text: 4 FIG. 3: Same as Figure 2 but for the models with the trun- FIG. 4: Same as Figure 2 but for the models with the trun- cated icosahedron (n = 60) grid. Solid, dashed, dot-dashed cated icosahedron (n = 60) grid and poloidal perturbations. and dotted lines correspond to the cases with f = 1 (purely Thick dot-dashed, thick solid and thick dashed lines corre- pl (cid:104) (cid:105) poloidal), 0.5, 0.1 and 0 (purely toroidal), respectively. spond to the cases with ∆B = 0.1, 0.2 and 0.4, respectively. ∝ − Thin lines are power laws N(>E) E1 γ with γ = 1.44 (dot-dashed), 1.57 (solid) and 1.74 (dashed). tron star. On the other hand, the distributions of the truncated icosahedron model and regular dodecahedron model are similar at low energies, where ﬁnite-size eﬀects are negligible. We show the dependence on the amplitude of perturba- (cid:104) (cid:105) tion ∆B in Figure 4, where the cumulative distributions are plotted for the models with the truncated icosahedron (n = 60) grid and poloidal perturbations. We ﬁnd that (cid:38) the distributions with E 3E are well ﬁtted with min power laws. The power-law index, γ, of the diﬀerential ∝ − distributions dN E γdE is related to the power-law ∝ − ﬁt of the cumulative distributions as N(>E) E1 γ. − According to the observations of SGR 1806 20 and SGR 1900+14, the energy distributions of bursts have been ﬁt- ∼ ted with γ = 1.43-1.76 [7, 8] and γ 1.55 [9]. In our CA (cid:104) (cid:105) simulation, γ depends on the value of ∆B and it is 1.44, (cid:104) (cid:105) 1.57 and 1.74 for ∆B = 0.1, 0.2 and 0.4, respectively. Here, Kolmogorov-Smirnov probabilities of the ﬁtting in the range of 3E < E < 100E are >0.999, >0.999 (cid:104) min(cid:105) min and 0.964 for ∆B = 0.1, 0.2 and 0.4, respectively. On the other hand, in the range of E > 3E , Kolmogorov- FIG. 5: Power-law index γ as a function of the amplitude of min (cid:104) (cid:105) Smirnov probabilities are 0.000 for all models due to the perturbation ∆B for the models with the truncated icosa- cutoﬀ. hedron (n = 60) grid and poloidal perturbations. From Figure 5, we can see that the power-law index (cid:104) (cid:105) γ increases with the amplitude of perturbation ∆B . The most natural interpretation for this is that, for large (cid:104) (cid:105) ∆B , the perturbation of one site can grow exclusively IV. CONCLUSION AND DISCUSSION and the site tends to become unstable before accumulat- ing perturbations of other sites. As a result, the number In conclusion, we have shown the SOC behavior of of large avalanches, i.e. with a large energy, is reduced. a new CA model with spherically closed grids so as to The lack of universality of the power-law index is an in- demonstrate the recurrent bursts of SGRs. We adopted teresting feature that was also observed in the model of the nodes of a regular dodecahedron and a truncated Olami et al. [4]. icosahedron as our CA grids. Perturbations were added ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: FIG.

text: Same

text: as

text: Figure

text: but

text: for

text: the

text: models

text: with

text: the

text: trun-

text: FIG.

text: Same

text: as

text: Figure

text: but

text: for

text: the

text: models

text: with

text: the

text: trun-

text: cated

text: icosahedron

text: (n

text: grid.

text: Solid,

text: dashed,

text: dot-dashed

text: cated

text: icosahedron

text: (n

text: grid

text: and

text: poloidal

text: perturbations.

text: and

text: dotted

text: lines

text: correspond

text: to

text: the

text: cases

text: with

text: f

text: (purely

text: Thick

text: dot-dashed,

text: thick

text: solid

text: and

text: thick

text: dashed

text: lines

text: corre-

text: pl

text: poloidal),

text: and

text: (purely

text: toroidal),

text: respectively.

text: spond

text: to

text: the

text: cases

text: with

text: ∆B

text: and

text: respectively.

text: Thin

text: lines

text: are

text: power

text: laws

text: N(>E)

text: γ

text: with

text: γ

text: (dot-dashed),

text: (solid)

text: and

text: (dashed).

text: tron

text: star.

text: On

text: the

text: other

text: hand,

text: the

text: distributions

text: of

text: the

text: truncated

text: icosahedron

text: model

text: and

text: regular

text: dodecahedron

text: model

text: are

text: similar

text: at

text: low

text: energies,

text: where

text: ﬁnite-size

text: eﬀects

text: are

text: negligible.

text: We

text: show

text: the

text: dependence

text: on

text: the

text: amplitude

text: of

text: perturba-

text: tion

text: ∆B

text: in

text: Figure

text: where

text: the

text: cumulative

text: distributions

text: are

text: plotted

text: for

text: the

text: models

text: with

text: the

text: truncated

text: icosahedron

text: (n

text: grid

text: and

text: poloidal

text: perturbations.

text: We

text: ﬁnd

text: that

text: the

text: distributions

text: with

text: E

text: are

text: well

text: ﬁtted

text: with

text: min

text: power

text: laws.

text: The

text: power-law

text: index,

text: γ,

text: of

text: the

text: diﬀerential

text: distributions

text: dN

text: E

text: γdE

text: is

text: related

text: to

text: the

text: power-law

text: ﬁt

text: of

text: the

text: cumulative

text: distributions

text: as

text: N(>E)

text: γ.

text: According

text: to

text: the

text: observations

text: of

text: SGR

text: and

text: SGR

text: the

text: energy

text: distributions

text: of

text: bursts

text: have

text: been

text: ﬁt-

text: ted

text: with

text: γ

text: and

text: γ

text: In

text: our

text: CA

text: simulation,

text: γ

text: depends

text: on

text: the

text: value

text: of

text: ∆B

text: and

text: it

text: is

text: and

text: for

text: ∆B

text: and

text: respectively.

text: Here,

text: Kolmogorov-Smirnov

text: probabilities

text: of

text: the

text: ﬁtting

text: in

text: the

text: range

text: of

text: E

text: are

text: min

text: and

text: for

text: ∆B

text: and

text: respectively.

text: On

text: the

text: other

text: hand,

text: in

text: the

text: range

text: of

text: E

text: Kolmogorov-

text: FIG.

text: Power-law

text: index

text: γ

text: as

text: a

text: function

text: of

text: the

text: amplitude

text: of

text: min

text: Smirnov

text: probabilities

text: are

text: for

text: all

text: models

text: due

text: to

text: the

text: perturbation

text: ∆B

text: for

text: the

text: models

text: with

text: the

text: truncated

text: icosa-

text: cutoﬀ.

text: hedron

text: (n

text: grid

text: and

text: poloidal

text: perturbations.

text: From

text: Figure

text: we

text: can

text: see

text: that

text: the

text: power-law

text: index

text: γ

text: increases

text: with

text: the

text: amplitude

text: of

text: perturbation

text: ∆B

text: The

text: most

text: natural

text: interpretation

text: for

text: this

text: is

text: that,

text: for

text: large

text: ∆B

text: the

text: perturbation

text: of

text: one

text: site

text: can

text: grow

text: exclusively

text: IV.

text: CONCLUSION

text: AND

text: DISCUSSION

text: and

text: the

text: site

text: tends

text: to

text: become

text: unstable

text: before

text: accumulat-

text: ing

text: perturbations

text: of

text: other

text: sites.

text: As

text: a

text: result,

text: the

text: number

text: In

text: conclusion,

text: we

text: have

text: shown

text: the

text: SOC

text: behavior

text: of

text: of

text: large

text: avalanches,

text: i.e.

text: with

text: a

text: large

text: energy,

text: is

text: reduced.

text: a

text: new

text: CA

text: model

text: with

text: spherically

text: closed

text: grids

text: so

text: as

text: to

text: The

text: lack

text: of

text: universality

text: of

text: the

text: power-law

text: index

text: is

text: an

text: in-

text: demonstrate

text: the

text: recurrent

text: bursts

text: of

text: SGRs.

text: We

text: adopted

text: teresting

text: feature

text: that

text: was

text: also

text: observed

text: in

text: the

text: model

text: of

text: the

text: nodes

text: of

text: a

text: regular

text: dodecahedron

text: and

text: a

text: truncated

text: Olami

text: et

text: al.

text: icosahedron

text: as

text: our

text: CA

text: grids.

text: Perturbations

text: were

text: added

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 6 goulis, Astron. Astrophys. 529, A101 (2011). mund, Astrophys. J. 412, 841 (1993). [21] E.P. Mazets, S. Golentskii, V.N. Il’inskii, R.L. Aptekar’, [28] M. Geolgoulis, R. Kluiving and L. Vlahos, Physica A and Y.A. Guryan, Nature 282, 587 (1979). 218, 191 (1995). [22] K. Hurley et al., Nature 397, 41 (1999). [29] D. Vassiliadis, A. Anastasiadis, M. Geolgoulis and [23] K. Hurley et al., Nature 434, 1098 (2005). V. Vlahos, Astrophys. J. Lett. 509, L53 (1998). [24] D.M. Palmer et al., Nature 434, 1107 (2005). [30] L.F. Morales and P. Charbonneau, Nonlin. Processes [25] T. Terasawa et al., Nature 434, 1110 (2005). Geophys. 17, 339 (2010). [26] Y. Masada, S. Nagataki, K. Shibata, and T. Terasawa, [31] S.W. McIntosh, P. Charbonneau, T.J. Bogdan, H.-L. Liu, Publ. Astron. Soc. Japan 62, 1093 (2010). and J.P. Norman, Phys. Rev. E 65, 046125 (2002). [27] E.T. Lu, R.J. Hamilton, J.M. McTiernan and K.R. Bro-

label: paragraph

text: goulis,

label: reference

text: Astron.

text: Astrophys.

text: mund,

text: Astrophys.

text: J.

text: E.P.

text: Mazets,

text: S.

