Phase diagram of an extended Agassi model

Background: The Agassi model is an extension of the Lipkin-Meshkov-Glick model that incorporates the pairing interaction. It is a schematic model that describes the interplay between particle-hole and pair correlations. It was proposed in the 1960's by D. Agassi as a model to simulate the properties of the quadrupole plus pairing model. Purpose: The aim of this work is to extend a previous study by Davis and Heiss generalizing the Agassi model and analyze in detail the phase diagram of the model as well as the different regions with coexistence of several phases. Method: We solve the model Hamiltonian through the Hartree-Fock-Bogoliubov (HFB) approximation, introducing two variational parameters that play the role of order parameters. We also compare the HFB calculations with the exact ones. Results: We obtain the phase diagram of the model and classify the order of the different quantum phase transitions appearing in the diagram. The phase diagram presents broad regions where several phases, up to three, coexist. Moreover, there is also a line and a point where four and five phases are degenerated, respectively. Conclusions: The phase diagram of the extended Agassi model presents a rich variety of phases. Phase coexistence is present in extended areas of the parameter space. The model could be an important tool for benchmarking novel many-body approximations.Comment: Accepted for publication in PR

An extended Agassi model: algebraic structure, phase diagram, and large size limit

The Agassi model is a schematic two-level model that involves pairing and monopole-monopole interactions. It is, therefore, an extension of the well known Lipkin-Meshkov-Glick (LMG) model. In this paper we review the algebraic formulation of an extension of the Agassi model as well as its bosonic realization through the Schwinger representation. Moreover, a mean-field approximation for the model is presented and its phase diagram discussed. Finally, a $1/j$ analysis, with $j$ proportional to the degeneracy of each level, is worked out to obtain the thermodynamic limit of the ground state energy and some order parameters from the exact Hamiltonian diagonalization for finite$-j$.Comment: Accepted in Physica Scripta. Focus on SSNET 201

Cotilo recubierto con hidroxiapatita de la artroplastia Omnifit: resultados a medio plazo

Las cúpulas acetabulares roscadas han mostrado malos resultados en períodos medios de evolución. Aunque la sujeción inmediata es excelente, con el paso del tiempo se ha comprobado la ausencia de fijación, mostrando un alto porcentaje de movilizaciones. Una alternativa para evitar esos fracasos es el recubrimiento de estos cotilos con hidroxiapatita. Hemos revisado clínica y radiográficamente 36 casos de artroplastias Omnifit con cotilo roscado y recubierto de hidroxiapatita, con un tiempo medio de seguimiento de 58 meses. Diez pacientes presentaban dolor en región inguinal. Sólo el 63,9% de los cotilos se encontraban radiográficamente integrados, habiendo sufrido una clara movilización seis que precisaron cirugía de revisión acetabular; otros siete mostraban signos radiológicos de probable inestabilidad. El recubrimiento con hidroxiapatita no asegura una buena fijación en los cotilos roscados Omnifit por lo que recomendamos componentes acetabulares implantados a presión con posibilidad de sujeción adicional con tornillos.The threaded acetabular cups have shown mid-term bad results. Though the immediate fixation is excellent, as time goes by it has been proven the absence of fixation, appearing a high percentage of mobilizations. An alternative to avoid those failures is the hydroxyapatite coated implant. We have analysed clinically and radiographically 36 cases of Omnifit arthroplasties with cup threaded and coated with hydroxyapatite with an average time of follow-up of 58 months. Ten patients had pain in the hip. Only 63.9% of the acetabular cups were found to be radiographically integrated, six having suffered a clear mobilization requiring surgical review. Seven showed radiological signs of probable instability. The hydroxytapatite coatings does not assure a good fixation in the Omnifit threaded cups, therefore we recommend press fit acetabular components with the possibility of additional fixation with screws

