Bose 2.2 User Manual download pdf (Page 36)

to diﬀerent layers reads

disc

, d

lat

) =

4π

1 − 3 d

lat

/(r

+ d

lat

)

+ d

lat

)

3/2

4π

− 2 d

lat

+ d

lat

)

5/2

(2.27)

Since we have chosen the extension of the discs in

-direction to be inﬁnitely small, it is

useful to deﬁne the 2D atomic density n

(ρ) =

(

πR

if ρ ≤ R

0 if ρ > R,

(2.28)

where

is the number of atoms in each disc and

the disc radius. In the

given geometry, we may introduce the cylindrical coordinates (

ρ, ϕ,

0) and (

, ϕ

, d

lat

) for

the ﬁrst and the second disc, respectively. We can then evaluate the mean-ﬁeld potential

disc

inter

(

= 0) at the center of the ﬁrst disc that is created by the presence of the second

disc, with

disc

inter

(ρ = 0) =

2π

disc

(ρ

, d

lat

) n

(ρ

) ρ

dρ

dϕ

4π

−2N

+ d

lat

)

3/2

(2.29)

which is valid in the cases

lat

= 0 and

R >

0. Considering the central mean-ﬁeld potential,

given by Eq.

(2.29)

, in diﬀerent limits for the separation

lat

and the disc radius

, we can

learn some basic properties of the dipolar inter-site interactions (we use the abbreviation

disc

inter

def

= Φ

disc

inter

(ρ = 0)):

(i) lim

lat

R

disc

inter

∝ (−2N)/d

lat

When the distance between the two discs is much larger than their radial extension,

we recover the

−3

scaling law of the two-body DDI potential (Eq.

(2.6)

). We further

recognize that the interaction of the two samples is enhanced by the population

of the discs, when compared to the case of only two dipoles in head-to-tail

conﬁguration.

(ii) lim

R d

lat

disc

inter

∝ (−2N)/R

At large disc radii or at small separations, the mean-ﬁeld potential does not depend

lat

anymore. This means, that the central part of two large and thin magnetic

discs does not contribute to a relative attraction, as there is no potential gradient in

the

-direction. However, we should be aware of edge eﬀects which we discuss below.

(iii) lim

R→∞

disc

inter

= 0

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Bose 2.2 User Manual Page 36

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