# Surface types

## 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.
• The positive side for a plane defined by three points is determined by the order in which the points are entered (see the right-hand rule).
Surface name Parameters Surface equation Description
px x0 $S(x) = x - x_0$ Plane perpendicular to x-axis at x = x0
py y0 $S(y) = y - y_0$ Plane perpendicular to y-axis at y = y0
pz 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

Notes:

• cyl is the same surface as cylz
• Infinite cylinder is assumed if three values are provided in the surface card for cylx, cyly, cylz or cyl. With five values the surface is assumed to be a truncated cylinder.
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 parallel to z-axis, base at (x0,y0,z0), base radius r, height h (distance from base to vertex)
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

Notes:

• Serpent can currently handle only circular torii. Radii R1 and R2 must be set equal (denoted in the surface equations as R).
Surface name Parameters Surface equation Description Notes
inf - $S(y,x,z) = -\infty$ All space Can not be used in root universe
torx x0, y0, z0, r, R1, R2 $S(x,y,z) = \left(R - \sqrt{(y - y_0)^2 + (z - z_0)^2}\right)^2 + (x - x_0)^2 - r^2$ Circular torus with major radius R perpendicular to x-axis, centred at (x0, y0, z0), minor radius r
tory x0, y0, z0, r, R1, R2 $S(x,y,z) = \left(R - \sqrt{(x - x_0)^2 + (z - z_0)^2}\right)^2 + (y - y_0)^2 - r^2$ Circular torus with major radius R perpendicular to y-axis, centred at (x0, y0, z0), minor radius r
torz x0, y0, z0, r, R1, R2 $S(x,y,z) = \left(R - \sqrt{(x - x_0)^2 + (y - y_0)^2}\right)^2 + (z - z_0)^2 - r^2$ Circular torus with major radius R perpendicular to z-axis, centred at (x0, y0, z0), minor radius r

## 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.
• Infinite cylinder is assumed if three values are provided in the surface card for cylx, cyly, cylz or cyl. With five values the surface is assumed to be a truncated cylinder.
Surface name Parameters Composed of Description
cylx y0, z0, r, x0, x1 Infinite cylinder + two planes Infinite cylinder parallel to x-axis, centred at (y0,z0), radius r, truncated between [x0, x1]
cyly x0, z0, r, y0, y1 Infinite cylinder + two planes Infinite cylinder parallel to y-axis, centred at (x0,z0), radius r, truncated between [y0, y1]
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
hexyc 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, centered 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 ("generalized cruciform prism", see figure below)
hexxap x0, y0, wd, dw, a Surface for simplifies modeling of hexagonal fuel assembly angle pieces. Consists of two infinite hexagonal prism parallel to z-axis, centered at (x0, y0), flat surface perpendicular to x-axis, with outer half-width wd and thickness of dw, and each half-section of each angle piece with width of a measured from the middle tip of the angle piece angle to the flat surface of the angle piece.
hexyap x0, y0, wd, dw, a Surface for simplifies modeling of hexagonal fuel assembly angle pieces. Consists of two infinite hexagonal prism parallel to z-axis, centered at (x0, y0), flat surface perpendicular to y-axis, with outer half-width wd and thickness of dw, and each half-section of each angle piece with width of a measured from the middle tip of the angle piece angle to the flat surface of the angle piece.

### Rounded corners

Infinite prisms:

• sqc
• hexxc
• hexyc
• cross

Allow defining rounded corners. The radius is then provided as the last surface parameter (s in figure below):

## MCNP-equivalent surfaces

Notes:

• Additional surfaces included to simplify input conversion between Serpent and MCNP.
• For description, see Chapter 3 of the MCNP5 User's Guide.[1]
Surface name Equivalent surface in MCNP
box BOX
ckx K/X
cky K/Y
ckz K/Z
mplane P (form defined by three points)
rcc RCC
x X
y Y
z Z

## User-defined surfaces

Notes:

• Remember to make a backup of your subroutine before installing new updates.
• If you have a working surface routine that might be useful for other users as well, contact the Serpent team and we'll include it in the next update as a built-in type.
Surface name Parameters Description
usr p1, p2, ... User-defined surface, implemented in source file "usersurf.c". The subroutine receives the number and list of surface parameters as input. Instructions are included as comments in the source file.

## 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.