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In order to enhance the Pd based anodic catalysts for direct formic acid fuel cells (DFAFCs), the research work includes increasing catalyst activity and preventing CO poison. In this study, various zirconium oxides-modified multi-walled carbon nanotubes (MWCNTs) were prepared as the supports of Pd catalysts for DFAFCs by adjusting the preparation parameters: metal adding, sintering temperature and atmospheres. The prepared pure zirconia has both monoclinic and tetragonal phases. The addition of MWCNTs depresses the growth of monoclinic phase. A small amount of Pd adding allows both monoclinic and tetragonal zirconia structures to appear again. Pd nanoparticles of 20 wt% synthesized on MWCNTs and tetragonal ZrO2/MWCNTs have similar particle size, while Pd/[Pd:ZrO2/AO-MWCNTs-300Air-900Ar] have more nanoparticles aggregation. The electrochemical surface area can be improved by adding zirconia which implies those zirconia modified Pd catalysts better electrocatalytic performance. By analyzing the maximum current density and the corresponding potential, Pd/AO-MWCNTs are inferred to undergo the formic acid direct oxidation initially. The Pd catalysts modified by tetragonal ZrO2 have higher current density. Those having both tetragonal and monoclinic ZrO2 modified Pd catalysts have lower potential of formic acid oxidation. All the Pd based catalysts with zirconia modification possess better CO resist ability and electrocatalytic activity. Pd/[ZrO2/AO-MWCNTs-300Air-900Ar] and Pd/[Pd:ZrO2/AO-MWCNTs-300Air-900Ar] which catalyze formic acid in direct oxidation path are the two best catalysts.

Palladium catalysts deposited on carbon nanotubes are used as the anode material in a direct formic acid fuel cell (DFAFC). In this paper we propose a simple method for enhancing the catalytic activity of Pd deposited on carbon nanotubes for formic acid electrooxidation by using ammonia solution. Multiwall carbon nanotubes (MWCNTs) used as the catalyst support are oxidized with concentrated nitric acid for 6 and 50 h. During the oxidation polynuclear aromatic compounds (PACs) are formed. We propose the use of ammonia solution after the oxidation process to remove the PACs from the carbon nanotubes surface. The properties of differently functionalized MWCNTs are studied and the results show that the use of ammonia solution removes the PACs formed during the oxidation process. The oxidation time of the carbon nanotubes has also a significant effect on the properties of the obtained catalysts. The palladium catalysts deposited on MWCNTs oxidized for a shorter time and rinsed with ammonia solution exhibit the highest initial activity in electrooxidation of formic acid in DFAFC (P = 216 mW mgPd−1).

The 20 wt% Pd supported on carbon Vulcan XC-72 (Pd/C) anode catalyst, used to formic acid (FA) electro-oxidation reaction, submitted to different high temperature treatment (HTT) procedures in the temperature range 200–400 °C was studied using X-ray diffraction (XRD) and X-ray photoelectron spectroscopy (XPS) aided with Quantitative Analysis of Surfaces by Electron Spectroscopy (QUASES). The HTT procedures increased the diameter of Pd crystallites influencing their electronic properties, led to formation of surface PdO and PdO2 from the respective bulk Pd oxides, decreased Pd oxide content and PdOx overlayer thickness. Carbon incorporation into Pd with PdCx composition of x

One of the drawbacks of low-temperature fuel cells is high price of platinum-based catalysts used for the electroreduction of oxygen at the cathode of the fuel cell. The aim of this work is to develop the palladium catalyst that will replace commonly used platinum cathode catalysts. A series of palladium catalysts for oxygen reduction reaction (ORR) were prepared and tested on the cathode of Direct Formic Acid Fuel Cell (DFAFC). Palladium nanoparticles were deposited on the carbon black (Vulcan) and on multiwall carbon nanotubes (MWCNTs) surface by reduction of palladium(II) acetate dissolved in ethanol. Hydrazine was used as a reducing agent. The effect of functionalization of the carbon supports on the catalysts physicochemical properties and the ORR catalytic activity on the cathode of DFAFC was studied. The supports were functionalized by treatment in nitric acid for 4 h at 80 °C. The structure of the prepared catalysts has been characterized by thermogravimetric analysis (TGA), X-ray diffraction (XRD), transmission electron microscope (TEM) and cyclic voltammetry (CV). Hydrophilicity of the catalytic layers was determined by measuring contact angles of water droplets. The performance of the prepared catalysts has been compared with that of the commercial 20 wt.% Pt/C (Premetek) catalyst. The maximum power density obtained for the best palladium catalyst, deposited on the surface of functionalized carbon black, is the same as that for the commercial Pt/C (Premetek). Palladium is cheaper than platinum, therefore the developed cathode catalyst is promising for future applications.