Passivity Breakdown of Titanium in LiBr solutions

The passive behavior of titanium and its susceptibility to undergo localized attack in different LiBr solutions at 25 degrees C have been investigated using different electrochemical techniques: potentiodynamic polarization curves, potentiostatic passivation tests, EIS measurements and Mott-Schottky analysis. In low and moderately concentrated LiBr solutions, the breakdown potential E-b decreased with increasing bromide concentrations, while in highly concentrated LiBr solutions, E-b increased with increasing LiBr concentration. In the most concentrated LiBr solution (11.42M) Ti did not undergo passivity breakdown even at 9 V-Ag/AgCl. This observation can be explained by a a decrease in the activity of water in highly concentrated LiBr solutions. (C) 2013 The Electrochemical Society.We wish express our gratitude to the Ministerio de Ciencia e Innovacion (Project CTQ2009-07518), and to Dr. M. Asuncion Jaime. for her translation assistance.Fernández Domene, RM.; Blasco-Tamarit, E.; García-García, D.; García Antón, J. (2014). Passivity Breakdown of Titanium in LiBr solutions. Journal of The Electrochemical Society. 161(1):25-35. https://doi.org/10.1149/2.035401jesS25351611Been J. Grauman J. S. , in: Uhlig's Corrosion Handbook, 2nd ed., Winston Revie R. (ed.), 863-885, Wiley Interscience, New York (2000).Blasco-Tamarit, E., Igual-Muñoz, A., García Antón, J., & García-García, D. (2007). Corrosion behaviour and galvanic coupling of titanium and welded titanium in LiBr solutions. Corrosion Science, 49(3), 1000-1026. doi:10.1016/j.corsci.2006.07.007Huang, Y. Z., & Blackwood, D. J. (2005). Characterisation of titanium oxide film grown in 0.9% NaCl at different sweep rates. Electrochimica Acta, 51(6), 1099-1107. doi:10.1016/j.electacta.2005.05.051Pan, J., Thierry, D., & Leygraf, C. (1996). Electrochemical impedance spectroscopy study of the passive oxide film on titanium for implant application. Electrochimica Acta, 41(7-8), 1143-1153. doi:10.1016/0013-4686(95)00465-3Assis, S. L. de, Wolynec, S., & Costa, I. (2006). Corrosion characterization of titanium alloys by electrochemical techniques. Electrochimica Acta, 51(8-9), 1815-1819. doi:10.1016/j.electacta.2005.02.121Birch, J. R., & Burleigh, T. D. (2000). Oxides Formed on Titanium by Polishing, Etching, Anodizing, or Thermal Oxidizing. CORROSION, 56(12), 1233-1241. doi:10.5006/1.3280511Peláez-Abellán, E., Rocha-Sousa, L., Müller, W.-D., & Guastaldi, A. C. (2007). Electrochemical stability of anodic titanium oxide films grown at potentials higher than 3V in a simulated physiological solution. Corrosion Science, 49(3), 1645-1655. doi:10.1016/j.corsci.2006.08.010Azumi, K., & Seo, M. (2001). Changes in electrochemical properties of the anodic oxide film formed on titanium during potential sweep. Corrosion Science, 43(3), 533-546. doi:10.1016/s0010-938x(00)00105-0Alves, V. A., Reis, R. Q., Santos, I. C. B., Souza, D. G., de F. Gonçalves, T., Pereira-da-Silva, M. A., … da Silva, L. A. (2009). In situ impedance spectroscopy study of the electrochemical corrosion of Ti and Ti–6Al–4V in simulated body fluid at 25°C and 37°C. Corrosion Science, 51(10), 2473-2482. doi:10.1016/j.corsci.2009.06.035Schmidt, A. M., Azambuja, D. S., & Martini, E. M. A. (2006). Semiconductive properties of titanium anodic oxide films in McIlvaine buffer solution. Corrosion Science, 48(10), 2901-2912. doi:10.1016/j.corsci.2005.10.013Sellers, M. C. K., & Seebauer, E. G. (2011). Measurement method for carrier concentration in TiO2 via the Mott–Schottky approach. Thin Solid Films, 519(7), 2103-2110. doi:10.1016/j.tsf.2010.10.071Jiang, Z., Dai, X., & Middleton, H. (2011). Investigation on passivity of titanium under steady-state conditions in acidic solutions. Materials Chemistry and Physics, 126(3), 859-865. doi:10.1016/j.matchemphys.2010.12.028Kong, D.