Physics Letters A 307 (2003) 65–68

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Charged particle with magnetic moment in the backgroundof line topological defect

Sérgio Azevedo1

Departamento de Física, ICEX, Universidade Federal de Minas Gerais, CP 702, 30123-970, Belo Horizonte, MG, Brazil

Received 25 October 2002; accepted 21 November 2002

Communicated by V.M. Agranovich

Abstract

Using a description of defects in solids in terms of three-dimensional gravity, we will consider a charged quantum particlewith spin placed in the field of a non-uniform magnetic flux in space with a topological defect, namely, a disclination or screwdislocation. We have found the exact expression for the energy eigenvalues and eigenfunctions for the quantum particle in amedium with magnetic field proportional to 1/ρ. 2002 Elsevier Science B.V. All rights reserved.

PACS: 61.72.Lk; 41.20.Cv; 41.20.-q

1. Introduction

Quantum effects on particles moving in crystallinemedia with topological defects have attracted consid-erable attention since the early 1950s [1]. More re-cently, a geometrical approach has been used [2–4]to study such effects. Geometry theory of defects de-scribes elastic deformations, dislocations, and discli-nations from a uniform point of view. This schemeincludes description which cannot be described inframework of classical elasticity theory.

The advantage of a geometric description of defectsin solids is twofold. Firstly, in contrast to the ordinary

E-mail address: sazevedo@uefs.br (S. Azevedo).1 On leave from Departamento de Física, Universidade Estadual

de Feira de Santana Km-03, BR116-Norte, 44031-460, Feira deSantana, BA, Brazil.

elasticity theory, this approach provides an adequatelanguage for continuous defects. Secondly, the mightymathematical machinery of differential geometry clar-ifies and simplifies any calculations.

The properties of electron interacting with sta-tic topological defects can be investigated by look-ing at their propagation in the background of thesedefects [5,6]. In this Letter, we consider a quantumparticle with spin placed in a region of non-uniformmagnetic field proportional to 1/ρ, in the backgroundof a topological linear defects, namely, a disclina-tion [7] (topological defect carrying curvature) andscrew dislocation [5] (topological defect carrying tor-sion). We explicitly obtain the expressions for eigen-values and eigenfunctions of this particle, and showedthe effects of the defect on the electron.

A topological defect consists of a core regioncharacterized by absence of order and a smooth

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01676-6

66 S. Azevedo / Physics Letters A 307 (2003) 65–68

far field region. In the continuum approximationthat we use, the core is shrunk until a singularity.Although not very realistic, this model is very usefulto show the appearance of global phenomenon, relatedrather to topology than to local geometry inducedby the defect. These defects, although formed duringphase transitions involving symmetry breaking, can beconceptually generated as a “cut-and-glue” process,known in the literature as the Volterra process [8].

We use the Katanaev and Volovich’s approach [8]which translates the theory of defects in solids to thelanguage of three-dimensional gravitation. A discli-nation is viewed [9] as the analogue of a cosmicstring [10]: a topological defect carrying curvature butnot torsion (in contrast to dislocations that carry tor-sion but not curvature). The disclination is obtainedconceptually by either removing (positive-curvaturedisclination) or inserting (negative-curvature disclina-tion) a wedge of dihedral angle 2π |α − 1| such thatthe total angle around thez-axis is 2πα instead of 2π .Therefore,α < 1 corresponds to an angle deficit, orpositive disclination, andα > 1 corresponds to anexcess angle, or negative disclination. The resultingspace has a null curvature tensor everywhere except atthe defect where it has a two-dimensional delta func-tion singularity. The dislocation is created cut the bodyalong any smooth surface having the curve at is bound-ary. Move the media in arbitrary ways on both sides ofsurface and the glue the sides together extracting oradding some media if needed. In the case of the screwdislocation, the Burger vector is parallel to the dislo-cation line.

This Letter is organized as follows. In Section 2we obtain the energy eigenvalues and eigenfunctionsfor the charged quantum particle in a region ofmagnetic field proportional to 1/ρ in a medium witha disclination; in Section 3 we study the particle in amedium with a screw dislocation; finally, in Section 4we summarize our main results.

2. Disclination

In this section we study the quantum particle withmagnetic moment in medium with a disclination. Thisdefect can be characterized by the metric

(1)ds2 = dz2 + dρ2 + α2ρ2 dθ2,

where 0� θ � 2π and 2π(1−α) is the angular deficitor excess.

The magnetic field is non-vanishing over a compactregion enclosing the defect, and can be written as

(2)�H(ρ) = Φ

2απRρΘ(R − ρ)z,

whereΦ is the total flux andΘ is the step function.The magnetic is uniform, in thez-direction, inside acylinder of radiusR.

The vector potential associated to this magneticfield is given by

(3)�A = Φ

2παρR

(ρ

RΘ(R − ρ) +Θ(ρ −R)

)θ .

In order to compute the modification on the en-ergy spectrum and eigenvalues due to disclination, oneneeds to write the time-independent Schrödinger equa-tion in the space described by metric (1). In this space,the one is

(4)H =[

1

2m

(h

i�∇α − e

c�A)2]

+µH(ρ).

The second term in (5) is the contribution due to theinteraction field magnetic moment withµ being givenby

(5)µ = gµB

hs,

whereµB = eh2mc

andg its gyromagnetic ratio. Note

that �∇α is the gradient operator in the backgrounddescribed by metric (1).

Due to spin conservation, the magnetic interac-tion (4) can be replaced by

(6)±1

2gµBH(ρ),

wherein± corresponds to the spin projection on theflux line. From here, we have restricted ourselves tothe minus sign in which the moment magnetic leads toa binding force.

