# STABLE PHYSICS V 3.0

The theoretical prediction of stable crystal structures is very difficult because of the need to search the very large space of possible structures, and the necessity of obtaining accurate energies for each of these structures. First-principles DFT methods have proved an efficient means of calculating quite accurate energies, and they have provided many insights into the properties of materials, including solid hydrogen5,7. At present, DFT offers the highest level of theoretical description at which we can carry out searches over many possible candidate structures.

## STABLE PHYSICS V 3.0

The small mass of the proton poses a significant problem for theoretical descriptions of hydrogen; the zero-point (ZP) motion of the nuclei is large enough to significantly affect the relative stabilities of structures and their vibrational properties. We have estimated ZP vibrational energies within DFT using the harmonic approximation. The harmonic approximation is likely to give useful estimates of the total ZP vibrational energy of the candidate structures at low pressures, as the calculated harmonic vibronic frequencies are similar to the measured values. There are, however, noticeable anharmonic effects in hydrogen, even at low pressures1,10. In monatomic structures, which are believed to be stable at very high pressures, the harmonic approximation fails completely11.

The structures formed by the heavier isotope deuterium are expected to be similar to those of hydrogen. Recent experiments3 indicate that phase II of deuterium has an incommensurate structure, although a periodic model of symmetry was suggested as having the same local ordering of molecules. We found this molecular structure to be poorly packed, and relaxation with the constraint that the symmetry is not lowered led to a structure of higher P63/m m c symmetry, which, however, is high in enthalpy, as shown in Fig. 1. We therefore conclude that the even less stable structure is unlikely to be related to phase II. (A similar conclusion was arrived at in ref. 14.) On the other hand, the neutron data for deuterium of ref. 3 were found to be incompatible with the P c a21 and P21/c structures.

If you consider placing a fifth charge at the origin it will be repelled by each of the 4 original charges, so it is not surprising that the forces acting on it in the xy plane push it back to the centre. It is, however, clearly unstable in the z direction as it is repelled by the entire existing arrangement of charges.

2) The passage states that upon dehydration of the cyclized product, a thermodynamic mixture of products is formed. Recall that there are two major factors influencing which products of a reaction are preferentially formed: thermodynamics and kinetics. Thermodynamic control will favor energetically stable products even if they require a higher activation energy and more time to produce. Kinetic control will favor products that form more quickly; these products will often be less energetically stable but are formed faster due to lower activation energies.

Like most problems in physics, this problem begins by identifying known and unknown information and selecting the appropriate equation capable of solving for the unknown. For this problem, the knowns and unknowns are listed below.

The potential energy for a particle undergoing one-dimensional motion along the x-axis is [latex] U(x)=2(x^4-x^2), [/latex] where U is in joules and x is in meters. The particle is not subject to any non-conservative forces and its mechanical energy is constant at [latex] E=-0.25\,\textJ [/latex]. (a) Is the motion of the particle confined to any regions on the x-axis, and if so, what are they? (b) Are there any equilibrium points, and if so, where are they and are they stable or unstable?

is negative at [latex] x=0 [/latex], so that position is a relative maximum and the equilibrium there is unstable. The second derivative is positive at [latex] x=\textx_Q [/latex], so these positions are relative minima and represent stable equilibria.SignificanceThe particle in this example can oscillate in the allowed region about either of the two stable equilibrium points we found, but it does not have enough energy to escape from whichever potential well it happens to initially be in. The conservation of mechanical energy and the relations between kinetic energy and speed, and potential energy and force, enable you to deduce much information about the qualitative behavior of the motion of a particle, as well as some quantitative information, from a graph of its potential energy.

Psychophysics Toolbox Version 3 (PTB-3) is a free set of Matlab and GNU Octave functions for vision and neuroscience research. It makes it easy to synthesize and show accurately controlled visual and auditory stimuli and interact with the observer. Some of its functionality is available as part of Python toolkits like PsychoPy. For commercial support and services visit www.psychtoolbox.net. Follow us on Twitter @psychtoolbox

Psychtoolbox has many active users, an active forum, and is widelycited. PTB-3 is based on the deprecated Psychophysics Toolbox Version 2with its Matlab C extensions rewritten to be more modular and to use OpenGL onthe back end. The current version supports at least Matlab R2022b, and Octave 5 andlater on Linux, and Octave 7.3 on macOS, and Windows.