-S., Lu, W.-H., Feng, Y.-Y., Yu, Z.-Y., Wu, J.-X., Fan, W.-J., & Liu, H.-Y. (2009). Studying on the Point-Defect-Conductive Property of the Semiconducting Anodic Oxide Films on Titanium. Journal of The Electrochemical Society, 156(1), C39. doi:10.1149/1.3021008Roh, B., & Macdonald, D. D. (2007). Effect of oxygen vacancies in anodic titanium oxide films on the kinetics of the oxygen electrode reaction. Russian Journal of Electrochemistry, 43(2), 125-135. doi:10.1134/s1023193507020012Sazou, D., Saltidou, K., & Pagitsas, M. (2012). Understanding the effect of bromides on the stability of titanium oxide films based on a point defect model. Electrochimica Acta, 76, 48-61. doi:10.1016/j.electacta.2012.04.158Roberge P. R. , Handbook of Corrosion Engineering, p. 756, McGraw-Hill, New York (2000).Basame, S. B., & White, H. S. (1995). Scanning electrochemical microscopy of native titanium oxide films. Mapping the potential dependence of spatially-localized electrochemical reactions. The Journal of Physical Chemistry, 99(44), 16430-16435. doi:10.1021/j100044a034Basame, S. B., & White, H. S. (2000). Pitting Corrosion of Titanium The Relationship Between Pitting Potential and Competitive Anion Adsorption at the Oxide Film/Electrolyte Interface. Journal of The Electrochemical Society, 147(4), 1376. doi:10.1149/1.1393364Dugdale, I., & Cotton, J. B. (1964). The anodic polarization of titanium in halide solutions. Corrosion Science, 4(1-4), 397-411. doi:10.1016/0010-938x(64)90041-1Virtanen, S., & Curty, C. (2004). Metastable and Stable Pitting Corrosion of Titanium in Halide Solutions. CORROSION, 60(7), 643-649. doi:10.5006/1.3287839Trompette, J. L., Massot, L., Arurault, L., & Fontorbes, S. (2011). Influence of the anion specificity on the anodic polarization of titanium. Corrosion Science, 53(4), 1262-1268. doi:10.1016/j.corsci.2010.12.021Casillas, N. (1994). Pitting Corrosion of Titanium. Journal of The Electrochemical Society, 141(3), 636. doi:10.1149/1.2054783Beck, T. R. (1973). Pitting of Titanium. Journal of The Electrochemical Society, 120(10), 1310. doi:10.1149/1.2403253Huo, S., & Meng, X. (1990). The states of bromide on titanium surface prior to pit initiation. Corrosion Science, 31, 281-286. doi:10.1016/0010-938x(90)90120-tFernández-Domene, R. M., Blasco-Tamarit, E., García-García, D. M., & García-Antón, J. (2011). Cavitation corrosion and repassivation kinetics of titanium in a heavy brine LiBr solution evaluated by using electrochemical techniques and Confocal Laser Scanning Microscopy. Electrochimica Acta, 58, 264-275. doi:10.1016/j.electacta.2011.09.034Srikhirin, P., Aphornratana, S., & Chungpaibulpatana, S. (2001). A review of absorption refrigeration technologies. Renewable and Sustainable Energy Reviews, 5(4), 343-372. doi:10.1016/s1364-0321(01)00003-xLee R. J. DiGuilio R. M. Jeter S. M. Teja A. S. , ASHRAE Tran., 96(1), (1990).Guiñon, J. L., Garcia-Anton, J., Pérez-Herranz, V., & Lacoste, G. (1994). Corrosion of Carbon Steels, Stainless Steels, and Titanium in Aqueous Lithium Bromide Solution. CORROSION, 50(3), 240-246. doi:10.5006/1.3293516Florides, G. A., Kalogirou, S. A., Tassou, S. A., & Wrobel, L. C. (2003). Design and construction of a LiBr–water absorption machine. Energy Conversion and Management, 44(15), 2483-2508. doi:10.1016/s0196-8904(03)00006-2Misra, R. D., Sahoo, P. K., & Gupta, A. (2005). Thermoeconomic evaluation and