Experiment and Theoretical Study of Critical Behavior in Magnetic Multilayers

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DOI 10.1007/s10948-012-1546-9 O R I G I N A L PA P E R

Experiment and Theoretical Study of Critical Behavior in Magnetic Multilayers

R. Masrour·M. Hamedoun·A. Benyoussef·H. Lassri

Received: 24 January 2012 / Accepted: 26 March 2012 / Published online: 14 April 2012

Abstract Ni/NM multilayers (with noble metal NM=Au, Ag and Cu) were prepared by the electron beam evaporation method under ultra high vacuum conditions. The magnetic properties of Ni/NM multilayers are examined as a function of Ni layer thickness t_Ni. The temperature dependence of the spontaneous magnetizationM(T )is well described by aT^3/2law in all multilayers. A spin-wave theory has been used to explain the temperature dependence of the magnetization and the approximate values for the bulk Exchange interactionJ_band surface exchange interactionJ_s for various Ni layer thicknesses have been obtained.

In the other hand we have used the high-temperature series expansion technique, to analyze the phase transition and the critical phenomena of a ferromagnetic a two-component multilayer, through three models: Ising,XYand Heisenberg.

The critical reduced temperatureτ_c(ν)is studied as function of the thickness of constituents in the unit cell of the mul-

R. Masrour (⁾

Laboratory of Materials, Processes, Environment and Quality, National School of Applied Sciences, Cady Ayyed University, Route Sidi Bouzid, BP 63, 46000 Safi, Morocco

e-mail:rachidmasrour@hotmail.com R. Masrour·A. Benyoussef

LMPHE, Faculté des Sciences, Université Mohamed V, Rabat, Morocco

M. Hamedoun·A. Benyoussef

Institute for Nanomaterials and Nanotechnologies, Rabat, Morocco

M. Hamedoun·A. Benyoussef

Academie Hassan II des Sciences et Techniques, Rabat, Morocco H. Lassri

LPMMAT, Faculté des Sciences Ain Chock, Université Hassan II, Casablanca, Morocco

tilayer. In the two-component multilayerτ_c(ν)is studied as function of the exchange interaction in each material and within the interfaceJs, Jb andJ_⊥, respectively. A critical value of the surface exchange interaction in the film and interface exchange interaction in the multilayer above which the surface and the interface magnetism appears is obtained.

The dependence of the reduced critical temperature on the thickness of the film and the unit cell of multilayer has been investigated. The effects of an amorphous magnetic surface on the critical properties of the film of simple cubic lattice have been studied. A number of characteristic behaviors, such as the possibility of the existence of a critical length of the unit cell thickness at which the temperature of the multilayer remains insensitive to the exchange coupling within interface, are reported.

In a defined range of the exchange interactions, the values ofγ are comparable to the universal ones and are indepen- dent of the film thickness. The asymmetry of the structure and the competition of the effects of the exchange coupling are important for the magnetic properties of the system.

Keywords Ni/(Cu, Au, Ag) multilayers·Magnetization· Spin wave excitations·High-temperature series

expansions·Non-equilibrium thermodynamics·Exchange interactions·Phase diagram

1 Introduction

During the last decade, considerable effort has been made to come to an understanding of magnetic films, layered structures and multilayers [1–12]. Within the advance of modern vacuum sciences, and in particular epitaxial techniques, it is now possible to grow in very controlled way, magnetic films with few atomic layers or even monolayer atop nonmagnetic