Impact of Pd/MWCNTs catalysts preparation method on the catalysts morphology and activity in a formic acid electrooxidation reaction was investigated. Three reduction methods of Pd precursor involving reduction in a high pressure microwave reactor (Pd1), reduction with NaBH4 (Pd2) and microwave-assisted polyol method (Pd3) were used in this paper. Crystallites size and morphology were studied using the scanning transmission electron microscopy (STEM), X-ray diffraction (XRD), whereas elemental composition, Pd chemical state and functional groups content by the X-ray photoelectron spectroscopy (XPS). The prepared catalysts were tested in a direct formic acid fuel cell (DFAFC) as an anode material. The catalytic activity was correlated with a mean fraction of the total Pd atoms exposed at the surface (FE). The value of FE was calculated from the crystallites size distribution determined by the STEM measurements. Non-linear dependence of a current density versus FE, approaching the maximum at FE≈0.25 suggests that the catalytic process proceeded at Pd nanocrystallites faces, with inactive edges and corners. Pd2 catalyst exhibited highest activity due to its smallest Pd crystallites (3.2 nm), however the absence of Pd crystallites aggregation and low content of carbon in PdCx phase, i.e. x = 4 at.% may also affect the observed.

One of the main problems of palladium based catalysts for a direct formic acid fuel cell (DFAFC) is their low stability during a long-term operation. In these studies, the Pd-ZrO2 catalyst supported on the multiwall carbon nanotubes (MWCNTs) was prepared and thermo-chemically treated. These catalysts were tested in a fuel cell for formic acid electrooxidation, and their chemical composition and structure were characterised by the XPS, STEM, HR-TEM and XRD techniques. It was found that the Pd-ZrO2/MWCNTs catalyst after synthesis causes oscillations of the cell voltage during operation resulting in significantly higher deactivation resistance than that of Pd/MWCNTs. This may be attributed to the "self-cleaning" mechanism of poisoned Pd catalyst by carbon monoxide through the electrochemical oxidation of COads (adsorbed) to CO2 (gas).

The performance of Pd electrocatalysts for formic acid electrooxidation was improved by application of metal oxide-multiwall carbon nanotubes composites as a catalyst support. Hybrid oxides/MWCNTs were synthesized by two different methods: chemical reduction method and impregnation method. Pd based catalysts were synthesized by polyol method on the MWCNTs or oxide/MWCNTs composites. The In2O3 was deposited on MWCNTs by impregnation method (In2O3/MWCNTs-IM support) and in the presence of NaBH4 (In2O3/MWCNTs-NaBH4 support). The physical properties of the Pd/In2O3/MWCNTs-IM, Pd/In2O3/MWCNTs-NaBH4, Pd/SnO2/MWCNTs, and Pd/MWCNTs catalysts were characterized and their electrocatalytical performance in formic acid oxidation was compared. During Pd deposition on In2O3/MWCNTs-NaBH4 support, InPd2 structure was formed as observed by XRD. The electrochemical tests indicate that the two Pd/ In2O3/MWCNTs electrocatalysts have higher electrocatalytic activity than those of Pd/SnO2/MWCNTs and Pd/MWCNTs. The best performance was observed for the catalyst obtained by In2O3 impregnation of MWCNTs denoted by Pd/In2O3/MWCNTs-IM.

The electrooxidation of formic acid was performed on Pd-based catalysts supported on hybrid TiO2-C materials prepared from different carbon origins by solvothermal and slurry synthesis. It has been found that carbonized hybrid TiO2-C supports with mesopore texture and high anatase:rutile ratio in the TiO2 framework led to about three times higher activity per Pd mass unit than the catalyst prepared on a commercial Vulcan XC-72 carbon black, which will allow to reduce considerably the amount of expensive noble metal at the anode of DFAFC. The conductivities of raw TiO2/C supports are several orders of magnitude lower than that for carbon Vulcan XC-72. However, the hybrid TiO2-C supports acquire high conductivity during palladium deposition. The rate of electrooxidation of formic acid on Pd/TiO2-C catalysts, calculated per one surface Pd atom (TOF), is for TOF

We present the synthesis of palladium nanocrystals self-assembled on helical carbon nanofibers functionalized with benzyl mercaptan (Pd-S-HCNFs) and their electrocatalytic activity toward the oxida ...