optimization of a double-effect H2O/LiBr vapour-absorption refrigeration system. International Journal of Refrigeration, 28(3), 331-343. doi:10.1016/j.ijrefrig.2004.09.006Hamer, W. J., & Wu, Y. (1972). Osmotic Coefficients and Mean Activity Coefficients of Uni‐univalent Electrolytes in Water at 25°C. Journal of Physical and Chemical Reference Data, 1(4), 1047-1100. doi:10.1063/1.3253108Prausnitz J. M. Lichtenthaler R. N. Azevedo E. G. , Molecular Thermodynamics of Fluid-Phase Equilibria, p. 517, Prentice Hall, Upper Saddle River, NJ (1999).Blandamer, M. J., Engberts, J. B. F. N., Gleeson, P. T., & Reis, J. C. R. (2005). Activity of water in aqueous systems; A frequently neglected property. Chemical Society Reviews, 34(5), 440. doi:10.1039/b400473fSelcuk, H., Sene, J. J., Zanoni, M. V. B., Sarikaya, H. Z., & Anderson, M. A. (2004). Behavior of bromide in the photoelectrocatalytic process and bromine generation using nanoporous titanium dioxide thin-film electrodes. Chemosphere, 54(7), 969-974. doi:10.1016/j.chemosphere.2003.09.016Muñoz, A. I., Antón, J. G., Guiñón, J. L., & Herranz, V. P. (2003). Corrosion Behavior and Galvanic Coupling of Stainless Steels, Titanium, and Alloy 33 in Lithium Bromide Solutions. CORROSION, 59(7), 606-615. doi:10.5006/1.3277591Muñoz-Portero, M. J., García-Antón, J., Guiñón, J. L., & Leiva-García, R. (2011). Pourbaix diagrams for titanium in concentrated aqueous lithium bromide solutions at 25°C. Corrosion Science, 53(4), 1440-1450. doi:10.1016/j.corsci.2011.01.013Davydov, A. . (2001). Breakdown of valve metal passivity induced by aggressive anions. Electrochimica Acta, 46(24-25), 3777-3781. doi:10.1016/s0013-4686(01)00664-8Lin, L. F. (1981). A Point Defect Model for Anodic Passive Films. Journal of The Electrochemical Society, 128(6), 1194. doi:10.1149/1.2127592Haruna, T. (1997). Theoretical Prediction of the Scan Rate Dependencies of the Pitting Potential and the Probability Distribution in the Induction Time. Journal of The Electrochemical Society, 144(5), 1574. doi:10.1149/1.1837643Macdonald, D. D. (1992). The Point Defect Model for the Passive State. Journal of The Electrochemical Society, 139(12), 3434. doi:10.1149/1.2069096Macdonald, D. D. (1999). Passivity–the key to our metals-based civilization. Pure and Applied Chemistry, 71(6), 951-978. doi:10.1351/pac199971060951Macdonald, D. D. (2011). The history of the Point Defect Model for the passive state: A brief review of film growth aspects. Electrochimica Acta, 56(4), 1761-1772. doi:10.1016/j.electacta.2010.11.005Macdonald, D. D., & Sun, A. (2006). An electrochemical impedance spectroscopic study of the passive state on Alloy-22. Electrochimica Acta, 51(8-9), 1767-1779. doi:10.1016/j.electacta.2005.02.103Park, K., Ahn, S., & Kwon, H. (2011). Effects of solution temperature on the kinetic nature of passive film on Ni. Electrochimica Acta, 56(3), 1662-1669. doi:10.1016/j.electacta.2010.09.077Macdonald, D. D. (2008). On the tenuous nature of passivity and its role in the isolation of HLNW. Journal of Nuclear Materials, 379(1-3), 24-32. doi:10.1016/j.jnucmat.2008.06.004Paola, A. D. (1989). Semiconducting properties of passive films on stainless steels. Electrochimica Acta, 34(2), 203-210. doi:10.1016/0013-4686(89)87086-0Gomes, W. P., & Vanmaekelbergh, D. (1996). Impedance spectroscopy at semiconductor electrodes: Review and recent developments. Electrochimica Acta, 41(7-8), 967-973. doi:10.1016/0013-4686(95)00427-0Da Cunha Belo, M., Hakiki, N. ., & Ferreira, M. G. . (1999). Semiconducting properties of passive films formed on nickel–base alloys type Alloy 600: influence of the alloying