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5d744cd73776a09002b7de9836fada15ec463896 | subsection | 25 | 32 | Discussion and conclusion | However, dimensional analysis yields P_{\rm cap}
\sim R\, Q_{\rm cap} so that the amplitude of
this capillary energy, which must vanish for large film thicknesses h\rightarrow \infty ,
has also a
too small numerical value to explain the experimental results reported in Ref. .We recall that for simplicity our calculatio... | {
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e4a49433a505dcddd7be56edb4cfd4e257abe29f | subsection | 26 | 32 | Corrections to the strong anchoring limit | Here we consider the asymptotic director field around a single colloid at the nematic–air
interface in the case of finite anchoring strength. For large distances from the colloid,
the overall deviations from the preferred director \mathbf {n} =(n_1,n_2,n_3)
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1c75e470e57c6c798947f2fc9129e551f29ab6b8 | subsection | 27 | 32 | Corrections to the strong anchoring limit | Thus, to leading order in 1/r, the boundary condition can only be met if
the leading multipole (n_i^{\ell }) fulfills the strong anchoring condition
n_i^{\ell }(z=0)=0 and is accompanied by a subleading multipole (n_i^{s\ell })
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f08937c7c001e93853693b9ed79bfd30e11cd940 | subsection | 28 | 32 | Force balance in a general configuration | For the benefit of the reader, we discuss in this Appendix some
previous results , concerning the force balance of a general
equilibrium configuration and demonstrate how the amplitude of an
asymptotic, logarithmically varying interfacial deformation is
determined solely by this mechanical condition of force balance.Fi... | {
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7d5436514602c597d80ea9d812e3cb28af678156 | subsection | 29 | 32 | Force balance in a general configuration | This tension can be expressed in terms of a line
integral involving the surface tension \gamma :
{\mathbf {F}}_{\rm contact} := \mbox{} - \gamma \oint _{C_0} d\ell \,{\mathbf {e}}_c ,
where {\mathbf {e}}_c is the unit vector tangent to the interface,
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919960c9425bf24e1d64d7e01b69ea5cb9f44d15 | subsection | 30 | 32 | Force balance in a general configuration | Thus there is a distance \xi beyond which the linear theory is applicable:\gamma \nabla ^2 u (\mathbf {r}) = \Pi _{zz}^1 (\mathbf {r}) - \Pi _{zz}^2(\mathbf {r}) \qquad ({\bf r} \in S_{\rm ext}) \;,where the piece of interface S_{\rm ext} is enclosed by the
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b7570eec922814fe544008e543a1138e79665065 | subsection | 31 | 32 | Force balance in a general configuration | (REF ) is proportional to the
force exerted by the upper and the lower fluid on the particle and on the meniscus:B_0 = \frac{1}{2\pi \gamma } {\bf e}_z \cdot \left[
{\bf F}_{\rm part} + {\bf F}_{\rm men} \right] .In obtaining Eq. (REF ) we have used that
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on... | {
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f98f4928a6039afc51d3d8842f5e8e023c5b4bf9 | abstract | 0 | 10 | Abstract | Deformed special relativity (DSR) is one of the possible realizations of a
varying speed of light (VSL). It deforms the usual quadratic dispersion
relations so that the speed of light becomes energy dependent, with preferred
frames avoided by postulating a non-linear representation of the Lorentz group.
The theory may ... | {
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c33a16125b86b647c33e2a3433d4066f8a71b2a2 | subsection | 1 | 10 | Introduction | In a recent Letter we proposed a mechanism for producing
scale-invariant density fluctuations of appropriate amplitude
based on a decaying speed of sound. The mechanism
is quite general and can be implemented using a variety of methods.
The examples of \kappa -essence ,
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9775ac39dce70da0c5571d9de7ac22d477e93f5e | subsection | 2 | 10 | Very basic DSR | We start with a quick review of DSR, cast in a formalism that can
be used here. It's possible that alternative
formulations , may be plugged into the fluctuations'
calculation that follows, but this has not been checked.Let us consider a deformed dispersion relation (DDR) of the form:E^2f_1^2-k^2 f_2^2=m^2where f_1 and... | {
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6d73ccdae80ee42cf8cf650d2bd3ad49bb45476a | subsection | 3 | 10 | An adaptation of the decaying speed of sound mechanism | It is straightforward to adapt the calculation in to
the present, slightly different context (this was already done,
with a different motivation, by , ). Here we
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1a62f1542911829545ea799b67026477849a3168 | subsection | 4 | 10 | An adaptation of the decaying speed of sound mechanism | Under (REF ) modes start inside the horizon (set by c
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... |
10604e3e59a1178edc66b3a0758152f2d1b5241f | subsection | 5 | 10 | An adaptation of the decaying speed of sound mechanism | In the range
under study (-1/3<w<1) it requires \gamma >2, for example for a
radiation background it requires \gamma =3. | {
"cite_spans": []
} | 10.1088/0264-9381/25/20/202002 | 0807.1854 | DSR as an explanation of cosmological structure | [
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9df6f205e7dd354c879d4270575eb6afdf60c7bb | subsection | 6 | 10 | The associated DSR | What can we learn about DSR from this calculation?
