US3843497A - Electrolytic production of boron - Google Patents

Abstract

1. A PROCESS FOR PRODUCING SUBSTANTIALLY PURE COHERENT BORON ON A CATHODE BY ELECTRODEPOSITION FROM A FUSED SALT BATH CONSISTING ESSENTIALLY OF AN ALKALI OR ALKALINE EARTH METAL FLUORIDE OR MIXTURE OF FLUORIDE COMPRISING: PURIFYING THE ALKALI OR ALKALINE EARTH METAL FLUORIDE OR MIXTURE OF FLUORIDES UNTIL THE RESIDUAL OXIANION CONTAMINATION LEVEL THEREIN IS NO GREATER THAN THAT AMOUNT WHICH WILL PRODUCE A CURRENT MAXIMUM OF 10 MA./CM.2 WHEN AN ANODIC POTENTIAL OF 1.5 VOLTS OF IMPRESSED ON A CARBON ELECTRODE THEREIN; HEATING THE FLUORIDE OR MIXTURE OF FLUORIDES TO A TEMPERATURE OF ABOUT 1000* TO 1400*F. IN A NONCONTAMINATING ATMOSPHERE; ESTABLISHING IN THE BATH A FLUOBORATE ION CONCENTRATION NOT EXCEEDING ABOUT 20 PERCENT BY WEIGHT; ESTABLISHING A CURRENT DENSITY AT THE CATHODE NOT EXCEEDING ABOUT 400 MILLIAMPERES PER SQUARE CENTIMETER; THE CURRENT DENSITY AT THE CATHODE BEING SELECTED WITH REFERENCE TO THE FLUOBORATE CONCENTRATION TO FALL WITHIN THE OPERATING LIMITS DEFINED BY THE AREA LMNO OF FIG.2; AND THE CURRENT DENSITY AT THE CATHODE FURTHER BEING SELECTED WITH REFERENCE TO THE BATH TEMPERATURE TO FALL WITHIN THE OPERATING LIMITS DEFINED BY THE AREA WXYZ OF FIG.3.

Description

2 Sheets-Sheet 1 5. RUSSELL ET AL.

ELECTROLYTIC PRODUCTION 0F BORON Oct. 22, 1974 Original Filed June 17. 1969 s. RUSSELL ETAL v Enmcrnomrrc Pnonucmon or onu Y 2 Sheets-Sheet 8 United States Patent O 3,843,497 ELECTROLYTIC PRODUCTION OF BORON Sid Russell, Suffield, 'Frederick A. Young, Simsbury, and Jordan D. Kellner, West Suffield, Conn., assignors to United Aircraft Corporation, East Hartford, Conn. Continuation of abandoned application Ser. No. 833,913, June 17, 1969. This application Mar. 13, 1972, Ser. No. 234,306

Int. Cl. C2311 5/00; C22d 3/00 U.S. Cl. 204-64 R 8 Claims ABSTRACT 0F THE DISCLOSURE A process for producing substantially pure coherent boron on a cathode by electrodeposition from a fused salt bath comprising an alkali or alkaline earth metal uoride or mixture of uorides -by establishing in the bath a iiuoborate ion concentration not exceeding about 20 percent by weight and heating the bath to a temperature of about 10001400 F. in a non-contaminating atmosphere and establishing a current density at the cathode not exceeding about 400 milliamperes per square centimeter.

This is a continuation of application Ser. No. 833,913, tiled June 17, 1969, now abandoned.

BACKGROUND OF TH'E INVENTION This invention relates to electrochemical processes and more particularly relates to an electrolytic process for depositing substantially pure coherent boron coatings from a fused haloborate bath.

It is known that coatings of essentially pure boron up to several mils in thickness may be produced by pyrolytic techniques wherein the boron is deposited on a resistively heated wire which is drawn through a gaseous reactant stream. Early investigations quickly revealed the utility of these fibers for a variety of applications including the production of fiber-reinforced articles having improved physical properties.

Although the technique of gas phase deposition of boron results in high density continuous coatings, there remain several inherent limitations therein. The process, in requiring the reaction of a boron containing gas with hydrogen gas, engenders problems associated with dual gas ow metering and mixing, ow patten control, and uniform temperature control. Further, the size and shape of substrates which can be coated are severely limited, and only very few substrate materials are compatible with the coating process due to the high temperatures involved.

