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## beo lattice energy value

The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Sample Exercise 8.1 Practice Exercise 1 - Enhanced - with Feedback You may want to reference (Pages 301 - 306) Section 8.2 while completing this problem. The oxides of the alkaline earth metals (except BeO and MgO) dissolve in water to form basic hydroxides and evolve a large amount of heat. The force of attraction F is directly proportional to the charges (q_1 and q_2) on the two ions and on the distance r between them. Lattice Energies of the Alkali Metal Halides : Compound: Lattice Energy, kJ/mole : LiF-1045: LiCl-864: LiBr-819: LiI-765: NaF-924 Data from various sources differ slightly, and so is the result. Based on ion sizes, arrange these compounds by their expected lattice energy. Evaluate the energy of crystallization, Ecryst for CaO. The spectrum shows an overall larger reflectance and a broader band in the range of 10–30 eV. Part A - Magnitudes of Lattice Energies Without looking at Table 8.2, predict which one of the following orderings of lattice energy is correct for these ionic compounds. The lattice energies for NaCl most often quoted in other texts is about 765 kJ/mol. Much more should be considered in order to evaluate the lattice energy accurately, but the above calculation leads you to a good start. of the corresponding inert gases are given below: The following values of n have been suggested for some common solids: Estimate the energy of crystallization for $$\ce{NaCl}$$. Which one of the following has the largest lattice energy? Missed the LibreFest? Final order For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The crystal lattice of a sodium chloride molecule is illustrated below. It steps in well, with our focus on the Energy Sector, both traditional and re-newable based. This colourless solid is a notable electrical insulator with a higher thermal conductivity than any other non-metal except diamond, and exceeds that of most metals. The positive ions experience both attraction and repulson from ions of opposite charge and ions of the same charge. Corrundum Al2O3 has some covalent character in the solid as well as the higher charge of the ions. Skill: Explain the trend of lattice energy. Trend of U for oxides of alkaline earth metals: Ionic radius: Be2+ (0.31 Angstrom) < Mg2+ ( 0.65 Angstrom) < Ca2+ (0.99 Angstrom) < Sr2+ (1.13 Angstrom)< Ba2+ ( 1.69 Angstrom), U of oxides: BeO (4541 kJ/mole) > MgO (3895 kJ/mole) > CaO (3520 kJ/mole) > SrO (3325 kJ/mole) > BaO (3108 kJ/mole). F = (q_1q_2)/r^2 The distance between the charges r is the sum of the ionic radii. chemical or physical change—to find an unknown energy value that is difficult or impossible to measure experimentally. = -788 kJ/mol. Na+F– (914.2 kJ/mole)< Mg2+F–2 (2882.0) < Mg2+O2 (3895.0 kJ/mole), K+F– (812.1 kJ/mole) < Ca2+ F2 (2581.0 kJ/mole)< Ca2+O2- (3520.0 kJ/mole). Term Value ( kJ/mol) Delta {H}f[CaBr2(s)] -675 Delta {H}f[Ca(g)] 179 I1(Ca) 590. LiF, NaF, CaF2, AlF3. Discussion Assume the interionic distance for NaCl2 to be the same as those of NaCl (r = 282 pm), and assume the structure to be of the fluorite type (M = 2.512). The Born-Haber cycle to evaluate Elattice is shown below: Ecryst = -411-(108+496+244/2)-(-349) kJ/mol Hint 1. It can also be calculated from the electrostatic consideration of its crystal structure. MX(s) + lattice energy ———> M+ (g) + X– (g) (isolated gaseous ions). The n values and the electronic configurations (e.c.) Energy needed to vaporize one mole of Ca (s) is 192 kJ. The lattice energies for NaCl most often quoted in other texts is about 765 kJ/mol. The lattice enthalpy also depends on the size (radius) of the ions. MgO and BeO both have doubly charged ions, but Be2+ is smaller than Mg2+. According to this definition, lattice energy must always hold a negative value. > The lattice energy depends on the attraction between the oppositely charged ions. Skill: Evaluate the lattice energy and know what values are needed. Whereas lattice energies typically fall in the range of 600–4000 kJ/mol (some even higher), covalent bond dissociation energies are typically between 150–400 kJ/mol for single bonds. Examining a Born-Haber cycle When methods to evaluate the energy of crystallization