4d mendel english september 2007

To the Problem of Four-Dimensional Symmetries

of the Properties of Chemical Elements and the Interpretation of this Phenomenon

F.Bogomolov, Y.Magarshak, On commuting operators related to asymptotic

symmetries in the atomic theory; Scientific Israel-Technological

Advantages, vol 8, issues 1-2, pp. 161-165 (2006)

• Y.Magarshak, "Four-Dimensional Pyramidal Structure of the Periodic

Properties of Atoms and Chemical Elements", Scientific Israel -

Technological Advantages vol. 7, No.1,2 , pp. 134-150 (2006)

• Yu. Magarshak, J.Malinsky,"A Three-Dimensional Periodic Table ", Nature,

vol.360: 113-114 (1992).

Орбитальное Рис. 1

Квантовое Порядок заполнения оболочек

число

Порядок заполнения термов (подоболочек) 14F 10D 6P 2S

1 2 3 4 5 6 7 8 Главное

квантовое число

),()( lmYrR

0)(2

2 rUE

m

EMPIRIC HUND RULE: Electronic configuration, which has the the largest (for that electronic configuration) spin value and the greatest possible at this value of the spin orbinal momentum has the lowest energy.

***

Experimentally proven order of atomic orbitals energy encrease: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f <5d <6p <7s <5f <6d <7p < 8s

5f < 6d < 7p < 8s … 4f < 5d < 6p < 7s < 4d < 5p < 6s < 3d < 4p < 5s < 3p < 4s < 2p < 3s < 2s < 1s <

5g 6F 7D 8P 8s

5f 6d 7p 8s

Восьмой уровень

Четвертый суперцикл

4f (14 элементов) 5d (10 элементов) 6p (6 элементов) 7s (2 элемента) Седьмой уровень

4d (10 элементов) 5p (6 элементов) 6s (2 элемента)

Шестой уровень

Третий суперцикл 3d (10 элементов) 4p(6 элементов) 5s (2 элемента)

Пятый уровень

3p (6 элементов) 4s (2 элемента)

Четвертый уровень

Второй суперцикл 2p (6 элементов) 3s (2 элемента)

Третий уровень

2s (2 элемента)

Второй уровень Первый суперцикл

1s (2 элемента) Первый уровень

Рис. 2

Copiright Yuri Magarshak All rights reserved

8s 7S 6 S 7P 5S 6P 4 S 5P 6D 3 S 4P 5D 2S 3P 4D 5F 1S 2P 3D 4F

1 2 3 4 5 6 7 8 Level

8S 7S 6S 7P 5S 6P 4S 5P 6D 3S 4P 5D 2S 3P 4D 5F 1S 2P 3D 4F

1 2 3 4 5 6 7 8 Level

8s 7s 6s 7p 5s 6p 4s 5p 6d 3s 4p 5d 2s 3p 4d 5f 1s 2p 3d 4f

1 2 3 4 5 6 7 8 Level

Fig. 7б


Fr Rb Na H

Ti Ga B

Lu Sc

La

Uut

Lr

Ac

Y

Al

Cs

In

K Li


Fr Rb Na H

Ti Ga B

Lu Sc

La

Uut

Lr

Ac

Y

Al

Cs

In

K Li

(1s)2 <(2s)2 <

(2p)6 < (3s)2 <(3p)6 < (4s)2 <

(3d)10 < (4p)6 < (5s)2 <(4d)10 < (5p)6 < (6s)2 <

(4f)14 < (5d)10 < (6p)6 < (7s)2 < (5f)14 < (6d)10 < (7p)6 < (8s)2

Порядок заполнения термов

элементами

Порядок Порядок заполнения заполнения периодов 3-7 периодов 2 и 3

термами термами


Li

H

Na

K

Cs

Rb

Fr

Al

B

Ga

In

Y

Sc

Lu

Lr

Ti

Uut

La

Ac

The Basic Features of the 4- Dimensional Pyramid

of Periodically Repeating Properties of Chemical Elements

All chemical elements can be subdivided by• Eight cycles having 2,2; 8,8;18,18, 32, 32

elements each• Four Supercycles having 4, 16, 36 и 64

elements each• Two clusters: M-cluster and F-cluster• Paired clusters, cycles, subshells and

elements

8S 7S 6S 7P 5S 6P 4S 5P 6F 3S 4P 5D 2S 3P 4D 5F 1S 2P 3D 4F

1 2 3 4 5 6 7 8 Level

8s 7s 6s 7p 5s 6p 4s 5p 6d

3s 4p 5d 2s 3p 4d 5f 1s 2p 3d 4f

1 2 3 4 5 6 7 8 Level

n (principal quantum number) 8S 7S 6S 7P 5S 6P 4S 5P 6F 3S 4P 5D 2S 3P 4D 5F CYCLE 1S 2P 3D 4F CYCLE

1 2 3 4 5 6 7 8 Level l (orbital quantum number)

