66

Рис. 1. Схематическое изображение зоны проводимости двух разных металлов (Масштабы не соблюдены).
    а) - вариант первый;
б) - вариант второй.

1. Расчеты по Ашкрофту и Мермину.

Э л е м е н т	Z	rs/а0	В теорич.	В измеренный
Cs	1	5.62	1.54	1.43
Cu	1	2.67	63.8	134.3
Ag	1	3.02	34.5	99.9
Al	3	2.07	228	76.0

2. Расчет по рассмотренным в работе моделям.

Э л е м е н т	Z	rs/а0	В теорич.	В измеренный
Cs	1	5.62	1.54	1.43
Cu	2	2.12	202.3	134.3
Ag	2	2.39	111.0	99.9
Al	2	2.40	108.6	76.0

ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THE DENSEST PACKINGS OF CHEMICAL ELEMENTS

© Henadzi Filipenka

Abstract

The literature generally describes a metallic bond as the one formed by means of mutual bonds between atoms' exterior electrons and not possessing the directional properties. However, attempts have been made to explain directional metallic bonds, as a specific crystal metallic lattice.

This paper demonstrates that the metallic bond in the densest packings (volume-centered and face-centered) between the centrally elected atom and its neighbours in general is, probably, effected by 9 (nine) directional bonds, as opposed to the number of neighbours which equals 12 (twelve) (coordination number).

Probably, 3 (three) "foreign" atoms are present in the coordination number 12 stereometrically, and not for the reason of bond. This problem is to be solved experimentally.

Introduction

At present, it is impossible, as a general case, to derive by means of quantum-mechanical calculations the crystalline structure of metal in relation to electronic structure of the atom. However, Hanzhorn and Dellinger indicated a possible relation between the presence of a cubical volume-centered lattice in subgroups of titanium, vanadium, chrome and availability in these metals of valent d-orbitals. It is easy to notice that the four hybrid orbitals are directed along the four physical diagonals of the cube and are well adjusted to binding each atom to its eight neighbours in the cubical volume-centered lattice, the remaining orbitals being directed towards the edge centers of the element cell and, possibly, participating in binding the atom to its six second neighbours /3/p. 99.

Let us try to consider relations between exterior electrons of the atom of a given element and structure of its crystal lattice, accounting for the necessity of directional bonds (chemistry) and availability of combined electrons (physics) responsible for galvanic and magnetic properties.

According to /1/p. 20, the number of Z-electrons in the conductivitiy zone has been obtained by the authors, allegedly, on the basis of metal's valency towards oxygen, hydrogen and is to be subject to doubt, as the experimental data of Hall and the uniform compression modulus are close to the theoretical values only for alkaline metals. The volume-centered lattice, Z=1 casts no doubt. The coordination number equals 8.

The exterior electrons of the final shell or subcoats in metal atoms form conductivity zone. The number of electrons in the conductivity zone effects Hall's constant, uniform compression ratio, etc.

Let us construct the model of metal - element so that external electrons of last layer or sublayers of atomic kernel, left after filling the conduction band, influenced somehow pattern of crystalline structure (for example: for the body-centred lattice - 8 'valency' electrons, and for volume-centered and face-centred lattices - 12 or 9).

ROUGH, QUALITATIVE MEASUREMENT OF NUMBER OF ELECTRONS IN CONDUCTION BAND OF METAL - ELEMENT. EXPLANATION OF FACTORS, INFLUENCING FORMATION OF TYPE OF MONOCRYSTAL MATRIX AND SIGN OF HALL CONSTANT.

(Algorithm of construction of model)

The measurements of the Hall field allow us to determine the sign of charge carriers in the conduction band. One of the remarkable features of the Hall effect is, however, that in some metals the Hall coefficient is positive, and thus carriers in them should, probably, have the charge, opposite to the electron charge /1/. At room temperature this holds true for the following: vanadium, chromium, manganese, iron, cobalt, zinc, circonium, niobium, molybdenum, ruthenium, rhodium, cadmium, cerium, praseodymium, neodymium, ytterbium, hafnium, tantalum, wolfram, rhenium, iridium, thallium, plumbum /2/. Solution to this  enigma must be given by complete quantum - mechanical theory of solid body. 
Roughly speaking, using the base cases of Born-Karman, let us consider a highly simplified case of one-dimensional conduction band. The first variant: a thin closed tube is completely filled with electrons but one. The diameter of the electron roughly equals the diameter of the tube.
With such filling of the area at local movement of the electron an opposite movement of the 'site' of the electron, absent in the tube, is observed, i.e. movement of non-negative sighting. The second variant: there is one electron in the
tube - movement of only one charge is possible - that of the electron with a negative charge. These two opposite variants
show, that the sighting of carriers, determined according to the Hall coefficient, to some extent, must depend on the
filling of the conduction band with electrons. Figure 1.