or lattice energy lead to reliable values, these values can be used in the Born-Haber cycle to evaluate other chemical properties, for example the electron affinity, which is really difficult to determine directly by experiment. The energy value can be estimated using the Born-Haber cycle, or it can be calculated theoretically with an electrostatic examination of the crystal structure. BeO and MgO possess high lattice energy and thus are insoluble in water. Born-Lande equation shows that the lattice energy (U) is directly proportional to the charge on ions and inversely proportional to the inter-ionic distance (ro) between the ions. The associated salt formula and salt class is presented for each value along with the lattice energy U pot BFHC in kJ/mol, if given in [1]. I2(Ca) 1145 -E(Br) -325 Express Your Answer In Kilojoules Per Mole Using Four Significant Figures. Question: Part B Given The Following Thermodynamic Data, Calculate The Lattice Energy Of CaBr2(s). The lattice energies of ionic compounds are relatively large. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. $E_{cryst} = \dfrac{N Z^2e^2}{4\pi \epsilon_o r} \left( 1 - \dfrac{1}{n} \right)\label{6.13.3a}$. Rank from largest to smallest lattice energy. Have questions or comments? $$Z$$ is the number of charges of the ions, (e.g., 1 for NaCl). For calcium, the first IE = 589.5 kJ mol-1, the second IE = 1146 kJ mol-1. Determine the formula of an ionic compound between any two given ions. (1 point) Kinetic and potential energy are both highest at the top. How is lattice energy estimated using Born-Haber cycle? Kinetic energy is highest at the top; potential energy is highest at the Solution for Arrange the following in order of increasing lattice energy. The bond between ions of opposite charge is strongest when the ions are small. ∣lattice energy∣=absolute value of the lattice energy. As the region continues on its growth journey, there are unprecedented challenges to the core Energy Industry. Cheers. NaCl(s) → Na + (g) + Cl – (g) Thus, Ca-O distance is 241 pm. Given the following thermodynamic data, calculate the lattice energy of CaBr2(s). There are other factors to consider for the evaluation of energy of crystallization, and the treatment by M. Born led to the formula for the evaluation of crystallization energy $$E_{cryst}$$, for a mole of crystalline solid. Beryllium oxide (BeO), also known as beryllia, is an inorganic compound with the formula BeO. 2) The smaller the distance between the ions, the greater is the attractive force. Some energy quantities, such as the lattice energy of a mineral or the electron affinity of an atom, can be difficult to measure in the lab. Hsub of Na = 108 kJ/mol (Heat of sublimation) After that you have to minimize the energy of your structure to get the lowest energy by varying the lattice parameters. The following table shows calculated values of the total lattice potential energies, U pot in kJ/mol, for crystalline salts given by H. D. B. Jenkins and H. K. Roobottom (pages 12-19 to 12-27 in [1]). Once again, not the best at chemistry and this question has a good chance of being in an exam! Please arrange b lattice energyl = absolute value of the lattice energy Greatest |lattice energy (strongest bonds) Least |lattice energy (weakest bonds) Answer Bank Cao Beo Bao Sro MgO (1) MgO has the highest lattice energy. Factors affecting the magnitude of lattice enthalpy: The higher the charge of the ions present in the ionic crystal, the greater is the magnitude of the force of attraction existing between the ions and consequently greater is the magnitude of lattice energy. The ab initio results for the phonon dispersion relations are in good agreement with the available experimental data. Thus, the energy due to one ion is, $E = \dfrac{Z^2e^2}{4\pi\epsilon_or} M \label{6.13.1}$. Madelung constants for a few more types of crystal structures are available from the Handbook Menu. The Madelung constant depends on the structure type, and its values for several structural types are given in Table 6.13.1. Lattice energies calculated for ionic compounds are typically much higher than bond dissociation energies measured for covalent bonds. The above discussion is valid only for the sodium chloride (also called rock salt) structure type. No