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substrate [13–15]. A multilayer in which the atoms vary from one monolayer to another can also be envisaged. In layered ferromagnetic materials, it has been found experimen- tally that one can obtain a rich variety of magnetic behavior depending on the materials, the thickness and the number of slabs and the applied field [16–19]. A number of theoretical works have been devoted to the magnetic and phase transition properties of magnetic thin film. Mean-field theories have been used [20,21], which are particularly simple to ap- ply for complicated systems, i.e., with non-equivalent lattice sites. However, these approaches completely neglect both thermal and quantum fluctuations, which are crucial for low- dimensional systems. Hu and Kawazoe [22] have treated the magnetic reorientation of ferromagnetic thin films within a continuum approach for the system and mean-field-like ex- pressions for the magnetization density. Saber et al. [23]

have examined the phase transition of a diluted spin-1/2 Ising film using the effective field theory with a probability distribution technique that account for the single-site spin correlations. Lin et al. [24] have developed an analytical method based on the variational cumulant expansion to calculate the critical temperature of the Ising film as a function of the film thickness. The semi-infinite Ising film with surface amorphization has been studied by Kaneyoshi [25] using the effective field theory and by Ilkovic [26] with the aid of reaction mean-field theory. These authors have found a number of characteristic behaviors for the surface magnetic properties, such as the possibility of the surface re-entrant phenomenon. The magnetic properties of anisotropic thin film have been studied using many-body Green’s functions, which allow calculations in the whole temperature range of interest [27–29]. Diep et al. [27] have shown how surface spin waves and surface anisotropy affect the critical temperature of ferromagnetic and antiferromagnetic thin film.

The condition for the appearance of the so-called “hard surface” and “pinned spin wave” is reported. Diep [28] has analysed the dependence of the surface magnetization and the critical temperature of ferromagnetic thin film with the free surfaces, on the surface exchange integrals. The same author has investigated the effects of quantum fluctuations on the layer magnetizations in the vicinity of the surface at an arbitrary temperature in bcc antiferromagnetic thin film with (001) surfaces [29]. Heneluis et al. [30] have used the quantum Monte Carlo method to study ferromagnetic thin film, described by a Heisenberg model including local anisotropies. They examined many interesting properties, of which we only mentioned the reorientation effect, in which the magnetization changes from out of plane to in plane.

On other hand, the theorem of Mermin and Wagner [31] proved that long-range magnetic order cannot ex- ist at any finite temperature in one-dimensional (1D) and two-dimensional (2D) magnet that is described purely by

the Heisenberg Hamiltonian with isotropic short-range interactions. The ferromagnetic order is destroyed by long- wavelength spin waves [32]. Recently, this theory has been extended to the case of magnetic thin films within three main many-body models, namely: Heisenberg, Hubbard and s–f model [33]. However, different anisotropy constraints acting on spins at the surface and in the bulk can induce the ordered phase in the thin film without any conflicting with the theorem of Mermin and Wagner. The phase transition is possible in thin Ising system, since the Ising model of magnetism uses a highly anisotropic spin-spin interaction. Ising spins do not rotate through all possible orientations, but instead are restricted to a particular axis and give rise to a finite magnetization below the critical temperature even in the absence of an external field [34]. Stanley [35] has obtained a phase transition atT_c=0 K for triangular and square lattices, using the high-temperature series expansions (HTSE) method and using a diagrammatic representation [36]. The diagrams whose contribution is nonvanishing are listed, in Fig.1, until the order 6. He argued that the magnetic order may occur at low-temperature state with zero spontaneous magnetization but with infinite zero-field susceptibility. In fact, the general expression of spin–spin correlation obtained by the semiclassical method of HTSE [36] takes into account the influence of the other neighbor spins. Then, the spin–spin correlation becomes different from the bulk to the surfaces and leads to an anisotropic behavior, which may be found in thermodynamic functions such as the magnetic susceptibil- ityχ. The consideration of some neighboring spins in the calculation of the correlation functions in the HTSE method can be understood as long-range interaction, which depress the effect of quantum and thermal fluctuations, and con- tribute to the ordered phase in the system with low dimen- sionality [37].

The so-called magnetic multilayers are defined as a periodic layered structure with alternating layers having different magnetic properties. Phase transitions in the multilayers and multilayers have their own behaviors, which are different from those in the bulk materials. They have been investigated by use of various theoretical methods. Fishman et al.