text: Golentskii,

text: V.N.

text: Il’inskii,

text: R.L.

text: Aptekar’,

text: M.

text: Geolgoulis,

text: R.

text: Kluiving

text: and

text: L.

text: Vlahos,

text: Physica

text: A

text: and

text: Y.A.

text: Guryan,

text: Nature

text: K.

text: Hurley

text: et

text: al.,

text: Nature

text: D.

text: Vassiliadis,

text: A.

text: Anastasiadis,

text: M.

text: Geolgoulis

text: and

text: K.

text: Hurley

text: et

text: al.,

text: Nature

text: V.

text: Vlahos,

text: Astrophys.

text: J.

text: Lett.

text: D.M.

text: Palmer

text: et

text: al.,

text: Nature

text: L.F.

text: Morales

text: and

text: P.

text: Charbonneau,

text: Nonlin.

text: Processes

text: T.

text: Terasawa

text: et

text: al.,

text: Nature

text: Geophys.

text: Y.

text: Masada,

text: S.

text: Nagataki,

text: K.

text: Shibata,

text: and

text: T.

text: Terasawa,

text: S.W.

text: McIntosh,

text: P.

text: Charbonneau,

text: T.J.

text: Bogdan,

text: H.-L.

text: Liu,

text: Publ.

text: Astron.

text: Soc.

text: Japan

text: and

text: J.P.

text: Norman,

text: Phys.

text: Rev.

text: E

text: E.T.

text: Lu,

text: R.J.

text: Hamilton,

text: J.M.

text: McTiernan

text: and

text: K.R.

text: Bro-


text: 2 Next, let us turn our attention to the magnetic be- In above, β = 1/T is the inverse temperature (we set haviour of the investigated spin-electron model at ﬁnite k = 1) and the unknown parameters A, J and H temperatures. In Fig. 2 we depict the magnetization pro- B eff eff are given by the expressions (cid:0) (cid:1) cess of the model with the ferromagnetic [Fig. 2(a)] as well as the antiferromagnetic Ising interaction [Fig. 2(b)], (cid:18) (cid:19) (cid:18) (cid:19) by assuming the ﬁxed hopping term t and various tem- − A = 2 [2 exp( βt) + exp(2βt)] V−V V 2 1/4, peratures. In agreement with the aforedescribed ground- + V−V V− state analysis, the total magnetization [m = (mI+me)/2] Jeff = T ln V 2+ , Heff = HI + T ln V , (3) of the system with the ferromagnetic Ising interaction + J < 0 takes the saturated value m = 1/2 in the whole sat ∓ ﬁeld region at T/J = 0, since all mobile electrons as where V∓ = cosh(βJ/2 βHe/2) and V = cosh(βHe/2). well as Ising spins are fully polarized into the magnetic Note that Eq. (2) allowFs an −exactZenumeration of the ﬁeld direction. On the other hand, the zero-temperature Helmholtz free e(cid:16)nergy(cid:17) = T ln(cid:16) o(cid:17)f the considered magnetization curve corresponding to the system with spin-electron tetrahedral chain and, subsequently, other the antiferromagnetic Ising interaction J > 0 exhibits thermodynamic quantities, e.g., the sublattice magneti- the abrupt jump from zero to the saturation magnetiza- − F − F (cid:0) (cid:1) | | zation mI = N1 ∂∂HI , me = N1 ∂∂He of the localized ttihoenﬁaetldth-iendcruicteicdalpﬁhealsdeHtrca/nJsit=ion1,bwethwicehencotiwncoidpeossswibitlhe Ising spins and mobile electrons, respectively, or the en- tropy per one magnetic particle s = − 1 ∂F . ground states AF and FM. At any ﬁnite temperature, 2N ∂T both magnetization curves start from zero in accordance III. RESULTS AND DISCUSSION with the one-dimensional character of the investigated spin-electron system. As usual, abrupt changes in low- Now, let us proceed to a discussion of the ground state, temperature magnetization curves observable around the magnetization process and magnetocaloric eﬀect of the | | ﬁelds H/J = 0 for J < 0 and H / J = 1 for J > 0 are spin-electron tetrahedral chain with the ferromagnetic c gradually smeared out upon increasing temperature [see (J < 0) or the antiferromagnetic (J > 0) Ising inter- the curves corresponding to T/J = 0.15, 0.2, 0.3, 0.4 in action. To reduce the number of free interaction param- | | Fig. 2(a) and T/ J = 0.05, 0.2, 0.4 in Fig. 2(b)]. eters, we will assume equal magnetic ﬁelds acting on the Ising spins and mobile electrons H = H = H . I e As expected, the investigated tetrahedral chain ex- hibits the trivial ferromagnetic (FM) ground-state order- T / |J| = 0 1.0 ing with a full alignment of all nodal Ising spins and mo- 0.2 bile electrons into the magnetic ﬁeld direction, if J < 0, 0.15 0.3 or it passes from two-fold degenerate antiferromagnetic (AF) ground stat|e t|o the ferromagnetic (FM) one at the msat 0.4 critical ﬁeld Hc/ J = 1, if J < 0. These two ground /m 0.5 states can be characterized through the following eigen- vectors and energies per one nodal lattice site: (cid:89) t / |J| = 1.0 0.0 | (cid:105) N |↑(cid:105) √1 † † † | (cid:105) 0.0 0.1 0.2 0.3 0.4 FM = σk 3 (ck1,↑ + ck2,↑ + ck3,↑) 0 k, (a) H / |J| k=1 J − − (cid:26) EFM = (cid:89)2 2t H; (4) 1.0 T / J = 0 |↑(cid:105) √1 † † † | (cid:105) 0.05 | (cid:105) N σk 3 (ck1,↓ + ck2,↓ + ck3,↓) 0 k 0.2 AF = kJ=1− |↓(cid:105)σk √13 (c†k1,↑ + c†k2,↑ + c†k3,↑)|0(cid:105)k, /m msat 0.5 0.4 E = 2t. (5) AF 2 0.0 t / J = 1.0 In above, the product runs over all primitive unit cells, | ↑(cid:105) | ↓(cid:105) 0.0 0.5 1.0 1.5 2.0 the ket vector ( ) determines the up (down) σk σk | (cid:105) (b) H / J state of kth localized Ising spin and 0 labels the vac- k uum state for the kth triangular plaquette occupied by FIG. 2. Magnetization normalized with respect to its sat- one mobile electron. It is apparent from Eq. (5) that the uration value as a function of the magnetic ﬁeld at various two-fold degenerate AF ground state corresponds to the temperatures for the model with the ﬁxed hopping term t antiferromagnetic order, in which the mobile electrons and the Ising interaction: (a) J < 0; (b) J > 0. are aligned in opposite to their Ising neighbours. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Next,

text: let

text: us

text: turn

text: our

text: attention

text: to

text: the

text: magnetic

text: be-

text: In

text: above,

text: β

text: is

text: the

text: inverse

text: temperature

text: (we

text: set

text: haviour

text: of

text: the

text: investigated

text: spin-electron

text: model

text: at

text: ﬁnite

text: k

text: and

text: the

text: unknown

text: parameters

text: A,

text: J

text: and

text: H

text: temperatures.

text: In

text: Fig.

text: we

text: depict

text: the

text: magnetization

text: pro-

text: B

text: eff

text: eff

text: are

text: given

text: by

text: the

text: expressions

label: equation

text: cess

text: of

text: the

text: model

text: with

text: the

text: ferromagnetic

text: [Fig.

text: as

text: well

text: as

text: the

text: antiferromagnetic

text: Ising

text: interaction

text: [Fig.

text: by

text: assuming

text: the

text: ﬁxed

text: hopping

text: term

text: t

text: and

text: various

text: tem-

text: A

text: exp(

text: βt)

text: V−V

text: V

text: peratures.

text: In

text: agreement

text: with

text: the

text: aforedescribed

text: ground-

text: V−V

text: V−

text: state

text: analysis,

text: the

text: total

text: magnetization

text: [m

text: Jeff

text: T

text: ln

text: V

text: Heff

text: HI

text: T

text: ln

text: V

text: of

text: the

text: system

text: with

text: the

text: ferromagnetic

text: Ising

text: interaction

text: J

text: takes

text: the

text: saturated

text: value

text: m

text: in

text: the

text: whole

text: sat

text: ﬁeld

text: region

text: at

text: T/J

text: since

text: all

text: mobile

text: electrons

text: as

text: where

text: V∓

text: and

text: V

text: well

text: as

text: Ising

text: spins

text: are

text: fully

text: polarized

text: into

text: the

text: magnetic

text: Note

text: that

text: Eq.

text: allowFs

text: an

text: −exactZenumeration

text: of

text: the

text: ﬁeld

text: direction.

text: On

text: the

text: other

text: hand,

text: the

text: zero-temperature

text: Helmholtz

text: free

text: T

text: the

text: considered

text: magnetization

text: curve

text: corresponding

text: to

text: the

text: system

text: with

text: spin-electron

text: tetrahedral

text: chain

text: and,

text: subsequently,

text: other

text: the

text: antiferromagnetic

text: Ising

text: interaction

text: J

text: exhibits

text: thermodynamic

text: quantities,

text: e.g.,

text: the

text: sublattice

text: magneti-

text: the

text: abrupt

text: jump

text: from

text: zero

text: to

text: the

text: saturation

text: magnetiza-

text: F

text: F

text: zation

text: mI

text: ∂∂HI

text: me

text: ∂∂He

text: of

text: the

text: localized

text: Ising

text: spins

text: and

text: mobile

text: electrons,

text: respectively,

text: or

text: the

text: en-

text: tropy

text: per

text: one

text: magnetic

text: particle

text: s

text: ∂F

text: ground

text: states

text: AF

text: and

text: FM.

text: At

text: any

text: ﬁnite

text: temperature,

text: ∂T

text: both

text: magnetization

text: curves

text: start

text: from

text: zero

text: in

text: accordance

text: III.

