PRESCRIPCIONES RELIGIOSAS EN TORNO A LA ALIMENTACIÓN

In order to calculate the resistivity of metal (1.7), one has to know the density of free electrons of valence and mobility of electrons. In one of the method of determina-tion of these values, the Hall effect (1879) is used. If a current i flows in the x direc-tion through a sample of metal placed in a uniform magnetic field of flux density B directed along the z axis (Figure 1.13), then the Lorentz force that acts on the elec-trons moving in the metal

Fm = −e (v × B) (1.11)

TABLE 1.4

Selected Important Electric, Mechanical, Thermal, and Chemical Properties of Aluminum

Properties Numerical Values Comments

Conductivity at temperature 20°C of purest annealed aluminum (99.997% Al)

38 × 10⁶ S/m Influence of admixtures, Figure 1.8 Conductivity at temperature 20°C of

aluminum conductors

(33–35) × 10⁶ S/m Polish standards PN/E-103 and 106 Thermal coefficient of resistivity at

temperature 0–150°C

0.004 1/K Mass density at temperature 20°C 2.70 g/cm³ Limit of tensile strength

Coefficient of thermal conductivity at temperature 20°C

209 W/(m K) Temperature coefficient of linear

expansion at temperature 20°C

Water, water vapor, CO, CO₂, nitric acid Does not affect aluminum at normal temperature In nitric and sulfur acids at heating Easy dissolves

In bases (even weak) Dissolves

On a humid contacts with copper It creates cell that causes strong corrosion of aluminum Source: Adapted from Handbook of Electrical Materials. (in Russian) Vol. 2, Moscow: Gosenergoizdat,

1960.

directed along the y axis in the negative direction. This force presses the electrons to the lower side of the conductor (Figure 1.13a). This results in a nonuniform distribution of electric charges, which causes the appearance of a transverse electric field E_H (Hall field) directed along the negative direction of the y axis and opposing the gathering of electrons in the lower part of the conductor. This phenomenon is called the Hall effect.

The electromotive force F = eUH acting on electrons due to the Hall effect should, in equilibrium state, compensate the magnetic force (1.11), which means

eUH= evB (1.12)

Introducing, according to Equation 1.6, the current density J = Nev, we obtain

eU^H = J BN (1.13)

The value

R U

H J BH

= −Ne1 = −

(1.14)

characterizing the body properties is called the Hall constant. Equation 1.14 allows the experimental determination of this constant, as well as the density of free elec-trons N in the conductor. Knowing the Hall constant and resistivity of the metal, it is possible then to determine, on the basis of Equation 1.7, the electron mobility μe. Considering the dimensions of the plate sample (Figure 1.13b), we can express Equation 1.14 in a more convenient form:

U R I B

H = H d (1.15)

The resistivity of the metal in the direction of the current flow r^H( )H U^x

= J (1.16)

i

(a) (b)

i

− +

+++ Ge I

B d

z y x

B E_H

UH

FIGURE 1.13 Illustration of the Hall effect: (a) vector relations; (b) scalar values.

is called magnetoresistivity. This value, in magnetic fields of flux density B not higher than about 10 T, grows with an increase of the magnetic flux intensity H ([1.22], p. 564) according to the dependence

∆rr = aH a²( ≈10⁻⁶m /A^{2 2}) (1.17)

and at stronger fields, this growth is directly proportional to H.

Since in most metals the density of free electrons N is on the order of 10²⁹ elec-trons/m³, the Hall constant R_H is not large. At room temperature, R_H equals, for example, 2.5 × 10⁻¹⁰ Vm³/(A ⋅ Wb) for Na and 0.55 × 10⁻¹⁰ Vm³/(A ⋅ Wb) for Cu (Wilkes [1.24], p. 272). In spite of that, the Hall effect in metals has been used for building direct current (DC) generators in the form of a copper disk rotating between magnet poles ([1.22], p. 563). Electrons in it are shifted to the edge of the rotating disk, which causes a creation of voltage between the axis and edge of the disk. This system is reversible and can also operate as a motor. The Hall effect is also utilized for building magnetohydrodynamic generators (Figure 2.6), or pumps of liquid met-als, and other equipment. The Hall effect, even more distinctively than in metmet-als, appears in semiconductors (Section 1.2.4).

1.2.2 SuPerconductivity

Superconductivity is referred to as a state in which a body sufficiently cooled down loses its resistivity, and an electric current—once excited—can flow in such conduc-tors without losses, for instance, for several years. Research work on liquefaction of gases opened up the way to superconductivity. The first liquefaction of SO₂ gas (around 1780) was achieved by French scientists J. Clouet and G. Monge.^*

However, the first liquefaction of permanent gases (air, oxygen, carbon monoxide, and nitrogen in static state, as well as hydrogen in hazy state) was achieved in 1883 by Poles K. Olszewski and Z. Wróblewski from Cracow Jagiellonian University.