Foremost we have constrained the dispersion relations of the
fluctuations. Specifically, if vacuum quantum fluctuations are
responsible for the structure of the Universe, then for all w
the DDRs should be of the form:\omega ^2-k^2(1+(\lambda k)^2)^2=m^2with \lambda \si... | {
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c9f3a2370dfc2fb7ee4925189d21fb914cbbf476 | subsection | 7 | 10 | The associated DSR | For example withU\circ (E,p)=(E,p(1+(\lambda p)^2))\;one gets the
non-linear representation of Lorentz transformations:E^{\prime }&=&\gamma [E-vp_x(1+(\lambda p)^2)]\\
p_x^{\prime }(1+(\lambda p^{\prime })^2)&=&\gamma [p_x (1+(\lambda p)^2) -v
E]\\
p_y^{\prime }(1+(\lambda p^{\prime })^2)&=&p_y (1+(\lambda p)^2) \\
p_z... | {
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} | 10.1088/0264-9381/25/20/202002 | 0807.1854 | DSR as an explanation of cosmological structure | [
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1a99e3bed34714e818a90daea4bbf20824f54fc5 | subsection | 8 | 10 | The associated DSR | This could be easily implemented by
choosing DDRs of the form:\frac{E^2-p^2(1+(\lambda p)^2)^2}{1-(L_{Pl}E)^2}=m^2\;.Although the DDRs are the
correct ones the ensuing HOD field theory doesn't comply with the
assumptions of the calculation. It is possible, however, that a
modified calculation could be carried out and l... | {
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} | 10.1088/0264-9381/25/20/202002 | 0807.1854 | DSR as an explanation of cosmological structure | [
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8b3e65d20a41488ae88f2f3e5ed83119d0a5ff4a | subsection | 9 | 10 | Conclusions | If the speed of light, seen as a function of the wavelength, has a pole
of degree 2 at the origin, then the simplest adaptation of the
varying speed of sound mechanism for structure formation leads
to scale-invariant fluctuations. Additional minimal technical
assumptions on DSR and its coupling to gravity have to be ma... | {
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2bc76a64bcd8b9784e28c6f01afbecebf6753200 | abstract | 0 | 16 | Abstract | We consistently incorporate Yang Mills matter fields into string corrected
(deformed), D=10, N=1 Supergravity. We solve the Bianchi identities within the
framework of the modified beta function favored constraints to second order in
the string slope parameter $\g$ also including the Yang Mills fields. In the
torsion, c... | {
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689ad20f56623151f63b6d084840371ee35ce9eb | subsection | 1 | 16 | Introduction | String corrections to quantum field theories are believed to
contribute Gauss Bonnet terms to the action. This invariant has
been studied in many different scenarios, for example Gauss-Bonnet
modified cosmology . These terms are introduced by hand
into the quantum gravity models. However it is also known that
Gauss-Bon... | {
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de94a2732d27e57acb46aed8fc8f061be5d3340f | subsection | 2 | 16 | The Solution | The Bianchi identities in Superspace are as follows:[[\nabla _{[A},\nabla _{B}\rbrace ,\nabla _{C)}\rbrace =0Here we have
extended the commutator in to include the Yang-Mills
field strengths, F_{AB}^{I}[\nabla _{A},\nabla _{B}\rbrace =T_{AB}{}^{C}\nabla _{C}+\frac{1}{2}R_{AB}{}^{de}M_{ed}+iF_{AB}^{I}t_{I}The t_{I} are ... | {
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8fbb842a8a40a0190128838c8c63211dc071428e | subsection | 3 | 16 | The Solution | We continue with this assumption.We have the H sector Bianchi identities with Yang-Mills fields as
follows :\frac{1}{6}\nabla _{(\alpha |}H_{|\beta \gamma \delta )}~-~\frac{1}{4}T_{(\alpha \beta |}{}^{M}
H_{M|\gamma \delta )}~=~-\frac{\gamma }{4}R_{(\alpha \beta |ef}R_{|\gamma \delta )}{}^{ef}
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28fe3656d5c60627a0c838d6b773f0b4992fbdb0 | subsection | 4 | 16 | The Solution | This has the effect of splitting the Bianchi
identities however we still have to be careful of cross terms. In
order to be cautious we will examine all contributions in the H
sector, in particular that for H^{(2)}{}_{ a \beta d}, for
thoroughness (see appendix). In all of the Bianchi identities we
encounter the spinor ... | {
"cite_spans": []
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fb877820afe53a7cd944a59346830dd39d560b8c | subsection | 5 | 16 | The Solution | We have the following modifications from :\nabla _{\gamma }T^{(0)}{}_{ef}{}^{\delta }=-\frac{1}{4}\sigma ^{mn}{}_{\gamma }{}^{\delta }R^{(0)}{}_{efmn}
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602a75d5d0df668689bde353f95febbf18ecf4aa | subsection | 6 | 16 | The Solution | Term by term we notice no
order \beta ^2 contributions occur. We also seek terms of the form
O(\gamma \beta ) and note that none appear other than those resulting
from the modification to A_{abc}. We also find H^{(2)}{}_{g \gamma d} as in , but we write it in a different way+\frac{i}{2}\sigma _{(\alpha \beta |}{}^{g}H^... | {
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b53e50a20a0a583211e58c066d1fd0892e4e2a5a | subsection | 7 | 16 | The Solution | We
also have, along with (33) of the first reference ,i\sigma ^{g}{}_{(\alpha \beta |}T^{(2)}{}_{|\gamma ) g}{}^{\delta } =
+2i\gamma \sigma ^{g}{}_{(\alpha \beta |} T^{(0)}{}^{ef}{}^{\delta }
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1f0a7409f6d17ad6bd98f1cc0fae3600d34df9ad | subsection | 8 | 16 | F Sector Bianchi Identities | We have seen that in the H and Torsion sectors the required minimal
Bianchi
identities have been fully satisfied by simply modifying the super current as
in equation (28). The fundamental identity (21) is then used to solve in each case.
This is the identity that enables such solutions to be obtained in the modified \b... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
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"hep-th"
] | 2,008 | en | Physics | [
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1f09afa14d374b07835855516584ab9601c56743 | subsection | 9 | 16 | F Sector Bianchi Identities | We propose
the following candidate and show that it works.F^{(2)}{}_{\alpha \beta }=-\frac{i\gamma }{12}\sigma ^{pqref}{}_{\alpha \beta }A^{(1)}{}_{pqr}F_{ef}At second order equation (29) becomesT^{(0)}{}_{(\alpha \beta |}{}^{\lambda }F^{(2)}{}_{|\gamma )\lambda }-\nabla _{(\alpha |}F^{(2)}{}_{|\beta \gamma )}
-i\sigma... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
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0bac64b6744920213bcd8496c8ffbb21f2d87b35 | subsection | 10 | 16 | F Sector Bianchi Identities | We also must use \begin{eqnarray}\sigma ^{pqref}{}_{(\alpha \beta |}\sigma _{e|\gamma )=~-~\sigma ^{pqref}{}_{\alpha |}\sigma _{e|\beta \gamma )}
}\end{eqnarray}Hence, we obtain
}\begin{eqnarray}-i\sigma ^{g}{}_{(\alpha \beta |}F^{(2)}{}_{|\gamma )g}+2i\gamma \sigma ^{g}{}_{(\alpha \beta |}\Omega ^{(1)}{}_{|\gamma )gef... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
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f67038ee8dade1526842e9ec35ed96a9306afefe | subsection | 11 | 16 | F Sector Bianchi Identities | We have
\end{eqnarray}\begin{eqnarray}\nabla _{(\alpha |}F_{|\beta )d}-T_{\alpha \beta }{}^{g}F_{gd}-T_{\alpha \beta }{}^{\lambda }F_{_|la d}+T_{(\alpha |d}{}^{g}F_{g|\beta )}+T_{(\alpha |d}{}^{\lambda }F_{\lambda |\beta )}=0\\
\end{eqnarray}and
\begin{eqnarray}\nabla _{\alpha }F_{bc} = \nabla _{[ b|}F_{\alpha |c]}+T_{... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
] | [
"hep-th"
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a226e7233201386b442095624a1e92390430c6ed | subsection | 12 | 16 | F Sector Bianchi Identities | We have
}}\begin{eqnarray}[\nabla _{\alpha },\nabla _{\beta }\rbrace =-~\frac{i\gamma }{12}\sigma ^{pqref}{}_{\alpha \beta }A^{(1)}{}_{pqr}[T_{ef}{}^{\gamma }\nabla _{\gamma }
+2
H^{(0)}{}_{ef}{}^{g}\nabla _{g}+\frac{1}{2}R^{(0)}{}_{ef}{}^{mn}M_{mn}+iF^{(0)}{}_{ef}{}^{I}t_{I}]\\~
\end{eqnarray}}But \begin{eqnarray}2 H^... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
] | [
"hep-th"
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fc8848570c47d07cb9244095f62b6f6ce41dd6a8 | subsection | 13 | 16 | F Sector Bianchi Identities | Our solution
to the Bianchi identities, obtrained within the framework of the
modified beta function favored constraints, holds to the second
order in the string slope parameter \gamma and includes also the
Yang Mills fields. We obtained as well a consistent
solution in the torsion, curvature and H sectors with a Yang ... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
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"hep-th"
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c6e8ae75bd8a0af2f12d778d0b8fbc8862095f84 | subsection | 14 | 16 | F Sector Bianchi Identities | Following a technique we developed in earlier papers, we also found a solution in the F sector and
gave an explicit formula for the modification induced in the commutator expression.