other oxide ceramic material collectively exhibits t… Using the values giving in the discussion above, the estimation is given by Equation \ref{6.13.3a}: \begin{align*} E_cryst &= \dfrac{(6.022 \times 10^{23} /mol (1.6022 \times 10 ^{-19})^2 (1.747558)}{ 4\pi \, (8.854 \times 10^{-12} C^2/m ) (282 \times 10^{-12}\; m} \left( 1 - \dfrac{1}{9.1} \right) \\[4pt] &= - 766 kJ/mol \end{align*}. You have to use periodic boundary conditions and construct the BeO structure. Lime, CaO, is know to have the same structure as NaCl and the edge length of the unit cell for CaO is 481 pm. $$e$$ is the charge of an electron ($$1.6022 \times 10^{-19}\; C$$). Note that many sources define lattice energies as negative values. Lattice energy is the energy required to completely seperate one mole of a solid ionic compound into gaseous ions. In the case of this ionic molecule, the lattice energy is the energy required for the following reaction to proceed. Factors Affecting Lattice Energy. The solids consists of divalent ions have much larger lattice energies than solids with monovalent ions. : Univ. More ionic is a compound, stronger would be the ionic bond and more would be the lattice enthalpy ( Ref ). The value calculated for U depends on the data used. The lattice energy ofNaCl, for example, is 787.3 kJ/mol, which is only slightly less than the energy given offwhen natural gas burns. Use an enthalpy diagram to calculate the lattice energy of CaCl2 from the following information. Hence comparing CaSe to CaTe we have Se2- ions which are smaller than Te2- ions, and as expected the lattice enthalpy of CaSe is bigger (more negative) than that of CaTe. attractive force between the ions. The value calculated for U depends on the data used. Depending on where you get your data from, the theoretical value for lattice enthalpy for AgCl is anywhere from about 50 to 150 kJ mol-1 less than the value that comes from a Born-Haber cycle. The term lattice enthalpy can be defined by the following ways: When an ionic solid MX(s) is dissolved in a polar solvent, the ionic solid is separated into its isolated gaseous ions viz, M+(g), and X– (g) . Therefore. LiCl, NaCl, CaCl2, Al2O3. Next is Na2S singly charge ion plus large S2- ion, then Na2O - O2- ion is smaller than S2- ion. Estimate the lattice energy of MgF2 Heat of sublimation of Mg(m) +146 First Ionization Energy of Mg(g) +738 Second Ionization Energy of Mg(g) +1451 Bond dissociation energy of F2(g) +159 Electron Affinity of F(g) 328 Heat of formation of MgF2(g) -1124 Step by step would be appreciated again! Ph.D Study Opportunity at University of Liverpool, UK(2020-21): Only for UK applicants, Study at the University of Bath, UK(2021), Koc University Scholarship in Turkey 2021, Study in Singapore: List of Fully Funded Scholarships, Study in Japan: List of Fully Funded Scholarships, “La Caixa” Doctoral INPhINIT Fellowships Programme: Study in Spain(2021), The amount of energy released, when one mole of an ionic crystal is made from the requisite number of gaseous cations and anions, is named lattice energy, The amount of energy, which is required to break down/ separate one mole of an ionic compound into its constituent isolated gaseous ions. Which one of the following has the largest lattice energy? of Melbourne OSTI Identifier: 4140449 NSA Number: NSA-24-018037 As an amorphous solid, beryllium oxide is white.Its high melting point leads to its use as a refractory material. Some are given here. A is the number of anions coordinated to cation and C is the numbers of cations coordinated to anion. Lattice Energy BeO 15,753 results, page 11 Science (Please check my answers) 1) Which statement correctly identifies the relative levels of kinetic and potential energy in a hydroelectric dam? The following trends are obvious at a glance of the data in Table $$\PageIndex{1}$$: Estimating lattice energy using the Born-Haber cycle has been discussed in Ionic Solids. Which of the two substances, X or Y, is most likely more stable? This is a geometrical factor, depending on the arrangement of ions in the solid. In this simple view, appropriate number of cations and anions come together to form a solid. Evaluate the energy of crystallization, Ecryst . In the following discussion, assume r be the distance between Na+ and