[38] investigated a periodic multilayer consisting of two ferromagnetic materials in a theory based on general Ginzburg- Landau formulation. They obtained the transition temperature and spin-wave spectra. The Landau formalism of Cam- ley and Tilley [39] has been applied to calculate the critical temperature in this system [40]. For more complicated multilayers with an arbitrary number of different layers in an elementary unit, Barnas [41] has derived some general dis- persion equations for the bulk and surface polaritons. These equations are then applied to magnetostatic modes and re- tarded wave propagation in the Voigt geometry [42]. Re- cently, Sy and Ow [43] using the mean-field approximation, and Seidov and Shaulov [44] using effective field theory with the differential operator technique, studied the phase

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Fig. 1 The diagrams of Stanley Kaplan [36] whose contribution is non vanishing, until the order 6

transition in an alternating magnetic multilayer. Saber et al.

[45] using the effective field theory with a probability distribution technique that accounts for the self-spin correlation functions, studied the critical properties in a magnetic multilayer consisting of two ferromagnetic materials with different bulk transition temperatures. The critical temperatures were obtained as functions of the site dilution and thickness of the multilayer with various exchange interactions in the same material and across the interface.

Furthermore, due to the immiscibility of the noble metal NM (NM=Au,Ag and Cu) and Ni, the Ni/NM system is an excellent one to investigate with nearly ideal artificially structures Ni/NM multilayers with flat and sharp layer interfaces are possible using the electron beam evaporation. In this paper we have applied the spin-wave theory to explain the temperature dependence of the magnetization and the approximate values for the bulk exchange interactionJ_band surface exchange interactionJs for various Ni layer thicknesses have been obtained.

In the other hand, we present a theoretical study of the critical properties of magnetic thin film, semi-infinite film and infinite alternating multilayer using the high- temperature series expansions (HTSE) extrapolated with

Padé approximants method (PA) [46]. We have considered three kinds of model namely Ising, XY and Heisenberg types. In the magnetic multilayer we have considered the (n.n.) exchange in the constituentsJs and Jb, and the exchange coupling between (n.n.) spins across the interface J_⊥(see Fig.2). Our intention is to study the effects of different exchange couplings on the critical temperature of the multilayer and on the critical exponent associated with the magnetic susceptibility. The technique of HTSE given by Stanley and Kaplan (SK) [47,48] has the following advan- tages: first, we can exactly deal with any symmetry of the magnetic structure; second, we can easily treat the Ising,XY and Heisenberg models in a unified way; third, this method is more fundamental than usual molecular field approach as it takes into consideration the spin–spin correlation. The technique of HTSE extrapolated with the PA method have been widely developed and applied to various magnetic systems. It provides valid estimations of critical temperature for real magnetic system [49,50]. The method considered here is more fundamental than usual molecular field approach as it partially takes into consideration spin–spin correlations.

The paper is organized as follows: In Sect.3, some discussions of the essentials the HTSE method are given.

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Fig. 2 Two-dimensional cross section of unit cell of infinite multilayer composed of two ferromagnetic materialsAandB, wheret=t_a+t_b is the thickness of the cell

In Sect.4, we present the results we have obtained, in the case of multilayer, for as set of the reduced exchange couplings R1= ^J_J^b_s andR2= ^J_J^⊥_s, thickness t_b. The phase diagram which describes the critical parameter of interface reduced couplingR₂^cin terms of the ratio of interactionsR1

in the two slabs is calculated and discussed. A discussion is given to explain the behavior of the critical temperature compared to the bulk one, taking into account the competition between the magnetism of the two constituents. The possibility of existence of critical length t^c above which the temperature of the system remains insensitive to the exchange coupling within the interface is reported. The critical exponentγ, associated with the magnetic susceptibility is estimated and is plotted as a function of the ratioR₂for one the simple cubic lattice ferromagnetic thin film. The vari- ation of γ is discussed in view of the structure symmetry.

Finally, a conclusion is given in Sect.5.