We present a single metal approach to produce highly active catalyst materials based on Pd-decorated helical carbon nanofibers. Helical carbon fibers are synthesized by a chemical vapor deposition ...

Influence of different treatments of Pd-Au/MWCNTs prepared by a modified polyol microwave-assisted method on the morphology and the catalytic activity in the formic acid electrooxidation was investigated. The following standard treatments were applied to purify the metal particles surface from the carbonaceous deposits formed from ethyl glycol adsorbed on the catalyst after the preparation: 1. annealing in air at 300 °C, 2. annealing in air at 300 °C and then reduction in 5%H2/Ar at 200 °C, 3. reduction in 5%H2/Ar at 200 °C. Surprisingly, the most catalytically active sample was that without additional purification treatments. The TEM, XRD and electron spectroscopy methods were applied to characterise the catalysts. Two Pdx Au1–x phases are present in all catalysts. The treatment number 1 and 2 caused: 1. partial removing of the carbonaceous deposits from the metal surface, 2. increases metal particle sizes, 3. decreases of Pd rich phase content, 4. increases ratio of Au to Pd in the catalyst surface layer. Treatment number 4 did not change significantly the sample morphology but decreased its catalytic activity in formic acid fuel cell. It can be due oligomerization of carbon deposits during thermal treatment in reducing (H2) atmosphere (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The catalytic activity in the formic acid (HCOOH) electrooxidation reaction of the Pd multiwall carbon nanotubes (MWCNTs) supported catalyst, prepared by a microwave-assisted polyol method, treated under different gas atmospheres, and temperatures (200 and 300 °C), was investigated. Their respective catalysts surface characterization proceeded in UHV using the electron spectroscopy methods. Content of constituents, i.e., Pd nanoparticles, C and O, was examined quantitatively. Content of Pd phases and chemical forms of C atoms was analyzed assuming respective binding energy (BE) values to Pd 3d and C 1s X-ray photoelectron spectra (XPS) spectra fitted to Gaussian–Lorentzian components. Surface structure was examined from the 3D nanostructure analysis by quantitative analysis of surfaces by electron spectroscopy (QUASES). Catalytic activity of Pd/MWCNTs after various surface treatments was attributed to different surface content and forms of Pd nanoparticles, and PdOx/C overlayer morphology, coverage, and thickness.

Multi-walled carbon nanotubes (MWCNTs) supported Pd–Au nanoparticles for electrooxidation of formic acid (FA) are prepared and compared with Pd/MWCNTs in this study. The nanocatalysts are prepared by a polyol method, and characterized using X-ray diffraction (XRD), thermogravimetry (TGA), SEM, and field emission transmission electron microscope (FE-TEM). Cyclic voltammetry (CV) is used to examine the catalytic activity towards FA electrooxidation. Controlling the contents of the precursors (PdCl2 and HAuCl4·4H2O), the different weight ratios of Pd–Au can be synthesized to form the various solid solution structures of Pd–Au. TEM image results indicate the average diameter of Pd–Au solid solution particles is about 10–30 nm. The results of electrochemical analysis show that the Au (10 wt.%)/[Pd/MWCNTs (1:9)](90 wt.%) hybrid catalysts exhibit the highest catalytic activity and better stability than that of Pd/MWCNTs in FA electrooxidation. The promoting of Pd-based catalyst with Au to form solid solution improves the performance of the catalyst in FA electrooxidation. The newly synthesized electrocatalyst is promising for application in direct formic acid fuel cell (DFAFC).