text: RESULTS

text: AND

text: DISCUSSION

text: with

text: the

text: one-dimensional

text: character

text: of

text: the

text: investigated

text: spin-electron

text: system.

text: As

text: usual,

text: abrupt

text: changes

text: in

text: low-

text: Now,

text: let

text: us

text: proceed

text: to

text: a

text: discussion

text: of

text: the

text: ground

text: state,

text: temperature

text: magnetization

text: curves

text: observable

text: around

text: the

text: magnetization

text: process

text: and

text: magnetocaloric

text: eﬀect

text: of

text: the

text: ﬁelds

text: H/J

text: for

text: J

text: and

text: H

text: J

text: for

text: J

text: are

text: spin-electron

text: tetrahedral

text: chain

text: with

text: the

text: ferromagnetic

text: c

text: gradually

text: smeared

text: out

text: upon

text: increasing

text: temperature

text: [see

text: (J

text: or

text: the

text: antiferromagnetic

text: (J

text: Ising

text: inter-

text: the

text: curves

text: corresponding

text: to

text: T/J

text: in

text: action.

text: To

text: reduce

text: the

text: number

text: of

text: free

text: interaction

text: param-

text: Fig.

text: and

text: T/

text: J

text: in

text: Fig.

text: eters,

text: we

text: will

text: assume

text: equal

text: magnetic

text: ﬁelds

text: acting

text: on

text: the

text: Ising

text: spins

text: and

text: mobile

text: electrons

text: H

text: H

text: H

text: I

text: e

text: As

text: expected,

text: the

text: investigated

text: tetrahedral

text: chain

text: ex-

text: hibits

text: the

text: trivial

text: ferromagnetic

text: (FM)

text: ground-state

text: order-

text: T

text: |J|

text: ing

text: with

text: a

text: full

text: alignment

text: of

text: all

text: nodal

text: Ising

text: spins

text: and

text: mo-

text: bile

text: electrons

text: into

text: the

text: magnetic

text: ﬁeld

text: direction,

text: if

text: J

text: or

text: it

text: passes

text: from

text: two-fold

text: degenerate

text: antiferromagnetic

text: (AF)

text: ground

text: stat|e

text: t|o

text: the

text: ferromagnetic

text: (FM)

text: one

text: at

text: the

text: msat

text: critical

text: ﬁeld

text: Hc/

text: J

text: if

text: J

text: These

text: two

text: ground

text: /m

text: states

text: can

text: be

text: characterized

text: through

text: the

text: following

text: eigen-

text: vectors

text: and

text: energies

text: per

text: one

text: nodal

text: lattice

text: site:

text: t

text: |J|

text: N

text: √1

text: FM

text: σk

text: k,

text: (a)

text: H

text: |J|

text: J

text: EFM

text: H;

text: T

text: J

text: √1

text: N

text: σk

text: k

text: AF

text: /m

text: msat

text: E

text: AF

text: t

text: J

text: In

text: above,

text: the

text: product

text: runs

text: over

text: all

text: primitive

text: unit

text: cells,

text: the

text: ket

text: vector

text: determines

text: the

text: up

text: (down)

text: σk

text: σk

text: (b)

text: H

text: J

text: state

text: of

text: kth

text: localized

text: Ising

text: spin

text: and

text: labels

text: the

text: vac-

text: k

text: uum

text: state

text: for

text: the

text: kth

text: triangular

text: plaquette

text: occupied

text: by

text: FIG.

text: Magnetization

text: normalized

text: with

text: respect

text: to

text: its

text: sat-

text: one

text: mobile

text: electron.

text: It

text: is

text: apparent

text: from

text: Eq.

text: that

text: the

text: uration

text: value

text: as

text: a

text: function

text: of

text: the

text: magnetic

text: ﬁeld

text: at

text: various

text: two-fold

text: degenerate

text: AF

text: ground

text: state

text: corresponds

text: to

text: the

text: temperatures

text: for

text: the

text: model

text: with

text: the

text: ﬁxed

text: hopping

text: term

text: t

text: antiferromagnetic

text: order,

text: in

text: which

text: the

text: mobile

text: electrons

text: and

text: the

text: Ising

text: interaction:

text: (a)

text: J

text: (b)

text: J

text: are

text: aligned

text: in

text: opposite

text: to

text: their

text: Ising

text: neighbours.

text: ##LTFigure##

text: ##LTFigure##

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text: 2 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA system moves along a circular geocentric orbit. Moreover, the equilibrium conﬁgurations of the system are obtained and the Lyapunov stability of conﬁgurations for two situations, ﬁrst when the station is composed of equal masses, second when masses at the ends of the station are diﬀerent are analyzed. For the “dumbbells–load” system with two unilateral connections, all rel- ative equilibria on the circular Keplerian orbit were established by Munitsina [22]. Recall that a relative equilibria of the system is a point of the phase space giving an evolution which is a one–parameter orbit of the action of the symmetry group of the system. These results were interpreted for studying the relative equilibria for which both connections are stretched in geometri- cal terms. The necessary and suﬃcient conditions for stability of the relative equilibria were stated. Celletti and Sidorenko [14] investigated the dumbbell satellite’s attitude dynamics, when the center of mass moves on a Keplerian trajectory. They found a stable relative equilibrium position in the case of circular orbits which disappears as far as elliptic trajectories are considered. They replaced the equilibrium position by planar periodic motions and they proved this motion is unstable with respect to out-of-plane perturbations. They also gave some numerical evidences of the existence of stable spatial periodic motions. Burov et al. [13] considered the motion of a dumbbell–shaped body in an attractive Newtonian central ﬁeld. They used the Poincare’s theory to determine the conditions for the existence of families of system periodic motions depending on the arising small parameter and passing into some stable radial steady–state motion of the unperturbed problem as the small parameter tends to zero. They also proved that, each of the radial relative equilibria generates one family of such periodic motions, for suﬃciently small parameter values. Furthermore, they studied the stability of the obtained periodic solutions in the linear approximation as well as these solutions were calculated up to terms of the ﬁrst order in the small parameter. Guirao et al. [17] gave suﬃcient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit. The statement of our main results is the following. Theorem 1. Consider a dumbbell satellite moving in a gravity ﬁeld gener- ated by an oblate body considering the eﬀect of the zonal harmonic parameter A. The pass trajectory of the mass center of the system is periodic and dif- ferent from the classical one. If A is equal to zero our solution coincides with the elliptical classical one. Finally, considering the motion in the satellite approximation we complete the classical theory, stating the following result.

text: E.I.

text: ABOUELMAGD,

text: J.L.G.

text: GUIRAO,

text: J.A.

text: VERA

text: system

text: moves

text: along

text: a

text: circular

text: geocentric

text: orbit.

text: Moreover,

text: the

text: equilibrium

text: conﬁgurations

text: of

text: the

text: system

text: are

text: obtained

text: and

text: the

text: Lyapunov

text: stability

text: of

text: conﬁgurations

text: for

text: two

text: situations,

text: ﬁrst

text: when

text: the

text: station

text: is

text: composed

text: of

text: equal

text: masses,

text: second

text: when

text: masses

text: at

text: the

text: ends

text: of

text: the

text: station

text: are

text: diﬀerent

text: are

text: analyzed.

text: For

text: the

text: “dumbbells–load”

text: system

text: with

text: two

text: unilateral

text: connections,

text: all

text: rel-

text: ative

text: equilibria

text: on

text: the

text: circular

text: Keplerian

text: orbit

text: were

text: established

text: by

text: Munitsina

text: Recall

text: that

text: a

text: relative

text: equilibria

text: of

text: the

text: system

text: is

text: a

text: point

text: of

text: the

text: phase

text: space

text: giving

text: an

text: evolution

text: which

text: is

text: a

text: one–parameter

text: orbit

text: of

text: the

text: action

text: of

text: the

text: symmetry

text: group

text: of

text: the

text: system.

text: These

text: results

text: were

text: interpreted

text: for

text: studying

text: the

text: relative

text: equilibria

text: for

text: which

text: both

text: connections

text: are

text: stretched

text: in

text: geometri-

text: cal

text: terms.

text: The

text: necessary

text: and

text: suﬃcient

text: conditions

text: for

text: stability

text: of

text: the

text: relative

text: equilibria

text: were

text: stated.

text: Celletti

text: and

text: Sidorenko

text: investigated

text: the

text: dumbbell

text: satellite’s

text: attitude

text: dynamics,

text: when

text: the

text: center

text: of

text: mass

text: moves

text: on

text: a

text: Keplerian

text: trajectory.

text: They

text: found

text: a

text: stable

text: relative

text: equilibrium

text: position

text: in

text: the

text: case

text: of

text: circular

text: orbits

text: which

text: disappears

text: as

text: far

text: as

text: elliptic

text: trajectories

text: are

text: considered.

text: They

text: replaced

text: the

text: equilibrium

text: position

text: by

text: planar

text: periodic

text: motions

text: and

text: they

text: proved

text: this

text: motion

text: is

text: unstable

text: with

text: respect

text: to

text: out-of-plane

text: perturbations.

text: They

text: also

text: gave

text: some

text: numerical

text: evidences

text: of

text: the

text: existence

text: of

text: stable

text: spatial

text: periodic

text: motions.

text: Burov

text: et

text: al.

text: considered

text: the

text: motion

text: of

text: a

text: dumbbell–shaped

text: body

text: in

text: an

text: attractive

text: Newtonian

text: central

text: ﬁeld.

text: They

text: used

text: the

text: Poincare’s

text: theory

text: to

text: determine

text: the

text: conditions

text: for

text: the

text: existence

text: of

text: families

text: of

text: system

text: periodic

text: motions

text: depending

text: on

text: the

text: arising

text: small

text: parameter

text: and

text: passing

text: into

text: some

text: stable

text: radial

text: steady–state

text: motion

text: of

text: the

text: unperturbed

text: problem

text: as

text: the

text: small

text: parameter

text: tends

text: to

text: zero.

text: They

text: also

text: proved

text: that,

text: each

text: of

text: the

text: radial

text: relative

text: equilibria

text: generates

text: one

text: family

text: of

text: such

text: periodic

text: motions,

text: for

text: suﬃciently

text: small

text: parameter

text: values.