From that moment, rapid development of the physics of low temperature followed, the so-called cryogenics.

In 1898, Scotsman J. Dewar liquefied hydrogen again. In 1908, Dutchman H.

Cammerlingh-Onnes liquefied helium, reaching temperature 4.2 K, and in 1911, he discovered the superconductivity effect (of mercury). It consists in it that in proxim-ity of temperature of absolute zero (−273.16°C), the resistivity ρ of metal begins to dramatically decrease zero, proportionally to the fifth power of the absolute temperature (ρ ~ T ⁻⁵) of the body (Figure 1.6). The vanishing of resistance occurs almost suddenly in the range of temperature differences of 0.01 K. Many metals can transition into the superconductivity state at temperatures higher than absolute zero. Such metals are called superconductors. They include about 27 pure metals and over 2000 known compounds and alloys. They are usually inferior metallic con-ductors, in which the free conductance electrons are able to join into pairs (Cooper

* After “General Encyclopedia,” Warsaw, 1976.

pairs—1957) as a result of electron–phonon interaction at certain temperatures (transition temperature). It happens in such a way that the resistivity caused by a collision of one electron with ions of crystal lattice is reduced exactly to zero due to rebound of a second counterpart electron, without loss of energy. Copper and fer-romagnetic materials are not superconductors. Superconductors lose superconduct-ing properties when the disruption of Cooper pairs occurs due to the surpasssuperconduct-ing of specific temperature T_c, called the critical temperature, or when the magnetic field intensity on the superconductor surface exceeds the critical value H_c (or B_c= μ0H_c) defined for given metal and given temperature, or when the critical current density J_c is exceeded in a superconductor. This current density is closely connected with the critical field intensity H_c.

According to the Silsbee hypothesis from 1916 ([1.22], p. 566), for a conductor of radius r, the critical current is

I r

c = 2Hc (1.18)

The critical field, per Ref. [1.22], is related to temperature by the dependence:

H H

T

r( )0 1 Tc 2

= −  

 (1.19)

The transition temperature at H_c= 0 and at Jc= 0 is called the critical tempera-ture Tc. Above the critical parameters, the electron pairs mentioned above disinte-grate and the resistivity of the metal returns to its normal value (Figure 1.14).

Superconductors are divided into two main classes:

1. Type I superconductors, ideal or “soft” (pure metals—Figure 1.15a) with strong diamagnetic properties, caused by the fact that the superconductivity current can flow in them merely in a thin surface layer

T

0

H J

FIGURE 1.14 The critical surface T–H–J embracing superconductivity region. (Adapted from Smolinski S.: Superconductivity. (in Polish) Warsaw: WNT 1983.)

2. Type II superconductors, nonideal, so-called “hard” (alloys and intermetal-lic compounds—Figure 1.15b) contain superconducting filaments distrib-uted in the whole mass of metal—thanks to the permanent magnetic field and current that occur in the entire superconductor cross section. These filaments can, similarly to miniature superconducting rings, “catch” the field, causing irreversible hysteresis effects (Figure 1.16) in superconduct-ing properties and two critical fields (Figure 1.16b). The hard superconduc-tors do not have a clear transition border, like soft superconducsuperconduc-tors (curves B_C in Figure 1.15)

Type I superconductors (elements) behave in the superconducting state like ideal diamagnetics.

Type II superconductors (alloys and intermetallic compounds) at fields lower than the first critical value (H < Hc1) behave similarly to Type I. At H_c1< H < Hc2, the field successively penetrates into the superconductor and due to that, in one sample, there can be superconducting zones and normal zones with different sets of values T, H, J (mixed state—Figure 1.16b). After exceeding the second critical value, H > Hc2, the superconductor passes to the normal state. For explanation of the phenomena that occur in superconductors, the so-called two-liquid model of electric conductivity is used. In such a model, one liquid represents the normal electrons of conductivity, and the second liquid represents the created doublet electrons of superconductivity (Cooper pairs). As long as the electrons of superconductivity exist, the electric field in a metal does not exist and normal electrons do not participate in the conduction of the current.

An evidence of quantum character of superconductivity is presented by the Josephson’s phenomenon (Nobel Prize, 1973) occurring on junctions SIS or SNS (S—superconductor, I—insulator, N—normal metal) in which through a thin barrier

B_c B_c

Nb–Ge J. Gavler, 1973 Ba–La–Cu–O, K. Müller, J. Bednorz, 1986 [1.24] (Nobel, 1987) V–Ba–Cu–O [1.23] La–Ba–Cu, P.CHU, 1987 [1.19]

FIGURE 1.15 Dependence of critical flux density Bc on the absolute temperature T, after different sources: (a) ideal, “soft” superconductors; (b) nonideal, “hard” superconductors (Kunzler 1962) and ceramic superconductors (30–100 K).

of thickness d ≈ 10⁻⁷ cm can penetrate both normal electrons and Cooper pairs, giv-ing the superconductgiv-ing state a strong nonlinearity i = f(u). These junctions have found applications in metrology of low voltages (e.g., 10⁻¹⁴ V) and in microelectron-ics [1.12].

