[301e096] | 1 | # Note: model title and parameter table are inserted automatically |
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[529b8b4] | 2 | r"""Calculate the interparticle structure factor for monodisperse |
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| 3 | spherical particles interacting through hard sphere (excluded volume) |
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| 4 | interactions. |
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[301e096] | 5 | |
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[529b8b4] | 6 | The calculation uses the Percus-Yevick closure where the interparticle |
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| 7 | potential is |
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[301e096] | 8 | |
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[eb69cce] | 9 | .. math:: |
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[529b8b4] | 10 | |
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| 11 | U(r) = \begin{cases} |
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| 12 | \infty & r < 2R \\ |
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| 13 | 0 & r \geq 2R |
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| 14 | \end{cases} |
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[301e096] | 15 | |
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[eb69cce] | 16 | where $r$ is the distance from the center of the sphere of a radius $R$. |
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[301e096] | 17 | |
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| 18 | For a 2D plot, the wave transfer is defined as |
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| 19 | |
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| 20 | .. math:: |
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| 21 | |
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[529b8b4] | 22 | q = \sqrt{q_x^2 + q_y^2} |
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[301e096] | 23 | |
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| 24 | |
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[eb69cce] | 25 | .. figure:: img/HardSphere_1d.jpg |
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[301e096] | 26 | |
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[eb69cce] | 27 | 1D plot using the default values (in linear scale). |
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[301e096] | 28 | |
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[eb69cce] | 29 | References |
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| 30 | ---------- |
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[301e096] | 31 | |
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| 32 | J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1 |
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| 33 | """ |
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| 34 | |
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[3c56da87] | 35 | from numpy import inf |
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[301e096] | 36 | |
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| 37 | name = "hardsphere" |
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| 38 | title = "Hard sphere structure factor, with Percus-Yevick closure" |
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| 39 | description = """\ |
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[3e428ec] | 40 | [Hard sphere structure factor, with Percus-Yevick closure] |
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[301e096] | 41 | Interparticle S(Q) for random, non-interacting spheres. |
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[3e428ec] | 42 | May be a reasonable approximation for other shapes of |
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| 43 | particles that freely rotate, and for moderately polydisperse |
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| 44 | systems. Though strictly the maths needs to be modified - |
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| 45 | which sasview does not do yet. |
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| 46 | effect_radius is the hard sphere radius |
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| 47 | volfraction is the volume fraction occupied by the spheres. |
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[301e096] | 48 | """ |
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[a5d0d00] | 49 | category = "structure-factor" |
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[301e096] | 50 | |
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[3e428ec] | 51 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 52 | parameters = [["effect_radius", "Ang", 50.0, [0, inf], "volume", |
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| 53 | "effective radius of hard sphere"], |
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| 54 | ["volfraction", "", 0.2, [0, 0.74], "", |
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| 55 | "volume fraction of hard spheres"], |
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| 56 | ] |
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[301e096] | 57 | |
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| 58 | # No volume normalization despite having a volume parameter |
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| 59 | # This should perhaps be volume normalized? |
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| 60 | form_volume = """ |
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| 61 | return 1.0; |
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| 62 | """ |
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| 63 | |
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| 64 | Iq = """ |
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[3e428ec] | 65 | double denom,dnum,alpha,beta,gamm,a,asq,ath,afor,rca,rsa; |
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| 66 | double calp,cbeta,cgam,prefac,c,vstruc; |
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| 67 | double struc; |
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| 68 | |
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| 69 | // compute constants |
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| 70 | denom = pow((1.0-volfraction),4); |
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| 71 | dnum = pow((1.0 + 2.0*volfraction),2); |
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| 72 | alpha = dnum/denom; |
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| 73 | beta = -6.0*volfraction*pow((1.0 + volfraction/2.0),2)/denom; |
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| 74 | gamm = 0.50*volfraction*dnum/denom; |
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| 75 | // |
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| 76 | // calculate the structure factor |
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| 77 | // |
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| 78 | a = 2.0*q*effect_radius; |
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| 79 | asq = a*a; |
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| 80 | ath = asq*a; |
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| 81 | afor = ath*a; |
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| 82 | SINCOS(a,rsa,rca); |
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| 83 | //rca = cos(a); |
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| 84 | //rsa = sin(a); |
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| 85 | calp = alpha*(rsa/asq - rca/a); |
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| 86 | cbeta = beta*(2.0*rsa/asq - (asq - 2.0)*rca/ath - 2.0/ath); |
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| 87 | cgam = gamm*(-rca/a + (4.0/a)*((3.0*asq - 6.0)*rca/afor + (asq - 6.0)*rsa/ath + 6.0/afor)); |
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| 88 | prefac = -24.0*volfraction/a; |
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| 89 | c = prefac*(calp + cbeta + cgam); |
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| 90 | vstruc = 1.0/(1.0-c); |
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| 91 | struc = vstruc; |
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| 92 | |
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| 93 | return(struc); |
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[301e096] | 94 | """ |
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| 95 | |
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| 96 | Iqxy = """ |
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| 97 | // never called since no orientation or magnetic parameters. |
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[529b8b4] | 98 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[301e096] | 99 | """ |
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| 100 | |
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| 101 | # ER defaults to 0.0 |
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| 102 | # VR defaults to 1.0 |
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| 103 | |
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[3e428ec] | 104 | demo = dict(effect_radius=200, volfraction=0.2, effect_radius_pd=0.1, effect_radius_pd_n=40) |
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[301e096] | 105 | oldname = 'HardsphereStructure' |
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| 106 | oldpars = dict() |
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| 107 | |
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