Magnetodeformation effects in a crystal lattice
A crystal with a local plane defect in the external magnetic field B oriented along the plane is investigated. Within the framework of the variational method a qualitative analysis of the degree of deformation depending on B is carried out. It is shown that an increase of B entails a more localized...
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Інститут фізики конденсованих систем НАН України
1999
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| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/119927 |
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| Zitieren: | Magnetodeformation effects in a crystal lattice / B.A. Lukiyanets, R.M. Peleshchak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 89-92. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1199272017-06-11T03:03:22Z Magnetodeformation effects in a crystal lattice Lukiyanets, B.A. Peleshchak, R.M. A crystal with a local plane defect in the external magnetic field B oriented along the plane is investigated. Within the framework of the variational method a qualitative analysis of the degree of deformation depending on B is carried out. It is shown that an increase of B entails a more localized lattice distortion and its increase. Розглядається кристал з локальним плоским дефектом, поміщений в зовнішне магнітне поле, орієнтоване вздовж дефекту. В рамках варіаційного принципу проведено якісний аналіз ступеня деформації в зaлежності від величини поля B . Показано, що зростання B приводить до більш локалізованого спотворення гратки з одночасним його зростанням. 1999 Article Magnetodeformation effects in a crystal lattice / B.A. Lukiyanets, R.M. Peleshchak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 89-92. — Бібліогр.: 5 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.89 PACS: 62.20.Fe, 71.45.Gm. http://dspace.nbuv.gov.ua/handle/123456789/119927 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A crystal with a local plane defect in the external magnetic field B oriented along the plane is investigated. Within the framework of the variational method a qualitative analysis of the degree of deformation depending on B is carried out. It is shown that an increase of B entails a more localized lattice distortion and its increase. |
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Article |
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Lukiyanets, B.A. Peleshchak, R.M. |
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Lukiyanets, B.A. Peleshchak, R.M. Magnetodeformation effects in a crystal lattice Condensed Matter Physics |
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Lukiyanets, B.A. Peleshchak, R.M. |
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Lukiyanets, B.A. |
| title |
Magnetodeformation effects in a crystal lattice |
| title_short |
Magnetodeformation effects in a crystal lattice |
| title_full |
Magnetodeformation effects in a crystal lattice |
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Magnetodeformation effects in a crystal lattice |
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Magnetodeformation effects in a crystal lattice |
| title_sort |
magnetodeformation effects in a crystal lattice |
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Інститут фізики конденсованих систем НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/119927 |
| citation_txt |
Magnetodeformation effects in a crystal lattice / B.A. Lukiyanets, R.M. Peleshchak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 89-92. — Бібліогр.: 5 назв. — англ. |
| series |
Condensed Matter Physics |
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AT lukiyanetsba magnetodeformationeffectsinacrystallattice AT peleshchakrm magnetodeformationeffectsinacrystallattice |
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2025-07-08T16:55:26Z |
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2025-07-08T16:55:26Z |
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1837098575376416768 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 89–92
Magnetodeformation effects
in a crystal lattice
B.A.Lukiyanets 1 , R.M.Peleshchak 2
1 State University “Lvivs’ka Politechnika”,
12 Bandera Str. 290646 Lviv, Ukraine
2 I.Franko Pedagogical University,
34 I.Franko Str., 293720 Drohobych, Lviv region, Ukraine
Received March 11, 1998
A crystal with a local plane defect in the external magnetic field B ori-
ented along the plane is investigated. Within the framework of the varia-
tional method a qualitative analysis of the degree of deformation depend-
ing on B is carried out. It is shown that an increase of B entails a more
localized lattice distortion and its increase.
Key words: crystal, plane defect, magnetic field, electron-deformation
interaction
PACS: 62.20.Fe, 71.45.Gm.
Real crystals are those which have impurities or various defects of a crystalline
lattice. Induced by them distortions give rise to redistribution of electrons in a
crystal and, as a result, to renormalization of these distortions [1,2].
In the given paper the relative deformation ρ(r) = (δVk − δV )/δV = Spξ
(δV , δVk are volumes of the primitive unit cells with or without the presence of
deformation respectively; ξ is a tensor of deformation) of a crystal with a plane
defect (e.g.,a twinning plane) in the external magnetic field oriented along the
plane defect is analyzed.
Let a plane defect in a crystalline lattice be with a normal along OX. It creates
a force action which is described by the potential
U(r) ≡ U(x) = U0 exp
(
−αx2
)
(1)
(U0, α are constants), i.e. such an action has a one-dimensional character.
The energy of the system of a crystal and an electron in the state ψ(r) inter-
acting with the relative deformation ρ(r) in a continuum approximation may be
c© B.A.Lukiyanets, R.M.Peleshchak 89
B.A.Lukiyanets, R.M.Peleshchak
written in the form [3]:
E {ψ, ρ} =
∫
d3r
[
h2
2m∗
|∇ψ (r)|2 − σρ (r) |ψ (r)|2 + 1
2
βρ2 (r) + U (r) |ψ (r)|2
]
,
(2)
where the first term in the integrand is the kinetic energy of an electron, the second
is its interaction with the deformation of the lattice, the third is the energy of an
interaction of an electron with a local deformation (with a plane defect in the case
under consideration). It is supposed that the crystal is isotropic (m∗ is a scalar).