Although it is known in the prior art to obtain elemental boron by electrolysis from molten salts, it has not been possible to obtain the boron as a pure coating of substantial thickness. Primarily, the prior electrolytic techniques have produced coatings of granular boron containing a high percentage of occluded electrolyte salts. In order to obtain substantially pure boron, secondary processing to leach out the salts has been necessary. Unfortunately, such processing destroys the coating and results in loose boron powder.

Thus, the great desirability of electrolytically producing a salt-free boron electroplate is well recognized but thus far has not been achieved by prior processes.

SUMMARY OF INVENTION In accordance with this invention, means have been formulated to provide by electrolytic deposition a substantially pure coherent electroplate of boron of essentially unlimited thickness upon a variety of substrates including those not readily amenable to the gas phase process. The present invention contemplates the precise control of critical processing variables such as temperature, current density and electrolyte composition in a fused salt electrolyte comprised of an alkali or alkaline earth metal fluoride or mixture of fluorides and containing tiuoborate ions. In a prefered embodiment, the invention contemplates a pretreatment of the electrolyte, prior to electrolysis, to reduce the concentration of contaminating agents to a level below that which has been found to interfere with the depositon process. In brief, the present invention permits the production of coherent boron coatings of from to 100% purity by the establishment of a fluoborate ion concentration up to 20% by weight of the bath, a current density up to 400 m./cm.2 and a temperature between l000 and 1400 F.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. l is a view of an electrolytic cell, taken in elevation and partially in cross section, which is suitable for use in the practice of the present invention;

FIG. 2 is a graph illustrating the relation between current density and BF3 concentration; and

FIG. 3 is a graph illustrating the relation between current density and temperature.

DESCRIPTION 'OF THE PREFERRED EMBODIMENTS Referring 'first to FIG. l, the electrolytic cell is designated generally -by the numeral l0 and includes a graphite or carbon Crucible 12 placed inside a heat resistant, metal shell or pot 14 made of nickel or other suitable metal. Th-e outer pot 14 is fitted with a covering lid 16 having an air-lock 17 so that the melt can ybe kept under an inert atmosphere during electrolysis as well as during a change of cathodes. The cell is externally heatable by an electric resistance furnace 18 surrounding the lower portion of the cell. The Crucible 12 is filled or partially filled with an electrolyte 20 in which are suspended an anode 22 and a cathode 24.

The fused salt electrolyte 20, sometimes referred to hereinafter as a fused salt bath, comprises preferably the uorides and fluoborates of the alkali and alkaline earth metals, such as lithium, sodium, potassium, barium, calcium, strontium and others which have a reduction potential more negative than that required for -boron deposition, with the bath containing up to 20 weight percent uoborate ions. In general, the temperature of the fused salt bath is maintained within the range of from approximately 1000 to l400 F., the exact temperature required being dependent upon the concentration of fluoborate in the electrolyte, the current density at which the boron is being plated, and the physical Configuration of the cell.

In a preferred embodiment of the process for plating pure boron coatings, a mixture of lithium fluoride and potassium uoride is used as an electrolyte base. A mixture comprising 40 to 60 mol percent lithium fluoride and 60 to 40 mol percent potassium uoride is satisfacory.

It is to be noted that electrolyte contaminants such as heavy metal ions must be kept out of the electrolyte. By

. utilizing the cell configuration shown in FIG. l, the electrolyte contacts carbon structural materials only and this objective is achieved.

In order to reduce the starting concentration of dissolved gases, the base halide salts are preferably subjected, before the addition of boron in any form, to a vacuum at a temperature just below their melting point for a period sufficient to achieve an equilibrium pressure of l0*4 mm. Hg. After the foregoing pretreatment, a fluoborate forming compound such as BF3 gas, itself purified of oxygen containing gases, is added to the base halide salts in appropriate amounts to produce the desired concentration of uoborate ions. A concentration up to 2O percent by weight of liuoborate ion content in the electrolyte is considered satisfactory, although a content between 2 and 10 percent by Weight is preferred. It will be appreciated by those skilled in the art that the heat-vacuum conditioning treatment of the base halide salts must be carried out prior to the introduction of the BFS and consequent formation of uoborate in the melt since, with lluoborate present, the decontamination of the electrolyte by vacuum treatment cannot be effectively accomplished due to the fact that the outgassing of EP3 from the electrolyte restricts the level of pressure to which the system can be evacuated.