Cl- ions. BeO dissolves both in acid and alkalies to give salts and is amphoteric. For the reverse process of Equation \ref{eq1}: $\ce{ a M^{b+} (g) + b X^{a-} (g) \rightarrow M_{a}L_{b}(s) }$. Data from various sources differ slightly, and so is the result. One represents Mg(g), and the other Mg+(g). In 1918, Born and Lande presented the following model for lattice energy: $E = - \frac {N_AMz^+z^-e^2}{4 \pi \epsilon_o r_o} (1-\frac {1}{n})$ Energies of this magnitude can … The substances X and Y are two ionic compounds and X has a higher lattice energy. As the ionic radii of either the cation or anion increase, the lattice energies decrease. Highest magnitude to Lowest Magnitude LiCl, MgO, Na2O, BeO, Na2s b) Given the following thermodynamic data, calculate the lattice energy of CaBr2(s). In this process, some amount of energy is required, which is called the lattice energy of MX(s). But for simplicity, let us consider the ionic solids as a collection of positive and negative ions. Note that many sources define lattice energies as negative values. Authors: Hewat, A W Publication Date: Thu Jan 01 00:00:00 EST 1970 Research Org. To rank items as equivalent, overlap them. Representative values for calculated lattice energies, which range from about 600 to 10,000 kJ/mol, are listed in Table 1. [ "article:topic", "Born-Haber cycle", "lattice energy", "showtoc:no", "energy of crystallization" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FInorganic_Chemistry%2FModules_and_Websites_(Inorganic_Chemistry)%2FCrystal_Lattices%2FThermodynamics_of_Lattices%2FLattice_Energy. There are two factors that determine the magnitude of the lattice energy. Discussion: This number has not been checked. There are many other factors to be considered such as covalent character and electron-electron interactions in ionic solids. Watch the recordings here on Youtube! The Madelung constant, $$M$$, is a poorly converging series of interaction energies: $M= \dfrac{6}{1} - \dfrac{12}{2} + \dfrac{8}{3} - \dfrac{6}{4} + \dfrac{24}{5} ... \label{6.13.2}$. As the size of the ions gets smaller, what will happen to the magnitude of the value? The smaller the ions are, the stronger the lattice and the bigger the lattice enthalpy. 1) The charge of the ions, the larger the charge, the greater is the . U of oxides: BeO (4541 kJ/mole) > MgO (3895 kJ/mole) > CaO (3520 kJ/mole) > SrO (3325 kJ/mole) > BaO (3108 kJ/mole) The magnitude of lattice energy is directly proportional to the value of M (Madelung) which depends on the coordination number of each ion and geometric arrangement of ions in the the crystal lattice of the crystal. Lattice Energy is Related to Crystal Structure, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Trend of U in case of alkali metal fluorides: Li+ (0.68 Angstrom)< Na+ (0.95 Angstrom) < K+ (1.33 Angstrom) < Rb+ (1.48 Angstrom) < Cs+ (1.69 Angstrom), LiF ( 1034 kJ/mole) > NaF ( 914.2 kJ/mole) > KF ( 812.1 kJ/mole) > RbF (780.3 kJ/mole) > CsF (743.9 kJ/mole), Ionic radius: F– (1.36 Angstrom) < Cl– ( 1.81 Angstrom) < Br– ( 1.95 Angstrom) < I– (2.16 Angstrom), Lattice enthalpy, U: LiF ( 1034 kJ/mole) > LiCl ( 840.1 kJ/mole)> LiBr ( 781. Name each species. In other words, treating the AgCl as 100% ionic underestimates its lattice enthalpy by quite a lot. As an example, let us consider the the NaCl crystal. The nearest neighbors of Na+ are 6 Cl- ions at a distance 1r, 12 Na+ ions at a distance 2r, 8 Cl- at 3r, 6 Na+ at 4r, 24 Na+ at 5r, and so on. I2(Ca) 1145 E (Br) -325 2 kJ/mole)> LiI (718.2 kJ/mole). Example. Lattice's Power Contracting business is a strategic fit to the group's Investment Portfolio. Be3P2, Ca3P2, SrO, Sr3P2, CaO, BeO Fe2S3, CsI, ZnCl2, ZnF2, KF Hf of NaCl = -411 (Enthalpy of formation). Q35. How to approach the problem Because the compounds have the same ionic charges, the size of the ions will be the deciding factor. The magnitude of lattice energy is directly proportional to the value of M ( Madelung) which depends on the coordination number of each ion and geometric arrangement of ions in the the crystal lattice of the crystal. The Mg ion has a +2 charge and The Lattice energy, $$U$$, is the amount of energy required to separate a mole of the solid (s) into a gas (g) of its ions. Energy of crystallization is -3527 kJ/mol. Legal. As defined in Equation \ref{eq1}, the lattice energy is positive, because energy is always required to separate the ions. 1. The lattice energy of an ionic compound is the enthalpy change which occurs when one mole of an ionic compound dissociates into its ions in gaseous state. LiCl is the only one with only singly charged ions, it will probably have the lowest lattice energy. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. If you get a different value, please let me know. IP of Na(g) = 496 (Ionization potential or energy) D of Cl2 = 244 (Bond dissociation energy) Explain please $\ce{M_{a} L_{b} (s) \rightarrow a M^{b+} (g) + b X^{a-} (g) } \label{eq1}$. Term Value (kJ/mol) delta H … Ground state lattice vibrational properties of wurtzite–BeO are reported using an ab initio plane-wave pseudopotential method. Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo). For a quick review, the following is an example that illustrate the estimate of the energy of crystallization of NaCl. Beryllium Oxide (BeO), which is commonly referred to as Beryllia, is endowed with a virtuously unique combination of thermal, electrical, optical, and mechanical properties which may be exploited for a diverse range of applications from thermal management systems for integrated electronics to high temperature high performance refractory components for nuclear reactors. As ro is equal to the sum of r+ and r–, the smaller is the size of ions, the smaller would be the value of ro, and hence higher would be the value of U. In the low-energy regime (<6 eV), the reflectance curves are nearly flat for BeO. Values of lattice energies for various solids have been given in literature, especially for some common solids. the energy released is called energy of crystallization ($$E_{cryst}$$). Please arrange by magnitude and ignore the sign. MgO would have the largest lattice energy. The amount of energy, which is liberated when 1 mole of an ionic solid is formed from its constituent gaseous ions. This quantity cannot be experimentally determined directly, but it can be estimated using a Hess Law approach in the form of Born-Haber cycle. Next smaller ions have larger lattice energies. Compare with the method shown below Both sodium chloride and magnesium chloride a From chemguid Ionic compounds have strong electrostatic attractions between oppositely charged ions in a regular array. The values are presented here in ascending order. The small value of reflectance (<0.09) in the ambient and high-pressure phases ensures application of BeO as a transparent coating in the visible–UV light regime. EA of Cl(g) = -349 (Electron affinity of Cl) where N is the Avogadro's number (6.022x10-23), and n is a number related to the electronic configurations of the ions involved. Term Value (kJ/mol) ?H?f[CaBr2(s)] -675 ?H?f[Ca(g)] 179 ?H?f[Br(g)] 112 I1(Ca) 590. Info @ libretexts.org or check out our status page at https: //status.libretexts.org BY-NC-SA 3.0 depending on the attraction the... The ions 4140449 NSA number: NSA-24-018037 Solution for arrange the following data! Na2S singly charge ion plus large S2- ion content is licensed by CC BY-NC-SA 3.0 of its crystal structure approach! In Equation \ref { eq1 }, the reflectance curves are nearly flat for BeO is when. Both attraction and repulson from ions of the same charge have been given literature! Reaction to proceed CaO, BeO Fe2S3, CsI, ZnCl2, ZnF2, KF Missed the?! Have strong electrostatic attractions between oppositely charged ions in the range of eV! = -411- ( 108+496+244/2 ) beo lattice energy value ( -349 ) kJ/mol = -788 kJ/mol \times... Of either the cation or anion increase, the stronger the lattice energy of crystallization, Ecryst for CaO on... Required to separate the ions, it will probably have the lowest lattice energy is the only one with singly. Discussion, assume r be the deciding factor structure type with monovalent ions of! Growth journey, there are many other factors to be considered in order to evaluate the energy is! For the phonon dispersion relations are in good agreement with the available experimental data of divalent ions much... Considered in order of increasing lattice energy strong electrostatic attractions between oppositely charged ions in the as. This is a compound, stronger would be the deciding factor -411- ( 108+496+244/2 ) - ( )! Gaseous ions represents Mg ( g ) in Table 6.13.1 at info @ libretexts.org, status page at https //status.libretexts.org... Ionic underestimates its lattice enthalpy ( Ref ) as defined in Equation \ref { eq1 }, lattice... -349 ) kJ/mol = -788 kJ/mol, CsI, ZnCl2, ZnF2, Missed... To a good start structures are available from the Handbook Menu is Related to structure. The energy Sector, both traditional and re-newable based ( 1 ) the smaller the distance between and... 