Figure 1. Schematic representation of the conduction band of two different metals. (scale is not observed).

a) - the first variant;
b) - the second variant.
The order of electron movement will also be affected by the structure of the conductivity zone, as well as by the temperature, admixtures and defects. Magnetic quasi-particles, magnons, will have an impact on magnetic materials.
Since our reasoning is rough, we will further take into account only filling with electrons of the conductivity zone. Let us fill the conductivity zone with electrons in such a way that the external electrons of the atomic kernel affect the formation of a crystal lattice. Let us assume that after filling the conductivity zone, the number of the external electrons on the last shell of the atomic
kernel is equal to the number of the neighbouring atoms (the coordination number) (5).
The coordination number for the volume-centered and face-centered densest packings are 12 and 18, whereas those
for the body-centered lattice are 8 and 14 (3).

The below table is filled in compliance with the above judgements.

Element	RH . 1010 (м3/K)	Z. (number)	Z kernel (number)	Lattice type
Na	-2,30	1	8	body-centered
Mg	-0,90	1	9	volume-centered
Al	-0,38	2	9	face-centered
Al	-0,38	1	12	face-centered
K	-4,20	1	8	body-centered
Ca	-1,78	1	9	face-centered
Ca	T=737K	2	8	body-centered
Sc	-0,67	2	9	volume-centered
Sc	-0,67	1	18	volume-centered
Ti	-2,40	1	9	volume-centered
Ti	-2,40	3	9	volume-centered
Ti	T=1158K	4	8	body-centered
V	+0,76	5	8	body-centered
Cr	+3,63	6	8	body-centered
Fe	+8,00	8	8	body-centered
Fe	+8,00	2	14	body-centered
Fe	Т=1189K	7	9	face-centered
Fe	Т=1189K	4	12	face-centered
Co	+3,60	8	9	volume-centered
Co	+3,60	5	12	volume-centered
Ni	-0,60	1	9	face-centered
Cu	-0,52	1	18	face-centered
Cu	-0,52	2	9	face-centered
Zn	+0,90	2	18	volume-centered
Zn	+0,90	3	9	volume-centered
Rb	-5,90	1	8	body-centered
Y	-1,25	2	9	volume-centered
Zr	+0,21	3	9	volume-centered
Zr	Т=1135К	4	8	body-centered
Nb	+0,72	5	8	body-centered
Mo	+1,91	6	8	body-centered
Ru	+22	7	9	volume-centered
Rh	+0,48	5	12	face-centered
Rh	+0,48	8	9	face-centered
Pd	-6,80	1	9	face-centered
Ag	-0,90	1	18	face-centered
Ag	-0,90	2	9	face-centered
Cd	+0,67	2	18	volume-centered
Cd	+0,67	3	9	volume-centered
Cs	-7,80	1	8	body-centered
La	-0,80	2	9	volume-centered
Ce	+1,92	3	9	face-centered
Ce	+1,92	1	9	face-centered
Pr	+0,71	4	9	volume-centered
Pr	+0,71	1	9	volume-centered
Nd	+0,97	5	9	volume-centered
Nd	+0,97	1	9	volume-centered
Gd	-0,95	2	9	volume-centered
Gd	T=1533K	3	8	body-centered
Tb	-4,30	1	9	volume-centered
Tb	Т=1560К	2	8	body-centered
Dy	-2,70	1	9	volume-centered
Dy	Т=1657К	2	8	body-centered
Er	-0,341	1	9	volume-centered
Tu	-1,80	1	9	volume-centered
Yb	+3,77	3	9	face-centered
Yb	+3,77	1	9	face-centered
Lu	-0,535	2	9	volume-centered
Hf	+0,43	3	9	volume-centered
Hf	Т=2050К	4	8	body-centered
Ta	+0,98	5	8	body-centered
W	+0,856	6	8	body-centered
Re	+3,15	6	9	volume-centered
Os	<0	4	12	volume-centered
Ir	+3,18	5	12	face-centered
Pt	-0,194	1	9	face-centered
Au	-0,69	1	18	face-centered
Au	-0,69	2	9	face-centered
Tl	+0,24	3	18	volume-centered
Tl	+0,24	4	9	volume-centered
Pb	+0,09	4	18	face-centered
Pb	+0,09	5	9	face-centered

Where Rh is the Hall's constant (Hall's coefficient) Z is an assumed number of electrons released by one atom to the conductivity zone. Z kernel is the number of external electrons of the atomic kernel on the last shell. The lattice type is the type of the metal crystal structure at room temperature and, in some cases, at phase transition temperatures (1).

