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For the doubly bridged tetracarbonyl, each iron has one terminal carbonyl ligand, two bridging carbonyl ligands and one cyclopentadienyl ligand. Neutral electron counting gives $8+5+2+2=17$ before direct bonding, so a single Fe–Fe bond gives 18 at each iron. For the triply bridged tricarbonyl, each iron instead has $8+5+3=16$ before direct bonding; formal completion to 18 requires a double Fe–Fe bond. Equivalently, the total fragment counts before metal–metal bonding are 34 and 32 respectively.The increased direct bond order explains the substantial contraction. The pentamethylcyclopentadienyl ligands also alter steric and electronic effects, and bridge geometry changes, so this is not a controlled measurement of bond order alone. Nevertheless, the markedly shorter distance in the less carbonyl-saturated dimer is consistent with stronger direct Fe–Fe bonding. The isolated triply bridged compound was characterized structurally and magnetically in the primary work.^[ref]

The dimer has two unpaired electrons and a triplet ground state, $S=1$. Its approximately threefold-symmetric Fe${}_2$(CO)${}_3$ core supplies a pair of degenerate frontier molecular orbitals. The last two electrons occupy separate members with parallel spins, favored by exchange, rather than pairing in one. A useful qualitative picture is a sigma bond plus two perpendicular, singly occupied pi half-bonds: together these support a formal Fe–Fe double bond while retaining paramagnetism. The spin-only magnetic moment for $S=1$ is $\sqrt{8}{\mu }_B\simeq 2.83{\mu }_B$; real moments can include orbital and spin–orbit effects.

The formal 18-electron count does not require every molecular orbital to be doubly occupied. It is therefore compatible with an open-shell dimer. The primary structural/magnetic study establishes this unusual triplet state; an isolated-atom high-spin $d^7$ diagram would not describe the coupled metal–ligand system.^[ref]

The metal is Cr(II), $d^4$. A chloride ligand produces a relatively weak ligand field, so the octahedral center is a high-spin complex with $t_{2g}^3{e}_g^1$, $S=2$. The singly occupied $e_g$ pair is orbitally degenerate: $d_{z^2}$ and $d_{x^2-y^2}$ cannot retain identical occupancies once the distortion selects one. This is a strong setting for the Jahn-Teller effect.

On tetragonal elongation, the two axial Cr–Cl distances increase. Repulsion is reduced most for $d_{z^2}$, so it falls relative to $d_{x^2-y^2}$; $d_{xz},d_{yz}$ also fall relative to $d_{xy}$. The single $e_g$ electron occupies the lower $d_{z^2}$-derived level. The electronic stabilization offsets an elastic cost until a distorted equilibrium geometry is reached. The high-spin $d^4$ occupancy drives an elongated, rather than regular, octahedron. The three lower-set electrons remain one per orbital in this simple diagram; it is primarily the unequal $e_g$ occupancy that produces the strong distortion.

For an octahedral complex, strong axial sigma donation raises the metal–ligand antibonding $e_g$-derived orbitals. A hydride is a very strong sigma donor: its occupied $1s$ orbital overlaps effectively along a metal–ligand axis and has no filled perpendicular p orbitals to act as a conventional pi donor. It can therefore produce a large octahedral splitting by raising $e_g$ relative to $t_{2g}$.

Triphenylphosphine also gives sigma donation, but its useful acceptor orbitals additionally receive metal pi backbonding. This lowers the metal $t_{2g}$-derived levels and enlarges their separation from $e_g$. The acceptors have substantial P–C antibonding character; invoking a dominant phosphorus $3d$ acceptor is unnecessary. Thus hydride is strong-field principally through sigma bonding, while PPh${}_3$ combines sigma donation with pi acceptance. The spectrochemical series describes the resulting splitting, not just a ranking of ligand basicity.

In the ionic assignment, both tetrahedral anions would require Mn(VII), $d^0$. Such a high oxidation state is strongly oxidizing and is best supported by strong bonding to oxygen. Short Mn–O bonds combine good orbital overlap with oxygen's resistance to further oxidation. This stabilizes permanganate relative to a hypothetical all-sulfide analogue.

sulfide is larger, more polarizable and much easier to oxidize than oxide. An attempted Mn(VII) sulfide would therefore combine an exceptionally oxidizing metal center with reducing ligands: internal redox can reduce manganese while oxidizing sulfur, for example toward S–S-bonded or elemental sulfur products. The larger sulfur donors also provide less favorable short multiple bonding to the small, highly charged manganese center. The oxygen compound supports Mn(VII); the sulfide combination is redox-incompatible. This predicts instability without requiring a specific unprovided decomposition product.