}\subsection *{Acknowledgements}
This work of has been supported in part by the European Community
Human Potential Program under contract... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
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cd7a78e33a99b81bd97efd44ca5cf40a34da0547 | subsection | 15 | 16 | F Sector Bianchi Identities | Also
there is no need to make modifications other than the adjustment of
A_{abc} in the H sector.
\end{eqnarray}}\begin{}{99}
\end{}\bibitem {10} Emilio Elizalde, Nikolaos Brouzakis, Nikolaos Tetradis, Eleftheria Tzavara,
Phys.Rev.D76:084029,2007;\\ Ishwaree P. Neupane: arXiv:0711.3234
[hep-th].
}\bibitem {5} S. Belluc... | {
"cite_spans": []
} | 10.1016/j.physletb.2008.07.004 | 0807.1855 | Inclusion of Yang-Mills Fields in String Corrected Supergravity | [
"S. Bellucci",
"D. O'Reilly"
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e0b70bdec56acb112609ff1c2079295ee5981b1b | abstract | 0 | 16 | Abstract | Kaon condensation may play an important role in the structure of hadronic
matter at densities greater than that of nuclear matter, as exist in the
interior of neutron stars. We present the results of the first lattice QCD
investigation of kaon condensation obtained by studying systems containing up
to twelve negatively... | {
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} | 10.1103/PhysRevD.78.054514 | 0807.1856 | Kaon Condensation with Lattice QCD | [
"William Detmold",
"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
] | [
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b63da426b06038fa082bcace1bd9911fa79322fe | subsection | 1 | 16 | Introduction | A key ingredient in determining whether supernovae evolve into
black-holes or neutron stars is the nuclear equation of state
(NEOS) . The NEOS depends upon the degrees of
freedom at a given density, which in turn are determined by the masses
of the various hadrons in the medium and their interactions. One
possibility i... | {
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8b9472345ed1800389da16798f97f28eb6c63e5d | subsection | 2 | 16 | Condensation of
Pseudo-Goldstone Bosons in | At LO in \chi PT, the Lagrange density describing the low-energy
dynamics of the pseudo-Goldstone bosons associated with the
spontaneous breaking of the approximate SU(3)_L\otimes SU(3)_R
chiral symmetry of QCD has the form{\cal L} & = &
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a11f3dbd3b46eaca76388c084df695e6578d802c | subsection | 3 | 16 | Condensation of
Pseudo-Goldstone Bosons in | (REF ) become covariant derivatives
, , \partial _\mu \Sigma \rightarrow D_\mu \Sigma = \partial _\mu \Sigma + i \delta _{\mu 0} \mu _I
\left[\hat{I}_z , \Sigma \right]\ +\ i \delta _{\mu 0} \mu _S \left[\hat{S} , \Sigma \right]
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a26be1a8d8d544ea73b546b1f0191ac716f46ed8 | subsection | 4 | 16 | Condensation of
Pseudo-Goldstone Bosons in | (REF ) is that, given the relatively
large expansion parameters of SU(3)_L\otimes SU(3)_R \chi PT, it is
natural to expect that there will be corrections to \rho _{K^-}, and
hence to the dependence of \mu _{K^-} on \rho _{K^-}, at the level
of m_K^2/\Lambda _\chi ^2 \sim 25\%, where \Lambda _\chi \sim 1 GeV is
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"William Detmold",
"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
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73b50af92e605590ee5a792558dac16711b976c6 | subsection | 5 | 16 | Multi-Meson Energies in a Finite Volume | In recent works , , , the
analytic volume dependence of the ground-state energy of n identical
bosons in a periodic volume of side L has been computed to {\cal O}(L^{-7}), extending the classic results of Bogoliubov
and Lee, Huang and Yang . The resulting shift in
ground-state energy of n particles of mass M due to th... | {
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"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
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b292d678112a935a4dbd55e797e62db941e2a6b1 | subsection | 6 | 16 | Multi-Meson Energies in a Finite Volume | (REF ) are&{\cal I}\ =\ -8.9136329\,, &{\cal J}\ =\ 16.532316\,, \qquad \qquad {\cal K}\ = \ 8.4019240\,,
\\
&{\cal L}\ = \ 6.9458079\,, &{\cal T}_0\ = -4116.2338\,,
\qquad \qquad {\cal T}_1\ = \ 450.6392\,, &and ^nC_m=n!/m!/(n-m)!. The three-body contribution to the
energy-shift given in eq. (REF ) is represented by t... | {
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"... | 10.1103/PhysRevD.78.054514 | 0807.1856 | Kaon Condensation with Lattice QCD | [
"William Detmold",
"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
] | [
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8d00e0140c7488aa8fd3f3140d0d78f33424e43b | subsection | 7 | 16 | Methodology and Details of the Lattice Calculation | The computation in this paper uses the mixed-action lattice QCD scheme
developed by LHPC , and described
fully in Ref. where multi-pion systems are
investigated in detail. The present calculations were performed
predominantly with the coarse,
asqtad-improved , MILC
configurations generated with rooted staggered sea
qu... | {
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"Andre Walker-Loud"
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994e083b37a48207c0643693c92e79dd335024e5 | subsection | 8 | 16 | Methodology and Details of the Lattice Calculation | This indicates that differences in the iterative (conjugate
gradient) propagator inversion for the two types of boundary
conditions are becoming significant.In order to determine the interaction energy in multi-meson systems,
both the single-meson, C_{1}(t), and n-meson, C_{n }(t),
correlation functions were computed, ... | {
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} | 10.1103/PhysRevD.78.054514 | 0807.1856 | Kaon Condensation with Lattice QCD | [
"William Detmold",
"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
] | [
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2e1bb19544475060b76e59033ed2588ad9b0558f | subsection | 9 | 16 | Methodology and Details of the Lattice Calculation | The interaction energy is related to the
n-particle energy eigenvalues, E_{n K}, and the kaon mass,\Delta E_{n K}\ \equiv \ E_{n K} - n\ m_K\, .A comprehensive review of the lattice techniques we have used in this
work can be found in Refs. , .