Рис. 9a




Порядок Порядок заполнения заполнения периодов 3-7 периодов 2 и 3

термам термами

Li

H

Na

K

Cs

Rb

Fr

Al

B

Ga

In

Y

Sc

Lu

Lr

Ti

Uut

La

Ac

Рис. 5




Порядок заполнения Циклы термов

каждого цикла

Li

H

Na

K

Cs

Rb

Fr

Al

B

Ga

In

Y

Sc

Lu

Lr

Ti

Uut

La

Ac

Ra Sr Ba Mg Ca He Be

Uub Hg Cd Zn

Uuo Ru Xe Cr Ar Ne

Yb No

O

СHEMISTRY NUMBERS. Each chemical element can be characterized by the set of four chemical numbers. CLUSTER NUMBER , having one of two values: either M or F SUPERCYCLE NUMBER (or, in short, super-number) , which determines to which pyramid super-level element belongs and on which super-cycle is disposed ( = 1, 2, 3, 4).LAYER NUMBER which determines the layer number of the cycle in which subshell is disposed. has integer values from -1 (bottom layer, which in the cycle is filled first) to 0 (surface layer, щелочные и щелочноземельные металлы, подоболочки которых заполняются в циклах последними). NODE NUMBER , which enumerates 2(2 + 1) elements of the subslell.

CERTAINITY PRICIPLE: TWO CHEMICAL ELEMENTS CAN NOT HAVE FOUR IDENTICAL CHEMICAL NUMBERS

),()( lmYrR

0)(2

2 rUE

m

lm

l

xmml

mlxP

mPml

mll

mPml

mll

mm

ml

mj

mj

,...,2,1,0

;,...,2,1,0

)1(!)!(

)!(

2

1)(

cos)(cos)!(2

)!(12

sin)(cos)!(2

)!(12

2/2

02)1(2

222

2

R

rE

mR

r

ll

dr

dR

rdr

Rd

1

2 22

2

lnn

n

mE

r

where n – radial quantum number, equal to the number of nonzero nodes of the radial part of the wave function at finite non-zero r.

Identity of the structure of the M-cluster (as well as that of the F-clusterl) with the dependence of n(l) for hydrogen atom, наводят на мысль о том, that D symmetry of the periodic elements properties on the charge of the nuclei all

elements can be considered as the sates of the single particles. However the necessity a) Getting two grids, which are rotated relative to each

other by /4; б) filling the nodes on the slide 1 (hydrogen atom) and

slide 2 (for the set of all chemical elements)in opposite directions, and

c) The presence of paired cycles, supercycles and clusters in periodically repeating elements properties

makes the derivation of the equation, which solution possess the needed properties, highly nontrivial.

Let the structures and symmetries that are described above and shown in Fig. 1 be generated by a pair of Hamiltonian-type operators, which naturally appear in the presence of two commutative Lie symmetry algebras in the quantum system. From the existence of simple commutative Lie symmetry algebras in a physical problem, it follows the existence of a single commuting pair of energy-type operators for localized quantum states.

Let us consider a finite-dimensional complex representation of the product of semisimple Lie algebras. Any such representation V of g1 g2 decomposes into a direct sum of tensor products Vk Wj where Vk

are irreducible representations of g1 and Wj are representations of g2.

Theorem: Assume that g1,g2 are simple Lie algebras. Then there is a

canonically defined pair of commuting operators 1, 2 such that both

i , i=1,2 commute also with the action of g1,g2 and have integer

nonnegative eigenvalues on V.

Note that the above decomposition may exist in the compexification of the symmetry algebra and not in the algebra itself. Since finite-dimensional representations of the complexified Lie algebra are the same as of it's real form and the above decomposition holds anyway. In our application the complexification of both real algebras are presumably isomoprhic to so(3,C)= sl(2,C) after a complexification and hence have rank 1. Thus there are two natural options:1) both algebras are coming from the independent rotation invariance of both the nucleus and the electron envelope.2) the local symmetry group is the Lorentz group SO(3,1) and though it's Lie algebra so(3,1) has no such a decompostion but the complexification so(3,1) x C = sl(2,C) + sl(2,C).

The actual Hamiltonian operator of the problem splits into a sum of two commuting energy time operators. In our opinion second case is physically more plausible

How could such a symmetry appear in reality? Currently the standard answer to such question is symmetry break. Namely the existence of symmetry within an ensemble of different particles is usually explained by considering them as different states of an "ideal object" with the above symmetry so that the particles are degenerate states of the above object where the symmetry is broken due to some process of natural degeneration.If we try to apply similar explanation we are brought to the idea to view the atom as one of the possible states of an ideal particle which we will be denoting as I-particle. This particle has Lorentz symmetry algebra within a bigger algebra of local symmetries and supersymmetries corresponding to other potential fields. The I-particle exists in this ideal state only 0-time which in practice means a very short time depending on the total energy of the I-particle according to the "uncertainty principle". While degenerating the I-particle creates an avalanche of intermediate particles which retain only pieces of the initial symmetry but previous symmetry is manifested in the set of possible degenerate states.

4d mendel english september 2007

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Transcript of 4d mendel english september 2007