For comparable octahedral environments with the same donor set, the expected qualitative order isAcross comparable first-row ions, the increasing effective attraction and changing metal–ligand distances tend to increase octahedral splitting, giving Mn(II) below Fe(II). Raising the charge on the same metal gives Fe(III) above Fe(II), because the ligand interaction generally strengthens and metal–ligand distances contract. Moving from $3d$ to more radially extended $4d$ orbitals gives larger metal–ligand overlap, placing Mo(III) above Fe(III); Rh(III) lies further along the same $4d$ series and typically gives the larger splitting. The $5d$ ion Pt(IV) combines the heavier row with high charge and gives the largest splitting here.

This is a matched-environment trend, not an intrinsic list of constants for bare ions. Actual $\Delta$ also depends strongly on donor identity, geometry and distance. It would be meaningless to compare an unspecified weak-field tetrahedral compound with a strong-field octahedral compound solely through this ordering.

iron remains Fe(II), $d^6$, in both states; the change is a spin crossover, rather than a change in oxidation state. At the higher temperature the high-spin complex has $t_{2g}^4{e}_g^2$: four unpaired electrons, $S=2$, andThis matches the observed moment. At low temperature a low-spin complex with $t_{2g}^6{e}_g^0$ has $S=0$ and shows diamagnetism. Cooling switches high-spin $d^6$ to low-spin $d^6$. Emptying the metal–ligand antibonding $e_g$ orbitals strengthens and shortens the Fe–N bonds, accounting for the contraction by roughly $0.2\text{A˚}$.

The octahedral splitting and electron pairing energy must be sufficiently close for both states to be accessible. The low-spin state gains electronic energy, while the high-spin state has greater spin degeneracy and usually greater vibrational entropy because of its softer bonds. The resulting Gibbs free energy balance favors high spin at higher temperature. The two endpoint observations do not by themselves specify hysteresis or transition cooperativity; those require a temperature series. Coupled structural and magnetic measurements distinguish this explanation from a mere bond-length anomaly.

Write $R$ for the current growing chain. Propene first binds at the available site of $[\mathrm{C}{\mathrm{p}}_2\mathrm{ZrR}{]}^{+}$, giving a coordinated alkene adjacent to the metal-bound chain. In the Cossee-Arlman mechanism, the chain migrates to the substituted alkene carbon while the terminal carbon forms the new Zr–C bond:The electron-pair arrows for insertion take the old Zr–R bond toward the substituted alkene carbon and the alkene pi bond toward the new Zr–terminal-carbon bond. A four-center arrangement permits both bonds to change in one migratory insertion; Zr remains Zr(IV). The product again has an accessible site, so coordination and insertion can repeat.

Predominant 1,2 insertion leaves a primary Zr–CH${}_2$ chain end and generates head-to-tail polypropylene. It avoids the more crowded alternative secondary Zr–CH(Me) end and unfavorable methyl contacts with the cyclopentadienyl ligands in the insertion geometry. This explains regioselectivity. An unbridged, unsubstituted zirconocene does not by itself establish a fixed tacticity; the schematic alignment of methyl groups in a polymer drawing is not evidence for an isotactic product.

The trityl cation acts as an electrophile and removes one metal-bound methyl by methyl abstraction. The balanced conversion isThus $C=[\mathrm{C}{\mathrm{p}}_2\mathrm{ZrMe}{]}^{+}[\mathrm{B}({\mathrm{C}}_6{\mathrm{F}}_5{)}_4{]}^{-}$ and $D=\mathrm{P}{\mathrm{h}}_3\mathrm{CMe}$, 1,1,1-triphenylethane. Transfer is formally of $\mathrm{M}{\mathrm{e}}^{-}$ to carbon, not transfer of a hydride or a methyl radical. zirconium remains Zr(IV), $d^0$. Ionic electron counting changes from $12+4=16$ in the dimethyl to $12+2=14$ at the cation, creating an accessible coordination site. The borate is a weakly coordinating anion, although ion pairing can still occur in solution.

The filled alkene $\pi$ molecular orbital donates into a vacant metal acceptor orbital directed toward the center of the C=C bond. In the reverse direction, a filled metal $d$ orbital can overlap with the alkene ${\pi }^{\ast }$ antibonding molecular orbital. These are the donation and pi backbonding components of the Dewar-Chatt-Duncanson model, illustrated in the lower part of the bonding schematic. The alkene is an ${\eta }^2$ alkene ligand, attached through both carbons. Increasing backbonding weakens and lengthens C=C and can push the bonding toward a metallacyclopropane description.