[Figure: The energy-differences (in lattice units), b\ \Delta E_{n K}, for ... | {
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d6909cc7f978811238ef6a82c867e5d78737c2e6 | subsection | 10 | 16 | Three-Meson interactions | The three-meson interactions extracted from our calculations, m_{\pi }
f_{\pi }^4\ \overline{\overline{\eta }}^L_{3\pi ^-} and m_{K} f_{K}^4\ \overline{\overline{\eta }}^L_{3K^-}, are shown in
Fig. REF , in units of the NDA estimate,
\overline{\overline{\eta }}^{L,(NDA)}_{3\pi ^-} = 1/(m_\pi f_\pi ^4)
and \overline{\ov... | {
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133788a457699cf5679c68f36239816671ed7e3b | subsection | 11 | 16 | The Equation of State and Chemical
Potentials | The energy of the n K^- system as a function of volume and of the
number of K^-'s is given explicitly in eq. (REF ) in
the large-volume expansion. From the equation of state, the K^-
chemical potential is\mu _{K^-} &=& \left.\frac{d E_{n K}}{d n}\right|_{V={\rm const}}
\ \ \ ,and can be constructed analytically from eq... | {
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4cba3750c494afc6d9cde4ff5d3bcf6bb4c71aa5 | subsection | 12 | 16 | The Equation of State and Chemical
Potentials | The lighter (green) rectanglesare obtained from the lattices with spatial extent L\sim 2.5~{\rm fm}, while the darker (red) rectangles are obtained from thelattices with L\sim 3.5~{\rm fm}. The dashed curves correspond totree-level \chi PT. The darker solid curve corresponds to theanalytic expression for the energy of ... | {
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e502dceb3c5a6bdeca11561f347785e2394713f7 | subsection | 13 | 16 | Conclusions | Kaon condensation may play an important role in the evolution of
supernovae . The theoretical analysis of the
condensation mechanism presently relies upon chiral perturbation
theory to determine both the interactions of kaons with the baryonic
matter and the kaon self-interactions. We have performed the first
Lattice Q... | {
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"Kostas Orginos",
"Martin J. Savage",
"Andre Walker-Loud"
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d0a14a2400e3598c4eb83b9c576e5a4f21b76eb4 | subsection | 14 | 16 | Effects of temporal boundaries | For temporally periodic gauge configurations and (anti-) periodic
quark propagators (or the P\pm A propagators), we are in a position
to account for the effects of the temporal boundary condition. As is
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584ed4091d3f835e42cf2c5ca8b2f9cf71c3c4e2 | subsection | 15 | 16 | Effects of temporal boundaries | Fig. REF
compares the two extractions of the \pi ^- and K^- chemical
potentials for the fine ensemble.In the case of a Dirichlet boundary in time, as employed in the
calculation of quark propagators on the coarse L\sim 2.5~{\rm fm}
ensembles, the nature of the states reflected from the walls is less
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c0c0387768d8058c8c4eacbce32fff48fad44c55 | abstract | 0 | 51 | Abstract | We have studied the structure of $^4$He droplets doped with magnesium atoms
using density functional theory. We have found that the solvation properties of
this system strongly depend on the size of the $^4$He droplet. For small drops,
Mg resides in a deep surface state, whereas for large size drops it is fully
solvate... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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1fb72820e891d185e43437c6c209beb8ecc6afe7 | subsection | 1 | 51 | Introduction | Optical investigations of atomic impurities in superfluid helium
nanodroplets have drawn considerable attention in recent
years,, as the shifts of the electronic transition
lines (atomic shifts) are a very useful observable to determine the
location of the foreign atom attached to a helium drop. In
this context, the st... | {
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"d... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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f8ee24db6979a669f5b92095493064aaf6a1c128 | subsection | 2 | 51 | Introduction | We calculate the absorption spectrum of a Mg atom attached to small
and large drops, finding a good agreement with the experiments
for the latter. We discuss in a qualitative way the effect of
the impurity angular momentum on the
electron-impact ionization yield and on the absorption spectrum.
We also address the struc... | {
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0d04b1ed70e75b9000e68cb521eb99c568917fd6 | subsection | 3 | 51 | Theoretical approach | In recent years, static and time-dependent density functional
methods, , have become increasingly popular to study
inhomogeneous
liquid helium systems because they provide an excellent compromise
between accuracy and computational effort, allowing to address
problems inaccessible to more fundamental approaches.