2 Experiment Methods

Ni/NM multilayers (with noble metal NM=Au,Ag and Cu) were grown by evaporation in ultrahigh vacuum under controlled conditions, and the pressure during the film deposition was maintained in the range 3–5×10⁻⁹ Torr. The deposition rate (about 0.3 Å/s) and the final thickness were

Fig. 3 XRD pattern of a Ni/Au multilayer witht_Ni=10 Å. The inset shows Kiessig’s interferences. The positions of reflections diffracted by the crystallographic planes (111) of Au and Ni are identified by arrows

monitored by precalibrated quartz oscillators. The Ni-layer thicknesstNi was varied from 10 to 55 Å and that oftAu, t_Ag and t_Cu were kept fixed at 15, 50 and 20 Å, respectively. Samples were deposited on glass substrates at 300 K on a NM buffer layer 100 Å thick. The top layer in all the samples was NM 20 Å thick. The number q of bilayers was varied from 10 to 30 and the growth parameters will be designated as(t_Ni/t_NM)_q. Low angle X-ray diffraction of all the samples revealed peaks typical of the modulated structure and theX-ray diffraction in the high angle range 30^◦<2θ <50^◦ showed the existence of the fcc Ni(111) peak (Fig.3). Magnetization M was measured using a vi- brating sample magnetometer under magnetic fields up to 1 T and in the temperature range 5 to 300 K.

3 Theoretical Method

3.1 Spin Waves

To understand better how the exchange coupling between neighboring Ni layers affects the magnetic behavior of these films, we extended the model for spin waves in ferromagnetic thin films proposed by Pinnettes and Lacroix [51,52]

to the ferromagnetic/nonmagnetic multilayers case. Here the multilayer (Xn/Y_m)_q is supposed to be formed by an al- ternate deposition of a magnetic layer (X) and nonmagnetic one (Y). The multilayer is characterized by the num- berq of bilayers (X/Y), the numbernof atomic planes in the magnetic layerμ and the numberm of atomic planes in the nonmagnetic layer. We chose the lattice unit vec- tors (eX,eY,eZ)so thateZ is perpendicular to the atomic

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planes. We denote by S_iαμ the spin operator of the atom i (i=1,2, . . . , N) in the plane α (α=1,2, . . . , n) of the magnetic layerμ (μ=1,2, . . . , q).

The system Hamiltonian is given byH=H_e+H_a, H_e describes the exchange interactions in the same magnetic layer (bulk and surface) as well as the exchange interactions between adjacent magnetic layers:

He= −Jb

_b

iαμ,j αμ

SiαμSj αμ+

iαμ,j αμ

SiαμS_{j α}μ

−Js

S iαμ,j αμ

SiαμSj αμ−JI

I iαμ,j αμ

SiαμS_{j α}μ

(1) whereJ_bandJ_s are the bulk and surface exchange interactions.J_⊥ is the interlayer coupling strength which depends on the number m of atomic planes in the nonmagnetic layer.

The contribution of the surface anisotropy is given by Ha=D^⊥

s iαμ

S_iα^Z2

+D s iαμ

S^X_iα2

− S_iα^Y2

(2) whereD^⊥andDare the surface anisotropy parameters for the uniaxial out of plane and in plane components. Further we denote by Σ^Ξ the summation on the sites of the bulk layer planes (Ξ =b), surface layer planes (Ξ =s) or the surfaces planes coupled via the nonmagnetic layer (Ξ=I).

The symbol denotes the pairs of nearest-neighbor atoms or adjacent magnetic planes.

In the Holstein–Primakoff formulation [53], the creation and annihilation operators (aiαμ anda_iαμ⁺ ) for each atomic spin are related to the spin operators by

S_iαμ^Z +iS_iαμ^X =(2S)¹²fiαμ(2S)aiαμ and

(3) S_iαμ^Z −iS_iαμ^X =(2S)¹²a_iαμ⁺ f_iαμ(2S).

In the frame work of noninteracting spin-wave theory, the linear approximation of the Holstein-Primakoff method is sufficient to describe the main magnetic behavior and the correction terms are quite small at low temperatures (T <

T_C/3) [54,55]. So, the value off_iαμ(2S)is fixed to 1.