In an alternative pretreatment technique, the electrolyte can be purified subsequent to the formation of uoborate. In this procedure, the base :halide salts are melted and a iluoborate forming compound such as BF3 gas or KBFg powder is added. The electrolyte is then conditioned by the addition of a reducing element such as elemental boron. The boron reacts with dissolved gases and vapors such as water, oxygen and carbon dioxide and also with oxianionic contaminants, such as carbonate and hydroxide to reduce their concentration to below interfering levels. It is to be noted that heavy metals in solution also are reduced out of solution by the elemental boron.

In the practice of the invention, a carbon Crucible 4 inches in diameter and inches in depth was used. Reagent grade anhydrous salts, KF (1385` grams) and LiF (615 grams), were weighed quickly in air to prevent excessive moisture pickup and then placed in a vacuum oven at 180 C. for several days. The hot, partially dried salts were then transferred to a similarly baked out carbon crucible and installed quickly within the electrolytic cell, which was then evacuated to 10F4 mm. Hg for at least 24 hours at 450 C. In general, subjecting the uorides to a temperature of from 110 to 480 C. at a pressure from 10*2 to 10H6 mm. Hg for up to 200 hours is satisfactory. Argon gas, which was passed through a magnesium perchlorate drier and a bed of hot copper turnings to remove oxygen, was then introduced over the salts and the temperature was raised to 700 C. to melt the salts. Tests showed that after the aforesaid hard vacuum-high temperature treatment, the residual dissolved moisture and gases, as determined by direct measurements of Water, carbon dioxide, and carbon monoxide in the argon purge gas, Was extremely small. Carbon dioxide and carbon monoxide have proved to be particularly effective indicators of contamination level since water and oxygen, which are the most important contaminants, react with the carbon Crucible to produce these gases. It has been determined that a satisfactorily purified melt exhibits less than 10 p.p.m./v. each of carbon monoxide and carbon dioxide in the purge gas at a tiow rate of 30 cm3/min.

The residual contamination level was also determined by voltammetry. It was found that the degree of residual oxianion contamination could be determined from the observed current when an anodic potential sweep mv./ min.) was made on a carbon electrode dipping into the conditioned melt. A current maximum was consistently observed when the potential reached about 1.5 volts more positive than a second carbon quasi reference electrode. The height of the current maximum could be related to the contamination level. Assignment of the current to oxianion discharge was possible from the observed elution of carbon dioxide and carbon monoxide from the working carbon electrode as measured in the argon purge gas. In a properly conditioned melt, that is, one containing contaminants at a non-interfering level, a maximum current of about 0.5 ma./cm.2 was observed, again with less than 10 p.p.m./v. each of carbon dioxide and carbon monoxide in the efliuent purge gas at a ow rate of 30 cm.3/min. Although a current maximlun of 0.5 ma./cm.2 or lower is preferred, satisfactory depositions have been made with current maximums up to l0 Ina/cm?.

When it was ascertained that the melt was sufliciently purified, commercially available BFS gas was bubbled into the melt through a hollow carbon tube until the selected concentration of tiuoborate ions was produced.` The gas was processed before entering the cell by passing it over steel wool heated to 450 F. to remove air, sulfur dioxide, and other oxygen containing gases which are generally present in the commercial supplies, and then passed through a particulate lter to take out the boric oxide smoke formed in that process. Almost all of the BPB added was absorbed by the melt, and therefore the fluoborate produced could be closely determined by the volurne of gas introduced. However, confirmation of the precise iiuoborate concentration was accomplished electrochemically by chronopotentiometry and by chemical analysis.