elements. Electrochimica Acta, 44(14), 2473-2481. doi:10.1016/s0013-4686(98)00372-7Hakiki, N. B., Boudin, S., Rondot, B., & Da Cunha Belo, M. (1995). The electronic structure of passive films formed on stainless steels. Corrosion Science, 37(11), 1809-1822. doi:10.1016/0010-938x(95)00084-wHamadou, L., Kadri, A., & Benbrahim, N. (2005). Characterisation of passive films formed on low carbon steel in borate buffer solution (pH 9.2) by electrochemical impedance spectroscopy. Applied Surface Science, 252(5), 1510-1519. doi:10.1016/j.apsusc.2005.02.135Wijesinghe, T. L. S. L., & Blackwood, D. J. (2008). Photocurrent and capacitance investigations into the nature of the passive films on austenitic stainless steels. Corrosion Science, 50(1), 23-34. doi:10.1016/j.corsci.2007.06.009Amri, J., Souier, T., Malki, B., & Baroux, B. (2008). Effect of the final annealing of cold rolled stainless steels sheets on the electronic properties and pit nucleation resistance of passive films. Corrosion Science, 50(2), 431-435. doi:10.1016/j.corsci.2007.08.013Li, D. G., Wang, J. D., & Chen, D. R. (2012). Influence of potentiostatic aging, temperature and pH on the diffusivity of a point defect in the passive film on Nb in an HCl solution. Electrochimica Acta, 60, 134-146. doi:10.1016/j.electacta.2011.11.024Fernández-Domene, R. M., Blasco-Tamarit, E., García-García, D. M., & García-Antón, J. (2013). Passive and transpassive behaviour of Alloy 31 in a heavy brine LiBr solution. Electrochimica Acta, 95, 1-11. doi:10.1016/j.electacta.2013.02.024Urquidi-Macdonald, M. (1989). Theoretical Analysis of the Effects of Alloying Elements on Distribution Functions of Passivity Breakdown. Journal of The Electrochemical Society, 136(4), 961. doi:10.1149/1.2096894Schmidt, A. M., & Azambuja, D. S. (2006). Electrochemical behavior of Ti and Ti6Al4V in aqueous solutions of citric acid containing halides. Materials Research, 9(4), 387-392. doi:10.1590/s1516-14392006000400008Brug, G. J., van den Eeden, A. L. G., Sluyters-Rehbach, M., & Sluyters, J. H. (1984). The analysis of electrode impedances complicated by the presence of a constant phase element. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 176(1-2), 275-295. doi:10.1016/s0022-0728(84)80324-1Valero Vidal, C., & Igual Muñoz, A. (2010). Study of the adsorption process of bovine serum albumin on passivated surfaces of CoCrMo biomedical alloy. Electrochimica Acta, 55(28), 8445-8452. doi:10.1016/j.electacta.2010.07.028Smart, N. G., & Bockris, J. O. (1992). Effect of Water Activity on Corrosion. CORROSION, 48(4), 277-280. doi:10.5006/1.3315933Frankel, G. S. (1998). Pitting Corrosion of Metals. Journal of The Electrochemical Society, 145(6), 2186. doi:10.1149/1.1838615Blasco-Tamarit, E., Igual-Muñoz, A., & García-Antón, J. (2007). Galvanic corrosion of high alloyed austenitic stainless steel welds in LiBr systems. Corrosion Science, 49(12), 4452-4471. doi:10.1016/j.corsci.2007.05.020Crozier, P. S., & Rowley, R. L. (2002). Activity coefficient prediction by osmotic molecular dynamics. Fluid Phase Equilibria, 193(1-2), 53-73. doi:10.1016/s0378-3812(01)00734-8Burstein, G. T. (1989). The Dissolution and Repassivation of New Titanium Surfaces in Alkaline Methanolic Solution. Journal of The Electrochemical Society, 136(5), 1313. doi:10.1149/1.2096913Banaś, J., Stypuła, B., Banaś, K., Światowska-Mrowiecka, J., Starowicz, M., & Lelek-Borkowska, U. (2008). Corrosion and passivity of metals in methanol solutions of electrolytes. 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Some results on the eigenfunctions of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra E6