We replace the atomic variables (aiαμ, a_iαμ⁺ ) by the magnon variables (bkαμ, b⁺_kαμ) after a two-dimensional Fourier transformation. It gives

H=H0+A S kαμ

b_kαμb₋_kαμ+b⁺

kαμb⁺

−kαμ

+ S kαμ

B_kb⁺

kαμb_kαμ + b kαμ

C_kb⁺

kαμb_kαμ

+

k(αμ,αμ)

Dkb⁺

kαμb_kα_μ+ I

k(αμ,αμ)

Ekb⁺

kαμb_kα_μ (4)

where A=S 2

D^⊥−D B_k=2S

J_s

n−λ_k

+J_bn^⊥_S +J_In +S

3D+D^⊥ C_k=2JbS

n−λ_k +n^⊥_V Dk= −JbSλ

k

E_k= −J_ISλ

k

H₀ is a constant term, the coefficients λ_k and λ_k depend on the crystallographic structure of the magnetic layer.n represents the number of nearest-neighbor sites in the same atomic plane, whilen^⊥_S andn^⊥_V are the numbers of surface and volume nearest neighbors in the adjacent plane in the same magnetic layer, respectively. For a given site in the surface plane of the magnetic layer,n represents the number of the nearest-neighbor sites in the adjacent layer across the nonmagnetic layer. For fcc (111) (n=6, n^⊥_S =3 and n^⊥_V =6) with the lattice constantaand in the case where the nonmagnetic layer does not disturb the succession order of the magnetic atomic planes (n =3):

λ_k=4 cos(akx√6/4)cos(aky√2/4) +2 cos(aky√2/2),

λ_k=4 cos(akx√6/12)cos(aky√2/4) +2 cos(akx√6/6)

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The motion equation is given by

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ i

∂b_kαμ

∂t = [b_kαμ, H] i

∂b⁺

kαμ

∂t = b⁺

−kαμ, H (6)

The spin system is characterized by 2nq×2nq equations, and then we have the resulting secular equation:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

(Ck+Bk+ωkαμ)bkαμ+Dkb_kαμ+Ekb_kαμ

+2Ab⁺_−kαμ=0

2Abkαμ+D_kb⁺₋_kα_μ+E_kb₋⁺_kα_μ +(Ck+Bk−ωkαμ)b⁺₋_kαμ=0

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We consider then×qpositive ones which correspond to the n×qmagnon excitation branchesω_k^r (r=1,2, . . . , n×q).

These branches can be classified intongroups ofq quasi- degenerate components in the usual case whereJ_I remain sufficiently small compared to the effective intralayer exchange strength (Fig.4). The reduced magnetization versus temperature is computed numerically from:

m(T )=1− 1 N_knqS

k,r

1 exp(^ω

r k

k_BT)−1

(8) The coefficientN_kindicates the number ofkpoints taken in the first Brouillin zone. The zero point fluctuations effects are neglected.

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Fig. 4 Spin-wave excitation spectrum vs. k_x (k_y=k_x√

2) for fcc (111) ferromagnetic multilayer witht_Ni=12 Å,S=0.3,J_s=80 K, J_b=250 K, J_I =0.01 K, D=0.1 K, D^⊥=0 K; in the case:

J_b/J_s=3

3.2 High-Temperature Series Expansions

The theoretical method used in this work has been developed in previous papers [49,50], here we give only a brief description of the essentials of the method.

The starting point of the semiclassical HTSE technique is the expansion of the zero-field static correlation function, between spins at site i and j, in power of the inverse of temperatureβ=_K¹_B_T [47,48]:

S_iS_j =TrS_iS_je⁻^βH Tre^−βH =

∞ l=0

(−1)^l

l! α_lβ^l (9)

where Tr means the trace over spin configurations andK_B is the Boltzmann constant.H= −2

ijJ_ijS_i· S_jstands for the isotropic Heisenberg Hamiltonian where Jij is the exchange coupling between spin at site i and spin at site j, andSis the spin operator with the magnitudeS.