In -an alternative technique, a less rigorous melt purification procedure is used and the B133 makeup gas and the argon sweep gas need not be purified as described above. During experimentation, it was discovered :that the concentration of dissolved contaminants can still be substantially controlled by treating the electrolyte containing the lluoborate with elemental boron. The boron additive reacts with interfering contaminants in the melt :to reduce their concentration, although the exact action of the elemental boron is incompletely understood at this time. When boron is rst added, evolution of 'a gas, suspected lto be hydrogen, is generally observed. This ceases after a short period of time. The boron addition is then characterized by la decrease in the concentration of iluo'borate in the electrolyte with an increase in the concentration of boron present as soluble ionic species other than uoborate. Boron chips subsequently retrieved from the electrolyte, at times, show that heavy metals such as iron also are being reduced on their surfaces. When the electrolyte becomes passive to the further addition of elemental boron, the treatment is considered complete. It is presently believed that the elemental boron combines with both the fiuoborate and dissolved contaminants to produce less interfering ionic boron oxiuoride species. It is desirable to use line lboron chips in the treatment, since the reaction with the electrolyte produces a coating on the boron surface which deactivates it, leading to long reaction time and material waste. The appropriate action of elemental boron i-s not accomplished under electrolysis xif the boron is being used as an anode.

In the practice of boron plating, although graphite anodes were found satisfactory, the working anode preferred was a boron-llcd carbon tube. The carbon tube used was one inch in diameter and stoppered at one end with a carbon plug and at the other end with a carbon plug and support rod. The tube used had approximately holes, 1/32 inch 'in diameter, -to permit the boron chips inside to be in contact with the electrolyte. The anodes were preferably assembled and baked out in an oven for several hours before use in order to remove absorbed oxygen and water. Since the amount of boron originally present in the anode was known, periodic checks on the amount of current passed were made as a means for determining the amount of boron remaining.

The cathodes most frequently used were generally pure copper strips 1A inch wide by 21/2 inches long. Other materials, including carbon and molybedum, and other shapes were also coated. In general, any metal and alloy inert to the fused bath salts and suitable for use as a cathode may be boron plated. The copper strips were suspended from above using copper rods and were anodically cleaned in an alkaline cyanide bath, follwed by rinses in water and dilute HC1, and iinally dried with acetone. At times, as an alternative to the electrolytic cleaning, the copper cathodes were cleaned in dilute HN-O3.

During the plating operation, a constant slow sweep of purified inert gas, such as argon, was maintained over the electrolyte at a pressure slightly above atmospheric. The argon sweep is advantageous in that is prevents the inflow of atmospheric air and also facilitates periodic analyses of the effluent gas mixture from the cell. Over Ia period of time, however, there vis la cumulative loss of BF3 gas from the system wherein the =BF3 gas must -be replenished.

Under varied combinations of operating parameters, boron coatings were obtained with five generally distinc- -tive surface finishes. As shown -in FIGS. 2 and 3, the coatings were producd under the operating conditions defined by the areas LMNO and WXYZ. All of the coatings were coherent, that is, they were continuous, homogeneous and in one piece. For convenience, the surface finishes have been classified as (A) a mirror finish, (B) a steel gray finish, (C) a matte finish, (D) a velvet finish, and (E) a sooty finish. The appearance of these coatings was characteristic of the geometry of the surface since the bulk of the coating appeared similar. The mirror finish coatings have very smooth flat surfaces while the steel gray finish coatings are characterized by scattered nodules which break up the reflection from the surface. The matte coatings have completely nodular surfaces. The velvet surface is covered with some microscopic granules, some of which canl be rubbed off. The sooty coatings have a more powdery surface and a more appreciable amount of boron can be rubbed off. The coating bulk, in all cases, when broken showed the typical conchoidal fractures of amorphous boron.

Some of the coatings were examined by X-ray diffraction and showed the halo pattern typical of amorphous boron :as is found for the pyrolytic-ally deposited material.

Chemical analysis of the various coating types have shown variations in coating purity from 85% boron to substantially 100% boron with the purity showing a general correlation with the surface finishes which were obtained. The mirror finish coatings generally exhibit the lowest purities, while essentially pure coatings are obtained when surfaces appear velvet or sooty. The lower purity of mirror finish coatings may be due to the formation of small microcracks and subsequent salt inclusion in openings, rather than to some intrinsic coating property. The coating type which presently promises wide utility is that which ranges between velvet and matte because it combines high purity withl good surface characteristics.

The plated coatings were normally one or two mils thick, although thicknesses up to 23 mils were prepared on occasion. Some preliminary physical properties were determined for the coatings. Values for elastic modulus in flexure range between 60 105 and 70 106 p.s.i., and -a Vickers hardness of approximately 3000 HV were measured. It will be appreciated that the later measurement is similar to the literature value for boron which is 2800 HV.