Although soft superconductors have been known for about 100 years, they have not found a broad application because of the low value of their critical magnetic field intensity (Figure 1.15a). Only the discovery and investigation of fibrous (hard) superconductors in 1960–1961, with the critical magnetic field intensity Hc exceed-ing 8 MA/m (Bc = 10 T) and critical current density up to 10⁴. . .10⁵ A/cm², caused more broad development of practical applications of superconductivity. It is dis-cussed later, in Section 2.6.

Until lately, the highest observed temperatures and critical fields were reached by compounds of Nb3Sn, V3Ga, and NbAlGe (Tc about 20 K). They are, however, fragile and expensive. Therefore, the broadest application found so far has been the alloy of niobium with zirconium Nb–Zr25% easily undergoing plastic working.

(a) (b)

M;B M;B

B₁

B B₀

M

0 0

c

H_c H

B

B⁰ B₀

B

2

1 M

M

H_c1 H_c2 H

c d e

2

FIGURE 1.16 Magnetization M and flux density B in superconductors: (a) Type I—soft, (b) Type II—hard (Adapted from Ashcroft N.W. and Mermin N.D.: Solid State Physics. Brooks/

Cole, 1976; Smolinski S.: Superconductivity. (in Polish). Warsaw: WNT, 1983; Wyatt O.H.

and Dew-Hughes D.: Metals, Ceramics & Polymers: An Introduction to the Structure and Properties of Engineering Materials. Cambridge University Press, 1974.) c—expulsion of field, Meissner effect (B = 0); d—progressive penetration of field, mixed state; e—normal state (H > Hc2); 1—reversible superconductor; 2—nonreversible superconductor (Adapted from Wyatt O.H. and Dew-Hughes D.: Metals, Ceramics & Polymers: An Introduction to the Structure and Properties of Engineering Materials. Cambridge University Press, 1974.);

B—magnetization characteristic of superconductor (B) and normal metal (B₁); with μ ≈ const.

The main difficulty in the broad application of superconducting technology, apart from the low critical parameters of superconductors known until the 1960s–1980s, was the high cost of cryogenic equipment and especially liquefiers of helium, whose costs reach tens or even hundreds of thousands of US$. Moreover, a high input power required to evacuate 1 W of power from the region of demanded temperature (0.6–3 kW/W) and the high price of helium were additional hindrances.

Achieving sufficiently low temperatures in the range of liquid helium (Table 1.5) was therefore very expensive and technically cumbersome (low heat of evaporation, materials, leak of welds, gaskets, etc.) Approaching the temperatures of liquid hydro-gen (Figure 1.15) in the 1970s promised significant cost reduction, but increased the risk of explosion.

Suddenly, in 1986, there appeared the long expected discovery of the so-called high-temperature superconductors (30–70 K), when the Polish-German scientist J. Bednorz [1.32] “advanced to the level of 30 K” on the basis of ceramic super-conductors. In April 1987, Bednorz together with Swiss K. A. Müller observed superconducting transition in oxide compounds based on rare earth materials (Y, La) of the type Ba–La–Cu–O, at 35 K. It was recognized as the opening of new era in the field of superconductivity,^* by awarding both research with the Nobel Prize in Physics for the year 1987.

Following this approach caused an avalanche development of new discoveries and popularization of the superconductivity research (also in the Technical University of Lodz—J. Turowski, J. Jackowski, and others).

In the Y–Ba–Cu–O system, superconductivity was determined at Tc ≈ 90 K [1.32]. In February 1987, Paul Chu (Houston, USA) discovered superconductivity in the composite of La–Ba–Cu oxide at 98 K [1.26]. Research of 100 K has been con-ducted [1.32] and the discovery of superconductors at transition temperature 240 K and even higher have been anticipated [1.38], [1.12]. Recently, there has been more and more talk of a discovery of superconductivity at room temperature and higher

* Yet, until 1978, some authors wrote ([1.22], p. 573): “Nowadays, it seems very improbable increas-ing critical temperature over 25 K.” It reminds the famous theoretical testimonies from the 1950s on

“impossibility to break away from Earth to the outer space. . .”