The calculation of the coefficientsα_l leads to a diagrammatic representation [47,48], which involves two separate phases:

(a) The finding and cataloguing of all diagrams or graphs which can be constructed from one dashed line connect- ing the siteiandj, andl straight lines, and the deter- mination of the diagrams whose contribution is nonvanishing. This step has already been accomplished in the Stanley and Kaplan (SK) work [47,48].

(b) Counting the number of times that each diagram can occur in the magnetic system.

In our case, we have to deal with nearest-neighbor coupling J_ij. The coefficientα_l may be expressed for each topological graph as

α_l= ¯S²

−2S¯²l J_ik^m¹

1J_k^m²

2k₃· · ·J_k^m^v

wj

[α_l] (10)

withS¯²=S(S+1)and the condition_v

r=1m_r=l for m_r=0,1, . . . , l. The “weight”[α_l]of each graph is tab- ulated and given in Ref. [47,48] and k1, k2, k, . . . , kw

represent the sites surrounding the sitesiandj. The method of SK is general and can be applied to any lattice. This semiclassical treatment is a simplification of the more complex procedure of Rushbrooke and Wood [55]

used for the calculation of the magnetic susceptibility in the quantum-mechanical case. SK argued that useful results in the critical region(T ∼=Tc)could be obtained from the clas- sical Heisenberg model, the errors in various critical properties of interest being small and decreasing rapidly with spin valueS[47,48].

Here HTSE method is developed for magnetic susceptibility per spin χ (T ) with arbitrary exchange interaction couplings (Jb, Js andJ_⊥) up to order 6 in β. We obtain the following relation:

χ (T )=gμ²_BβS(S+1) N

6 n=0

_n

p=0

n q=0

a(p, q, n)R^p₁R^q₂

τ⁻ⁿ (11) whereR₁=J_b/J_s, R₂=J_⊥/J_s, τ =k_BT /2S(S+1)Js in the case of multilayer.

The conditionp+q≤nmust be satisfied.

For the Ising andXYmodels, we considered the new values of[α_l]by using the transformation [56] depending only on the dimensionν of the spin (ν=1 for Ising type,ν=2 forXY type andν=3 for Heisenberg type).

For some of multilayer unit cell thickness(t_a=3,5,12), a(p, q, n) are given in Table 1 for the three model(ν= 1,2,3).

In order to evaluate the reduced critical temperatureτ_c(ν) at which the magnetic susceptibility diverges, and the critical exponentγ (ν)associated with the magnetic suscepti- bilityχ, we use the well-known Padé approximants method [46].

4 Results and Discussions

The magnetization decreases with a decrease in Ni layer thickness and the analysis of the data at 5 K indicates that is no Ni dead layer present in these systems. This result shows that the interfaces are quite sharp with no significant inter- mixing. All the samples studied show a negative effective anisotropy and a small contribution to the surface anisotropy indicating that the magnetization is lying in the plane.

Figures5a,5b, 5c shows the magnetization versus temperature for several values of tNi thicknesses. It gives ev- idence that the Curie temperature T_C decreases when t_Ni decreases. The low-temperature magnetization was studied in detail for a few samples. For three-dimensional magnetic

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Table 1 The nonzero coefficientsa(p, q, n)from Eq. (11) in the text, with different thickness (t_a=3,5,12), for Ising,XY and Heisenberg models

(p, q, n) Ising a(p, q, n)

XY a(p, q, n)

Heisenberg a(p, q, n)