Further to the above properties, it was determined that the coatings produced by the presen-t process were tightly adherent -to the cathode. The boron plate formed an excellent bond to Vthe copper, and coatings produced directly on carbon and molybdenum were also found t0 be highly adherent.

As indicated, the operating conditions for obtaining the various coatings are shown in FIGS. 2 and 3. From these graphs, it can be seen that, in general, by increasing the concentration of B-F3 in the electrolyte, the same type coating is achieved at increasing current densities. By the same token, an increase in temperature has the effect of permitting the achievement of the same type coating at a higher current density. For example, at temperatures between 1080" and 1350 F., current densities of ma./cm.2 to 100 ma./cm.2, depending on the concentration of BF( in the electrolyte, are necessary to produce smooth metallic coherent coatings of boron. Concentrations of about 1-3 weight percent BFf, produce smooth coatings in the current density range of 15- 50 ma./cm.2 at temperatures above 1200 F. At concentrations of 3-7 weight percent BFf, smooth coatings are obtained up to 100 ma./cm.2 current density and down to 1100 F. temperature. With appropriate control over electrolyte BF4 concentration, temperature, and cathode operating current density, it is possible to select and produce the desired type boron coating over a wide range of plating conditions.

While the present invention has been described with reference to the application of |boron coatings directly to a material to be coated, it is appreciated that the described coating technique admits of wider application. It is recognized for example that the boron coating need not be direct and that copper, for example, may be used as a tie coat to promote `bonding to other metal substrates. This technique has, for example, actually been used successfully to effect bonding to titanium and steel. It is also recognized that the process may be used in the production of such products as two dimensionally isotropic composite structures, which may be built up by plating successive coatings of boron and another material on each other. Such a composite was actually made using alternate coatings of boron and copper. Additionally, composite structures have been made using epoxy adhesive between successive sheets of boron coated copper foil.

For the sake of clarity, the foregoing description has been made with reference to particular materials, embodiments and operating parameters. It will be understood that these examples are illustrative only and that alternative materials, arrangements and operating conditions will be evident to those skilled in the art. Accordingly, the true scope of the invention will be measured, not by the illustrative material but rather, in the spirit of the invention, by the appended claims.

What is claimed is:

1. A process for producing substantially pure coherent boron on a cathode by electrodeposition from a fused salt bath consisting essentially of an alkali or Aalkaline earth metal fluoride or -mixture of fluoride comprising:

purifying the alkali or alkaline earth metal fluoride or mixture of fluorides until the residual oxianion contamination level therein is no greater than that amount which will produce a current maximum of 10 ma./cm.2 when an anodic potential of 1.5 volts is impressed on a carbon electrode therein;

heating the fluoride or mixture of fluorides to a temperature of about l000 to 1400" F. in a noncontaminating atmosphere; establishing in the bath a fluoborate ion concentration not exceeding about 20 percent by weight;

establishing a current density at the cathode not exceeding about 400 milliamperes per square centimeter; the current density at the cathode being selected with reference to the fluoborate concentration to fall within the operating limits defined by the area LMNO of IFIG. 2; and

the current density at the cathode further being selected with reference to the bath temperature to fall within the operating limits defined by the area WXYZ of FIG. 3.

2. The invention of claim 1 wherein the current density at the cathode is selected with reference to the fluoborate concentration to fall within the operating limits defined by the area LPQ of FIG. 2 and wherein the current density at the cathode is further selected with reference to the bath temperature to fall within the operating limits defined bythe area UVW of FIG. 3.

3. The invention of claim 1 wherein said fused salt bath comprises a mixture of 40 to 60 mol percent potassium fluoride and 60 to 40 mol lpercent lithium fluoride.

4. The invention of claim 3 wherein the fluoborate ion concentration is established by bubbling boron trifluoride gas into the fluoride mixture.

5. The invention of claim 4 wherein the bath is purified prior to the establishment of the luoborate concentration by subjecting the fluorides to a heat and vacuum treatment.

6. The invention of claim 5 wherein said fluorides are subjected to a temperature of from to 480 C. at a 7 pressure of from 10*2 to 10-5 mm. Hg for up to 200 hours. l

7. A boron coated product obtained by the method of claim 2, the boron coating having a hardness of approximately 3000 HV, an elastic modulus in exure of 60x106 to 70x106 p.s.i. and beingcoherent and amorphous.