TABLE 1.5

Features of Cryogenic Liquids

Properties Nitrogen Hydrogen Helium Neon

Chemical symbol N₂ H₂ He Ne

Molecular mass 28.2 2.02 4.00 20.18

Inflammability at contents in air in % Incombustible 4–74 Incombustible 27.25

Mass of 1 L of liquid in gram 815 64.8 125 24.70

Boiling point in K at 1 atm ≈ 0.1 MPa 77.32 20.37 4.216

Freezing point in K at 1 atm 63.15 13.96 —

Cubical expansion at 1 atm in boiling point 697 777 682 Latent heat of vaporization in cal/kg 47.7 107.5 5.72

in organic material. See http://www.physorg.com/news134828104.html.^* Exceeding the temperature of liquid nitrogen (77 K) is in fact a real technical revolution, because this fluid is easily available and safe. A worse situation is with the critical current in which oxide superconductors based on yttrium Y and lanthanum La are very small in comparison with the NbTi or Nb₃Sn superconductors [1.38]. Therefore, some years will pass until new high-temperature superconductors will be widely introduced into industrial practice. But first, the 240 MVA grid autotransformer was designed (Sykulski [1.54]).

Presently, for building experimental synchronous generators, wires made of Nb–

Ti are used as superconductors.

Casual or operational changes of temperature, field, or current can cause sudden, uncontrolled transitions of the superconductor into a normal state, what can create a hazard of explosion from the energy accumulated in the magnetic field and hence, the destruction of the whole device. That is why a fundamental problem of design-ing superconductdesign-ing devices is the assurance of stability of superconductors. As per S. Smolinski ([1.12, p. 94]), a superconducting wire is internally stable when its thickness or diameter does not exceed

x J

where J_c is the critical current density (A/m²) and C_sc is the volume thermal capacity (J/(K m²)). For instance, in Nb–Ti, x < 50 μm.

Technically, the stability of conductors in superconducting windings (cryogenic stability) is assured by fusion in, rolling in, or splicing of the thread of superconduc-tors into copper surrounding. This hazard is especially high at AC currents, due to the hysteresis losses in superconductor and eddy-current losses in the stabilization envelope. In a case when a step change of flux destroys superconductivity in some section, the conduction of current is taken over by a surrounding normal conductor (Cu) with a sufficiently high cross section. The Cu stabilizer is in turn seated in a material with much higher resistivity, such as Cu–Ni.

In order to limit magnetic couplings between particular fibers, the pitch of the twist of conductors must not be bigger than the permissible value ([1.12], p. 96).

In the rotor of the cryogenerator of Mitsubishi company (Ueda [1.44]), there were used, for instance, conductors in the form of a multicore cable (17 wires) with the dimensions of the conductors 2.1 × 9.3 mm, Cu/Cu–Si/Nb–Ti = 2/0.4/1, supercon-ducting threads of diameter 14 μm with pitch of twist 15 mm and wires of diameter 1.12 mm and length of lay 70 mm. The critical current was 5000 A at the flux density 6.5 T [1.44]. See also Sykulski [1.54].

The later developed multicore conductors for AC 50/60 Hz superconducting coils (Tanaka [1.40]) have the diameter of 0.1 mm, twist pitch of 0.9 mm, threads of super-conductors Nb–Ti of diameter ca. 0.5 μm, in a Cu–Ni envelope, the current 31.8 or 100 A, at the critical magnetic flux density 0.8 to 0.5 T (see Figures 1.14 and 1.15).

* Room temperature superconductivity: One step closer to the Holy Grail of physics. Nature, 9 July 2008.

S. Smolinski ([1.12], p. 97) divides technical superconducting conductors in two classes, from the point of view of their crystalline structure:

Class 1: Alloys of a regular space-centered structure (Figure 1.5a), like Nb–Zr and Nb–Ti, with good ductility and parameters: T_c= 10 K, B_c(T = 4 K) = 10 T; Jc(B = 0) = (4 to 6) × 10⁹ A/m².

Class 2: Compounds of a special structure, for example, Nb₃, Sn, V, Ga, frag-ile, used mainly for spraying of strips made of Cu or Al, with parameters:

T_c= 18 K, Bc(T = 4 K) = 22 T; Jc(B = 0) = 5 × 10¹⁰ A/m².

A broader description of the processing and technical data of superconducting conductors is given in the works of Smolinski [1.12] and Sykulski [1.54]. However, that latest discoveries may have significantly changed some of this information.

It is necessary to mention that besides superconductors there are also the so-called cryoconductive conductors ([1.12], p. 168), working over the transition tem-perature. For instance, a cable made of pure Al in temperatures of 20–30 K has a resistivity 250–100 times lower than resistivity of copper at room temperature. It can also be copper, pure beryllium in temperature of liquid nitrogen, and sodium (Na).

More recently, superconductors have found broader application in superconduct-ing coils for the production of strong magnetic field [10.15]. However, the eddy-current concentrators of AC field (Bessho [1.29])^* still compete with them to obtain 60 Hz flux densities even up to 16 T.

In document Hábitos, comportamientos y actitudes de los  adolescentes inmigrantes sobre nutrición.   (página 158-162)