Zr(IV) is $d^0$, so alkene-to-metal donation predominates. It has suitable empty orbitals but no occupied metal $d$ set to provide substantial conventional backbonding. An electron-deficient cationic zirconocene can consequently bind and activate an alkene largely through its Lewis acidity; it should not be drawn as a strongly backbonded late-metal alkene merely because its formal coordination mode is ${\eta }^2$.

A plausible explanation is a stronger, less labile Hf–C interaction, reflecting the different bonding of the heavier metal. Breaking and reorganizing that interaction in migratory insertion can raise the propagation barrier and reduce the insertion rate. At the same time, stronger retention of the metal-bound alkyl can suppress beta-hydride elimination or other chain transfer in chain polymerization processes even more strongly.

The chain length depends on competing rates, not on propagation speed alone. In a simple stationary competition,where the denominator includes the operative chain-transfer or chain-stopping rates per active chain. The Hf catalyst can propagate more slowly yet make longer chains if transfer is suppressed by a still larger factor. Slower insertion alone would not explain longer polymers. Distinguishing particular transfer pathways would require polymer end-group analysis and the concentration dependence of molecular mass and polymerization rate.

The methyl nucleophile attacks an electrophilic ring boron. That boron becomes tetrahedral, bearing its original methyl, the incoming methyl, and its two ring nitrogen neighbors. This interrupts its p-orbital participation, so the metal attaches to the remaining five-atom conjugated segment. $A$ is the anionic ${\eta }^5$ heptamethylated-ring Cr(CO)${}_3$ complex, paired formally with $[\mathrm{MgBr}{]}^{+}$; solvent ligation and ion pairing at magnesium are not specified by the spectrum. The original connectivity, with the attacked boron labeled $B_1$, is shown in the first panel below.

There are seven methyl groups. The two attached to tetrahedral $B_1$ are inequivalent because the chromium-bound face distinguishes their environments: one signal of relative integral one for each. The other two B–Me groups are equivalent to one another, giving a signal of integral two. The two N–Me groups adjacent to $B_1$ form another equivalent pair, while the N–Me opposite $B_1$ is unique. The predicted methyl-group multiplicities are therefore $1,1,2,2,1$, the same unordered set as the observed ratio. Multiplying by three gives proton integrals totaling 21. This assignment assumes no fast process interchanging the two $B_1$ methyl environments.

The six-electron neutral ${\eta }^6$ ring donor has become an anionic six-electron ${\eta }^5$ donor, so Cr(0) retains an 18-electron count: $6_d+6_{\mathrm{ring}}+3(2{)}_{\mathrm{CO}}=18$. This borazine dearomatization explains both the changed hapticity and the extra methyl without requiring attack at chromium or loss of a carbonyl.

oxidative addition inserts Pt into one B–Br bond. All three B–Br positions of the starting ring are equivalent, so there is no distinct choice among them before the first substitution. The product is the square-planar Pt(II) boryl bromide $\mathrm{trans}-[\mathrm{Pt}(\mathrm{PC}{\mathrm{y}}_3{)}_2(\mathrm{Br})({\mathrm{B}}_3\mathrm{B}{\mathrm{r}}_2{\mathrm{N}}_3{\mathrm{H}}_3)]$, with the two phosphines trans, the Pt–B bond trans to Pt–Br, and the six-membered ring intact. The third panel shows this connectivity. This is a metal–boron sigma bond, not attachment of the whole ring as an ${\eta }^6$ pi ligand. Formal electron counting gives Pt(II), $d^8$, with $8+4_{\mathrm{P}}+2_{\mathrm{Br}}+2_{\mathrm{boryl}}=16$ electrons, appropriate for square-planar coordination. The primary synthesis establishes this trans-bromo(borazinyl) structure.^[ref]

The electron-rich Pt(0) reacts at the polarized B–Br bond, whose electrophilic boron and departing bromide provide a favorable bond-activation pathway. This accounts for B functionalization without needing cleavage of a ring B–N bond or selection of a nitrogen-bound product. It does not imply that arbitrary N–H activation is impossible under every condition.