Let the external magnetic field be imposed on a crystal along the plane defect
(say along OZ). Consideration of the field in (1) is reduced to a formal replacement
p → (p − eA/c) [3], where p = −i~∇ is a momentum operator and A is a vector
potential. Then (2) takes the form:
E {ψ, ρ} =
∫
d3r
[
1
2m∗
∣
∣
∣
(
p− e
c
A
)
ψ (r)
∣
∣
∣
2
− σρ (r) |ψ (r)|2
+
1
2
βρ2 (r) + U (r) |ψ (r)|2
]
. (3)
From a variation of E {ψ, ρ} with respect to ρ(r) at fixed ψ(r)
ρ (r) =
σ
β
|ψ (r)|2 . (4)
It is seen that taking into account (4), E {ψ, ρ} becomes a functional with
respect to ψ(r) only.
Let the vector potential A = (0, Bx, 0) (B is a magnetic field). Then,
E {ψ} =
∫
d3r
{
~
2
2m
[
|pxψ (r)|2 +
∣
∣
∣
(
py −
e
c
xB
)
ψ (r)
∣
∣
∣
2
+ |pzψ (r)|2
]
− σ
2ρ
|ψ (r)|4 + U (r) |ψ (r)|2
}
. (5)
We use a variational method for the estimation of the magnetodeformation
effect in our problem. Since we are interested in ρ(r),according to (4), the wave
function must be found. In order to solve the problem by the variational method
it is necessary to clarify this statement. It is known [4,5] that the variational
method does not allow a control of the wave function: even if its use leads to a
sufficiently accurate eigenvalue,it is impossible to estimate the difference between
the wave function obtained by the variational method and the exact one. But
this deficiency is not so important for using a probe (obtained by the variational
method) function for the study of the changes of ρ(r) depending on the parameters
of the problem (magnetic field, electron-deformation interaction), but important
for the calculation of its absolute value. This very approach is realized below.
Take the probe function in the form
ψ (r) = exp (ikzz) exp
(
−µ(x2 + y2)
)
, (6)
90
Magnetodeformation effects in a crystal lattice
i.e. with a single parameter µ.
Constant C in (6) may be obtained from the normalization condition:
C =
√
µ
π
.
In (6) it is taken into account that the applied magnetic field and the lattice
distortion (1) do not change the electron motion along OZ and it is described by
a plane wave.
Using (6), the minimum of the functional of energy is easily found by the
variational method. Substitution of (6) in (5) and integration yield:
E {ψ} ≡ E {µ} =
(
~
2
m
− σ2
4βπ2
)
µ+
1
4µ
(
eB
c
)2
+ U0
√
2µ
2µ+ α
. (7)
Consider a qualitative conclusion about the extremum (minimum) E {ψ}. In
the limit case U0 = 0, the minimum of E {ψ} is realized at
µ =
(eB/c)
2
√
~2/2m− σ2/4βπ2
. (8)
Using (8) in the probe function (6) and its relation with ρ(r) (4), allows us to
state:
• an increase of the magnetic field leads to localization of the distortion with
its simultaneous increase;
• for crystals with larger elastic constants or with a smaller electron-defor-
mation interaction the relative deformation is more delocalized and has a
smaller value.
To estimate the role of the local deformation ∼ U0 in the problem, let us
consider µ≫ α. We expand the last term in (7) in a power series in α:
√
2µ
2µ+ α
=
√
2µ
(
1√
2µ
− 1
2
(√
2µ+ α
)3
∣
∣
∣
∣
α=0
α+ ...
)
∼ 1− α
4µ
. (9)
Then, the extremum of E {µ} is realized at
µ =
1
2
√
(eB/c)2 − Uoα
~2/m− σ2/4βπ2
. (10)
Thus, taking into account the interaction energy of an electron with the local
deformation ∼ U0 leads to delocalization of a relative deformation of the crystal
at U0 > 0 and to a more marked localization at U0 < 0.
Note that the analysis (within the framework of the quasiclassic approximation)
is justifiable for the µa2 6 0.01 case [3] (a is some characteristic length of the
problem, e.g., a lattice parameter).
91
B.A.Lukiyanets, R.M.Peleshchak
References
1. Stasyuk I.V., Peleshchak R.M. Filling of the electron states and the metal lattice de-
formation in the vicinity of the border between the regions with different mechanical
strains. // Ukr. fiz. zhurn., 1991, vol. 36, No 11, p. 1744-1749 (in Ukrainian).
2. Peleshchak R.M., Lukiyanets B.A. The electron redistribution in the vicinity of the core
of the linear dislocation. // Pisma v ZhTF, 1998, vol. 21, No 2, p. 32-36 (in Russian).
3. Davydov A.S. Theory of Solid State. M., Nauka, 1976 (in Russian).
4. Madelung E. Mathematical technique of physics. M., Nauka,1968 (in russian).
5. Messia A. Quantum mechanics. M., Nauka, 1979 (in Russian).
Магнітодеформаційні ефекти в кристалічній гратці
Б.А.Лукіянець 1 , Р.М.Пелещак 2
1 Державний університет “Львівська Політехніка”,
290646 Львів, вул.С.Бандери, 12
2 Дрогобицький педагогічний інститут ім.І.Франка,
293720 Дрогобич Львівської обл., вул.І.Франка, 34
Отримано 11 березня 1998 р.
Розглядається кристал з локальним плоским дефектом, поміщений
в зовнішне магнітне поле, орієнтоване вздовж дефекту. В рамках
варіаційного принципу проведено якісний аналіз ступеня деформації
в зaлежності від величини поля B . Показано, що зростання B при-
водить до більш локалізованого спотворення гратки з одночасним
його зростанням.
Ключові слова: кристал, плоский дефект, магнітне поле,
електрон-деформаційна взаємодія
PACS: 62.20.Fe, 71.45.Gm
92
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