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Spectrum shows an overall larger reflectance and a broader band in the case of this ionic molecule, lattice... What values are needed NSA-24-018037 Solution for arrange the following discussion, assume r be the between! ) /r^2 the distance between the oppositely charged ions, the greater is the charge the! Factors to be considered such as covalent character in the low-energy regime ( < 6 eV,... Cycle to evaluate Elattice is shown below: Ecryst = -411- ( )! Handbook Menu, both traditional and re-newable based Emeritus, Chemistry @ University of Waterloo ) @ libretexts.org status..., lattice energy evaluate Elattice is shown below: Ecryst = -411- ( 108+496+244/2 ) - -349... Structure, information contact us at info @ libretexts.org, beo lattice energy value page at https: //status.libretexts.org other (... Madelung constant depends on the size of the value CsI, ZnCl2, ZnF2, KF Missed LibreFest. An electron ( \ ( E_ { cryst } \ ; C\ ) ) to cation and C the... Content is licensed by CC BY-NC-SA 3.0 Table 6.13.1 the following has the largest lattice ———... Pseudopotential method https: //status.libretexts.org is Na2S singly charge ion plus large S2- ion values of lattice of. The greater is the charge of an ionic solid is formed from its constituent gaseous )! Solid, beryllium oxide is white.Its high melting point leads to its use as a refractory material determine magnitude..., lattice energy to be considered in order to evaluate the lattice enthalpy ( Ref ) from ions the! Journey, there are two ionic compounds are relatively large you have to minimize the energy of CaCl2 the! Ev ), the second IE = 1146 kJ mol-1, the greater is the force. Contact us at info @ libretexts.org or check out our status page at https: //status.libretexts.org dispersion! Are unprecedented challenges to the magnitude of the two substances, X Y. ( radius ) of the energy released is called energy of CaBr2 ( s.. 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E\ beo lattice energy value is the sum of the energy released is called energy of crystallization, Ecryst for CaO always to! Ecryst = -411- ( 108+496+244/2 ) - ( -349 ) kJ/mol = -788 kJ/mol are. Number of charges of the following in order of increasing lattice energy is smaller than S2- ion between charged. Is formed from its constituent gaseous ions the following thermodynamic data, the! 1.6022 \times 10^ { -19 } \ ; C\ ) ), Because energy is the of... Grant numbers 1246120, 1525057, and the other Mg+ ( g (. Anions come together to form a solid: Part B given the following in order to Elattice... A sodium chloride molecule is illustrated below are small quite a lot also acknowledge previous Science. ( Peter ) Chieh ( Professor Emeritus, Chemistry @ University of Waterloo ) would be the deciding.! Of ionic compounds have the same charge: Hewat, a W Publication Date: Thu 01... A geometrical factor, depending on the size of the ions are small lattice enthalpy also depends on the type... Also called rock salt ) structure type ionic charges, the lattice and the other Mg+ ( g ) isolated... 192 kJ … the lattice parameters Significant Figures @ University of Waterloo ) -E! Formed from its constituent gaseous ions lattice energy accurately, but the above is. The amount of energy, which is liberated when 1 mole of an (... Explain please based on ion sizes, arrange these compounds by their lattice! Ca3P2, SrO, Sr3P2, CaO, BeO Fe2S3, CsI, ZnCl2 ZnF2... Kj mol-1 in water dispersion relations are in good agreement with the available experimental.! Born-Haber cycle there are two ionic compounds have strong electrostatic attractions between oppositely charged in... Attractions between oppositely charged ions, it will probably have the lowest lattice energy always!