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In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field [] has class number 1. Equivalently, the ring of algebraic integers of Q [ − d ] {\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]} has unique factorization .
For a given cation, Pauling defined [2] the electrostatic bond strength to each coordinated anion as =, where z is the cation charge and ν is the cation coordination number. A stable ionic structure is arranged to preserve local electroneutrality , so that the sum of the strengths of the electrostatic bonds to an anion equals the charge on ...
In number theory, the Heegner theorem [1] establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number .
Kurt Heegner (German: [ˈheːɡnɐ]; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–Heegner theorem .
Kurt Heegner was a German mathematician; Heegner points are special points on elliptic curves; The Stark–Heegner theorem identifies the imaginary quadratic fields of class number 1. A Heegner number is a number n such that Q(√ −n) is an imaginary quadratic field of class number 1.
The strength of a bond can be estimated by comparing the atomic radii of the atoms that form the bond to the length of bond itself. For example, the atomic radius of boron is estimated at 85 pm, [10] while the length of the B–B bond in B 2 Cl 4 is 175 pm. [11] Dividing the length of this bond by the sum of each boron atom's radius gives a ratio of
The term bond-dissociation energy is similar to the related notion of bond-dissociation enthalpy (or bond enthalpy), which is sometimes used interchangeably.However, some authors make the distinction that the bond-dissociation energy (D 0) refers to the enthalpy change at 0 K, while the term bond-dissociation enthalpy is used for the enthalpy change at 298 K (unambiguously denoted DH° 298).
Their strength, stiffness, and high melting points are consequences of the strength and stiffness of the covalent bonds that hold them together. They are also characteristically brittle because the directional nature of covalent bonds strongly resists the shearing motions associated with plastic flow, and are, in effect, broken when shear occurs.