The quantum trigonometric Calogero-Sutherland models related to Lie algebras admit a parametrization in which the dynamical variables are the characters of the fundamental representations of the algebra. We develop here this approach for the case of the exceptional Lie algebra E6.Comment: 17 pages, no figure

La producción científica española en cirugía ortopédica y traumatología: su presencia en una base de datos internacional

Una manera de evaluar la calidad de la investigación desarrollada en un país es estudiar su presencia en una base de datos bibliográfica internacional. Se han analizado el número y el factor de impacto de los trabajos producidos por España en Cirugía Ortopédica y Traumatología que han sido recogidos por Science Citation Index durante los años 1986 a 1989. De un total de 12706 documentos que se habían originado en nuestro país, sólo 41 (0,3%) pertenecían al área temática referida, siendo 34 artículos y siete comunicaciones a congresos. El factor de impacto medio ha sido de 1,2 y el número medio de autores, 4. Nuestro país se encuadra dentro de los países con baja actividad y baja atracción en Cirugía Ortopé- dica y Traumatología. Estos hallazgos confirman, además de su escasa presencia en un índice bibliográfico internacional, el reducido valor que los autores extranjeros asignan a nuestra producción científica.The study of the presence in an international bibliographic data-base seems to be a suitable assessment of the quality of the research developed in a country. In this paper we analize the number and the impact-factor of the works produced in Spain in the field of orthopaedic surgery and traumatology which have been included in the Science Citation Index from 1986 to 1989. Out of 12706 documents produced in our country, only 41 (0,3%) corresponded to the field of orthopaedics. There were 34 articles and 7 congress contributions. The mean impact factor was 1,2 and the average of authors was 4. From these date, Spain can be considered to have low research activity and low atraction of the publications on orthopaedic surgery and traumatology. Apart from the scarce presence in an international citation, these findings confirm the limited value that international authors fine to our scientific production in this area

Giant magnetic anisotropy at nanoscale: overcoming the superparamagnetic limit

It has been recently observed for palladium and gold nanoparticles, that the magnetic moment at constant applied field does not change with temperature over the range comprised between 5 and 300 K. These samples with size smaller than 2.5 nm exhibit remanence up to room temperature. The permanent magnetism for so small samples up to so high temperatures has been explained as due to blocking of local magnetic moment by giant magnetic anisotropies. In this report we show, by analysing the anisotropy of thiol capped gold films, that the orbital momentum induced at the surface conduction electrons is crucial to understand the observed giant anisotropy. The orbital motion is driven by localised charge and/or spin through spin orbit interaction, that reaches extremely high values at the surfaces. The induced orbital moment gives rise to an effective field of the order of 103 T that is responsible of the giant anisotropy.Comment: 15 pages, 2 figures, submitted to PR

Evolución de fascies abanico aluvial-fluvial-lacustre en el Plioceno de la Depresión de Guadix-Baza

El corte del Río de Gor muestra la evolución lateral de los materiales de la Fm. Guadix (fluvial) a la Fm. Gorafe-Huelago (lacustre). Desde el borde de la cuenca hacia el interior (en 6 km) se reconocen, en los materiales pliocenos, los siguientes medios sedimentarios yuxtapuestos: A. Abanico aluvial medio. B. Abanico aluvial medio. C. Ríos trenzados (braided). D. Meandriforme proximal. E. Meandriforme distal. F. Llanura lutítica. G. Medio lacustr

Frustration free gapless Hamiltonians for Matrix Product States

For every Matrix Product State (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state (the so-called non-injective case), we construct another 'uncle' Hamiltonian which is local and frustration free but gapless, and its spectrum is $\R^+$. The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the parent Hamiltonian has a unique ground state (the so-called injective case) we also build such uncle Hamiltonian with the same properties in the thermodynamic limit.Comment: 36 pages, new version with some contents rearranged, and a correction in the injective cas

Ensayo de fertilización foliar sobre regenerado de pino rodeno en Cuéllar (Segovia)

El regenerado de Pinus pinaster Ait. en la tierra de pinares segoviana presenta frecuentemente, en estado de monte bravo, un desarrollo deficiente de la guía y de las ramas del tercio superior de la copa. El objetivo del presente estudio es tratar de determinar si este deficiente desarrollo está relacionado con alguna carencia nutricional. Para ello se realiza un ensayo de fertilización foliar y se analiza su influencia sobre la Longitud de las acículas y la Longitud de los ramillos desarrollados en el periodo vegetativo siguiente a la fertilización. Se consideran cuatro localizaciones diferentes y tres tratamientos distintos (Testigo, T, sin fertilización; fertilización simple, F; y fertilización doble, FF)