electron Li) is a valence electron , therefore the optical properties of Na are similar to those of Li. From Z=12 there is a sequential filling of the M-shell. Ar (Z = 18) turns out to be similar to He and Ne: in its outer shell all s- and p-states are filled. Ag is chemically inert and completes the third period of the Periodic table.
The nineteenth electron K (Z=19) would have to occupy the 3d state in the M shell. However, both optically and chemically, the K atom is similar to the Li and Na atoms, which have an outer valence electron in the s-state. Therefore, the 19th valence electron of K must also be in the s-state, but this can only be the s-state of the new shell (N-shell), i.e., filling the N-shell for K begins when the M-shell is unfilled. This means that as a result of the interaction of electrons, the state n = 4, l=0 has less energy than the n=3 state, l=2. Spectroscopic and chemical properties of Ca (Z=20) show that its 20th electron is also in the 4s state of the N-shell. In subsequent elements, the M-shell is filled (from Sc (Z=21) to Zn (Z=30)). Next, the N-shell is filled to Kr (Z = 36), for which, again, as in the case of Ne and Ar, s - and p-states of the outer shell are completely filled. Krypton ends the IV period of the Periodic Table. Similar reasoning applies to other elements of the periodic table, but this data can be found in reference books. Let us only note that the initial elements of subsequent periods Rb, Cs, Fr are alkali metals, and their last electron is in the s-state. In addition, the atoms of inert gases (He, Ne, At, Kr, Xe, Rn) occupy a special position in the table - in each of them the s- and p-states of the outer shell are completely filled and they complete the next periods of the Periodic System.
| Period | Z | Element | K | L | M | N | Period | Z | Element | K | L | M | N | ||||||||||||
| 1s | 2s | 2p | 3s | 3p | 3d | 4s | 4p | 4d | 4f | 1s | 2s | 2p | 3s | 3p | 3d | 4s | 4p | 4d | 4f | ||||||
| H He | IV | K Ca Sc Ti V Cr Mn Fe Co Ni | - - | ||||||||||||||||||||||
| III | Na Vg Al Si P S Cl Ar | Cu Zn Ga Ge As Se Br Kr |
Table 7
Each of the two groups of elements - lanthanides (from lanthanum (Z = 57) to lutetium (Z = 71)) and actinides (from actinium (Z = 89) to lawrencium (Z = 103)) - must be placed in one cell of the table, so as the chemical properties of the elements within these groups are very similar. This is explained by the fact that for lanthanides the filling of the 4f subshell, which can contain 14 electrons, begins only after the 5s, 5p and 6s subshells are completely filled . Therefore, for these elements the outer P-shell (6s 2) turns out to be the same. Similarly, the Q-shell (7s 2) is the same for actinides.
Thus, the periodicity discovered by Mendeleev in chemical properties elements is explained by the repeatability in the structure of the outer shells of atoms of related elements. Thus, inert gases have identical outer shells of 8 electrons (filled s- and p-states); in the outer shell of alkali metals (Li, Na, K, Rb, Cs, Fr) there is only one s-electron; in the outer shell of alkaline earth metals (Be, Mg, Ca, Sr, Ba, Ra) there are two s-electrons; halogens (F, C1, Br, I, At) have outer shells in which one electron is missing to the inert gas shell, etc.
X-ray spectra
A major role in elucidating the structure of the atom, namely the distribution of electrons among shells, was played by radiation discovered in 1895 by the German physicist W. Roentgen (1845-1923) and called X-ray. The most common source of X-ray radiation is an X-ray tube, in which highly accelerated electric field electrons bombard the anode (a metal target made of heavy metals, for example W or Pt), experiencing sharp braking on it. This produces X-ray radiation, which is electromagnetic waves with a wavelength of approximately 10 12 -10 -8 m. The wave nature of X-ray radiation is proven by experiments on its diffraction, discussed in § 182.
A study of the spectral composition of X-ray radiation shows that its spectrum has a complex structure (Fig. 306) and depends both on the energy of the electrons and on the anode material. The spectrum is a superposition of a continuous spectrum, limited on the short wavelength side by a certain boundary l min, called the boundary of the continuous spectrum, and a line spectrum - a collection of individual lines appearing against the background of the continuous spectrum.