Eq. (4), using (2), (3), and (5), can be rewritten as

(7)

[1

2m

(h

i�∇α − q

c�A)2

− gµB

2

Φ

2παRρ

]ψ = Eψ,

where

(8)�∇2α = 1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)+ 1

α2ρ2

∂2

∂θ2 + ∂2

∂z2 .

S. Azevedo / Physics Letters A 307 (2003) 65–68 67

Using the ansatz

(9)ψ(ρ, θ, z) = eikzei!θR(ρ),

and substituting in (7), using (8), we obtain

1

ρ

d

dρ

(ρ

d

dρ

)R− 1

ρ2

[(!′ − δ

ρ

R

)]2

R

(10)+ g

2δ

1

ρRR−K2R= 0,

whereδ = Φq2πhc

. The constants!′, andK are given by

(11)!′ = !2

α2 ,

(12)K2 = −2mE

h2 + k2.

The solution of this equation, regular inρ = 0, is givenby

R(ρ) = ρ|!|/α

× 1F1[!α

(12 + |!|

α− !

αδ

δ

) − g4δ

δ,1+ 2|!|

α;2δ ρ

R

](13)× e−δρ/R,

where

(14)δ2 = δ2 −K2R2.

Note thatα = 1, we obtain the same solution as that inthe conventional space.

From (12), we can determine the normalizationcondition of the wavefunction as

(15)!

α

(1

2+ |!|

α− !

α

δ

δ

)− g

4

δ

δ= −n,

wheren = 0,1,2,3 . . . .Using (12), (13), and (15), the energy eigenvalues

can be written as

(16)En,l = h2

2m

δ2

R2

[1− [(!/2)2 + g/4]2[

!α

(12 + !

α

) + n]2

]+ h2k2

2m.

Note that the eigenvalues depend on the topologicalparameter and gyromagnetic factor.

The ground state,!/α = 1 andn = 0, is given by

(17)En,! = h2

2m

δ2

R2

[(2− g)(g + 10)

36

]+ h2k2

2m.

From (16), we can see thatg = 2 does not correspondsto bound state; on the other hand, ifg < 2 we obtain

scattering. Finally,g > 2 corresponds to bound state.Note that the quantum number! is modified by thefactor topologicalα, which carries information on thedefect.

3. Screw dislocation

The screw dislocation is described by the followingmetric in cylindrical coordinates [12]

(18)ds2 = (dz + β dθ)2 + dρ2 + ρ2dθ2,

whereβ is a parameter related to the Burger vector�bby β = b

2π . This topological defect, carries torsion butno curvature.

The time-independent Schrödinger equation in thespace described by metric (17) is given by

(19)

[1

2m

(h

i�∇β − e

c�A)2

− gµBH(ρ)

2

]ψ = Eψ,

where �H(ρ) and �A(ρ) are the same as that describedby (2) and (6), without parameterα.

The Laplace–Beltrami operator�∇2β , using the met-

ric (1), is given by

�∇2βψ = ∂2

∂z2ψ + 1

ρ2

(∂

∂θ− β

∂

∂z

)2

ψ

(20)+ 1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)ψ.

The solution of Eq. (19), using (9) and (20), can bewritten as

R= ρ|!−βk|

× 1F1[(!− βk)

(12 + |!− βk| − (!− βk) δ

δ

)− g

4δ

δ1+ |!− βk|;2δ ρ

R

](21)× e−δρ/R.

The eigenvalues are given by

E!,n = h2

2m

δ2

R2

[1− [(!− βk)2 + g/4]2[

(!− βk)(1

2 + |!− βk| + n)]2

]

(22)+ h2k2

2m.

Note that, in this case, the quantum number! is shiftedby parameterβk. This fact is due to match between thecoordinatesz andθ , because of the topological defect.

68 S. Azevedo / Physics Letters A 307 (2003) 65–68

The ground state,!′ = ! − βk = 1, andn = 0, canbe written as

(23)E = h2

2m

δ2

R2

[(2− g)(g + 10)

36

]+ h2k2

2m,

which is the same as that obtained in (16).

4. Conclusions

In this Letter, we have obtained the energy spec-trum and eigenfunctions of a quantum particle, in aspace with a linear topological defect, disclinationand screw dislocation, placed in a medium of a non-uniform magnetic field. We showed that in the case ofa disclination (defect carrying curvature) the angularmomentum is multiplied by factor 1/α. On the otherhand, in the case of a screw dislocation (defect carry-ing torsion), the angular momentum is shifted by thefactorβk. In the case of the disclination, the topologi-cal factor 1/α appear as a multiplicative one of correc-tion to !. This is quite natural, since the effect of thechange of topology introduced by disclination is to al-ter the total angle around thez-axis from 2π to 2πα.However, in the case of the screw dislocation, the num-ber quantum! is shifted by a additive factorβk. Thisis due to the fact that change of topology caused bydefect, alter thez-coordinate fromz to z − βθ .

We have demonstrated that the binding energy, of aparticle is affected by the existence of disclination andscrew dislocation. It is important to remark that the re-

gion outside defect has zero curvature. Besides, it wasinvestigated the case where the magnetic flux coincidewith defectR → 0, the well-known Aharonov–Bohmeffect. In this case, the bound states energies tend toinfinity, therefore this results coincide as that obtainedin [11]. We showed, in the casen = 0 and! = 1/α(disclination), or! = ! − βk = 1 (screw dislocation)that only exists bound state forg > 2.

Acknowledgement

This work was partially supported by CNPq.

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