Replacing an electronegative bromine substituent by a metal fragment changes the boron's electron demand. Relative to the B–Br sites, the metal-bound boron is less effective at accepting N-to-B pi donation, so its adjacent B–N bonds are expected to have less multiple-bond character and be longer; donation toward the remaining B–Br sites produces compensating unequal ring bonds. The original threefold equivalence is lost. The detailed lengths depend on the full electronic structure; a localized drawing with three rigid alternating double bonds would conceal the redistribution that the unequal lengths reveal.

In the chromium complex, Cr(0) is $d^6$ and receives substantial donation from four carbonyl ligands. Its occupied orbitals can give pi backbonding into the boron-rich acceptor orbitals of the B${}_2$N${}_2$ ring, while the nitrogen-rich occupied ring orbitals donate to the metal. This favors binding across the ring face, approximately ${\eta }^4$, with all four ring atoms participating. A nearly planar ring maintains the p-orbital alignment needed for that interaction and accounts for comparable Cr–B and Cr–N distances. A four-electron ring donor also gives the useful count $6_d+8_{\mathrm{CO}}+4_{\mathrm{ring}}=18$.

Ti(IV) in TiCl${}_4$ is $d^0$ and strongly electron deficient. It is a good acceptor for the two nitrogen lone pairs, but has no filled $d$ set for comparable backbonding into the boron acceptors. chromium therefore favors a face-bound pi complex, while titanium favors N,N sigma coordination. Donation makes the nitrogen environments more nearly tetrahedral and permits the 2,4-Diazadiboretidine ring to pucker. The borons are not directly bound in the same way, explaining $d(\mathrm{Ti}\cdots \mathrm{B})>d(\mathrm{Ti}-\mathrm{N})$. The distinction follows from electron availability, rather than simply calling every short metal–ring contact a separate ordinary sigma bond.

With spin degeneracy, each energy band contains two electrons per primitive unit cell. Divalency therefore permits a filled first band, but only if its maximum lies below the second band's minimum. With negligible ionic periodic potential, the circular Fermi surface and the rectangular zone have equal enclosed area, not equal shape. Electrons occupy some states outside the first zone while some higher-energy first-zone states remain empty. Folded bands overlap, and the material is a metal.

A periodic potential mixes plane waves differing by a reciprocal lattice vector. At a two-wave Bragg point, the nearly-free electron model gives a direct gap $2|V_{\mathbf{G}}|$. Increasing appropriate Fourier coefficients can remove the band overlap. With the first band filled, the system is then a band insulator, provided the temperature is small compared with the insulating band gap in energy units.

The rectangular free-electron band-overlap scale follows from two different points in reciprocal space. Put $C={\hslash }^2/(2m_e)$. Before gap opening, the highest first-zone energy is at the corner, while the lowest second-zone energy is at the midpoint of the nearer face:Thus the downward shift ${\delta }_v$ of the first-band maximum plus the upward shift ${\delta }_c$ of the second-band minimum must overcomeIn a rigid-splitting estimate, an opening $\Delta$ lowers the first band by $\Delta /2$ and raises the second by $\Delta /2$, giving the familiar threshold $\Delta >{\hslash }^2{\pi }^2/(2m_e{b}^2)$. Equality is only band touching, not a positive insulating gap. This expression is the overlap that must be removed, under that explicit band-shift approximation.

Positive direct gaps can coexist with band overlap: a local direct splitting and the global indirect band gap are not interchangeable. The shape and Fourier coefficients of the ionic potential are unspecified, so there is no universal exact critical direct-gap width determined by $a,b$ alone. For a concrete demonstration, a separable potential $2V_x\cos (2Kx)+2V_y\cos (2Ly)$ has axial direct gaps ${\Delta }_x=2|V_x|$ and ${\Delta }_y=2|V_y|$. To first order, the corner's lowest level is $C(K^2+L^2)-|V_x|-|V_y|$, and the two candidate second-band minima are $C{K}^2+|V_x|$ and $C{L}^2+|V_y|$. Therefore, within this model and neglecting higher-order band shifts, nonoverlap requiresEqual axial gaps would give the first-order estimate $\Delta >2C{L}^2/3$, rather than the rigid-splitting estimate. Such a large opening can exceed the quantitative range of weak-potential perturbation theory, but the example identifies the missing model assumption: corner levels feel both couplings. The exact general answer is the positive indirect-gap criterion above; a geometry-only direct-gap threshold would claim more than the supplied potential information supports.

gas heats when compression, shocks or absorbed radiation increase its internal energy. Important sources include accretion shocks, stellar winds, supernovae, photoionization, photoelectric emission from interstellar dust, cosmic rays and energy deposited by active galactic nuclei. photoionization heating uses the photoelectron energy above the ionization threshold; compression contributes mechanical work rather than photon absorption.