text: Furthermore,

text: they

text: studied

text: the

text: stability

text: of

text: the

text: obtained

text: periodic

text: solutions

text: in

text: the

text: linear

text: approximation

text: as

text: well

text: as

text: these

text: solutions

text: were

text: calculated

text: up

text: to

text: terms

text: of

text: the

text: ﬁrst

text: order

text: in

text: the

text: small

text: parameter.

text: Guirao

text: et

text: al.

text: gave

text: suﬃcient

text: conditions

text: for

text: the

text: existence

text: of

text: periodic

text: solutions

text: of

text: the

text: perturbed

text: attitude

text: dynamics

text: of

text: a

text: rigid

text: dumbbell

text: satellite

text: in

text: a

text: circular

text: orbit.

text: The

text: statement

text: of

text: our

text: main

text: results

text: is

text: the

text: following.

text: Theorem

text: Consider

text: a

text: dumbbell

text: satellite

text: moving

text: in

text: a

text: gravity

text: ﬁeld

text: gener-

text: ated

text: by

text: an

text: oblate

text: body

text: considering

text: the

text: eﬀect

text: of

text: the

text: zonal

text: harmonic

text: parameter

text: A.

text: The

text: pass

text: trajectory

text: of

text: the

text: mass

text: center

text: of

text: the

text: system

text: is

text: periodic

text: and

text: dif-

text: ferent

text: from

text: the

text: classical

text: one.

text: If

text: A

text: is

text: equal

text: to

text: zero

text: our

text: solution

text: coincides

text: with

text: the

text: elliptical

text: classical

text: one.

text: Finally,

text: considering

text: the

text: motion

text: in

text: the

text: satellite

text: approximation

text: we

text: complete

text: the

text: classical

text: theory,

text: stating

text: the

text: following

text: result.


text: DYNAMICS OF A DUMBBELL SATELLITE... 19 Acknowledgements This work has been partially supported by MICINN/FEDER grant num- ber MTM2011–22587. References [1] E.I. Abouelmagd, S.M. El–Shaboury, Periodic orbits under combined eﬀects of oblateness and radiation in the restricted problem of three bodies, Astrophys Space Sci. 341 (2012), 331–341 [2] E.I. Abouelmagd, Existence and stability of triangular points in the restricted three– body problem with numerical applications, Astrophys Space Sci. 342 (2012), 45–53 [3] E.I. Abouelmagd, Stability of the triangular points under combined eﬀects of radia- tion and oblateness in the restricted three-body problem, Earth Moon and Planets 110 (2013), 143–155 [4] E.I. Abouelmagd, Existence and stability of triangular points in the restricted three– body problem with numerical applications, Astrophys Space Sci. 342 (2012), 45–53 [5] E.I. Abouelmagd, The eﬀect of photogravitational force and oblateness in the per- turbed restricted three-body problem, Astrophys Space Sci. 346 (2013), 321–332 [6] E.I. Abouelmagd, M.A. Sharaf, The motion around the libration points in the restricted three-body problem with the eﬀect of radiation and oblateness, Astrophys Space Sci. 344 (2013), 321–332 [7] E.I. Abouelmagd, H.M. Asiri, M.A. Sharaf, The eﬀect of oblateness in the per- turbed restricted three-body problem, Meccanica 48 (2013), 2479–2490 [8] E.I. Abouelmagd, M.E. Awad, E.M.A. Elzayat, I.A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem, Astrophys Space Sci. 350 (2014), 495–505 [9] V.V. Beletsky, Motion of an artiﬁcial satellite about its center of mass, Israel pro- gram for scientiﬁc translations. Jerusalem (1966) [10] V.V. Beletsky and D.V. Pankova, Connected bodies in the orbit as dynamic bil- liard, Regular Chaot. Dynam. 1 (1996), 87–103 [11] R.C. Brereton and V.J. Modi, On the stability of planar librations of a dumbbell satellite in an elliptic orbit, Aeronaut. J. 70 (1966), 1098–1102 [12] A. Burov and A. Dugain, Planar oscillations a vibrating dumbbell–like body in a central ﬁeld of forces, Aeronaut. J. 49(4) (2011), 353–359 [13] A. Burov, I.I. Kosenko and H. Troger, On periodic motions of an orbital dumbbell–shaped body with a cabin elevator, Mechanics of Solids 47(3) (2012), 269–284 [14] A. Celletti and V. Sidorenko, Some properties of the dumbbell satellite attitude, Celestial Mech. Dyn. Astr. 101 (2008), 105–126 [15] A. Elipe, M. Palacios and H. Pretka–Ziomek, Equilibria of the two–body problem with rigid dumbbell satellite, Chaos Soliton & Fractals 35 (2008), 830–842 [16] C.D. Murray and S.F. Dermott, Solar system dynamics, Cambridge University Press, (1999) [17] J.L.G. Guirao, J.A. Vera and B.A. Wade, On the periodic solutions of a rigid dumbbell satellite in a circular orbit, Astrophys Space Sci. 346 (2013), 437–442 [18] U. Kirchgraber, U. Manz and D. Stoffer, Rigorous proof of chaotic behavior in a dumbbell satellite model, J. Math. Anal. Appl. 251 (2000), 897–911 [19] M. Krupa, A. Steindl and I.I. Troger, Stability of relative equilibria. Part II: Dumbbell satellites, Meccanica 35 (2001), 353–371 [20] A.J. Maciejewski , About attitude motion of a satellite with time dependent mo- ments of inertia, Artiﬁcial Satellites 17(2-3) (1982), 49–60

text: DYNAMICS

text: OF

text: A

text: DUMBBELL

text: SATELLITE...

text: Acknowledgements

label: section

text: This

text: work

text: has

text: been

text: partially

text: supported

text: by

text: MICINN/FEDER

text: grant

text: num-

text: ber

text: References

text: E.I.

text: Abouelmagd,

text: S.M.

text: El–Shaboury,

text: Periodic

text: orbits

text: under

text: combined

text: eﬀects

text: of

text: oblateness

text: and

text: radiation

text: in

text: the

text: restricted

text: problem

text: of

text: three

text: bodies,

text: Astrophys

text: Space

text: Sci.

text: E.I.

text: Abouelmagd,

text: Existence

text: and

text: stability

text: of

text: triangular

text: points

text: in

text: the

text: restricted

text: three–

text: body

text: problem

text: with

text: numerical

text: applications,

text: Astrophys

text: Space

text: Sci.

text: E.I.

text: Abouelmagd,

text: Stability

text: of

text: the

text: triangular

text: points

text: under

text: combined

text: eﬀects

text: of

text: radia-

text: tion

text: and

text: oblateness

text: in

text: the

text: restricted

text: three-body

text: problem,

text: Earth

text: Moon

text: and

text: Planets

text: E.I.

text: Abouelmagd,

text: Existence

text: and

text: stability

text: of

text: triangular

text: points

text: in

text: the

text: restricted

text: three–

text: body

text: problem

text: with

text: numerical

text: applications,

text: Astrophys

text: Space

text: Sci.

text: E.I.

text: Abouelmagd,

text: The

text: eﬀect

text: of

text: photogravitational

text: force

text: and

text: oblateness

text: in

text: the

text: per-

text: turbed

text: restricted

text: three-body

text: problem,

text: Astrophys

text: Space

text: Sci.

text: E.I.

text: Abouelmagd,

text: M.A.

text: Sharaf,

text: The

text: motion

text: around

text: the

text: libration

text: points

text: in

text: the

text: restricted

text: three-body

text: problem

text: with

text: the

text: eﬀect

text: of

text: radiation

text: and

text: oblateness,

text: Astrophys

text: Space

text: Sci.

text: E.I.

text: Abouelmagd,

text: H.M.

text: Asiri,

text: M.A.

text: Sharaf,

text: The

text: eﬀect

text: of

text: oblateness

text: in

text: the

text: per-

text: turbed

text: restricted

text: three-body

text: problem,

text: Meccanica

text: E.I.

text: Abouelmagd,

text: M.E.

text: Awad,

text: E.M.A.

text: Elzayat,

text: I.A.

text: Abbas,

text: Reduction

text: the

text: secular

text: solution

text: to

text: periodic

text: solution

text: in

text: the

text: generalized

text: restricted

text: three-body

text: problem,

text: Astrophys

text: Space

text: Sci.

text: V.V.

text: Beletsky,

text: Motion

text: of

text: an

text: artiﬁcial

text: satellite

text: about

text: its

text: center

text: of

text: mass,

text: Israel

text: pro-

text: gram

text: for

text: scientiﬁc

text: translations.

text: Jerusalem

text: V.V.

text: Beletsky

text: and

text: D.V.

text: Pankova,

text: Connected

text: bodies

text: in

text: the

text: orbit

text: as

text: dynamic

text: bil-

text: liard,

text: Regular

text: Chaot.

text: Dynam.

text: R.C.

text: Brereton

text: and

text: V.J.

text: Modi,

text: On

text: the

text: stability

text: of

text: planar

text: librations

text: of

text: a

text: dumbbell

text: satellite

text: in

text: an

text: elliptic

text: orbit,

text: Aeronaut.

text: J.

text: A.

text: Burov

text: and

text: A.

text: Dugain,

text: Planar

text: oscillations

text: a

text: vibrating

text: dumbbell–like

text: body

text: in

text: a

text: central

text: ﬁeld

text: of

text: forces,

text: Aeronaut.

text: J.

text: A.

text: Burov,

text: I.I.

text: Kosenko

text: and

text: H.

text: Troger,

text: On

text: periodic

text: motions

text: of

text: an

text: orbital

text: dumbbell–shaped

text: body

text: with

text: a

text: cabin

text: elevator,

text: Mechanics

text: of

text: Solids

text: A.

text: Celletti

text: and

text: V.

text: Sidorenko,

text: Some

text: properties

text: of

text: the

text: dumbbell

text: satellite

text: attitude,

text: Celestial

text: Mech.