Althoug... | {
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"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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01464d8e18d853b86b63fb1d3e16a785f1237932 | subsection | 4 | 51 | Theoretical approach | Specifically, we minimize the expressionE + \frac{\lambda _C}{2} [{\cal Z} - {\cal Z}_0]^2 \; ,where
{\cal Z} is the average distance between the
impurity and the center of mass of the helium droplet{ \cal Z} =
\int d \mathbf {r}^3 \, z \, |\Phi (\mathbf {r})|^2 -
\frac{1}{N} \int d \mathbf {r}^3 \, z \,\rho (\mathbf {... | {
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} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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c33e1a0eb279f2f944c9fc8765cfdfcc6d23d44c | subsection | 5 | 51 | Theoretical approach | The main
reason
is that these coordinates allow to use fast Fourier transformation
techniques to efficiently compute the convolution integrals
entering the definition of {\cal E}(\rho ), i.e. the mean field helium
potential and the coarse-grained density needed to compute the
correlation term in the He density function... | {
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... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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f1504f04ca74d945bcbf79073a860f3187a05aa6 | subsection | 6 | 51 | Structure and energetics of Mg-doped helium nanodroplets | Equations (REF -REF ) have been solved for \lambda _C=0 and
several N values, namely N=30, 50, 100, 200, 300, 500, 1000, and
2000. This yields the ground state of the drop-impurity complex, and
will allow us to study the atomic shift for selected cluster
sizes.Figure REF shows the energy of a magnesium atom in a drop,
... | {
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... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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c9bfbf20e475024b8e8d545c517589a7c2ecd132 | subsection | 7 | 51 | Structure and energetics of Mg-doped helium nanodroplets | This causes changes in the total energy of the system
by less that 1 %, but has a large effect on the value of the
atomic shift. We will address this important issue, and its consequencies
on the computed spectral properties of Mg@He_N in the next Sections.Figure REF shows the helium configurations
for the four station... | {
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0bf3467710600ae83c747681e410e1e2d2cf37c4 | subsection | 8 | 51 | Excitation spectrum of a Mg atom attached to a | Lax method offers a realistic way to study the
absorption spectrum of a foreign atom embedded in liquid drops. It
makes use of the Franck-Condon principle within a semiclassical
approach, and it has been widely employed to study the absorption
spectrum of atomic dopants attached to ^4He drops, see e.g. Ref.
Her08 and r... | {
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... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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83fd9b8d5bff93af8c59a4c487a263de544966ff | subsection | 9 | 51 | Excitation spectrum of a Mg atom attached to a | Taking into account that
\mathrm {e}^{-\frac{it}{\hbar }\mathcal {H}_{gs}}|\Psi ^{gs}\rangle =
\mathrm {e}^{-it\omega _{gs}}|\Psi ^{gs}\rangle and projecting on
eigenstates of the orbital angular momentum of the excited electron
| m \rangle one obtainsI(\omega ) \propto |D_{\mathrm {ge}}|^2\sum _{m}\int \mathrm {d}t~\m... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
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"Marti Pi",
"Francesco Ancilotto"
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9545b9aea6941310791c20a2597b74985875106d | subsection | 10 | 51 | Excitation spectrum of a Mg atom attached to a | In this case we obtain the expressionI(\omega )&\propto &
\sum _{m}\int \mathrm {d}^n[\alpha ]\int \mathrm {d}^3\mathbf {r}~|\psi _X^{\mathrm {gs}}(\mathbf {r},[\alpha ])|^2
\delta (\omega +\omega ^{\mathrm {gs}}_X-V^{\mathrm {ex}}_m(\mathbf {r},[\alpha ])/\hbar )
\\
&=& \hbar \int \mathrm {d}^n[\alpha ] \int _{\Omega ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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e9036b8d3ac238aece27a36a9ff3093cebb9e61b | subsection | 11 | 51 | Excitation spectrum of a Mg atom attached to a | In cartesian coordinates, and assuming that the He-impurity spin-orbit
interaction is negligible for magnesium, the eigenvalues of the
symmetric matrixU_{ij}(\mathbf {r},[\alpha ])=
\int \mathrm {d}^3\mathbf {r^{\prime }}\rho (\mathbf {r^{\prime }}+\mathbf {r},[\alpha ])
\left\lbrace V_\Pi (r^{\prime })\delta _{ij}+[V_... | {
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11be193b632fb7090dca885c3d73758f27ab660d | subsection | 12 | 51 | Results for the absorption spectrum of magnesium atoms | The problem of obtaining the atomic shift of magnesium in a helium
drop has been thus reduced to that of the dopant in the 3D trapping
potentials corresponding to the ground state and P excited states.
We consider first the [\alpha ]=0 case (i.e. no zero-point
deformations of the helium cavity hosting the impurity). Th... | {
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"raw": "F. Stienkemeier, J. Higgins, C. Callegari, S. I. Kanorsky, W. E. Ernst, and G. Scoles, Z. Phys. D 38, 253 (1996).",
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"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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8f4a8326401e71a80e880a10af984ff7b032c687 | subsection | 13 | 51 | Results for the absorption spectrum of magnesium atoms | The surface location of Ca, Sr, and Ba in these drops has been
further confirmed by DFT calculations.,
It is also interesting to recall that
LIF experiments on Ca atoms in liquid ^4He and ^3He have found a broad line in the
region of the 4s4p ^1P_1 \leftarrow 4s^2 ^1S_0 transition with
no apparent splitting, contraril... | {
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"Marti Pi",
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59ac374ab9784767c328314786eecfa55c20f671 | subsection | 14 | 51 | Thermal motion and angular momentum effects | To describe the displacement of the helium bubble inside the
droplet, we have fitted the E({\cal Z}_0) curve
of the Mg@^4He_{1000} system to a parabola, and have
obtained the excitation energy
\hbar \omega for this 3D isotropic harmonic oscillator.
The hydrodynamic mass of the impurity atom has been estimated by
its bu... | {
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} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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dfe0d0cdf8987510bca8e0397f8ac45a61d406a8 | subsection | 15 | 51 | Thermal motion and angular momentum effects | The radial probability density w
isw(R) = \frac{\mathrm {d}W}{\mathrm {d}R} = Q^{-1} R^2 \sum _{n\ell }
(2\ell +1) e^{-E_{n\ell }/k_BT} |\phi _{n\ell }(R)|^2 \; .This expression has been evaluated for N=50 and 1000 by solving the
Schrödinger equation for the Hamiltonian Eq. (REF )
to obtain the orbitals \phi _{n\ell } ... | {
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... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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c6dbf74fd91fabc69731bed4253277b5c9ac6614 | subsection | 16 | 51 | Thermal motion and angular momentum effects | REF shows that for Mg@^4He_{1000}
and Mg@^4He_{2000}, if thermal motion is taken into account and
the impurity retains some of the pick-up angular
momentum, the maximum density probability of Mg is at \sim 15 Å
beneath the drop surface in both cases. To obtain it, we have taken for
M^* the bulk value 40 a.u. As seen fr... | {
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"raw": "The mean value \\langle L^2\\rangle is calculated as Q^{-1}Tr\\left(L^2e^{-{\\cal H}/k_BT}\\right)=Q^{-1}\\sum _{n\\ell }\\ell (\\ell +1)(2\\ell +1)e^{-E_{n\\ell }/k_BT} in the quantal approach, and... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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1d171d730874c55f298ac14d9ae56c5bd99ca10e | subsection | 17 | 51 | Homogeneous width from shape deformations of the helium bubble | We have shown that for large drops, the Mg atom is fully solvated and
its thermal motion only produces small changes in the absorption shift.