For specific internal energy $u$, an idealized thermal balance isHere $\mathcal{H},\mathcal{C}$ are powers per volume and $Z$ describes abundances. The electron-hydrogen normalization of the cooling function must be specified: some conventions instead factor out $n_H^2$. For a monatomic gas with total particle density $n$,when heating is negligible. Compare this with the dynamical time: rapid cooling permits loss of pressure support, whereas long cooling times allow a hot atmosphere to persist.

collisional excitation followed by spontaneous emission cools partially ionized gas; radiative recombination also produces continuum cooling. metal lines enhance cooling over substantial intermediate-temperature ranges, while thermal bremsstrahlung gives a roughly $\Lambda \propto T^{1/2}$ high-temperature tail. Atomic hydrogen cooling becomes inefficient below roughly $10^4\mathrm{K}$, where molecular line cooling and interstellar dust can be important. inverse Compton scattering cools hot electrons against a colder radiation field. A tabulated cooling curve assumes a composition and ionization model; photoionization or nonequilibrium ionization can change it.^[ref]

X-ray line ratios and continuum shape constrain temperature and ionization, while the emission measure constrains $\int n_e{n}_{H}dV$. Turning that into a density and radiative gas cooling time also requires a geometry; unresolved clumping, foreground absorption and multiphase gas are important controls.

An organic substance may also contain nitrogen, sulfur, phosphorus, halogens and, in some cases, metals or other elements. “Organic” does not mean “made of only carbon, hydrogen and oxygen”. Detection generally requires converting a covalently bound element into a species that an appropriate inorganic test can recognize.

A historically available general method is the sodium fusion test. Fusion converts suitable nitrogen-containing material into cyanide, sulfur into sulfide, and chlorine-, bromine- or iodine-containing material into the corresponding soluble halides. The cooled reaction material is worked up to an aqueous extract, with separate portions reserved for separate tests. The transformations are analytical conversion principles, not claims that every nitrogen or sulfur compound must give quantitative recovery under any fusion conditions. Blanks and a known positive material are essential when a negative result is to be interpreted.^[ref]

For nitrogen, a suitable cyanide-containing extract is treated with a fresh iron(II) reagent and then brought through the appropriate oxidation and acidification sequence to obtain Prussian blue. Formation of ferrocyanide is represented byThe eventual ferric ferrocyanide material provides the blue observation. It is not sufficient simply to add any iron salt and assume that a positive blue must appear. Another classical principle converts nitrogen in suitable organic materials into Ammonia by heating with alkaline reagents, then recognizes the ammonia in a receiver. That test has substrate-dependent limitations and is not a universal replacement for complete nitrogen analysis. A later specialized nitrogen-digestion protocol must not be silently attributed to 1865.

For sulfur, a sulfide-containing alkaline extract can give black lead sulfide with a suitably introduced lead reagent:The strongly acidic form of the reagent conditions changes the available sulfide, so conditions must be controlled. Oxidizing the material to sulfate and identifying a persistent barium sulfate precipitate supplies another principle, using sulfur-free reagents and excluding pre-existing sulfate contamination. If nitrogen and sulfur occur together, thiocyanate formation can alter a fusion test; failure of the expected cyanide reaction is not conclusive evidence that nitrogen is absent.

For chlorine, bromine and iodine, remove interfering cyanide and sulfide by an appropriate controlled treatment of a separate extract portion, then use silver nitrate under suitable nitric-acid conditions. The precipitates are ordinarily white silver chloride, cream silver bromide and yellow silver iodide. Their differing behaviour with aqueous Ammonia provides corroboration: chloride dissolves relatively readily, bromide requires stronger conditions, and iodide is resistant. Do not use hydrochloric acid to prepare the portion in which original chloride is being sought. Fluorine requires a different recognition method: soluble silver fluoride means this precipitation scheme does not provide an equivalent fluoride test.

For phosphorus, oxidize a suitable portion to phosphate, then apply a phosphate test such as yellow ammonium phosphomolybdate formation under the proper acidic conditions. Arsenic species and other interferences must be excluded when relevant. Mineral ash can reveal metal constituents through their specific precipitation or flame reactions, but oxidation during ashing can change their original chemical form. Absence of a visible ash does not prove absence of every metallic or otherwise volatile element.

A positive result identifies an element or an operationally generated ion; it does not establish which functional group contained it in the original molecule. Conversely, an incomplete conversion can give a false negative. This is why the three-element premise used in the preceding combustion calculation should be checked rather than accepted merely because a substance is described as organic.

