Das amerikanische Regierungssystem

Fraenkel, Ernst

doi:10.1007/978-3-322-91771-3

Cited by 16 publications

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“…Large quantities of gas molecules can be encapsulated in zeolite cavities if their molecular kinetic diameters are somewhat larger than the effective diameters of the zeolite windows. This can be accomplished by heating the zeolite and gas at high pressure, followed by quenching to ambient temperature while the high pressure is maintained. − These encapsulated gas molecules can sustain high-pressure concentrations without leakage at room temperature; in this way zeolites may be used as storage media. Controlled decapsulation can be achieved by relaxing the window blockage by reheating the zeolite, and/or by exposing the zeolite to small polar molecules. − …”

Section: Introductionmentioning

confidence: 99%

“…To further characterize the confinement effect, argon atoms were encapsulated in the cavities of fully dehydrated Cs 3 −A 29-31 and their positions were observed crystallographically. Both the Cs + and Na + ions were expected to contribute to the encapsulation of Ar atoms by blocking the eight- and six-rings, respectively. ,,,− ,, Although only weak interactions were expected, the precise coordinates of encapsulated Ar atoms, sensitive to the electrostatic field in a relatively unperturbed zeolite, would be seen. Perhaps interesting clusters of Ar atoms would be found.…”

Section: Introductionmentioning

confidence: 99%

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Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

1996

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The positions of Ar atoms encapsulated in the cavities of fully dehydrated Cs 3 Na 8 H-A (Cs 3 -A) have been determined. Cs 3 -A was exposed to 660 atm of argon gas at 400°C, followed by cooling at pressure to encapsulate Ar atoms; a second crystal was treated similarly at 1000 atm. The resulting crystal structures, of Cs 3 -A(5Ar) (a ) 12.253(2) Å, R 1 ) 0.055, R 2 ) 0.056) and Cs 3 -A(6Ar) (a ) 12.253(2) Å, R 1 ) 0.064, R 2 ) 0.065), have been determined by single-crystal X-ray diffraction techniques in the cubic space group Pm3 hm at 21(1)°C and 1 atm. The five and six Ar atoms per unit cell, respectively, are distributed over three crystallographically distinct positions: one Ar atom at Ar(1) lies opposite a six-ring in the sodalite unit, two Ar atoms at Ar(3) occupy a similar position in the large cavity, and two or three Ar atoms in Cs 3 -A(5Ar) and Cs 3 -A(6Ar), respectively, are at Ar(2) opposite four-rings in the large cavity. Relatively strong interactions of Ar(1) and Ar(3) atoms with six-ring Na + ions are observed: Na-Ar(1) ) 3.26 and Na-Ar(3) ) 3.00 Å. The four Ar atoms in the large cavity of Cs 3 -A(5Ar) form a rhombus with Ar(2)-Ar(3) ) 4.75(8) Å and Ar(2)-Ar(3)-Ar(2) ) 88(1)°. The five Ar atoms in the large cavity of Cs 3 -A(6Ar) have a trigonal bipyramidal arrangement with three Ar(2) atoms at equatorial positions and two Ar(3) atoms at axial positions; Ar(2)-Ar(3) ) 4.63(9) Å. Both of the above arrangements are stabilized by alternating dipoles induced on Ar(2) and Ar(3) by four-ring oxygens and six-ring Na + ions, respectively.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

1996

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“…Fraenkel has check that all positive integers less than 2 50 have a finite stopping time and the conjecture is still not erroneous [7].…”

Section: The Specified Calculationsmentioning

confidence: 99%

The Analysis of Convergence for the 3<i>X</i> + 1 Problem and Crandall Conjecture for the a<i>X</i> + 1 Problem

Hu¹

2021

APM

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The 3X + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3X + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value E(n) of energy contraction index. The corresponding convergence analysis explicitly shows that a = 3 leads to a difficult problem, while a > 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3X + 1 problem, but also shed some light on the future research.

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“…Both the Cs + and Na + ions of Cs 3 -A were expected to contribute to the effective encapsulation of Xe atoms by blocking the eight-and six-rings, respectively. 19,[27][28][29][32][33][34][35][36] Although weak interactions were expected, the precise coordinates of the encapsulated Xe atoms, sensitive to the electrostatic field in a relatively unperturbed zeolite, would be seen. Perhaps interesting clusters of Xe atoms in the large cavity, and a Xe atom despite its size in the sodalite unit, would be seen.…”

