# Surface types

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## Elementary surfaces

Notes:

• Elementary surfaces refer here to surfaces that can be represented by a single equation.

### Planes

Notes:

• Parametric form of the general plane is assumed if four values are provided in the surface card. With six values the plane is assumed to be defined by three points.
Surface name Parameters Surface equation Description
py x0 $S(x) = x - x_0$ Plane perpendicular to x-axis at x = x0
pz y0 $S(y) = y - y_0$ Plane perpendicular to y-axis at y = y0
px z0 $S(z) = z - z_0$ Plane perpendicular to z-axis at z = z0
plane A, B, C, D $S(x,y,z) = Ax+ By + Cz - D$ General plane in parametric form
plane x1, y1, z1, x2, y2, z2, x3, y3, z3 General plane defined by three points

### Second-order quadratic surfaces

Notes:

• cyl is the same surface as cylz
Surface name Parameters Surface equation Description
cylx y0, z0, r $S(y,z) = (y - y_0)^2 + (z - z_0)^2 - r^2$ Infinite cylinder parallel to x-axis, centred at (y0,z0), radius r
cyly x0, z0, r $S(x,z) = (x - x_0)^2 + (z - z_0)^2 - r^2$ Infinite cylinder parallel to y-axis, centred at (x0,z0), radius r
cylz, cyl x0, y0, r $S(x,y) = (x - x_0)^2 + (y - y_0)^2 - r^2$ Infinite cylinder parallel to z-axis, centred at (x0,y0), radius r
cylv x0, y0, z0, u0, v0, w0, r $S(x,y,z) = (1-u_0^2)(x - x_0)^2 + (1-v_0^2)(y - y_0)^2 + (1-w_0^2)(z - z_0)-r^2$ Infinite cylinder, parallel to (u0,v0,w0), centred at (x0,y0,z0), radius r
sph x0, y0, z0, r $S(x,y,z) = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 - r^2$ Sphere, centred at (x0,y0,z0), radius r
cone x0, y0, z0, r, h $(x - x_0)^2 + (y - y_0)^2 - \left(1 - (z - z_0)/h\right)r^2$ Half cone on z-axis, centred at (x0,y0,z0), base radius r, height h
quadratic A, B, C, D, E, F, G, H, I, J, K $S(x,y,z) = Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx + Gx + Hy + Jz + K$ General quadratic surface in parametric form

### Non-quadratic surfaces

Surface name Parameters Surface equation Description
inf - $S(y,x,z) = -\infty$ All space
torx x0, y0, z0, r, R $S(x,y,z) = \left(R - \sqrt{(y - y_0)^2 + (z - z_0)^2}\right)^2 + (x - x_0)^2 - r^2$ Circular torus with major radius R perpendicular to x-axis, centred at (x0, y0, z0), minor radius r
tory x0, y0, z0, r, R $S(x,y,z) = \left(R - \sqrt{(x - x_0)^2 + (z - z_0)^2}\right)^2 + (y - y_0)^2 - r^2$ Circular torus with major radius R perpendicular to y-axis, centred at (x0, y0, z0), minor radius r
torz x0, y0, z0, r, R $S(x,y,z) = \left(R - \sqrt{(x - x_0)^2 + (y - y_0)^2}\right)^2 + (z - z_0)^2 - r^2$ Circular torus with major radius R perpendicular to z-axis, centred at (x0, y0, z0), minor radius r

### MCNP-equivalent surfaces

• Additional surfaces included to simplify input conversion between Serpent and MCNP.
• Includes cones and axisymmetric surfaces defined by points as used by MCNP.
• For description, see Chapter 3 of the MCNP5 User's Guide.[1]
Surface name Equivalent in MCNP
ckx K/X
cky K/Y
ckz K/Z
x X
y Y
z Z

## Derived surface types

Notes:

• Derived surfaces refer here to surfaces composed of two or more elementary types.

### Truncated cylinders

Notes:

• Truncated cylinders use the same names as the infinite cylinders above, with two additional values determining the height.
Surface name Parameters Composed of Description
cylx y0, z0, r, z0, z1 Infinite cylinder + two planes Infinite cylinder parallel to x-axis, centred at (y0,z0), radius r, truncated between [z0, z1]
cyly x0, z0, r, z0, z1 Infinite cylinder + two planes Infinite cylinder parallel to y-axis, centred at (x0,z0), radius r, truncated between [z0, z1]
cylz, cyl x0, y0, r, z0, z1 Infinite cylinder + two planes Infinite cylinder parallel to z-axis, centred at (x0,y0), radius r, truncated between [z0, z1]

### Regular prisms

Notes:

Surface name Parameters Composed of Description
sqc x0, y0, d four planes Infinite square prism parallel to z-axis, centred at (x0,y0), half-width d
rect x0, x1, y0, y1 four planes Infinite rectangular prism parallel to z-axis, between [x0, x1] and [y0, y1]
hexxc x0, y0, d six planes Infinite hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to x-axis, half-width d
hexxy x0, y0, d six planes Infinite hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to y-axis, half-width d
hexxprism x0, y0, d, z0, z1 eight planes Truncated hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to x-axis, half-width d, truncated between [z0, z1]
hexyprism x0, y0, d, z0, z1 eight planes Truncated hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to y-axis, half-width d, truncated between [z0, z1]
octa x0, y0, d1, d2 eight planes Infinite octagonal prism parallel to z-axis, centred at (x0,y0), half-widths d1 and d2
dode x0, y0, d1, d2 twelve planes Infinite dodecagonal prism parallel to z-axis, centred at (x0,y0), half-widths d1 and d2

### 3D polyhedra

Notes:

• The description of parallelepiped may be wrong.
Surface name Parameters Composed of Description
cube x0, y0, z0, d six planes Cube, centred at (x0,y0,z0), half-width d
cuboid x0, x1, y0, y1, z0, z1 six planes Cuboid, between [x0, x1], [y0, y1] and [z0, z1]
ppd x0, y0, z0, Lx, Ly, Lz, $\alpha$x, $\alpha$y, $\alpha$z six planes Parallelepiped, with corner at (x0, y0, z0) and edges of length Lx, Ly and Lz at angles $\alpha$x, $\alpha$y and $\alpha$z (in degrees) with respect to the coordinate axes

### Other derived surface types

Surface name Parameters Description
pad x0, y0, r1, r2, $\alpha$1, $\alpha$2 Sector from $\alpha$1 to $\alpha$2 (in degrees) of a cylinder parallel to z-axis, centred at (x0,y0), between radii r1 and r2
cross x0, y0, l, d Cruciform prism parallel to z-axis, centred at (x0,y0), half-width l, half-thickness d
gcross x0, y0, d1, d2, ... Prism parallel to z-axis, centred at (x0,y0), formed by planes at distances dn from the center (see figure below)

## References

1. ^ X-5 Monte Carlo Team. "MCNP — A General Monte Carlo N-Particle Transport Code, Version 5, Volume II: User’s Guide." LA-CP-03-0245, Los Alamos National Laboratory, 2003.