A new carbon black supported palladium catalyst for direct formic acid fuel cell applications has been prepared and characterized by X-ray diffraction. Bi-modal distribution of Pd crystallite sizes was observed. The average Pd size for crystallites in small size and large size ranges were about 2.7 nm and 11.2 nm, respectively. The initial activity of the catalyst in the oxidation of formic acid tested in a fuel cell was similar to a commercial well dispersed 20 wt.% Pd/Vulcan. The rates of the fuel cell power decay were measured for formic acid of two purities for various current loadings. The results showed that various mechanisms contribute to the decrease of cell power with time. In direct formic acid fuel cell (DFAFC) fed with a very pure HCOOH accumulation of CO(2) gas bubbles in anode catalyst layer is responsible for observed power decay. In DFAFC fed with a pure for analysis (p.a.) grade formic acid the formation of CO(ads) poison from the formic acid impurities is the main deactivation reason. (C) 2011 Elsevier B.V. All rights reserved.

The Pd–Au multiwall carbon nanotubes (MWCNTs) supported catalyst exhibits higher power density in direct formic acid fuel cell (DFAFC) than similar Pd/MWCNTs catalyst. The Pd–Au/MWCNTs catalyst also exhibits higher activity and is more stable in electrooxidation reaction of formic acid during cyclic voltammetry (CV) measurements. After preparation by polyol method, the catalyst was subjected to two type of treatments: (I) annealing at 250 °C in 100% of Ar, (II) reducing in 5% of H2 in Ar atmosphere at 200 °C. It was observed that the catalyst after treatment I was completely inactive, whereas after treatment II exhibited high activity. In order to explain this effect the catalysts were characterized by electron spectroscopy methods. The higher initial catalytic activity of Pd–Au/MWCNTs catalyst than Pd/MWCNTs catalyst in reaction of formic acid electrooxidation was attributed to electronic effect of gold in Pd–Au solution, and larger content of small Au nanoparticles of 1 nm size. The catalytic inactivity of Pd–Au/MWCNTs catalysts annealed in argon is attributed to carbon amorphous overlayer covering of Pd oxide shell on the metallic nanoparticles.

The multiwall carbon nanotubes (MWCNTs) supported Pd catalyst, prepared by the polyol method and reduced in 5% of H2 in Ar atmosphere at temperature of 200 °C, exhibits high activity in electrooxidation of formic acid, whereas, the same catalyst calcinated at temperature of 250 °C in 100% of Ar is inactive. The structural and chemical properties of the catalyst surfaces after reducing and calcination treatments were investigated using electron spectroscopy. The catalytically active sample indicated smaller O content, with the same content of Pd and C at the surface. Simultaneously, the content of carbon–oxygen–hydrogen groups, i.e., carboxyl, carbonyl, ester, ether, and acid anhydride, was lower. In the surface layer of both catalysts, three chemical Pd forms were observed: Pd metallic crystallites, PdO and Pd in organic matrix. In active catalyst, similar content of Pd crystallites and smaller content of PdO were observed. The differences in activity of the investigated Pd/MWCNTs catalysts after reduction and calcination result from the amorphous carbon covering the PdO/Pd nanoparticles, where the PdO layer is of larger thickness for inactive catalyst. This overlayer is removed after reduction in 5% of H2 in Ar atmosphere at temperature of 200 °C.

It is known that palladium-based catalysts are initially very active in direct formic acid oxidation but they suffer from fast deactivation caused by a strongly adsorbed CO intermediate. Reactivation of the catalysts involving application of anodic potential may cause palladium dissolution. The aim of the present study is to increase the stability and performance of palladium-based catalysts in direct formic acid fuel cells (DFAFCs). Preparation and characterization of palladium/multiwalled carbon nanotubes (Pd/MWCNTs) and towards formic acid oxidation via different treatments are described. The catalysts were characterized by thermogravimetric analysis (TGA), X-ray diffraction (XRD), transmission electron microscopy (TEM) and cyclic voltammetry (CV). It was shown that the Pd and Pd–Au MWCNTs supported catalysts after reduction in H2–Ar at 200 °C (R200 treatment) were highly active in formic acid electro-oxidation, whereas the catalysts after heating in argon at 250 °C (C250 treatment) were inactive. The catalysts after hydrogen treatment have smaller metal particles and better contact with MWCNTs support. CV, simulating reactivation of the catalysts, showed that the Pd catalyst suffers from severe Pd dissolution, whereas for the Pd–Au selective leaching of Pd is considerably slower.