This allows us to decouple the effect of the translational motion
of the helium bubble on the absorption line, from that of
its shape fluctuation. Moreover, we can address shape ... | {
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"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
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06a9d0e0172c98403b309db61cad8faa888f4703 | subsection | 18 | 51 | Homogeneous width from shape deformations of the helium bubble | The dipole mode amplitude \alpha _{1\mu } is absent since, for an incompressible fluid,
it corresponds to a translation of the bubble, and this has been considered in the
previous Subsection.If S[R(\Omega )] represents the bubble surface and \sigma the
surface tension, the energy for a large drop can be written asE&=& ... | {
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"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
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6bb9c186b784d975d9566f4c317c37ab655dd714 | subsection | 19 | 51 | Homogeneous width from shape deformations of the helium bubble | Wil64,Rin80,Kin96,Fow68,
namely, if \rho _0(r) is the helium spherical ground state density,
deformations are introduced as
\rho (\mathbf {r},t)=\rho _0[R(\mathbf {r},t)]\,{\cal K}^{-1},
withR(\mathbf {r},t)=r+\alpha _0(t)+
\sum _{\lambda =2}^\infty \sum _{\mu =-\lambda }^\lambda \alpha _{\lambda \mu }(t)Z_{\lambda \mu... | {
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"Alberto Hernando",
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"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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4e9d9bfc4eed86aa6039ed385c2f070f7f15a4c2 | subsection | 20 | 51 | Homogeneous width from shape deformations of the helium bubble | This equation represents the Hamiltonian of a set of
uncoupled harmonic oscillators, whose quantization yields a ground state to
whose energy each mode contributes with
\varepsilon _{\lambda \mu } =\frac{1}{2}\hbar \omega _\lambda , with
\omega _\lambda =\sqrt{
\frac{4 \pi E^{(2)}_\lambda }{(2\lambda +1)M^*_{\lambda }}... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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466de31cd76611921784cd095e50cf230868442d | subsection | 21 | 51 | Homogeneous width from shape deformations of the helium bubble | The details are given in Appendix B, where we show that
the breathing mode affects the shift and shape of the line, whereas
quadrupole modes only affect the shift.Consequently, we restrict in Eq. (REF ) the deformation
parameters needed to properly describe the homogeneous broadening of the absorption
dipole line, name... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
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a20d9f01d8a53af14504f8f270cda1f1522846e5 | subsection | 22 | 51 | Homogeneous width from shape deformations of the helium bubble | (REF ) as probability density. Next,
for each set we have found the eigenvalues
V^{\mathrm {ex}}_m(r,\alpha _0^i,[\alpha _2]^i)
of the U_{i,j} matrix that define the potential energy surfaces
and have used a trapezoidal rule to evaluate the integralsI_m(\omega ,[\alpha ]^i)=4\pi \int \mathrm {d}r~|r~\Phi (r,\alpha _0^i... | {
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"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
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9cc33eec53ae7973e46494aaca08a69b9c355264 | subsection | 23 | 51 | Two magnesium atoms attached to a | The attachment of magnesium atoms in ^4He droplets has been recently
addressed using resonant two-photon-ionization.
In particular, the authors of Ref. Prz07 have obtained the
absorption spectrum for drops doped with different, selected numbers of Mg atoms.
From their measurements it appears that two main features
cont... | {
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254,
235079,
7,
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133291,
101935,
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177,
72350,
135179,
52... | [
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cc4eaf02dd0c68795ef9b4ec2f31872a0fdaffd8 | subsection | 24 | 51 | Two magnesium atoms attached to a | To obtain the structure of two Mg atom in a ^4He drop, we have minimized the
energy of the system written asE[\Psi , \Phi _1,\Phi _2] &=& \frac{\hbar ^2}{2\,m_{He}}
\int \mathrm {d}^3 \mathbf {r}\, |\nabla \Psi (\mathbf {r}\,)|^2 +
\int \mathrm {d}^3 \mathbf {r} \, {\cal E}(\rho )
\\
&+&
\frac{\hbar ^2}{2\,m_{Mg}}\int ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
113054,
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... | |
35a836f0d9c82d7ee14d4d4d37d9e995c5e53505 | subsection | 25 | 51 | Two magnesium atoms attached to a | The correction to the leading term of the
long-range dispersion interaction, -C_6/r^6,
due to three-body correlation effects
can be written to first order as
-C_6(1-2\pi n\alpha /3)/r^6, \alpha being the static polarizability
of the host fluid (\alpha _{He}=1.39\,a_0^3).
Such correction is of the order of only 1% in ou... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 272,
"openalex_id": "",
"raw": "M. J. Renne and B. R. A. Nijboer, J. Phys. C 6, L10 (1973).",
"source_ref_id": "fa09374161d1188d5cb66df7218d04e2a43f9ea3",
"start": 0
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{
"arxiv_id": "",
"doi": "",
... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.17407226... | |
1713ae7c60508f7b6ec6863ff5d85ec2ab1fe155 | subsection | 26 | 51 | Two magnesium atoms attached to a | Note that the energy of the Mg+Mg system increases as d does because
the two Mg atoms in a drop form a state more bound than that
of a pure drop with the impurities well apart.Since the height of the barrier is larger than
the experimental temperatures, T\sim 0.4\,K, two solvated
Mg atoms will not easily merge into a d... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 836,
"openalex_id": "",
"raw": "M. Lewerenz, B. Schilling, and J.P. Toennies, J. Chem. Phys. 102, 8191 (1995).",
"source_ref_id": "6bbc1da6c8408b929cbe52f96ca5681fb987183e",
"start": 583
},
{
"arxiv_id": ... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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8e3ab6d4afee1842a802b173074c7c5bc05a5fa9 | subsection | 27 | 51 | Two magnesium atoms attached to a | (REF ). We are led to conclude that
there is no barrier in the case of liquid ^3He.
The configuration corresponding to the closest d we have calculated
is shown in Fig. REF .We are now in the position to determine the effect of these
weakly-bound systems on the LIF
and R2PI experiments on ^4He droplets containing more ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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d6d3b2176f1344ebdaf1744c562f602a188d85ac | subsection | 28 | 51 | Summary | We have obtained, within DFT, the structure of ^4He droplets doped with Mg atoms
and have discussed in detail the magnesium solvation properties.
