Journal of Materials Science

, Volume 44, Issue 1, pp 40–46 | Cite as

Real and reciprocal space order parameters for porous arrays from image analysis

  • Forrest H. KaatzEmail author
  • Adhemar Bultheel
  • Takeshi Egami


A real space technique based on the pair distribution function (PDF) and a reciprocal space method utilizing a 2D fast Fourier transform (FFT) quantify the order in porous arrays. Porous arrays fabricated from nanoscience technology are analyzed. The PDFs are fit with a series of Gaussian curves and the widths of the Gaussian peaks are used to model the linear strain in the array. An order parameter is defined from the PDF and takes values from [0,1], where the value 1 represents an ideal array. The radial distribution function (RDF) is also determined for the porous arrays. The FFT of the porous arrays is used to generate an order parameter as a ratio of intensity to the full width at half maximum (σ) of the peaks. Defined as relative intensity \( I_{\text{r}} /\sigma \), this parameter takes values from [0,∞], where larger values represent more order in the array. We use a variety of available software to generate this data.


Fast Fourier Transform Real Space Anodize Aluminum Oxide Radial Distribution Function Pair Distribution Function 



An iMac @ 2.4 GHz running Mac OS X 10.5.5 was used. Image SXM is a free download for the Mac OS available from Ref. [17]. MATLAB, the Curve Fitting Toolbox, Excel, and Kaleidagraph complete the software tools necessary to create and analyze the data. In Figs. 1 and 4 we have reprinted with permission from Ref. [1, 7], copyright [2003, 1998], American Institute of Physics and reprinted with permission from Ref. [3], copyright [2005], IOPP Publishing.


  1. 1.
    Choi J, Luo Y, Wehrspohn RB, Hillebrand R, Schilling J, Gösele U (2003) J Appl Phys 94(8):4757CrossRefGoogle Scholar
  2. 2.
    Asoh H, Nishio K, Nakao M, Tamamura T, Masuda H (2001) J Electrochem Soc 148(4):B152CrossRefGoogle Scholar
  3. 3.
    Krishnan R, Nguyen HQ, Thompson CV, Choi WK, Foo YL (2005) Nanotechnology 16:841CrossRefGoogle Scholar
  4. 4.
    Hulteen JC, Treichel DA, Smith MT, Duval ML, Jensen TR, Van Duyne RP (1999) J Phys Chem B 103:3854CrossRefGoogle Scholar
  5. 5.
    Blanford CF, Carter CB, Stein A (2004) J Microsc 216(3):263CrossRefGoogle Scholar
  6. 6.
    Campbell M, Sharp DN, Harrison MT, Denning RG, Turberfield AJ (2000) Nature 404(6773):53CrossRefGoogle Scholar
  7. 7.
    Li AP, Müller F, Birner A, Nielsch K, Gösele U (1998) J Appl Phys 84:6023CrossRefGoogle Scholar
  8. 8.
    Masuda H, Fukuda K (1995) Science 238:1466CrossRefGoogle Scholar
  9. 9.
    Sun F, Cai W, Li Y, Cao B, Lei Y, Zhang L (2004) Adv Funct Mater 14(3):283CrossRefGoogle Scholar
  10. 10.
    Shang XF, Wang M, Qu SX, Zhao R, Zhou JJ, Xu XB, Tan MQ, Li ZH (2008) Nanotechnology 19:065708CrossRefGoogle Scholar
  11. 11.
    Sellmyer DJ, Zheng M, Skomski R (2001) J Phys Condens Matter 13:R433CrossRefGoogle Scholar
  12. 12.
    Lei Y, Cai W, Wilde G (2007) Prog Mater Sci 52:465CrossRefGoogle Scholar
  13. 13.
    Li J, Papadopoulos C, Xu JM, Moskovits M (1999) Appl Phys Lett 75:367CrossRefGoogle Scholar
  14. 14.
    Park KH, Lee S, Koh KH, Lacerda R, Teo KBK, Milne WI (2005) J Appl Phys 95:024311CrossRefGoogle Scholar
  15. 15.
    Shingubara S, Okino O, Sayama Y, Sakaue H, Takahagi T (1997) Jpn J Appl Phys 36:7791CrossRefGoogle Scholar
  16. 16.
    Egami T, Billinge SJL (2003) Underneath the Bragg peaks: structural analysis of complex materials. Pergamon, AmsterdamCrossRefGoogle Scholar
  17. 17.
    Barrett S (2008) The website of Image SXM is at Accessed November 2008
  18. 18.
    Kaatz FH (2006) Naturwissenschaften 93:374CrossRefGoogle Scholar
  19. 19.
    Kaatz FH, Bultheel A, Egami T (2008) Naturwissenschaften 95:1033CrossRefGoogle Scholar
  20. 20.
    Kodama K, Iikubo S, Taguchi T, Shamoto S (2006) Acta Crystallogr A 62:444CrossRefGoogle Scholar
  21. 21.
    Mason G (1968) Nature (London) 217:733CrossRefGoogle Scholar
  22. 22.
    Kashi MA, Ramazani A (2005) J Phys D Appl Phys 38:2396CrossRefGoogle Scholar
  23. 23.
    Ba L, Li WS (2000) J Phys D Appl Phys 33:2527CrossRefGoogle Scholar
  24. 24.
    Rao YL, Anandan V, Zhang G (2005) J Nanosci Nanotechnol 5:2070CrossRefGoogle Scholar
  25. 25.
    Kashi MA, Ramazani A, Rahmandoustand M, Noormohammadi M (2007) J Phys D Appl Phys 40:4625CrossRefGoogle Scholar
  26. 26.
    Sulka GD, Parkoaa KG (2006) Thin Solid Films 515:338CrossRefGoogle Scholar
  27. 27.
    Frigo M, Johnson SG (2005) Proc IEEE 93(2):216CrossRefGoogle Scholar
  28. 28.
    Reis PM, Ingale RA, Shattuck MD (2006) Phys Rev Lett 96:258001CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Forrest H. Kaatz
    • 1
    Email author
  • Adhemar Bultheel
    • 2
  • Takeshi Egami
    • 3
    • 4
    • 5
  1. 1.Department of Mathematics and Life/Natural SciencesOwens Community CollegeToledoUSA
  2. 2.Department of Computer ScienceK.U.LeuvenHeverleeBelgium
  3. 3.Department of Materials Science and EngineeringThe University of TennesseeKnoxvilleUSA
  4. 4.Department of Physics and AstronomyThe University of TennesseeKnoxvilleUSA
  5. 5.Materials Science and Technology DivisionOak Ridge National LaboratoryOak RidgeUSA

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