8. A process for producing substantially pure coherent boron on a cathode by electrodeposition from a fused salt bath consisting essentially of a mixture of 40 to 60 mol percent potassium iiuoride and 60 to 40 mol percent lithium uoride comprising the steps of:

heating the bath to a temperature of about 1000 to 1400 F. in a noncontaminating atmosphere;

establishing in the bath a uoborate concentration `of approximately 1 to 2O percent by Weight by adding potassium uoborate thereto;

purifying the bath by nonanodically introducing therein tine chips of elemental boron until the electrolyte becomes passive thereto;

establishing a current density at the cathode not exceeding about 400 milliamperes per square centimeter; the current density at the cathode being selected with reference to the Huoborate concentration to fall 8 within the operating limits defined by the area LMNO of FIG. 2; and A the current density at the cathode further being selected with reference to the bath temperature to fall within the operating limits defined by the area WXYZ of FIG. 3. Y l References Cited UNITED STATES PATENTS 1,795,512 3/1931 Schmidt er a1. 204.49 1,845,978 2/1932 Hosenfeld 204- 39 2,918,417 12/1959 Cooper et al. 204-60 2,984,605 5/1961 Cooper 204-60 3,030,284 4/1962 siem v1204-60 `FOREIGN' PATENTS 989,807 4/1965 Great Britain 204-39 JOHN H. MACK, Primary Examiner D. R. VALENTINE, Assistant Examiner U.S. Cl. X.R. 204-39 P04050 UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION (5l/Es) Dated October 22, l21

Patent No. 3,8%,497

Inventor@ SID RUSSELL ET AL It is certified that error appears in 1e above-identified patent and that said Letters Patent are hereby corrected as shown below:

Column l, line 45' "Patten" should be pattern.

Column 2, line 3 "prefered" should be "preferred".

line ll "m. /cm should be wma/cm Column 3, lne'SI "mv. /mn' should be -I-mV/min--n Column 4, line 61+ "follwed" should be "followed". line 7l, Y "S'w' `SeConclMoccurrence Should be o Signed and sealed this 24th day of December 1974.

SEAL) Attest:

C. MARSHALL DANN Commissioner of Patents MCCOY M. GIBSON JR. Attestng Officery

Claims (1)

1. A PROCESS FOR PRODUCING SUBSTANTIALLY PURE COHERENT BORON ON A CATHODE BY ELECTRODEPOSITION FROM A FUSED SALT BATH CONSISTING ESSENTIALLY OF AN ALKALI OR ALKALINE EARTH METAL FLUORIDE OR MIXTURE OF FLUORIDE COMPRISING: PURIFYING THE ALKALI OR ALKALINE EARTH METAL FLUORIDE OR MIXTURE OF FLUORIDES UNTIL THE RESIDUAL OXIANION CONTAMINATION LEVEL THEREIN IS NO GREATER THAN THAT AMOUNT WHICH WILL PRODUCE A CURRENT MAXIMUM OF 10 MA./CM.2 WHEN AN ANODIC POTENTIAL OF 1.5 VOLTS OF IMPRESSED ON A CARBON ELECTRODE THEREIN; HEATING THE FLUORIDE OR MIXTURE OF FLUORIDES TO A TEMPERATURE OF ABOUT 1000* TO 1400*F. IN A NONCONTAMINATING ATMOSPHERE; ESTABLISHING IN THE BATH A FLUOBORATE ION CONCENTRATION NOT EXCEEDING ABOUT 20 PERCENT BY WEIGHT; ESTABLISHING A CURRENT DENSITY AT THE CATHODE NOT EXCEEDING ABOUT 400 MILLIAMPERES PER SQUARE CENTIMETER; THE CURRENT DENSITY AT THE CATHODE BEING SELECTED WITH REFERENCE TO THE FLUOBORATE CONCENTRATION TO FALL WITHIN THE OPERATING LIMITS DEFINED BY THE AREA LMNO OF FIG.2; AND THE CURRENT DENSITY AT THE CATHODE FURTHER BEING SELECTED WITH REFERENCE TO THE BATH TEMPERATURE TO FALL WITHIN THE OPERATING LIMITS DEFINED BY THE AREA WXYZ OF FIG.3.

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