Research has shown that the nature of the continuous spectrum is completely independent of the anode material, but is determined only by the energy of the electrons bombarding the anode. A detailed study of the properties of this radiation showed that it is emitted by electrons bombarding the anode as a result of their deceleration during interaction with target atoms. The continuous X-ray spectrum is therefore called the bremsstrahlung spectrum. This conclusion is in agreement with the classical theory of radiation, since when moving charges are decelerated, radiation with a continuous spectrum should actually arise.
From classical theory However, the existence of a short-wave boundary of the continuous spectrum does not follow. From experiments it follows that the more kinetic energy electrons causing X-ray bremsstrahlung, the smaller l min. This circumstance, as well as the presence of the border itself, is explained quantum theory. Obviously, the limiting energy of a quantum corresponds to the case of braking in which all the kinetic energy of the electron is converted into quantum energy, i.e.
where U is the potential difference due to which the energy E max is imparted to the electron, v max is the frequency corresponding to the boundary of the continuous spectrum. Hence the cutoff wavelength
which is fully consistent with experimental data. By measuring the boundary of the X-ray continuous spectrum, using formula (229.1), one can determine the experimental value of Planck’s constant h, which most closely matches modern data.
When the energy of the electrons bombarding the anode is sufficiently high, individual sharp lines appear against the background of a continuous spectrum - a line spectrum determined by the anode material and called the characteristic X-ray spectrum (radiation).
Compared to optical spectra, the characteristic X-ray spectra of elements are completely uniform and consist of several series, designated K, L, M, N and O . Each series, in turn, contains a small set of individual lines, designated in descending order of wavelength by the indices a, b, g ... (K a, K b, K g,.... L a, L b, L g , ...). When moving from light to heavy elements, the structure of the characteristic spectrum does not change, only the entire spectrum shifts towards shorter waves. The peculiarity of these spectra is that the atoms of each chemical element, regardless of whether they are in a free state or included in chemical compound, have a certain line spectrum of characteristic radiation inherent only to this element. Thus, if the anode consists of several elements, then the characteristic X-ray radiation is a superposition of the spectra of these elements.
Consideration of the structure and features of characteristic X-ray spectra leads to the conclusion that their occurrence is associated with processes occurring in the internal, built-up electronic shells of atoms, which have a similar structure.
Let us analyze the mechanism of the occurrence of x-ray series, which is shown schematically in Fig. 307.
Let us assume that, under the influence of an external electron or a high-energy photon, one of the two electrons of the it-shell of the atom is ejected. Then an electron from shells L, M, N,..., more distant from the nucleus, can take its place. Such transitions are accompanied by the emission of X-ray quanta and the appearance spectral lines K-series: K a (L®K), K b (M®K), K g (N®K), etc. The longest wavelength line of the K-series is the K a line . The line frequencies increase in the series K a ® K b ® K g, since the energy released during the transition of an electron to the K-shell from more distant shells increases. On the contrary, the line intensities in the series К a ®К b ®К g decrease, since the probability of electron transitions from the L-shell to the K-shell is greater than from the more distant shells M and N. The K-series is necessarily accompanied by other series, since when emission of its lines, vacancies appear in the shells L, M,..., which will be filled by electrons located at higher levels.
Other series arise similarly, observed, however, only for heavy elements. The considered lines of characteristic radiation can have a fine structure, since the levels determined by the principal quantum number are split according to the values of the orbital and magnetic quantum numbers.
By studying the X-ray spectra of elements, English physicist G. Moseley (1887-1915) established in 1913 a relationship called Moseley's law:
(229.2)
where v is the frequency corresponding to a given line of characteristic x-ray radiation, R is the Rydberg constant, s is the screening constant, m = 1,2, 3, ... (defines the x-ray series), n takes integer values starting from +1 (defines a separate line of the corresponding series). Moseley's law (229.2) is similar to the generalized Balmer formula (209.3) for the hydrogen atom.
The meaning of the screening constant is that an electron undergoing a transition corresponding to a certain pinnium is not affected by the entire charge of the nucleus Ze, and the charge (Z - s)e , weakened by the shielding effect of other electrons. For example, for K a -line s = 1, and Moseley’s law will be written in the form