Novel multi-wall carbon nanotubes (MWCNTs) supported Pd–Au catalyst for electrooxidation of formic acid was prepared and compared with a similarly prepared Pd/MWCNTs and with a commercial Pt–Ru/Vulcan catalyst. The catalysts were prepared by a two-stage polyol method, followed by H2–Ar annealing and were characterised by using X-ray diffraction, FE-TEM and EDX. Cyclic voltammetry (CV) was used to test their catalytic activity towards formic acid electrooxidation and accessible metal surface in the catalyst layer. In the case of the precursor of Pd–Au/MWCNTs catalyst (before H2–Ar annealing), subsequent deposition of Pd and Au led to a material of the core-shell structure, catalytically inactive. Annealing of the core shell Pd–Au/MWCNTs precursor in H2–Ar resulted in the formation of the novel Pd–Au/MWCNTs catalyst. A highly dispersed Pd–Au solid solution (average XRD particle size 4.8 nm) is formed, and the novel catalyst is more active than the Pd/MWCNTs one. Both the Pd–Au/MWCNTs and the Pd/MWCNTs catalysts turned out to be more active than a commercial, highly dispersed Pt–Ru/Vulcan catalyst.

Developments in the last three decades of kinetics of selective hydrogenation of ethyne in ethene‐rich streams on palladium catalysts are reviewed. Most of the studies can be described comprehensively by a model that assumes carbonaceous deposits (i) create irreversibly on the palladium surface small A types of active site (selective to ethene) and large E types of active site (selective to ethane), and (ii) are involved in hydrogenation of ethene on E s sites on the support. The relative importance of these sites, with varying (i) reaction conditions, (ii) palladium dispersion, (iii) process modifiers, and (iv) promoters, is discussed.

Novel metal catalysts are easily poisoned by CO adsorption or leaching in oxidation of formic acid that leads to decrease the performances of catalyst. To increase the catalyst activity and poison tolerance in fuel cells, novel metal nanoparticles are usually supported on modified support materials to enhance its performance. In this study, TiO2/MWCNTs are synthesized by sol-gel method. Also, ammonium is used to dope nitrogen into TiO2 to modify its electrical and chemical property. MWCNTs, TiO2/MWCNTs and N-doped TiO2/MWCNTs are three supporters using in this study to examine the effects of supporters on the electrocatalytic performance of Pd and AuPd catalysts. The synthesized metal nanoparticles are uniformly dispersed on the surfaces of MWCNTs, TiO2 and N-doped TiO2 modified MWCNTs. The electrochemical analysis illustrate that Pd/N-doped TiO2/MWCNTs (molar ratio of NH4OH:TiO2=4:1) catalyst exhibits higher activity and better stability than that of Pd/MWCNTs either/or Pd/TiO2/MWCNTs catalyst in formic acid electrooxidation. Same results are observed in AuPd series of catalyts. It indicates that suitable N-doping TiO2 improves dramatically on the performance of Pd or AuPd-based catalysts in electrochemical reaction. Thus, hybrid AuPd/N-doped TiO2/MWCNTs (molar ratio of NH4OH:TiO2=4:1) materials have potentially to be used in the direct formic acid fuel cells (DFAFCs) in the future. Keywords—Fuel cells, Formic Acid, Pd, AuPd, N-doped TiO2, catalyst, MWCNTs

The effect of various changes to the palladium catalyst during its stabilization in the course of the selective hydrogenation of ethyne‐ethene mixtures (formation of hydride and carbide phases and of carbonaceous deposits) was reviewed. The deposits in the form of a carbonaceous overlayer on the palladium surface create sites at which selective ethyne hydrogenation to ethene can occur. The carbonaceous deposit on the support increases the selectivity to ethane formation by increasing the rate of ethene hydrogenation on support sites (spillover of hydrogen from metal to the support surface) and by decreasing the effective diffusivity of ethyne in the pores.

The kinetics of hydrogenation reactions of acetylene–ethylene mixtures over a Pd/α-Al 2 O 3 commercial catalyst was studied. Reaction rates were measured by using a gradientless reactor. The kinetic expressions derived, based upon a mechanism involving two types of active sites produced on the palladium surface by carbonaceous deposits, were used for the interpretation of experimental results. The kinetic parameters in these equations and in recently published kinetic expressions were estimated by using multiresponse technique. Statistical analysis of regression showed that the proposed kinetic model describes the process more adequately than do the competitive models.