In agreement with previous DMC calculations,,
we have found that Mg is not fully solvated in small ^4He drops, whereas
it becomes fully solvated in large droplet.As a conseq... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 309,
"openalex_id": "",
"raw": "M. Mella, G. Calderoni, and F. Cargnoni, J. Chem. Phys. 123, 054328 (2005).",
"source_ref_id": "f6ffbd6986b816ef7e56f1fd95e9f6d78c87a86d",
"start": 146
},
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"arxiv_id": "",... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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44d05c23f988e22b36c9fa2923afdf690751bc49 | subsection | 29 | 51 | Summary | We
attribute to this static quadrupole moment the origin of the low energy
peak
in the absorption line, and confirm the suggestion made by the Rostock
group that the splitting of the absorption line, rather than being
due to a dynamical (Jahn-Teller) deformation of the helium bubble, is
due to the presence of more than... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 361,
"openalex_id": "",
"raw": "A. Przystawik, S. Göde, J. Tiggesbäumker, and K-H. Meiwes-Broer, contribution to the XXII International Symposium on Molecular Beams, University of Freiburg (2007); A. Przystawik, S. Göde, T. Döppne... | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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c49c7811058e7c28875f0761ee2c62c2a36b196f | subsection | 30 | 51 | Body | In this Appendix we obtain the energy of the doped drop up to
second order in the deformation parameters and the hydrodynamic
mass of the helium bubble. To this end, the helium
order parameter and Mg wave function are expressed as
\Psi (\mathbf {r},t)=\sqrt{\rho (\mathbf {r},t)}
\exp [i \frac{m_{He}}{\hbar }S(\mathbf {... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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4dd92981049c7aeb33559ec0892124aef71d9e1e | subsection | 31 | 51 | Body | Neglecting the velocity-dependent terms of the Orsay-Trento
functional that mimic backflow effects,
the total energy of the system is written as&E& =
\frac{1}{2}m_{He}
\int \mathrm {d}^3 \mathbf {r}\, \rho (\mathbf {r},t)|\nabla S(\mathbf {r},t)|^2 +
\frac{1}{2}m_{Mg}\int \mathrm {d}^3 \mathbf {r}\,
|\Phi (\mathbf {r},... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1378,
"openalex_id": "",
"raw": "F. Dalfovo, A. Lastri, L. Pricaupenko, S. Stringari, and J. Treiner, Phys. Rev. B 52, 1193 (1995).",
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"start": 0
}
]
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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5829187a1490fcdec66ebabe09d6d0ea6a6e0738 | subsection | 32 | 51 | Body | Thus,E =T+V= \int \mathrm {d}^3 \mathbf {r}
\left\lbrace t[\rho ,S,|\Phi |,\varphi ]+v[\rho ,|\Phi |]\right\rbrace \; .In the adiabatic approximation, the dynamics of the system
requires the following steps:
i) introduce a set of collective variables (or deformation
parameters) [\alpha (t)]) that define the helium dens... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
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d357112aa4cb45b095fea132f55c8063a39afcd0 | subsection | 33 | 51 | Body | To first order, the density can be written as\rho (\mathbf {r},t)\simeq \rho _0(r)+\rho _0^{\prime }(r)
\sum _{\lambda =0}^\infty \sum _{\mu =-\lambda }^\lambda \alpha _{\lambda \mu }(t)Z_{\lambda \mu }(\widehat{r}) \; ,where from now on, the prime will denote the derivative of the function
with respect to its argument... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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a88b35d881f37d2e4d76228e841507c8013a51dd | subsection | 34 | 51 | Body | \\
&&\left.-4\pi \int \mathrm {d}r~|r~\Phi (r,\alpha _0)|^2
\delta ^{\prime }[\omega +\omega (\alpha _0)-V^{\mathrm {ex}}_i(r,\alpha _0)/\hbar ]\frac{1}{\hbar }\epsilon _{\lambda \mu }^i(r)
\right\rbrace \alpha _{\lambda \mu } \; ,where \epsilon _{\lambda \mu }^i(r)=\left.\partial V^{\mathrm {ex}}_i(r)/\partial \alpha ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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055e954c1e2cc9b6383bfca116d11f44c1a76e4b | subsection | 35 | 51 | Body | Taken into account that the PES have a stationary point
at r=0 due to the spherical geometry –the first order term is zero– we can
safely stop the expansion at the zeroth order term,
since the wave function \Phi (r,\alpha _0) is very narrow&&I(\omega ,[\alpha ])\simeq I(\omega ,\alpha _0)
\\
&&-4\pi \sum _{i}\sum _{\la... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.04022216796875,
... | |
e908c2ba2cfe53a5e19bd65831395dace3a723a6 | subsection | 36 | 51 | Body | \\
&&\left.
+\frac{1}{\sqrt{3}}[V_\Sigma (r^{\prime })-V_\Pi (r^{\prime })]\left(
\begin{array}{ccc}
-\frac{1}{\sqrt{3}}Z_{20}(\hat{r})+Z_{22}(\hat{r})&Z_{2-2}(\hat{r})&Z_{21}(\hat{r})\\
Z_{2-2}(\hat{r})&-\frac{1}{\sqrt{3}}Z_{20}(\hat{r})-Z_{22}(\hat{r})&Z_{2-1}(\hat{r})\\
Z_{21}(\hat{r})&Z_{2-1}(\hat{r})&\frac{2}{\sqr... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
18991,
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0.002... | |
42936452484a597960737e9baf9ab5dbdfdf5f0f | subsection | 37 | 51 | Body | (REF ).