text: Dyn.

text: Astr.

text: A.

text: Elipe,

text: M.

text: Palacios

text: and

text: H.

text: Pretka–Ziomek,

text: Equilibria

text: of

text: the

text: two–body

text: problem

text: with

text: rigid

text: dumbbell

text: satellite,

text: Chaos

text: Soliton

text: Fractals

text: C.D.

text: Murray

text: and

text: S.F.

text: Dermott,

text: Solar

text: system

text: dynamics,

text: Cambridge

text: University

text: Press,

text: J.L.G.

text: Guirao,

text: J.A.

text: Vera

text: and

text: B.A.

text: Wade,

text: On

text: the

text: periodic

text: solutions

text: of

text: a

text: rigid

text: dumbbell

text: satellite

text: in

text: a

text: circular

text: orbit,

text: Astrophys

text: Space

text: Sci.

text: U.

text: Kirchgraber,

text: U.

text: Manz

text: and

text: D.

text: Stoffer,

text: Rigorous

text: proof

text: of

text: chaotic

text: behavior

text: in

text: a

text: dumbbell

text: satellite

text: model,

text: J.

text: Math.

text: Anal.

text: Appl.

text: M.

text: Krupa,

text: A.

text: Steindl

text: and

text: I.I.

text: Troger,

text: Stability

text: of

text: relative

text: equilibria.

text: Part

text: II:

text: Dumbbell

text: satellites,

text: Meccanica

text: A.J.

text: Maciejewski

text: About

text: attitude

text: motion

text: of

text: a

text: satellite

text: with

text: time

text: dependent

text: mo-

text: ments

text: of

text: inertia,

text: Artiﬁcial

text: Satellites


text: 10 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA The continuation method enables us to construct a periodic solution for (cid:54) A = 0. If we consider that the angular velocity changes due to the non– linear terms, the asymptotic solution u(θ, A) and the angular velocity ω of the dynamical system can be expanded as u(θ, A) = u (θ) + Au (θ) + A2u (θ) + . . . (21) 0 1 2 ω = 1 + Aω + A2ω + . . . 1 2 To construct a uniformly valid solution, we will introduce a stretched variable τ = ωθ, therefore d d = ω , dθ dτ (22) d2 d2 = ω2 . dθ2 dτ2 Substituting equations (22) into (17) we obtain d2u k 3 (23) ω2 + u = (1 + Au2). dτ2 h2 2 Now, under the following conditions 1 u(0, A) = , r p u (0, A) = 0, τ u(τ + 2π, A) = u(τ, A), we insert the series expansion (21) into (23) and equating terms of the same order in A with keeping the terms up to ﬁrst order of A, we obtain the following: • The coeﬃcient of A0 gives a homogeneous equation in the form d2u k 0 + u = dτ2 0 h2 where u (0) = 1 , du0(0) = 0 and u (τ + 2π, A) = u (τ, A) with 0 rp dτ 0 0 a solution k (24) u (τ) = (1 + e cos τ), 0g h2 0 being h2/k (25) r = . 1 + e cos τ 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: E.I.

text: ABOUELMAGD,

text: J.L.G.

text: GUIRAO,

text: J.A.

text: VERA

text: The

text: continuation

text: method

text: enables

text: us

text: to

text: construct

text: a

text: periodic

text: solution

text: for

font: default

text: A

text: If

text: we

text: consider

text: that

text: the

text: angular

text: velocity

text: changes

text: due

text: to

text: the

text: non–

text: linear

text: terms,

text: the

text: asymptotic

text: solution

text: u(θ,

text: A)

text: and

text: the

text: angular

text: velocity

text: ω

text: of

text: the

text: dynamical

text: system

text: can

text: be

text: expanded

text: as

text: u(θ,

text: A)

text: u

text: (θ)

text: Au

text: (θ)

text: (θ)

text: ω

text: Aω

text: To

text: construct

text: a

text: uniformly

text: valid

text: solution,

text: we

text: will

text: introduce

text: a

text: stretched

text: variable

text: τ

text: ωθ,

text: therefore

text: d

text: d

text: ω

text: dθ

text: dτ

text: ω2

text: Substituting

text: equations

text: into

text: we

text: obtain

text: k

text: ω2

text: u

text: Now,

text: under

text: the

text: following

text: conditions

text: A)

text: r

text: p

text: u

text: A)

text: τ

text: u(τ

text: A)

text: u(τ,

text: A),

text: we

text: insert

text: the

text: series

text: expansion

text: into

text: and

text: equating

text: terms

text: of

text: the

text: same

text: order

text: in

text: A

text: with

text: keeping

text: the

text: terms

text: up

text: to

text: ﬁrst

text: order

text: of

text: A,

text: we

text: obtain

text: the

text: following:

text: The

text: coeﬃcient

text: of

text: gives

text: a

text: homogeneous

text: equation

text: in

text: the

text: form

text: k

text: u

text: where

text: u

text: and

text: u

text: (τ

text: A)

text: u

text: (τ,

text: A)

text: with

text: rp

text: dτ

text: a

text: solution

text: k

text: u

text: (τ)

text: e

text: cos

text: τ),

text: being

text: r

text: e

text: cos

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • Minimize useless pixels thanks to a better/easier arrangement of light into the detector (conversely to, e.g., lenslet and masked multi-object spectrographs like TIGER11); • Allow mass production of optical components at a lower cost for the future highly multiplex instrumenta- tion. Astrophotonics has the potential to provide the solution to the continuous increase in complexity of future optical/infra-red ground-based spectrographs, while reducing both size and cost of the instruments. The com- pactness of these instruments is one of the most obvious advantages in the context of space missions, and these have already found important applications on rockets, balloons and drones.12 2. PHOTONIC SPECTROMETERS The Wiener-Khinchin theorem states that the power spectral density of a stationary random process is the Fourier transform of the corresponding autocorrelation function. It is the basis of spectrometry, and can be summarized as an autocorrelation of light propagation in time. This autocorrelation can be obtained with at least three diﬀerent concepts: • Diﬀracting elements, like gratings, which are the most common type of spectrographs in operation today; • Interferential ﬁlters like Fabry-P´erots, where only few speciﬁc spectral regions are transmitted; • Fourier Transform Spectrographs, where we measure the signal autocorrelation, and the Fourier transform operation is performed numerically. A fourth way is now under investigation with energy sensitive detectors. They are not considered in the following. Today, astrophotonics technologies are mostly single-mode (SM) devices, i.e we can couple at best a source with an optical etendue SΩ = λ2. This corresponds roughly to the diﬀraction limit of a telescope (whatever SM its size): in this case, the coupling eﬃciency reaches a maximum value of 82%. But in most cases, the source is spatially extended, either because it is spatially resolved (e.g. galaxies or nebulae), and/or because of the atmospheric turbulence that spreads the stellar light into a large speckle pattern. Although in the latter case, Adaptive Optics systems can signiﬁcantly reduce the etendue for point sources, SΩ is still high as long as we do not make use of Extreme AO systems. In all other situations, the optical etendue can reach values up to ∼ several 1000 λ2 for ELTs, especially in visible wavelengths (Tab. 1), i.e. only 1/1000 of the light gathered by the telescope is actually coupled to the instrument. Thanks to the high number of modes M that Multi-mode (MM) waveguides can accept, the optical etendue ∼ of the system is greatly improved (SΩ Mλ2). However, considering a ﬁnal photonic spectrograph –mostly SM devices in the current framework–, the coupling from the MM waveguide to the spectrograph will be similar to the direct injection into a SM waveguide, i.e. very low. MM waveguides can in addition limit spectral resolution because of modes propagating at diﬀerent speeds, then generating a series of shifted and stretched spectra on the detector. The photonic lantern9,10 becomes a key component in this context: by converting an initial MM beam into multiple SM ones, the initial etendue can be preserved and we make a correct use of the large aperture of the ∼ Table 1. Number of modes M SΩ/λ2 of point sources according to the expected performance of MCAO/LTAO systems on the 42-m E-ELT (former design) in standard conditions (seeing = 0.8”).13,14 EE stands for Encircled Energy. Band V I J H K M N Point + AO (EE > 50%) > 1500 > 500 250 12 3 2 1 (cid:29) Point + AO (EE > 85%) 1500 > 500 > 250 > 150 > 80 > 16 > 4 Resolved (θ = 50mas) 375 125 62 38 20 4 1 Resolved (θ = 100mas) 1500 500 250 150 78 16 4 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: Minimize

text: useless

text: pixels

text: thanks

text: to

text: a

text: better/easier

text: arrangement

text: of

text: light

text: into

text: the

text: detector

text: (conversely

text: to,

text: e.g.,

text: lenslet

text: and

text: masked

text: multi-object

text: spectrographs

text: like

text: Allow

text: mass

text: production

text: of

text: optical

text: components

text: at

text: a

text: lower

text: cost

text: for

text: the

text: future

text: highly

text: multiplex

text: instrumenta-

text: tion.

text: Astrophotonics

text: has

text: the

text: potential

text: to

text: provide

text: the

text: solution

text: to

text: the

text: continuous

text: increase

text: in

text: complexity

text: of

text: future

text: optical/infra-red

text: ground-based

text: spectrographs,

text: while

text: reducing

text: both

text: size

text: and

text: cost

text: of

text: the

text: instruments.

text: The

text: com-

text: pactness

text: of

text: these

text: instruments

text: is

text: one

text: of

text: the

text: most

text: obvious

text: advantages

text: in

text: the

text: context

text: of

text: space

text: missions,

text: and

text: these

text: have

text: already

text: found

text: important

text: applications

text: on

text: rockets,

text: balloons

text: and

text: PHOTONIC

text: SPECTROMETERS

text: The

text: Wiener-Khinchin

text: theorem

text: states

text: that

text: the

text: power

text: spectral

text: density

text: of

text: a

text: stationary

text: random

text: process

text: is

text: the

text: Fourier

text: transform

text: of

text: the

text: corresponding

text: autocorrelation

text: function.