Conclusions

In spite of the rough reasoning the table shows that the greater number of electrons gives the atom of the element to the conductivity zone, the more positive is the Hall's constant. On the contrary the Hall's constant is negative for the elements which have released one or two electrons to the conductivity zone, which doesn't contradict to the conclusions of Payerls. A relationship is also seen between the conductivity electrons (Z) and valency electrons (Z kernel) stipulating the crystal structure. 
The phase transition of the element from one lattice to another can be explained by the transfer of one of the external electrons of the atomic kernel to the metal conductivity zone or its return from the conductivity zone to the external shell of the kernel under the
influence of external factors (pressure, temperature).
We tried to unravel the puzzle, but instead we received a new puzzle which provides a good explanation for the physico-chemical properties of the elements. This is the "coordination number" 9 (nine) for the face-centered and volume-centered lattices.
This frequent occurrence of the number 9 in the table suggests that the densest packings have been studied insufficiently.
Using the method of inverse reading from experimental values for the uniform compression towards the theoretical calculations and the formulae of Arkshoft and Mermin (1) to determine the Z value, we can verify its good agreement with the data listed in Table 1.
The metallic bond seems to be due to both socialized electrons and "valency" ones - the electrons of the atomic kernel.

Literature:

1) Solid state physics. N.W. Ashcroft, N.D. Mermin. Cornell University, 1975
2) Characteristics of elements. G.V. Samsonov. Moscow, 1976
3) Grundzuge der Anorganischen Kristallchemie. Von. Dr. Heinz Krebs. Universitat Stuttgart, 1968
4) Physics of metals. Y.G. Dorfman, I.K. Kikoin. Leningrad, 1933
5) What affects crystals characteristics. G.G.Skidelsky. Engineer N 8, 1989

Appendix 1

Metallic Bond in Densest Packing (Volume-centered and face-centered)

It follows from the speculations on the number of direct bonds ( or pseudobonds, since there is a conductivity zone between the neighbouring metal atoms) being equal to nine according to the number of external electrons of the atomic kernel for densest packings that similar to body-centered lattice (eight neighbouring atoms in the first coordination sphere). Volume-centered and face-centered lattices in the first coordination sphere should have nine atoms whereas we actually have 12 ones. But the presence of nine neighbouring atoms, bound to any central atom has indirectly been confirmed by the experimental data of Hall and the uniform compression modulus (and from the experiments on the Gaase van Alfen effect the oscillation number is a multiple of
nine.
In Fig.1,1. d, e - shows coordination spheres in the densest hexagonal and cubic packings.


Fig.1.1. Dense Packing.
It should be noted that in the hexagonal packing, the triangles of upper and lower bases are unindirectional, whereas in the hexagonal packing they are not unindirectional.

Literature:

1. Introduction into physical chemistry and chrystal chemistry of semi-conductors. B.F. Ormont. Moscow, 1968.

Appendix 2

Theoretical calculation of the uniform compression modulus (B).

B = (6,13/(r_s/a_o))⁵* 10¹⁰ dyne/cm²

Where B is the uniform compression modulus a_o is the Bohr radius r_s - the radius of the sphere with the volume being equal to
the volume falling at one conductivity electron. 
r_s=(3/4p n)^1/3,
Where n is the density of conductivity electrons.

Table 1. Calculation according to Ashcroft and Mermine Element Z rs/ao theoretical calculated

	Z	rs/a0	B theoretical	B calculated
Cs	1	5.62	1.54	1.43
Cu	1	2.67	63.8	134.3
Ag	1	3.02	34.5	99.9
Al	3	2.07	228	76.0

Table 2. Calculation according to the models considered in this paper

	Z	rs/a0	B theoretical	B calculated
Cs	1	5.62	1.54	1.43
Cu	2	2.12	202.3	134.3
Ag	2	2.39	111.0	99.9
Al	2	2.40	108.6	76.0

Of course, the pressure of free electrons gases alone does not fully determine the compressive strenth of the metal, nevertheless in the second calculation instance the theoretical uniform compression modulus lies closer to the experimental one (approximated the experimental one) this approach (approximation) being one-sided. The second factor the effect of "valency" or external electrons of the atomic kernel, governing the crystal lattice is evidently required to be taken into consideration.

Literature:

1. Solid state physics. N.W. Ashcroft, N.D. Mermin. Cornell University, 1975

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介绍固体物理。

传导电子有助于低热金属（法的独龙族-佩蒂特） 。理论计算的同族模型表明，作出贡献的电子热应该是相当可观的。金属原子密集便携，但不是一个，而是好几种类型的软件包-晶格。因此，除了茂密的包装的形成晶格的金属，也发挥了作用和化学特性的原子（核骨架） 。金属键是由于协会的几个原子外层电子的金属一般来说，这些电子导电区。存在的区域显示在众所周知的经验，将其作为短期制动目前期间出现以前促进线圈，以及一些传导电子确定实验大厅。如何定义一个“化学”属性的原子骨架？要做到这一点，确定了一些混合轨道原子骨架，吸引周围区的电导率。钻石堆积密度中的原子晶格等于百分之三十四，协调号码（最接近的原子数目为tsentralnoizbrannogo ）等于4 。一个杂化轨道原子钻石占34除以4等于8.5 protsentov.Po类推的钠原子68除以8等于8.5 protsentov.Otsyuda一些混合轨道原子厚的软件包将等于74除以8,5 ravno9电脑。 （轨道） 。阐述了“关于金属键密集填料的化学元素” http://kristall.lan.krasu.ru/Science/publ_grodno.html http://sciteclibrary.ru/eng/catalog/pages/5216.html （ inEnglish ） 外壳电子，或填写podobolochek第一款混合动力轨道，其余的电子被安置在区的电导率。据推测，在真正的空间，区的电导率应设在附近的细胞表面的Wigner - Zeyttsa 。大体上，它类似于梳子。因此，传导电子有助于低发热的金属，如他们实际上是两个三维空间的复杂曲面。频率的传导电子在晶体连接不仅与晶格常数，但立体几何混合（价）壳牌公司的原子轨道。更多ostsilyatsii在实验中日哈斯车尔芬研究费米表面。鉴于上述规定，很显然，这些机制的电子报税和支付水平，原子壳牌和区电导率必须有所不同。一个很好的文章看出，计算材料特性可以立即向化学元素，而不是空洞的古巴出生，卡门。所有这一切都可能dikovato ，以量子力学，所以将容忍不同意见。 超导金属单晶 为什么决定链接的出现超导电性的晶格热振动的原子？由于材料具有不同的同位素的组成部分转变温度的超导状态。当然，这种依赖，但重要的是微不足道的。 Sverhrovodimost不是取决于类型的网格。超导铌在表中的许多著名指挥，但不会超过。热原子振动几乎相同。为什么其他金属的超导找不到？热波动不是原子的主要机制超导！电导率取决于温度。但是，随着铜，银，由于某种原因，在最低的高温超导不遵守，铌和指挥，这是远不如铜，银高温超导是。是变得更加困难，并导致这类晶格铜。这意味着，没有热波动主要在这里，一些进程中区的电导率。审议你需要知道的一些电子，每个原子的晶格优先区的电导率。波克塞作者认为，超导参与每十个电子，并根据理论刚体在一个简单的传导参与一至三个电子从一个原子或大约每10或100电子。然而，电流的超导更正常传导电流！有事的电子传导区！确定的任务。区的导电性在我看来，细胞表面的Wigner - Zeyttsa ，这是位于原子晶格。大型电子传导性，并没有留下来，再次对这个表面。过渡到超导国家在该地区的传导电子必须组成一个团队或依赖于对方。因此，在区的传导电子使原子应该比较大的铜，镍或银，这是不超导体。人数传导电子的金属元素是在工作，网址： Http ： / / kristall.lan.krasu.ru /科学/ publ_grodno.html ü钒，铌，钽和5传导电子的原子，因此，温度转换因子= 5月30日。 .. 9.26和4.48光ü ，铪，钛，锆， 3电子，和TC = 0.09 ... 0.39和0.65光让正确的内容，表外，还有铅，锡， 4月5日电子和铝， galy ，铟，铊有2-3电子，和TC = 1196 ... 1091 ... 3.40 ... 2第39条，分别为。我们铅和锡因子= 7.19和3.72 ，分别。所需要的证明。由于导热面积，并有电子自旋，然后在我的组织的传导电子社区是通过他的背部。 -------------------------------------------------- ------------------------------我在这里说，传导电子当然是作为一个统一，但不作为波克塞，因为他们开始发挥在距离几千原子之间有更多的电子，然后“队友” 。同样清楚的是，在一些能源各级区的电导率不等于一些传导电子（如量子力学） ，这是因为金额数目相等的传导电子从原子晶格，即1-5个或更多一点。 -------------------------------------------------- ------------------------------传导电子有助于低热金属（法的独龙族-佩蒂特） 。理论计算的同族模型表明，作出贡献的电子热应该是相当可观的。据推测，在真正的空间，区的电导率应设在附近的细胞表面的Wigner - Zeyttsa 。大体上，它类似于梳子。因此，传导电子有助于低发热的金属，如他们实际上是两个三维空间的复杂曲面。这个错误德鲁德。频率的传导电子在晶体连接不仅与晶格常数，但立体几何混合（价）壳牌公司的原子轨道。更多ostsilyatsii在实验中日哈斯车尔芬研究费米表面。 •约瑟夫森效应？有许多报告超导磁现象有关。因此，看来有趣的地方两国超导体薄薄的一层铁磁（如铁）或铜diamagnetics和分析结果。不作任何这些三明治较高的技术合作？ •增加热电偶。根据上述规定。为了提高热电偶金属可以提供以下。负电荷的金属样品，并进行测试。