DPD angles

7.2. DPD angles#

Hide code cell content

Equation (A1) from [2]:

Hide code cell source
angles = [
    formulate_scattering_angle(1, 2),
    formulate_scattering_angle(2, 3),
    formulate_scattering_angle(3, 1),
]
display_latex(dict(angles))
θ12=acos(2σ3(m12m32+σ2)(m02m32σ3)(m12m22+σ3)λ(m02,m32,σ3)λ(σ3,m12,m22))θ23=acos(2σ1(m12m22+σ3)(m02m12σ1)(m22m32+σ1)λ(m02,m12,σ1)λ(σ1,m22,m32))θ31=acos(2σ2(m22m32+σ1)(m02m22σ2)(m12+m32+σ2)λ(m02,m22,σ2)λ(σ2,m32,m12))

Equation (A2):

for i in [1, 2, 3]:
    _, θii = formulate_theta_hat_angle(i, i)
    assert θii == 0

Equation (A3):

Hide code cell source
angles = [
    formulate_theta_hat_angle(3, 1),
    formulate_theta_hat_angle(1, 2),
    formulate_theta_hat_angle(2, 3),
]
display_latex(dict(angles))
θ^3(1)=acos(2m02(m12m32+σ2)+(m02+m12σ1)(m02+m32σ3)λ(m02,m12,σ1)λ(m02,σ3,m32))θ^1(2)=acos(2m02(m12m22+σ3)+(m02+m12σ1)(m02+m22σ2)λ(m02,m22,σ2)λ(m02,σ1,m12))θ^2(3)=acos(2m02(m22m32+σ1)+(m02+m22σ2)(m02+m32σ3)λ(m02,m32,σ3)λ(m02,σ2,m22))

Equations (A4-5):

Hide code cell content
θ31hat = formulate_theta_hat_angle(3, 1)[1]
θ13hat = formulate_theta_hat_angle(1, 3)[1]
θ12hat = formulate_theta_hat_angle(1, 2)[1]
θ21hat = formulate_theta_hat_angle(2, 1)[1]
θ23hat = formulate_theta_hat_angle(2, 3)[1]
θ32hat = formulate_theta_hat_angle(3, 2)[1]
assert θ31hat == -θ13hat
assert θ12hat == -θ21hat
assert θ23hat == -θ32hat

Equations (A6):

for i in [1, 2, 3]:
    for k in [1, 2, 3]:
        _, ζi_k0 = formulate_zeta_angle(i, k, 0)
        _, ζi_ki = formulate_zeta_angle(i, k, i)
        _, ζi_kk = formulate_zeta_angle(i, k, k)
        assert ζi_ki == ζi_k0
        assert ζi_kk == 0

Equations (A7):

Hide code cell source
angles = [
    formulate_zeta_angle(1, 1, 3),
    formulate_zeta_angle(1, 2, 1),
    formulate_zeta_angle(2, 2, 1),
    formulate_zeta_angle(2, 3, 2),
    formulate_zeta_angle(3, 3, 2),
    formulate_zeta_angle(3, 1, 3),
]
display_latex(dict(angles))
ζ1(3)1=acos(2m12(m02m22+σ2)+(m02+m12σ1)(m12m22+σ3)λ(m02,m12,σ1)λ(σ3,m12,m22))ζ2(1)1=acos(2m12(m02m32+σ3)+(m02+m12σ1)(m12m32+σ2)λ(m02,m12,σ1)λ(σ2,m12,m32))ζ2(1)2=acos(2m22(m02m32+σ3)+(m02+m22σ2)(m22m32+σ1)λ(m02,m22,σ2)λ(σ1,m22,m32))ζ3(2)2=acos(2m22(m02m12+σ1)+(m02+m22σ2)(m12m22+σ3)λ(m02,m22,σ2)λ(σ3,m22,m12))ζ3(2)3=acos(2m32(m02m12+σ1)+(m02+m32σ3)(m12m32+σ2)λ(m02,m32,σ3)λ(σ2,m32,m12))ζ1(3)3=acos(2m32(m02m22+σ2)+(m02+m32σ3)(m22m32+σ1)λ(m02,m32,σ3)λ(σ1,m32,m22))

Equations (A8):

Hide code cell content
ζ1_12 = formulate_zeta_angle(1, 1, 2)[1]
ζ1_21 = formulate_zeta_angle(1, 2, 1)[1]
ζ2_23 = formulate_zeta_angle(2, 2, 3)[1]
ζ2_32 = formulate_zeta_angle(2, 3, 2)[1]
ζ3_31 = formulate_zeta_angle(3, 3, 1)[1]
ζ3_13 = formulate_zeta_angle(3, 1, 3)[1]
assert ζ1_12 == -ζ1_21
assert ζ2_23 == -ζ2_32
assert ζ3_31 == -ζ3_13

Equations (A10):

Hide code cell source
angles = [
    formulate_zeta_angle(1, 2, 3),
    formulate_zeta_angle(2, 3, 1),
    formulate_zeta_angle(3, 1, 2),
]
display_latex(dict(angles))
ζ2(3)1=acos(2m12(m22+m32σ1)+(m12m22+σ3)(m12m32+σ2)λ(σ2,m32,m12)λ(σ3,m12,m22))ζ3(1)2=acos(2m22(m12+m32σ2)+(m12m22+σ3)(m22m32+σ1)λ(σ1,m22,m32)λ(σ3,m12,m22))ζ1(2)3=acos(2m32(m12+m22σ3)+(m12m32+σ2)(m22m32+σ1)λ(σ1,m22,m32)λ(σ2,m32,m12))

Equations (A11):

Hide code cell content
ζ1_32 = formulate_zeta_angle(1, 3, 2)[1]
ζ1_23 = formulate_zeta_angle(1, 2, 3)[1]
ζ2_13 = formulate_zeta_angle(2, 1, 3)[1]
ζ2_31 = formulate_zeta_angle(2, 3, 1)[1]
ζ3_21 = formulate_zeta_angle(3, 2, 1)[1]
ζ3_12 = formulate_zeta_angle(3, 1, 2)[1]
assert ζ1_32 == -ζ1_23
assert ζ2_13 == -ζ2_31
assert ζ3_21 == -ζ3_12