text: It

text: is

text: the

text: basis

text: of

text: spectrometry,

text: and

text: can

text: be

text: summarized

text: as

text: an

text: autocorrelation

text: of

text: light

text: propagation

text: in

text: time.

text: This

text: autocorrelation

text: can

text: be

text: obtained

text: with

text: at

text: least

text: three

text: diﬀerent

text: concepts:

text: Diﬀracting

text: elements,

text: like

text: gratings,

text: which

text: are

text: the

text: most

text: common

text: type

text: of

text: spectrographs

text: in

text: operation

text: today;

text: Interferential

text: ﬁlters

text: like

text: Fabry-P´erots,

text: where

text: only

text: few

text: speciﬁc

text: spectral

text: regions

text: are

text: transmitted;

text: Fourier

text: Transform

text: Spectrographs,

text: where

text: we

text: measure

text: the

text: signal

text: autocorrelation,

text: and

text: the

text: Fourier

text: transform

text: operation

text: is

text: performed

text: numerically.

text: A

text: fourth

text: way

text: is

text: now

text: under

text: investigation

text: with

text: energy

text: sensitive

text: detectors.

text: They

text: are

text: not

text: considered

text: in

text: the

text: following.

text: Today,

text: astrophotonics

text: technologies

text: are

text: mostly

text: single-mode

text: (SM)

text: devices,

text: i.e

text: we

text: can

text: couple

text: at

text: best

text: a

text: source

text: with

text: an

text: optical

text: etendue

text: SΩ

text: This

text: corresponds

text: roughly

text: to

text: the

text: diﬀraction

text: limit

text: of

text: a

text: telescope

text: (whatever

text: SM

text: its

text: size):

text: in

text: this

text: case,

text: the

text: coupling

text: eﬃciency

text: reaches

text: a

text: maximum

text: value

text: of

text: But

text: in

text: most

text: cases,

text: the

text: source

text: is

text: spatially

text: extended,

text: either

text: because

text: it

text: is

text: spatially

text: resolved

text: (e.g.

text: galaxies

text: or

text: nebulae),

text: and/or

text: because

text: of

text: the

text: atmospheric

text: turbulence

text: that

text: spreads

text: the

text: stellar

text: light

text: into

text: a

text: large

text: speckle

text: pattern.

text: Although

text: in

text: the

text: latter

text: case,

text: Adaptive

text: Optics

text: systems

text: can

text: signiﬁcantly

text: reduce

text: the

text: etendue

text: for

text: point

text: sources,

text: SΩ

text: is

text: still

text: high

text: as

text: long

text: as

text: we

text: do

text: not

text: make

text: use

text: of

text: Extreme

text: AO

text: systems.

text: In

text: all

text: other

text: situations,

text: the

text: optical

text: etendue

text: can

text: reach

text: values

text: up

text: to

text: several

text: λ2

text: for

text: ELTs,

text: especially

text: in

text: visible

text: wavelengths

text: (Tab.

text: i.e.

text: only

text: of

text: the

text: light

text: gathered

text: by

text: the

text: telescope

text: is

text: actually

text: coupled

text: to

text: the

text: instrument.

text: Thanks

text: to

text: the

text: high

text: number

text: of

text: modes

text: M

text: that

text: Multi-mode

text: (MM)

text: waveguides

text: can

text: accept,

text: the

text: optical

text: etendue

text: of

text: the

text: system

text: is

text: greatly

text: improved

text: (SΩ

text: However,

text: considering

text: a

text: ﬁnal

text: photonic

text: spectrograph

text: –mostly

text: SM

text: devices

text: in

text: the

text: current

text: framework–,

text: the

text: coupling

text: from

text: the

text: MM

text: waveguide

text: to

text: the

text: spectrograph

text: will

text: be

text: similar

text: to

text: the

text: direct

text: injection

text: into

text: a

text: SM

text: waveguide,

text: i.e.

text: very

text: low.

text: MM

text: waveguides

text: can

text: in

text: addition

text: limit

text: spectral

text: resolution

text: because

text: of

text: modes

text: propagating

text: at

text: diﬀerent

text: speeds,

text: then

text: generating

text: a

text: series

text: of

text: shifted

text: and

text: stretched

text: spectra

text: on

text: the

text: detector.

text: The

text: photonic

text: becomes

text: a

text: key

text: component

text: in

text: this

text: context:

text: by

text: converting

text: an

text: initial

text: MM

text: beam

text: into

text: multiple

text: SM

text: ones,

text: the

text: initial

text: etendue

text: can

text: be

text: preserved

text: and

text: we

text: make

text: a

text: correct

text: use

text: of

text: the

text: large

text: aperture

text: of

text: the

text: Table

text: Number

text: of

text: modes

text: M

text: of

text: point

text: sources

text: according

text: to

text: the

text: expected

text: performance

text: of

text: MCAO/LTAO

text: systems

text: on

text: the

text: E-ELT

text: (former

text: design)

text: in

text: standard

text: conditions

text: (seeing

text: EE

text: stands

text: for

text: Encircled

text: Energy.

text: Band

text: V

text: I

text: J

text: H

text: K

text: M

text: N

text: Point

text: AO

text: (EE

text: Point

text: AO

text: (EE

text: Resolved

text: (θ

text: Resolved

text: (θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 3. Summary of identiﬁed Fabry-P´erot and ﬁlter technologies. ZZS20 MEMS21 PPSI22 Spectral domain [nm] 1300-1800 1000-1785 400-700 R Spectral resolution 50 100 5 Modicity SM, MM MM MM Simultaneous spectra 1 1000 106 3D multiplexing Good Good Very good Table 4. Summary of identiﬁed FTS technologies. µSPOC23 SWIFTS24 LLIFTS25 AMZI26 Spectral domain [nm] 3300-5000 400-1000 1200-1900 R Spectral resolution 1000 100000 256 3600 Modicity MM SM, M-SM SM M-SM Simultaneous spectra > 300 1 1 1 3D multiplexing No Possible Bad Possible (cid:113) S S = λ , (1) N δ2 + δ2 det phot where: • S is the measured signal on a given spectral channel, which expresses as λ · · · S = ρ T F t, (2) λ λ ≤ with ρ the coupling eﬃciency ( 1 for ﬁbered instruments, =1 for bulk ones). F denotes the spectral ﬂux λ density of the target in one spectral channel. T is the optical throughput of the instrument, including the detector quantum eﬃciency η. It also includes fringe contrast in the case of an FTS (due e.g. to beams imbalance). t is the integration time. • δ is the detector noise linked to detector read-out-noise (RON). det • δ is the photon noise, including the source and the background. Tab. 5 gives typical values of the phot sky brightness in the optical bands. The sky background increasing in proportion of the observed optical ´etendue, AO are very useful by concentrating the source energy on the smallest possible area of the sky. This formula has been adapted to take into account the speciﬁc source noise of each group of spectrometer: Disperser – In a disperser, a pixel is equivalent to a spectral channel. It is then only sensitive to the light from the source F and the background B at the given wavelength. The ﬂux of a spectral channel is in practice λ λ Table 5. Typical sky background ﬂux in visible and IR.27,28 Band V I J H Ks L M − B [mag.arsec 2] 21.6 19.6 16.5 14.4 13.0 3.9 1.2 λ − · · · B for SΩ = λ2 [ph.s 1] 0.2 4 77 730 2 103 9 106 8 107 λ 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Summary

label: caption

text: of

text: identiﬁed

text: Fabry-P´erot

text: and

text: ﬁlter

text: technologies.

label: table

text: Spectral

text: domain

text: [nm]

text: R

text: Spectral

text: resolution

text: Modicity

text: SM,

text: MM

text: MM

text: MM

text: Simultaneous

text: spectra

text: multiplexing

text: Good

text: Good

text: Very

text: good

text: Table

text: Summary

text: of

text: identiﬁed

text: FTS

text: technologies.

text: Spectral

text: domain

text: [nm]

text: R

text: Spectral

text: resolution

text: Modicity

text: MM

text: SM,

text: M-SM

text: SM

text: M-SM

text: Simultaneous

text: spectra

text: multiplexing

text: No

text: Possible

text: Bad

text: Possible

text: S

text: S

text: λ

text: N

text: δ2

text: δ2

text: det

text: phot

text: where:

text: S

text: is

text: the

text: measured

text: signal

text: on

text: a

text: given

text: spectral

text: channel,

text: which

text: expresses

text: as

text: λ

text: S

text: ρ

text: T

text: F

text: t,

text: λ

text: λ

text: with

text: ρ

text: the

text: coupling

text: eﬃciency

text: for

text: ﬁbered

text: instruments,

text: for

text: bulk

text: ones).

text: F

text: denotes

text: the

text: spectral

text: ﬂux

text: λ

text: density

text: of

text: the

text: target

text: in

text: one

text: spectral

text: channel.

text: T

text: is

text: the

text: optical

text: throughput

text: of

text: the

text: instrument,

text: including

text: the

text: detector

text: quantum

text: eﬃciency

text: η.

text: It

text: also

text: includes

text: fringe

text: contrast

text: in

text: the

text: case

text: of

text: an

text: FTS

text: (due

text: e.g.

text: to

text: beams

text: imbalance).

text: t

text: is

text: the

text: integration

text: time.

text: δ

text: is

text: the

text: detector

text: noise

text: linked

text: to

text: detector

text: read-out-noise

text: (RON).

text: det

text: δ

text: is

text: the

text: photon

text: noise,

text: including

text: the

text: source

text: and

text: the

text: background.

text: Tab.

text: gives

text: typical

text: values

text: of

text: the

text: phot

text: sky

text: brightness

text: in

text: the

text: optical

text: bands.

text: The

text: sky

text: background

text: increasing

text: in

text: proportion

text: of

text: the

text: observed

text: optical

text: ´etendue,

text: AO

text: are

text: very

text: useful

text: by

text: concentrating

text: the

text: source

text: energy

text: on

text: the

text: smallest

text: possible

text: area

text: of

text: the

text: sky.

