Theory of the vibrations of sodium chloride

The interest in the frequency spectrum of the thermal vibrations in a crystal arose chiefly in connection with the problem of the specific heat of crystals at low temperatures, Debye' s theory of the specific heat, however, has been so successful that the actual determination of the frequency spectrum according to Born and v.Karman (1912) has been pushed into the background. But recent investigations, especially those of Blackman (1938, 1935, 1937) have shown that appreciable deviations from Debye's theory should occur according to the correct atoruistic treatment. These deviations appear to be most pronouneed _ear the absolute zero of temperature. It, therefore, seemed desirable to calculate the exact frequency spectrum of a crystal.The first attempt to calculate the frequency spectrum of a crystal was made iIty Born and v.Karrlian in their original paper. They assumed only quasi -elastic forces between neighbouring particles. Later calculations have been made for ionic lattices, assurlinghereal forces in the crystal. The chAif difficulty in that calculation has always been the long range of the Joulonrb force which makes a direct summation over all lattice points impossible.Born and Thompson (1934) suggested a way of transforming these stuns into more rapidly convergent expressions using a method developed by Ewald (1921) and Thompson (1935) has given the final formulae for the coupling coefficients due to the Coulomb force in the equation of motion, but in his paper a slight mistake occurred in the definition of the coefficients and so far no numerical results of these calculations have been published. Broch (1937) has given formulae for the case of an one dimensioonnaall lattice mating use of Epstein's Zeta functions; Herzfeld and Lyddane (1938) have used an extension of Madelung' s method (1918) and they have given some numerical results; but their formulae are rather complicated so that one cannot expect to compute the whole frequency spectrum by this method. Moreover, the problem. of the thermal oscillations of an ionic lattice is not a purely electrostatic problem and this point has not been made sufficiently clear by Herzfeld and Lyddane. This applies especially to the case of the residual rays and the question whether the potential, from which the coupling coefficients are obtained, satisfies the Laplace equation or Poisson's equation.In this paper we have used Ewald's method mentioned above, but interpreted and extended by him in a recent paper (1938). By this method one obtains comparatively simple and quickly convergent expressions for the coupling coefficients in the equation of motion which allow a numerical calculation to an arbitrary degree of accuracy. Because of the good convergence it has not been too laborious to compute numerical values for 48 different modes of vibration,In sections III III we give the derivation of these expressions by treating the problem as an electrostatic problem, neglecting the retardation; but the proper way of solving our problem is, to find a solution of :._axwell's equation for the electromagnetic field in the crystal, 'this will be done in Section IV, From this field the force exerted on a particle and the coupling coefficients can be obtained (Section V), We shall see that in this proper treatment the case of infinitely long waves plays a special role and must be considered separately, In all the other cases this treatrIent leads to the same result as the electrostatic derivation, If we define a potential function from which the coupling coefficients are obtained as second derivatives, this potential satisfies in general Laplace's equation; but in the special case of infinitely long waves it satisfies I'oisson's equation,In Section VI the coupling coefficients for the ra Cl lattice are given and in Sec :ion VII the contribution due to the repulsive forces is calculatèd, In Section VIII the equations for the coupling coefficients are checked by deriving from them formulae for the elastic constants,Finally, in Section IX the coefficients have been calculated and in section X the frequencies for 17 modes of vibrations, and the spectrum has been discussed

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements

Searches for the ZγZ\gamma decay mode of the Higgs boson and for new high-mass resonances in pppp collisions at s=13\sqrt{s} = 13 TeV with the ATLAS detector

International audienceThis article presents searches for the Zγ decay of the Higgs boson and for narrow high-mass resonances decaying to Zγ, exploiting Z boson decays to pairs of electrons or muons. The data analysis uses 36.1 fb$^{−1}$ of pp collisions at $\sqrt{s}=13$ recorded by the ATLAS detector at the CERN Large Hadron Collider. The data are found to be consistent with the expected Standard Model background. The observed (expected — assuming Standard Model pp → H → Zγ production and decay) upper limit on the production cross section times the branching ratio for pp → H → Zγ is 6.6. (5.2) times the Standard Model prediction at the 95% confidence level for a Higgs boson mass of 125.09 GeV. In addition, upper limits are set on the production cross section times the branching ratio as a function of the mass of a narrow resonance between 250 GeV and 2.4 TeV, assuming spin-0 resonances produced via gluon-gluon fusion, and spin-2 resonances produced via gluon-gluon or quark-antiquark initial states. For high-mass spin-0 resonances, the observed (expected) limits vary between 88 fb (61 fb) and 2.8 fb (2.7 fb) for the mass range from 250 GeV to 2.4 TeV at the 95% confidence level

Search for direct top squark pair production in final states with two leptons in s=13\sqrt{s} = 13 TeV pppp collisions with the ATLAS detector

International audienceThe results of a search for direct pair production of top squarks in events with two opposite-charge leptons (electrons or muons) are reported, using $36.1~\hbox {fb}^{-1}$ of integrated luminosity from proton–proton collisions at $\sqrt{s}=13$ TeV collected by the ATLAS detector at the Large Hadron Collider. To cover a range of mass differences between the top squark $\tilde{t}$ and lighter supersymmetric particles, four possible decay modes of the top squark are targeted with dedicated selections: the decay $\tilde{t} \rightarrow b \tilde{\chi }_{1}^{\pm }$ into a b-quark and the lightest chargino with $\tilde{\chi }_{1}^{\pm } \rightarrow W \tilde{\chi }_{1}^{0}$ , the decay $\tilde{t} \rightarrow t \tilde{\chi }_{1}^{0}$ into an on-shell top quark and the lightest neutralino, the three-body decay $\tilde{t} \rightarrow b W \tilde{\chi }_{1}^{0}$ and the four-body decay $\tilde{t} \rightarrow b \ell \nu \tilde{\chi }_{1}^{0}$ . No significant excess of events is observed above the Standard Model background for any selection, and limits on top squarks are set as a function of the $\tilde{t}$ and $\tilde{\chi }_{1}^{0}$ masses. The results exclude at 95% confidence level $\tilde{t}$ masses up to about 720 GeV, extending the exclusion region of supersymmetric parameter space covered by previous searches