Ernest Rutherford discovered that all the positive charge of an atom was located in a tiny dense object at the center of the atom. By the 1930s, it was known that this object was a ball of positively charged protons and electrically neutral neutrons packed closely together. Protons and neutrons are callednucleons. The nucleus is a quantum object. We cannot understand its properties and behaviors using classical physics. We cannot track the individual protons and neutrons inside a nucleus. Nevertheless, experiments have shown that the "volume" of a nucleus is proportional to the number of nucleons that make up the nucleus. We define the volume of the nucleus (and also the volume of any other quantum particle) as the volume of the region over which its interaction with the outside world differs from that of a point particle, i.e. a particle with no size. With the above definition of the volume and size of a quantum particle we find that protons and neutrons are each about 1.4*10-15 m in diameter, and the size of a nucleus is essentially the size of a ball of these particles. For example, iron 56, with its 26 protons and 30 neutrons, has a diameter of about 4 proton diameters. Uranium 235 is just over 6 proton diameters across. One can check, for example, that a bag containing 235 similar marbles is about six marble diameters across.Most nuclei are approximately spherical. The average radius of a nucleus with A nucleons is R = R0A1/3, where R0 = 1.2*10-15 m. The volume of the nucleus is directly proportional to the total number of nucleons. This suggests that all nuclei have nearly the same density. Nucleons combine to form a nucleus as though they were tightly packed spheres.

While the attractive nuclear force must be stronger than the electrostatic force to hold the protons together in the nucleus, it is not a long range 1/r2 force like the electrostatic force and gravity. It drops off much more rapidly than 1/r2, with the result that if two protons are separated by more than a few proton diameters, the electric repulsion becomes stronger than the nuclear attraction. The separation D0 at which the electric repulsion becomes stronger than the nuclear attraction is about 4 proton diameters. This distance D0, which we will call therange of the nuclear force, can be determined by looking at the stability of atomic nuclei. If we start with a small nucleus, and keep adding nucleons, for a while the nucleus becomes more stable if we add the right mix of protons and neutrons. By more stable, we mean more tightly bound. The more stable a nucleus is, the more energy is required, per nucleon, to pull the nucleus apart. This stability is caused by the attractive nuclear force between nucleons.

Iron 56 is the most stable nucleus. It is most efficiently bound and has the lowest average mass per nucleon. Nickel 62, Iron 58 and Iron 56 are the most tightly bound nuclei. It takes more energy per nucleon to take one of these nuclei completely apart than it takes for any other nucleus. If a nucleus gets bigger than these nuclei, it becomes less stable. If a nucleus gets too big, bigger than a Lead 208 or Bismuth 209 nucleus, it becomes unstable and decays by itself. The stability of Iron 56 results from the fact that an Iron 56 nucleus has a diameter about equal to the range of the nuclear force. In an Iron 56 nucleus every nucleon is attracting every other nucleon. If we go to a nucleus larger than Iron 56, then neighboring nucleons still attract each other, but protons on opposite sides of the nucleus now only repel each other. This repulsion between distant protons leads to less binding energy per particle and instability. We usually give the binding energy of a nucleus as a positive number. It then is the energy that is needed from an external source to separate the nucleus into its constituent protons and neutrons.

Physicists map the inventory of known nuclei on a "chart of nuclides." On the chart shown on the right, the vertical axis represents the number of protons a nucleus contains and the horizontal axis represents the number of neutrons it possesses. The region of stable nuclei is roughly found on a diagonal line, where the neutron number approximately equals proton number. Below this diagonal is a jagged line called the "neutron dripline" and above this diagonal is another jagged line called the "proton dripline." Nuclei found above the proton dripline and below the neutron dripline tend to be highly unstable and undergo radioactive decay immediately.

Based on the results, the camera was stable within 3 % over an 18-month time period. The daily flood source acquisitions can be a reliable source for tracking camera stability and may provide information on updating the calibration factor for quantitative imaging. 041b061a72