The hydrogenation of acetylene−ethylene mixtures over a Pd/α-Al2O3 catalyst was studied. Kinetic investigations were carried out in a flow system equipped with a gradientless microreactor over a wide range of acetylene and ethylene partial pressures. Three types of active sites were thus identified: A1 and A2 sites which are highly selective in the hydrogenation of acetylene to ethylene via a mechanism involving the competitive (A1) and noncompetitive (A2) adsorption of acetylene and hydrogen, and E sites which are active only for the hydrogenation of ethylene. E sites are able to adsorb acetylene, an effect which strongly inhibits ethane formation. The above three types of active sites are very likely created on the palladium surface by carbonaceous deposits formed during the stabilization process. The identified sites determine the selectivity of the catalyst and manifest themselves in kinetic investigations depending on the process conditions.

Pd catalysts are very active towards formic acid oxidation, but are being gradually poisoned, mainly by formic acid impurities. Pt-Ru catalysts are stable, but considerably less active than Pd. The Pd-Ru materials are expected to combine high activity and stability. In the present work, such alloys were obtained by high-energy ball milling and by a rapid co-reduction of Pd and Ru salts. Using mechanical alloying we have obtained single phase a fcc solid solution of Ru in Pd (xRu=0.5). The new catalysts appeared to be more stable and about two times more active in the electrooxidation of formic acid than a commercial high surface Pd black. Catalysts obtained by co-reduction of Pd2+ and Ru3+ ions consisted of two phases: hex solid solution of Pd in Ru and fcc solid solution of Ru in Pd. This catalyst was less stable and had a slightly lower activity than the commercial Pd black.

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A novel-designed camera for simultaneous X-ray and kinetic studies was developed. The camera was used for studying the phase transformation and reaction kinetics in the hydrogenation of acetylene-ethylene mixture on Pd/Al 2 O 3 catalysts.

The kinetics of gasoil cracking over an HY zeolite catalyst have been studied at atmospheric pressure and temperatures of 482°, 503° and 524°C, using a fixedbed integral reactor. The total conversion of gas-oil, the yield of gasoline, of various gaseous products, and the influence of temperature on conversion, gasoline and minor product yield are presented here. The various primary reaction rate constants, the constants describing the decay of catalyst activity, and the refractoriness of the feed, have been determined for this system and are compared with those obtained using a LaY zeolite catalyst. The observed differences between these two catalysts are discussed in terms of the nature and number of active sites present on the two catalysts. We conclude that the initial reaction of a saturated gas-oil feed occurs on a range of Lewis acid sites of various strengths. Such sites are more numerous on LaY than on HY, and this disparity is larger for weak sites than it is for strong sites. On a etudie la cinetique du craquage du gas-oil sur un catalyseur forme de zeolite HY, sous la pression atmospherique et a des temperatures de 482°C, 503 °C et 524°C; on a employe un reacteur integral a lit fixe. On presente la conversion totale du gas-oil, le rendement en essence et en differents produits gazeux, et l'influence de la temperature sur la conversion ainsi que sur le rendement en essence et en certains produits d'importance mineure. On a determine, pour ce systeme, les diverses constantes de reaction primaire, les constantes qui decrivent la decroissance de l'activite du catalyseur et le caractere refractaire du produit d'alimentation; on a compare ces resultats aux donnees correspondantes obtenues en employant un catalyseur forme de zeolite LaY. On discute les differences qu'on a observees, lors de l'emploi des deux catalyseurs precites, en fonction de la nature et du nombre des sites actifs sur chaque catalyseur. On conclut que la reaction initiale d'un produit d'alimentation sature en gas-oil se produit sur une serie de sites acides de Lewis de differentes force acides; ces sites sont plus nombreux sur LaY que sut HY et cette disparite est plus prononcee dans le cas des sites faibles que dans celui des sites forts.

Universal mathematical relations between the mean relative size of metallic crystallites (drel(VS)) and their dispersion (FE) in supported metal catalysts have been developed. The relations are: drel(VS) = 5.01/FE for FE 0.2 the commonly used definition of dVS should be modified. The developed relations apply well to various thermodynamically probable crystallite shapes including fcc cubooctahedra of varying geometry, icosahedra, surface-reconstructed cubooctahedra, bcc rhombic dodecahedra, and hcp truncated bipyramids. The new relations are more precise than the commonly used equation dVS = a/FE, where a is a constant specific to a given metal.