• 통합자연과학논문집
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• 제10권2호
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• pp.110-114
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• 2017

The connected and simply connected four-dimensional matrix solvable Lie group $Sol^4_1$ is the four-dimensional geometry. A crystallographic group of $Sol^4_1$ is a discrete cocompact subgroup of $Sol^4_1{\rtimes}D(4)$. In this paper, we geometrically describe the crystallographic groups of $Sol^4_1$.

• 충청수학회지
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• 제34권1호
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• pp.17-26
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• 2021

The purpose of this paper is to determine the structure of the crystallographic group 𝚷 of the 4-dimensional solvable Lie group Sol ⁴_m,n that the translation subgroup of 𝚷, 𝚪 := 𝚷; ∩ Sol⁴_m,n, is generated by the particular elements.

• 대한수학회지
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• 제52권6호
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• pp.1209-1251
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• 2015

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$, and more generally, the crystallographic groups of $Sol_1^4$. The maximal compact subgroup of Isom($Sol_1^4$) is $D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold of $Sol_1^4$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2$ (5 families), and ${\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$ (2 families).

• 통합자연과학논문집
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• 제14권3호
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• pp.139-146
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• 2021

The purpose of this paper is to classify the torsion-free extensions 1→ℤ³→𝛱→ℤ_𝜱→1 with injective abstract kernel 𝜙 : ℤ_𝜱→Aut(ℤ³). From this classification, we handle the sufficient conditions so as to classify the crystallographic groups of Sol⁴_m,n.

• 한국결정학회지
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• 제14권2호
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• pp.110-114
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• 2003

LiAl/sub 0.24/Mn/sub 1.76/O/sub 4-y/S/sub y/ (y=0, 0.02) were synthesized by the sol-gel method. Both structures were refined by Rietveld method, its structure refined as a cubic spinel, space group Fd-3m, a=8.17937(30) Å and 8.18331(19) Å respectively. Though it has been made a charge/discharge experiment above 20 times, there was no change of 3 V/4 V capacity degradation. It was considered that the volume change of MnO/sub 6/ octahedron induced by sulfur substitution plays a key role in keeping the 3 V/4 V capacity. The refined composition of the compound could be confirmed with the ICP analysis.

• 한국자기학회지
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• 제20권5호
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• pp.187-190
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• 2010

$Sn_{0.99}{^{57}Fe}_{0.01}O_2$ 분말을 졸-겔(sol-gel) 방법으로 제조하였으며, 결정학적 및 자기적 특성을 x-선 회절(x-ray diffractometer), 진동시료 자화율 측정기(vibrating sample magnetometer)과 초전도양자간섭장치(Superconducting quantum interference devices) 및 뫼스바우어 분광기(M$\ddot{o}$ssbauer spectroscopy)을 이용하여 연구하였다. $Sn_{0.99}{^{57}Fe}_{0.01}O_2$의 경우 rutile tetragonal 결정구조에 공간그룹은 $P4_2$/mnm이며, 5.6 % 정도의 산소결핍 현상이 있음을 Rietveld 정련법으로 분석하였다. 상온에서 자기화 값은 $1.95{\times}10^{-2}{\mu}_B/Fe$을 가지며 상자성과 강자성적 특성을 나타내고 있었으며, 큐리-바이스 온도 ${\theta}_{cw}$= 18 K임을 확인할 수 있었다. 뫼스바우어 측정으로 부터 상온으로부터 극저온(4.2 K)까지 Sextet이 존재하며, 이성질체 이동치의 값은 전 온도구간에서 0.18~0.36 mm/s로서 $^{57}Fe$ 이온은 모두 +3로 존재함을 알 수 있었다. $Sn_{0.99}{^{57}Fe}_{0.01}O_2$의 강자성 특성의 발현은 산소결핍으로 인한 전자를 매개로 하여 이웃하고 있는 $Fe^{3+}$ 이온들이 강자성 결합에 기인하는 것으로 해석할 수 있다.