At variance with the approach of Ref. Kin96, where the
above matrix is approximated by its diagonal expression, implying that
only the \mu =0 and 2 components of the quadrupole deformation are
considered, our approach incorporates all five components.The relation between the eigenvalues \lambda _i(r,[\alpha ]) ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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... | |
4526fa0f4ac60993789726f8a8183475d7a432f6 | subsection | 38 | 51 | Body | If \beta _0\gg \beta _{\lambda >0} \, (\Rightarrow \gamma _0\gg \gamma _{\lambda >0}), we can
compute the line shape to first order in \beta _{\lambda >0}
(\gamma _{\lambda >0}); in analogy with Eq. (REF ) we write&&I(\omega ,[\beta ])\simeq I(\omega ,\beta _0)
\\
&&-4\pi \sum _{i}\sum _{\lambda =1}^\infty \int \mathrm... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.04522705078125... | |
9d3dc77d3b1d68966c4b803a6e9895b0958845d1 | subsection | 39 | 51 | Body | (REF ), to first
order this matrix becomesU(\mathbf {r},[\beta ],[\alpha ])&\simeq &
U(r,\beta _0,\alpha _0)
+\frac{4\pi }{5\sqrt{3}}
\int \mathrm {d} r^{\prime } \, r^{\prime 2} \, \beta _0\,\tilde{\rho }^{\prime }_0(r^{\prime },\alpha _0)
[V_\Sigma (r^{\prime })-V_\Pi (r^{\prime })]
\\
&&\times \left(
\begin{array}{c... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.0279998779... | |
9a73885c081f6eb7c4e9588348aa175f0bee708b | subsection | 40 | 51 | Body | Hin03;circles connected with a dotted line, fromRef. Par01.][Figure: Energy of the Mg atom as a function of the number of atomsin the drop, obtained using the Mg-He potential of Ref.Hin03 (squares). The values given in Ref.Her07are also displayed (dots). The lines are drawn to guide the eye.][Figure: (Color online)Bott... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0... | |
2c35ff4eee9a7bd1c19aaca444d5a5eced0125b5 | subsection | 41 | 51 | Body | The solidline represents the probability density of the N=10\,000 drop.][Figure: (Color online)Total absorption spectrum of one Mg atom attached to^4He_{1000} in the vicinity of the3s3p ^1P_1 \leftarrow 3s^2 ^1S_0 transition.The line has been decomposed into its two \Pi and one \Sigma components, the former one is the ... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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27f8a86264736a3258d2d280148d41e14c57a089 | subsection | 42 | 51 | Body | Also indicated is the corresponding wavelength \lambda . The value of thetransition energy in the gas phase is 35 051 cm^{-1}.] | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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a83ee35c04e1f1ee2277fc1e3e9473f421ff53d7 | subsection | 43 | 51 | Impurity wave function | To first order, the wave function |\Phi (\mathbf {r},[\alpha ])| is
written as|\Phi (\mathbf {r},t)|\simeq \Phi _0(r)+
\sum _{\lambda =0}^\infty \sum _{\mu =-\lambda }^\lambda \alpha _{\lambda \mu }(t)\Phi ^{(1)}_{\lambda \mu }(\mathbf {r})\; .The amplitudes \Phi ^{(1)}_{\lambda \mu }(\mathbf {r}) are determined
in fir... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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c3078795846c0295bba2913cbe7561e2a8e50f23 | subsection | 44 | 51 | Impurity wave function | We obtain\Phi ^{(1)}_{\lambda \mu }(\mathbf {r})&=&\sum _{n\ell m}
\frac{\langle \Phi _0|U^{(1)}_{\lambda }Z_{\lambda \mu }|\Phi _{n\ell }Z_{\ell m}\rangle }{\varepsilon _0-\varepsilon _{n\ell m}}
\Phi _{n\ell }(r)Z_{\ell m}(\widehat{r})
\\
&=&\left[\sum _{n}\frac{4\pi }{2\lambda +1}
\frac{\langle \Phi _0|U^{(1)}_{\lam... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.07525634... | |
67e751d649d97896bba8c8a23a37c562cfafbb32 | subsection | 45 | 51 | Impurity wave function | We can evaluate the second order contribution to the collective potential energy asV^{(2)}&=&\sum _{\lambda \mu }\sum _{\lambda ^{\prime }\mu ^{\prime }}\frac{1}{2}\int \mathrm {d}^3\mathbf {r}
\left.\frac{\partial ^2 v}{\partial \alpha _{\lambda \mu }\partial \alpha _{\lambda ^{\prime } \mu ^{\prime }}}\right|_{\rho _... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.... | |
a131af78a383ae9b46251f26077e0995cf27df35 | subsection | 46 | 51 | Velocity field potentials | Introducing the expansion
S(\mathbf {r},t)\equiv \sum _{\lambda \mu }\dot{\alpha }_{\lambda \mu }(t) \tilde{S}_{\lambda }(r) Z_{\lambda \mu }(\widehat{r}),
where the dot denotes the time-derivative,
the continuity equation for the liquid helium is, to first order,-\sum _{\lambda \mu }\dot{\alpha }_{\lambda \mu }~
Z_{\l... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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0.09... | |
510445d28f566db489a1d32bbfca83a583318a6c | subsection | 47 | 51 | Velocity field potentials | (REF ) reduces to the radial part
of the Laplace equation0=\frac{\mathrm {d}^2\tilde{S}_{\lambda }}{\mathrm {d}r^2}+
\frac{2}{r}\frac{\mathrm {d}\tilde{S}_{\lambda }}{\mathrm {d}r}
-\frac{\lambda (\lambda +1)}{r^2}\tilde{S}_{\lambda }whose general solution is\tilde{S}_{\lambda }(r)=A_\lambda ~r^\lambda +\frac{B_\lambda... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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cfb1b2b531205cee3eaa1c23adce5fc28015e4fc | subsection | 48 | 51 | Velocity field potentials | (REF ) is obtained as\tilde{S}_\ell (r)=
\tilde{S}_\lambda ^\mathrm {inh}(r)+C\tilde{S}_\lambda ^{\mathrm {h}}(r)withC=-\frac{
1+\left.
\frac{\mathrm {d}\tilde{S}_{\lambda }^{\mathrm {inh}}}{\mathrm {d}r}\right|_{r_i}
}{
\left.\frac{\mathrm {d}\tilde{S}_{\lambda }^{\mathrm {h}}}{\mathrm {d}r}\right|_{r_i}
} \; .The fie... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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575fc198fe0c341edb3c568d86d9191dfaec0969 | subsection | 49 | 51 | Kinetic energy | Once the velocity fields S(\mathbf {r},t) and \varphi (\mathbf {r},t)
have been determined, the collective kinetic energy can be easily
calculated to second order in the collective parameters:T&=& \int \mathrm {d}^3 \mathbf {r}~t[\rho ,S,|\Phi |,\varphi ]
\\
&=&
\frac{1}{2}m_{He}\int \mathrm {d}^3 \mathbf {r}\,
\rho (\... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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4d48bdaefb5740750697250a3c0eef4fc1097e0e | subsection | 50 | 51 | Kinetic energy | \\
&&\left.+
\frac{1}{2}m_{Mg}
\int \mathrm {d}r~r^2~|\Phi _0(r)|^2~\left[
\left(\frac{\mathrm {d}\tilde{\varphi }_{\lambda }}{\mathrm {d}r}\right)^2+
\frac{\lambda (\lambda +1)}{r^2}\tilde{\varphi }_{\lambda }^2
\right]
\right\rbrace \sum _{\mu =-\lambda }^\lambda \dot{\alpha }_{\lambda \mu }^2
\\
&\equiv &\frac{1}{2}... | {
"cite_spans": []
} | 0807.1857 | Magnesium doped helium nanodroplets | [
"Alberto Hernando",
"Manuel Barranco",
"Ricardo Mayol",
"Marti Pi",
"Francesco Ancilotto"
] | [
"cond-mat.other"
] | 2,008 | en | Physics | [
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