Section: Introductionmentioning

confidence: 99%

“…In this work, xenon atoms were encapsulated at various pressures in the cavities of fully dehydrated Cs 3 -A 26-29 in order to observe their positions and to further characterize the confinement effect 28,29 crystallographically. Both the Cs + and Na + ions of Cs 3 -A were expected to contribute to the effective encapsulation of Xe atoms by blocking the eight- and six-rings, respectively. ,− ,− Although weak interactions were expected, the precise coordinates of the encapsulated Xe atoms, sensitive to the electrostatic field in a relatively unperturbed zeolite, would be seen. Perhaps interesting clusters of Xe atoms in the large cavity, and a Xe atom despite its size in the sodalite unit, would be seen.…”

Section: Introductionmentioning

confidence: 99%

Crystal Structures of Encapsulates within Zeolites. 3. Xenon in Zeolite A

et al. 1999

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The positions of Xe atoms encapsulated in the cavities of fully dehydrated zeolite A of unit-cell composition Cs 3 Na 8 HSi 12 Al 12 O 48 (Cs 3 -A) have been determined. Cs 3 -A was exposed to 255 atm of xenon at 400 °C for 7 days, followed by cooling at pressure to encapsulate Xe atoms; a second crystal was treated similarly at 450 atm, and a third at 1020 atm. The resulting crystal structures of Cs 3 -A(2.5Xe) (crystal 1, a ) 12.245(2) Å, R 1 ) 0.056, R 2 ) 0.059), Cs 3 -A(4.5Xe) (crystal 2, a ) 12.258(2) Å, R 1 ) 0.061, R 2 ) 0.058), and Cs 3 -A(5.25Xe) (crystal 3, a ) 12.236(2) Å, R 1 ) 0.061, R 2 ) 0.057) were determined by single-crystal X-ray diffraction techniques in the cubic space group Pm3m at 21(1) °C and 1 atm. The approximately 2.5, 4.5, and 5.25 Xe atoms per unit cell, respectively, are distributed over three crystallographically distinct positions. At the center of each sodalite unit in crystals 1, 2, and 3, respectively, are 0.5, 0.75, and 0.75 Xe atoms at Xe(1); in addition, off center in crystal 3 are 0.25 Xe atoms at Xe(4), opposite a six-ring in the sodalite unit. Opposite four-rings in the large cavity at Xe(2) are 1, 2, and 2.25 Xe atoms, respectively, for the three crystals. Opposite six-rings in the large cavity at Xe(3) are 1, 1.75, and 2 Xe atoms, respectively. Relatively strong interactions of Xe(3) atoms with six-ring Na + ions are observed: Na-Xe(3) ) 3.18(2), 3.22(2), and 3.38(2) Å, respectively. In Cs 3 -A(2.5Xe), two Xe atoms are found in each large cavity: Xe 2 with a 4.54(4) Å distance may exist. Cs 3 -A(4.5Xe) may be viewed as a mixture of two kinds of unit cells: in most of their large cavities, four Xe atoms form a rhombus with Xe(2)-Xe(3) ) 4.51(3) Å and Xe(2)-Xe(3)-Xe(2) ) 95.1(5)°. Cs 3 -A(5.25Xe) may also be viewed as having two kinds of large cavities: 75% contain a rhombus of four Xe atoms, Xe(2)-Xe(3) ) 4.45(2) Å and Xe(2)-Xe(3)-Xe(2) ) 97.3(4)°. The remaining 25% have five Xe atoms arranged trigonal-bipyramidally with three Xe(2)'s equatorial and two Xe(3)'s axial (Xe(2)-Xe(3) ) 4.45(2) Å). These arrangements of encapsulated Xe atoms in the large cavity are stabilized by alternating dipoles induced on Xe(2) and Xe(3) atoms by four-ring oxygens and six-ring Na + ions, respectively.

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Das amerikanische Regierungssystem

Abstract: Buchbinderei: W. Langeliiddeci

Cited by 16 publications

References 0 publications

Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

The Analysis of Convergence for the 3<i>X</i> + 1 Problem and Crandall Conjecture for the a<i>X</i> + 1 Problem

Crystal Structures of Encapsulates within Zeolites. 3. Xenon in Zeolite A

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Das amerikanische Regierungssystem

Abstract: Buchbinderei: W. Langeliiddeci

Cited by 16 publications

References 0 publications

Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

Crystal Structures of Encapsulates within Zeolites. 2. Argon in Zeolite A

The Analysis of Convergence for the 3&lt;i&gt;X&lt;/i&gt; + 1 Problem and Crandall Conjecture for the a&lt;i&gt;X&lt;/i&gt; + 1 Problem

Crystal Structures of Encapsulates within Zeolites. 3. Xenon in Zeolite A

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The Analysis of Convergence for the 3<i>X</i> + 1 Problem and Crandall Conjecture for the a<i>X</i> + 1 Problem