text: This

text: formula

text: has

text: been

text: adapted

text: to

text: take

text: into

text: account

text: the

text: speciﬁc

text: source

text: noise

text: of

text: each

text: group

text: of

text: spectrometer:

text: Disperser

text: In

text: a

text: disperser,

text: a

text: pixel

text: is

text: equivalent

text: to

text: a

text: spectral

text: channel.

text: It

text: is

text: then

text: only

text: sensitive

text: to

text: the

text: light

text: from

text: the

text: source

text: F

text: and

text: the

text: background

text: B

text: at

text: the

text: given

text: wavelength.

text: The

text: ﬂux

text: of

text: a

text: spectral

text: channel

text: is

text: in

text: practice

text: λ

text: λ

text: Table

text: Typical

text: sky

text: background

text: ﬂux

text: in

text: visible

text: and

text: Band

text: V

text: I

text: J

text: H

text: Ks

text: L

text: M

text: B

text: [mag.arsec

text: λ

text: B

text: for

text: SΩ

text: λ2

text: [ph.s

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [4] Bland-Hawthorn, J. and Kern, P., “Molding the ﬂow of light: Photonics in astronomy,” Physics Today 65(5), 31–37 (2012). [5] Bland-Hawthorn, J., Ellis, S., Haynes, R., and Horton, A., “Photonic OH suppression of the infrared night sky: ﬁrst on-sky results,” Anglo-Australian Observatory Epping Newsletter 115, 15 (Feb. 2009). [6] Benisty, M., Berger, J., Jocou, L., Labeye, P., Malbet, F., Perraut, K., and Kern, P., “An integrated optics beam combiner for the second generation VLTI instruments,” Astronomy & Astrophysics 498, 601–613 (May 2009). [7] Huby, E., Perrin, G., Marchis, F., Lacour, S., Kotani, T., Duchˆene, G., Choquet, E., Gates, E. L., Woillez, J. M., Lai, O., F´edou, P., Collin, C., Chapron, F., Arslanyan, V., and Burns, K. J., “FIRST, a ﬁbered aperture masking instrument. I. First on-sky test results,” Astronomy & Astrophysics 541, A55 (May 2012). [8] Jovanovic, N., Tuthill, P. G., Norris, B., Gross, S., Stewart, P., Charles, N., Lacour, S., Ams, M., Lawrence, J. S., Lehmann, A., Niel, C., Robertson, J. G., Marshall, G. D., Ireland, M., Fuerbach, A., and Withford, M. J., “Starlight demonstration of the Dragonﬂy instrument: an integrated photonic pupil-remapping in- terferometer for high-contrast imaging,” Monthly Notices of the Royal Astronomical Society 427, 806–815 (Nov. 2012). [9] Leon-Saval, S. G., Birks, T. A., Bland-Hawthorn, J., and Englund, M., “Multimode ﬁber devices with single-mode performance,” Opt. Lett. 30, 2545–2547 (Oct 2005). [10] Noordegraaf, D., Skovgaard, P. M., Nielsen, M. D., and Bland-Hawthorn, J., “Eﬃcient multi-mode to single-mode coupling in a photonic lantern,” Optics Express 17, 1988–1994 (Feb. 2009). [11] Bacon, R., Adam, G., Baranne, A., Court`es, G., Dubet, D., Dubois, J.-P., Georgelin, Y., Monnet, G., Pecontal, E., and Urios, J., “The Integral Field Spectrograph TIGER,” in [European Southern Observatory Conference and Workshop Proceedings], Ulrich, M.-H., ed., European Southern Observatory Conference and Workshop Proceedings 30, 1185 (1988). [12] Fogarty, L. M. R., Xiao, S. Z. ., Funamoto, J., Cairns, I. H., Bland-Hawthorn, J., Wu, X., Betters, C. H., Leon-Saval, S. G., and Monger, A. G., “The i-inspire satellite: a university pico-satellite project,” 84421B– 84421B–7 (2012). [13] Diolaiti, E., Conan, J.-M., Foppiani, I., Lombini, M., Marchetti, E., Petit, C., Robert, C., Schreiber, L., Baruﬀolo, A., Bellazzini, M., Bregoli, G., Ciliegi, P., Cosentino, G., Biliotti, V., D’Odorico, S., Fusco, T., Hubin, N., Meimon, S., and Sauvage, J.-F., “Towards the phase A review of MAORY, the multi-conjugate adaptive optics module for the E-ELT,” in [Adaptative Optics for Extremely Large Telescopes], (2010). [14] Fusco, T., Meimon, S., Clenet, Y., Cohen, M., Schnetler, H., Pauﬁque, J., Michau, V., Amans, J.-P., Gratadour, D., Petit, C., Robert, C., Jagourel, P., Gendron, E., Rousset, G., Conan, J.-M., and Hubin, N., “ATLAS: the E-ELT laser tomographic adaptive optics system,” in [Proc. of SPIE], 7736 (July 2010). [15] Betters, C. H., Leon-Saval, S. G., Robertson, J. G., and Bland-Hawthorn, J., “Beating the classical limit: A diﬀraction-limited spectrograph for an arbitrary input beam,” Opt. Express 21, 26103–26112 (Nov 2013). [16] Grabarnik, S., Wolﬀenbuttel, R., Emadi, A., Loktev, M., Sokolova, E., and Vdovin, G., “Planar double- grating microspectrometer,” Optics Express 15, 3581–3588 (Mar. 2007). [17] Avrutsky, I., Chaganti, K., Salakhutdinov, I., and Auner, G., “Concept of a miniature optical spectrometer using integrated optical and micro-optical components,” Appl. Opt. 45, 7811–7817 (Oct 2006). [18] Lupu, A., Cassan, E., Laval, S., Melhaoui, L. E., Lyan, P., and Fedeli, J., “Experimental evidence for superprism phenomena in soi photonic crystals,” Opt. Express 12, 5690–5696 (Nov 2004). [19] Momeni, B., Hosseini, E. S., Askari, M., Soltani, M., and Adibi, A., “Integrated photonic crystal spectrom- eters for sensing applications,” Optics Communications 282(15), 3168 – 3171 (2009). [20] Murry, S. and Lin, C., “Reﬂectively coupled zigzag waveguide device for wavelength locking,” (Dec. 20 2001). US Patent App. 10/029,058. [21] Wolﬀenbuttel, R. F., “Mems-based optical mini- and microspectrometers for the visible and infrared spectral range,” Journal of Micromechanics and Microengineering 15(7), S145 (2005). [22] Laux, E., Genet, C., Skauli, T., and Ebbesen, T., “Plasmonic photon sorters for spectral and polarimetric imaging,” Nature Photonics 2(3), 161–164 (2008). 8

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text: ∗ † Miguel A. Caro1,1, (Dated: May 26, 2020) I propose a straightforward generalization of the projection scheme for elastic tensors introduced by Moakher and Norris [J. Elasticity 85, 215 (2006)] that takes into account also rotations. The “closest” tensor of any desired symmetry to the original tensor of lower symmetry is “closer” in this generalized scheme. The method has an important application in the context of the special quasir- andom structure (SQS) method for the computational modeling of alloys, whereby the supercell’s symmetry, and therefore that of the tensors representing its properties, is reduced with respect to the material’s underlying symmetry. The approach allows to extract the tensor components most representative of the macroscopic symmetry of the material. Although the approach is general, in the present case I apply it to the elastic tensor and give numerical examples. Simple approximate analytical expressions for cubic materials are also provided. Keywords: SQS; elasticity; projection; tensor; rotation In the context of material science, many material prop- one arises from the speciﬁc cubic symmetry of the crystal erties are represented by tensors of diﬀerent ranks. The class, whereby all three Cartesian directions are equiva- electric polarization is given by a rank-1 tensor P , strain lent within this preferential reference frame (I shall come i is represented by a rank-2 tensor (cid:15) , the piezoelectric back to this later on), and therefore C = C = C , ij 12 13 23 response is given by a rank-3 tensor e , elasticity can C = C = C and C = C = C . ijk 11 22 33 44 55 66 be described with a rank-4 tensor C , and so on. The When using the SQS approach, e.g. to study the elastic ijkl geometrical particularities of the diﬀerent materials give properties of a cubic alloy, one is left using a “pseudo- rise to the diﬀerent symmetries of the tensor represent- cubic” unit cell which could have its symmetry reduced ing their properties, as determined by the material’s to anything lower than cubic, including triclinic (i.e. no point group: hexagonal, cubic, etc. In some cases, these symmetry at all). Tasn´adi et al.4 used the SQS approach symmetries are “slightly” broken, and materials with an together with Moakher and Norris’s projector scheme,5 underlying expected symmetry might present deviations that I shall discuss in more detail in the following, to × due to imperfections, impurities, alloying eﬀects, and study cubic Ti Al N alloys. One of the triclinic 4 × 0.5 0.5 similar. Within the frame of computational materials 3 2 supercells gave the following stiﬀness tensor (in science it is common practice, in part due to compu- GPa):   tational limits, to use ﬁnite size supercells to represent   actual materials. The special quasirandom structure    436 161 160 12 11 25  (SQS) method1 is a common approach when modeling    453 160 4 15 1  alloys in the context of ab initio calculations, such as C ≡  428 13 3 8  density functional theory (DFT).2,3 The use of these tric , (2) 188 12 9 SQS leads to supercells whose properties slightly deviate SYM 186 9 from the ones expected from the crystallographic class 189 of the macroscopic alloy. These are due to the small de- viations of the ﬁnite-size cell with respect to the perfect which resembles a cubic stiﬀness tensor [Eq. (1)] but supercell geometry of the macroscopic material. Take strictly speaking has inherited the triclinic symmetry (no as an example the case of the stiﬀness tensor of a cubic symmetry) of the parent supercell. In order to extract material which, employing Voigt notation for the sake of the cubic “part” of Eq. (2), Tasn´adi et al. resorted to convenience, can be written in matrix form as the projection scheme introduced by Moakher and Nor-   ris. This method relies on projector operators Psym that   C  C C C 0 0 0  project any given tensor, for instance tric in Eq. (2),  11 12 12   C C C 0 0 0  onto the closest tensor of a higher symmetry of choice C ≡  C12 C11 C12 0 0 0  Csym, by means of minimizing the Euclidean distance be- cub 12 12 11 , (1) (cid:107)C − C (cid:107) 0 0 0 C 0 0 tween the two, tric cub in this case. The reader is 44 0 0 0 0 C 0 referred to Ref. 5 for the details of the method. Moakher 44 0 0 0 0 0 C and Norris’s approach makes for an elegant and powerful 44 formalism. In the particular case of Eq. (2), the projec- C C where the elements cub (with uppercase indices in- tion onto cub is done using Pcub (see Refs. 4 and 5 for IJ dicating Voigt notation) are the elastic constants of the its speciﬁc form) as material. There are two sets of symmetries applicable to C cub. The ﬁrst one is the general intri↔nsic symmetry of Ccub = PcubCtric, (3) the stiﬀness tensor upon exchange of I J. The second

text: Miguel

label: author

text: A.