t_a=3 t_a=5 t_a=12 t_a=3 t_a=5 t_a=12 t_a=3 t_a=5 t_a=12

(000) 1 1 1 1 1 1 1 1 1

(001) 2 14/5 70/17 1 7/5 35/17 2/3 14/15 70/51

(101) 7/2 14/5 28/17 7/4 7/5 14/17 7/6 14/15 28/51

(011) 1/2 2/5 4/17 1/4 1/5 2/17 1/6 2/15 4/51

(002) 35/4 13 20 35/16 13/4 5 35/36 13/9 20/9

(202) 65/4 13 130/17 65/16 13/4 65/34 65/36 13/9 130/153

(012) 5/2 2 20/17 5/8 1/2 5/17 5/18 2/9 20/153

(112) 5/2 2 20/17 5/8 1/2 5/17 5/18 2/9 20/153

(013) 21/2 42/5 84/17 21/16 21/20 21/34 7/18 14/45 28/153

(113) 25/2 10 100/17 25/16 5/4 25/34 25/54 10/27 100/459

(003) 227/6 898/15 4904/51 75/16 297/40 1623/136 152/81 1478/675 1616/459

(303) 449/6 898/15 1796/51 297/32 297/40 297/68 13/324 1478/675 2956/2295

(033) −1/6 −2/15 −4/51 −1/32 −1/40 −1/68 −1/54 −2/225 −4/765

(023) 1 4/5 8/17 1/8 1/10 1/17 739/270 4/135 8/459

(213) 21/2 42/5 84/17 21/16 21/20 21/34 7/18 14/45 28/153

(123) 1 4/5 8/17 1/8 1/10 1/17 1/27 4/135 8/459

(014) 271/6 548/15 1096/51 179/64 181/80 181/136 89/162 4/9 40/153

(004) 470/3 799/3 22816/51 615/64 131/8 1873/68 152/81 1297/405 7424/1377

(404) 3995/12 799/3 470/3 655/32 131/8 655/68 1297/324 1297/405 2594/1377

(224) 3 12/5 24/17 3/16 3/20 3/34 1/27 4/135 8/459

(034) −5/6 −2/3 −20/51 −5/64 −1/16 −5/136 −1/54 −2/135 −4/459

(134) −5/6 −2/3 −20/51 −5/64 −1/16 −5/136 −1/54 −2/135 −4/459

(314) 137/3 548/15 1096/51 181/64 181/80 181/136 5/9 4/9 40/153

(114) 105/2 42 420/17 105/32 21/8 105/68 35/54 14/27 140/459

(214) 105/2 42 420/17 105/32 21/8 105/68 35/54 14/27 140/459

(124) 16 64/5 128/17 1 4/5 8/17 16/81 64/405 128/1377

(024) 13/4 12/5 24/17 13/64 3/20 3/34 13/324 4/135 8/459

(035) −7/2 −14/5 −28/17 −21/128 −21/160 −21/272 −7/270 −14/675 −28/2295

(415) 395/2 158 1580/17 97/16 97/20 97/34 1919/2430 3838/6075 7676/20655

(325) 29/3 116/15 232/51 19/64 19/80 19/136 47/1215 188/6075 376/20655

(015) 189 158 1580/17 371/64 97/20 97/34 917/1215 3838/6075 7676/20655

(025) 38/3 116/15 232/51 25/64 19/80 19/136 62/1215 188/6075 376/20655

(005) 9679/15 88798/75 106096/51 7475/384 8617/240 103243/1632 21337/8505 28202/6075 33848/4131 (505) 44399/30 88798/75 177596/255 8617/192 8617/240 8617/408 14101/2430 28202/6075 56404/20655

(055) 1/15 4/75 8/255 1/192 1/240 1/408 1/945 4/4725 8/16065

(045) −2/3 −8/15 −16/51 −1/32 −1/40 −1/68 −2/405 −8/2025 −16/6885

(235) −7/2 −14/5 −28/17 −21/128 −21/160 −21/272 −7/270 −14/675 −28/2295

(145) −2/3 −8/15 −16/51 −1/32 −1/40 −1/68 −2/405 −8/2025 −16/6885

(215) 441/2 882/5 1764/17 441/64 441/80 441/136 49/54 98/135 196/459

(115) 1355/6 548/3 5480/51 895/128 181/32 905/272 445/486 20/27 200/459

(125) 163/2 316/5 632/17 81/32 157/80 157/136 269/810 1564/6075 184/1215

(315) 685/3 548/3 5480/51 905/128 181/32 905/272 25/27 20/27 200/459

(225) 79 316/5 632/17 157/64 157/80 157/136 391/1215 1564/6075 184/1215

(135) −1/6 −2/15 −4/51 −13/128 −13/160 −13/272 −17/810 −34/2025 −4/405