Iodine is a dark, lustrous crystalline halogen which gives a violet vapour; historically, an important source was the ash of iodine-rich seaweeds. Elemental iodine commonly occurs as diatomic ${\mathrm{I}}_2$. It melts at about $114{}^{\circ }\mathrm{C}$ and boils at about $184{}^{\circ }\mathrm{C}$ at ordinary atmospheric pressure. It also undergoes sublimation below its melting point, depositing crystals again on a cooler surface.^[ref]

Iodine is sparingly soluble in water, giving a brownish solution, and dissolves more freely in ethanol and in an aqueous iodide solution. In the latter case the chemical equilibriumforms triiodide. Iodine with starch, under suitable conditions, gives a sensitive dark-blue colour; this is a useful recognition test, but oxidants that liberate iodine from added iodide must be distinguished from iodine initially present. Reaction with reactive metals gives iodides; in such compounds the iodine usually has oxidation state $-1$.

For the historical seaweed route, collect and burn iodine-accumulating seaweeds to produce kelp ash, then extract the soluble salts with water. Crystallization removes much of the common salt content, leaving an iodide-enriched mother liquor. Oxidation releases iodine, which can be distilled or sublimed and collected on a cool surface. One representative net chemical equation for liberation from iodide isThe manganese dioxide is the oxidizing agent; the seaweed does not contain a supply of ready-made violet iodine crystals. This describes the chemical principle of a historical extraction, rather than prescribing one unique factory recipe.

Iodine also occurs in natural brines and in iodate-bearing deposits associated with nitrate ores. Modern industrial supply relies heavily on brines and iodate ores; it should not be equated with the seaweed source emphasized in nineteenth-century teaching.^[ref]

The Pauli exclusion principle confines thermally excited conduction electrons to an energy interval of order $k_{B}T$ around the Fermi surface. Their excess energy is therefore proportional to $T^2$, giving an electronic heat capacity $\gamma T$. More precisely, $\gamma =({\pi }^2/3)k_B^2{g}_{\mathrm{mol}}({ϵ}_F)$. In a three-dimensional Debye model, acoustic phonons have density of states proportional to ${\omega }^2$; their thermal energy scales as $T^4$, givingper mole of atoms. This assumes a normal metal without a lower-temperature superconducting or magnetic anomaly.

For ${\theta }_D\ll T\ll T_F$, the lattice approaches the Dulong-Petit value, so $C_V\simeq 3R+\gamma T$, usually dominated by $3R$. Each classical vibrational mode contributes $k_B$ through kinetic and potential equipartition. This is a high-temperature solid-state regime, not a claim that the solid persists to arbitrarily high temperature; anharmonicity and $C_P-C_V$ can matter.

The body-centred cubic cell has two atoms, so $n_{\mathrm{at}}=2/a^3$. Counting three acoustic branches gives $k_D^3=6{\pi }^2{n}_{\mathrm{at}}$. By the Debye-temperature definition,This is an effective Debye sound speed; separate longitudinal and transverse speeds are not determined. For isotropic branches, $3/v_D^3=1/v_L^3+2/v_T^3$.

A Fresnel zone plate replaces a conventional refracting surface by concentric alternating transmitting and absorbing annuli. Successive annuli span half a wavelength of optical-path excess to the chosen focus; blocking alternate zones prevents their approximately opposite wave phases from cancelling. For a normally incident plane wave,The paraxial radii grow as $\sqrt{j}$, so the zones become narrower outwards.

Using the photon energy relation $\lambda =hc/E$ gives $\lambda =4.133\times 10^{-11}\mathrm{m}=0.04133\mathrm{nm}$. For $f=10^{-2}\mathrm{m}$,The first three boundaries are approximately $0.6429$, $0.9092$ and $1.1135\mu \mathrm{m}$. This fixes the zone pattern but does not fix a unique outside diameter: the number of zones is a design choice. For example, $N=100$ gives radius $6.429\mu \mathrm{m}$, diameter $12.86\mu \mathrm{m}$ and outermost zone width about $32.2\mathrm{nm}$. A larger usable diameter requires finer outer zones.

One could pattern absorbing metal annuli on a thin X-ray-transparent support, choosing absorber thickness from its attenuation at the selected photon energy; that thickness cannot be determined from energy and focal length alone. The design is chromatic: for fixed radii $f\propto 1/\lambda$, so a narrow energy band is needed for a sharp common focus.