text: (Dated:

text: May

label: date

text: I

label: abstract

text: propose

text: a

text: straightforward

text: generalization

text: of

text: the

text: projection

text: scheme

text: for

text: elastic

text: tensors

text: introduced

text: by

text: Moakher

text: and

text: Norris

text: [J.

text: Elasticity

text: that

text: takes

text: into

text: account

text: also

text: rotations.

text: The

text: “closest”

text: tensor

text: of

text: any

text: desired

text: symmetry

text: to

text: the

text: original

text: tensor

text: of

text: lower

text: symmetry

text: is

text: “closer”

text: in

text: this

text: generalized

text: scheme.

text: The

text: method

text: has

text: an

text: important

text: application

text: in

text: the

text: context

text: of

text: the

text: special

text: quasir-

text: andom

text: structure

text: (SQS)

text: method

text: for

text: the

text: computational

text: modeling

text: of

text: alloys,

text: whereby

text: the

text: supercell’s

text: symmetry,

text: and

text: therefore

text: that

text: of

text: the

text: tensors

text: representing

text: its

text: properties,

text: is

text: reduced

text: with

text: respect

text: to

text: the

text: material’s

text: underlying

text: symmetry.

text: The

text: approach

text: allows

text: to

text: extract

text: the

text: tensor

text: components

text: most

text: representative

text: of

text: the

text: macroscopic

text: symmetry

text: of

text: the

text: material.

text: Although

text: the

text: approach

text: is

text: general,

text: in

text: the

text: present

text: case

text: I

text: apply

text: it

text: to

text: the

text: elastic

text: tensor

text: and

text: give

text: numerical

text: examples.

text: Simple

text: approximate

text: analytical

text: expressions

text: for

text: cubic

text: materials

text: are

text: also

text: provided.

text: Keywords:

text: SQS;

text: elasticity;

text: projection;

text: tensor;

text: rotation

text: In

text: the

text: context

text: of

text: material

text: science,

text: many

text: material

text: prop-

text: one

text: arises

text: from

text: the

text: speciﬁc

text: cubic

text: symmetry

text: of

text: the

text: crystal

text: erties

text: are

text: represented

text: by

text: tensors

text: of

text: diﬀerent

text: ranks.

text: The

text: class,

text: whereby

text: all

text: three

text: Cartesian

text: directions

text: are

text: equiva-

text: electric

text: polarization

text: is

text: given

text: by

text: a

text: tensor

text: P

text: strain

text: lent

text: within

text: this

text: preferential

text: reference

text: frame

text: (I

text: shall

text: come

text: i

text: is

text: represented

text: by

text: a

text: tensor

text: the

text: piezoelectric

text: back

text: to

text: this

text: later

text: on),

text: and

text: therefore

text: C

text: C

text: C

text: ij

text: response

text: is

text: given

text: by

text: a

text: tensor

text: e

text: elasticity

text: can

text: C

text: C

text: C

text: and

text: C

text: C

text: C

text: ijk

text: be

text: described

text: with

text: a

text: tensor

text: C

text: and

text: so

text: on.

text: The

text: When

text: using

text: the

text: SQS

text: approach,

text: e.g.

text: to

text: study

text: the

text: elastic

text: ijkl

text: geometrical

text: particularities

text: of

text: the

text: diﬀerent

text: materials

text: give

text: properties

text: of

text: a

text: cubic

text: alloy,

text: one

text: is

text: left

text: using

text: a

text: “pseudo-

text: rise

text: to

text: the

text: diﬀerent

text: symmetries

text: of

text: the

text: tensor

text: represent-

text: cubic”

text: unit

text: cell

text: which

text: could

text: have

text: its

text: symmetry

text: reduced

text: ing

text: their

text: properties,

text: as

text: determined

text: by

text: the

text: material’s

text: to

text: anything

text: lower

text: than

text: cubic,

text: including

text: triclinic

text: (i.e.

text: no

text: point

text: group:

text: hexagonal,

text: cubic,

text: etc.

text: In

text: some

text: cases,

text: these

text: symmetry

text: at

text: all).

text: Tasn´adi

text: et

text: used

text: the

text: SQS

text: approach

text: symmetries

text: are

text: “slightly”

text: broken,

text: and

text: materials

text: with

text: an

text: together

text: with

text: Moakher

text: and

text: Norris’s

text: projector

text: underlying

text: expected

text: symmetry

text: might

text: present

text: deviations

text: that

text: I

text: shall

text: discuss

text: in

text: more

text: detail

text: in

text: the

text: following,

text: to

text: due

text: to

text: imperfections,

text: impurities,

text: alloying

text: eﬀects,

text: and

text: study

text: cubic

text: Ti

text: Al

text: N

text: alloys.

text: One

text: of

text: the

text: triclinic

text: similar.

text: Within

text: the

text: frame

text: of

text: computational

text: materials

text: supercells

text: gave

text: the

text: following

text: stiﬀness

text: tensor

text: (in

text: science

text: it

text: is

text: common

text: practice,

text: in

text: part

text: due

text: to

text: compu-

text: GPa):

text: tational

text: limits,

text: to

text: use

text: ﬁnite

text: size

text: supercells

text: to

text: represent

text: actual

text: materials.

text: The

text: special

text: quasirandom

text: structure

text: (SQS)

text: is

text: a

text: common

text: approach

text: when

text: modeling

text: alloys

text: in

text: the

text: context

text: of

text: ab

text: initio

text: calculations,

text: such

text: as

text: C

text: density

text: functional

text: theory

text: The

text: use

text: of

text: these

text: tric

text: SQS

text: leads

text: to

text: supercells

text: whose

text: properties

text: slightly

text: deviate

text: SYM

text: from

text: the

text: ones

text: expected

text: from

text: the

text: crystallographic

text: class

text: of

text: the

text: macroscopic

text: alloy.

text: These

text: are

text: due

text: to

text: the

text: small

text: de-

text: viations

text: of

text: the

text: ﬁnite-size

text: cell

text: with

text: respect

text: to

text: the

text: perfect

text: which

text: resembles

text: a

text: cubic

text: stiﬀness

text: tensor

text: [Eq.

text: but

text: supercell

text: geometry

text: of

text: the

text: macroscopic

text: material.

text: Take

text: strictly

text: speaking

text: has

text: inherited

text: the

text: triclinic

text: symmetry

text: (no

text: as

text: an

text: example

text: the

text: case

text: of

text: the

text: stiﬀness

text: tensor

text: of

text: a

text: cubic

text: symmetry)

text: of

text: the

text: parent

text: supercell.

text: In

text: order

text: to

text: extract

text: material

text: which,

text: employing

text: Voigt

text: notation

text: for

text: the

text: sake

text: of

text: the

text: cubic

text: “part”

text: of

text: Eq.

text: Tasn´adi

text: et

text: al.

text: resorted

text: to

text: convenience,

text: can

text: be

text: written

text: in

text: matrix

text: form

text: as

text: the

text: projection

text: scheme

text: introduced

text: by

text: Moakher

text: and

text: Nor-

text: ris.

text: This

text: method

text: relies

text: on

text: projector

text: operators

text: Psym

text: that

text: C

text: C

text: C

text: C

text: project

text: any

text: given

text: tensor,

text: for

text: instance

text: tric

text: in

text: Eq.

text: C

text: C

text: C

text: onto

text: the

text: closest

text: tensor

text: of

text: a

text: higher

text: symmetry

text: of

text: choice

text: C

text: Csym,

text: by

text: means

text: of

text: minimizing

text: the

text: Euclidean

text: distance

text: be-

text: cub

text: C

text: C

text: tween

text: the

text: two,

text: tric

text: cub

text: in

text: this

text: case.

text: The

text: reader

text: is

text: C

text: referred

text: to

text: Ref.

text: for

text: the

text: details

text: of

text: the

text: method.

text: Moakher

text: C

text: and

text: Norris’s

text: approach

text: makes

text: for

text: an

text: elegant

text: and

text: powerful

text: formalism.

text: In

text: the

text: particular

text: case

text: of

text: Eq.

text: the

text: projec-

text: C

text: C

text: where

text: the

text: elements

text: cub

text: (with

text: uppercase

text: indices

text: in-

text: tion

text: onto

text: cub

text: is

text: done

text: using

text: Pcub

text: (see

text: Refs.

text: and

text: for

text: IJ

text: dicating

text: Voigt

text: notation)

text: are

text: the

text: elastic

text: constants

text: of

text: the

text: its

text: speciﬁc

text: form)

text: as

text: material.

text: There

text: are

text: two

text: sets

text: of

text: symmetries

text: applicable

text: to

text: C

text: cub.

text: The

text: ﬁrst

text: one

text: is

text: the

text: general

text: intri↔nsic

text: symmetry

text: of

text: Ccub

text: PcubCtric,

text: the

text: stiﬀness

text: tensor

text: upon

text: exchange

text: of

text: I

text: J.

text: The

text: second