View source for Delta- and surface-tracking

This a brief description on the delta-tracking based transport routine used in Serpent. The method is also used by other Monte Carlo codes, most notably in the HOLE geometry package in MONK and MCBEND.
The original delta-tracking algorithm was introduced by Woodcock in 1965,<ref>Woodcock, E. R., Murphy, T., Hemmings, P. J., and Longworth, T. C. ''"Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry."'' ANL-7050, Argonne National Laboratory, 1965.</ref> and a mathematical verification was derived by Coleman in 1968.<ref>Coleman, W. A. ''"Mathematical verification of a certain Monte Carlo sampling technique and applications of the technique to radiation transport problems."'' Nucl. Sci. Eng., '''31''' (1968) 76–81.</ref> Delta-tracking is well described in a text book by Lux and Koblinger,<ref>Lux, I. and Koblinger, L. ''"Monte Carlo Particle Transport Methods: Neutron and Photon Calculations."'' CRC Press, Inc. (1991).</ref> and a description of the methodology used in Serpent is found in an article in Annals of Nuclear Energy from 2010.<ref>Leppänen, J. ''"Performance of Woodcock delta-tracking in lattice physics applications using the Serpent Monte Carlo reactor physics burnup calculation code."'' Ann. Nucl. Energy [http://www.sciencedirect.com/science/article/pii/S0306454910000320 '''37''' (2010) 715–722].</ref>

The input parameters related to delta-tracking are:
* [[input syntax manual#set dt|set dt]] - sets the probability threshold used for selecting between surface- and delta-tracking
* [[input syntax manual#set forcedt|set forcedt]] - enforces the use of delta-tracking in a given list of materials
* [[input syntax manual#set blockdt|set blockdt]] - enforces the use of surface-tracking in a given list of materials
* [[input syntax manual#set minxs|set minxs]] - definse the mean-free-path of collisions used to score the collision flux estimator

The output parameters are:
* TODO


== Transport algorithms in Monte Carlo simulation ==

The Monte Carlo simulation consists of a large number particle histories, in which the random
walk of an individual particle is followed, or tracked, through the geometry from its birth to eventual
absorption or escape.
Tracking is most typically carried out in a [[Constructive solid geometry type|constructive solid geometry (CSG)]], composed of homogeneous material cells, which are defined by combinations of elementary and derived surface types. Serpent 2 also has two advanced geometry types, based on [[CAD-based geometry type|STL format CAD models]] and an [[Unstructured mesh-based geometry type|unstructured polyhedral mesh]]. 

The transport simulation follows a random walk from one interaction to the next. The procedure can be described as follows:
#Sample path length (distance to next collision)
#Transport particle to the collision point
#Sample interaction
If the sampled interaction is scattering, the procedure restarts from beginning by sampling the
distance to the next collision. The direction and energy are changed in each scattering event.

By definition, the interaction probability per traveled path length is given by the macroscopic total
cross section (denoted here as <math>\Sigma</math>).  If it is assumed that the particle travels through an infinite
homogeneous medium characterized by constant total cross section, it can be shown that the free path length follows an exponential distribution. This distribution can be sampled using the inverse method, and the distance to the next collision site is given by:

<math>
l = -\frac{1}{\Sigma}\log\xi
</math>

where <math>\xi</math> is a uniformly distributed random variable on the unit interval.

The prerequisite of using this simple formula for sampling the distance to the next
collision site is that the material is infinite and homogeneous. This is the case when the particle remains within a single material region, but not if its track crosses the boundary between two materials.

=== Surface-tracking ===

If it is known that the particle makes it to the next material boundary and interacts somewhere
beyond the other side, the point of crossing can be taken as the starting point of a new path. This
is the general idea in the surface-tracking algorithm, in which the particle track is stopped at each
boundary crossing, and a new path sampled using the cross section of the next material.

The algorithm requires calculating the distance to the nearest boundary surface. The only way
to accomplish this is to loop over all candidate surfaces and pick the shortest value. 

Surface tracking is considered the standard tracking algorithm and it is used by virtually every
Monte Carlo transport calculation code. The method has a few drawbacks related to its efficiency in
complex geometries:
*Determining the distance to the nearest boundary can become computationally expensive if the cells are comprised of a large number of surfaces
*The fact that the particle track has to be stopped at each boundary crossing becomes a computational bottleneck when the mean-free-path is long compared to the dimensions

=== Delta-tracking ===

An alternative to surface-tracking is the Woodcock delta-tracking
algorithm, which is based on the rejection sampling of particle path lengths.
The procedure relies on the concept of a ''virtual collision'',
which is a fictive interaction that preserves the energy and direction
of the transported particle.

Since virtual collisions do not change the 
random walk in any way, the material total cross section <math>\Sigma</math>
can be adjusted with an arbitrary virtual collision cross section <math>\Sigma_0</math>:

<math>
\Sigma'(\mathbf{r}, E) = \Sigma(\mathbg{r},E) + \Sigma_0(\mathbf{r},E)
</math>

without changing the outcome of the simulation.
It is then possible to adjust the cross sections of all 
material regions 1, 2, ... in the
system such that:

<math>
\Sigma_1'(E) = \Sigma_2'(E) = \Sigma_3'(E) \dots = \Sigma_\mathrm{m}(E)
\</math>

where <math>\Sigma_\mathrm{m}</math> is called the ''majorant'' cross section. 

In practice, it is not necessary
to define the virtual collision cross sections at all if the majorant is simply
taken as the maximum of all material totals at each energy point:

<math>
\Sigma_\mathr{m}(E) = \max{\big[\Sigma(\mathbf{r},E)\big]}
</math>

Unlike the physical total cross section, which depends on the material 
located at the particle position, the majorant cross section is completely
independent of the spatial coordinates.

The point of having a macroscopic cross section that is uniform
throughout the geometry is that when used for sampling path lengths:

<math>
l = -\log(\xi)/\Sigma_\mathrm{m}
</math>

the values are statistically valid regardless of the number of material
boundaries crossed. 

At the end point of the sampled path the tracking routine
performs rejection sampling. The probability to accept the collision is given
by ratio of the physical total cross section to the majorant:

<math>
P = \frac{\Sigma(\mathbf{r},E)}{\Sigma_\mathrm{m}(E)}
</math>

If the collision is rejected, a new path length is sampled using \Sigma_\mathrm{m}
and the particle is moved to the next collision site candidate.

Since the majorant cross section is always larger than or equal to the
total cross section, the path lengths sampled in delta-tracking
are, on the average, shorter than those sampled with surface-tracking.
The average physical distance between two collisions is preserved, as some
paths are extended over multiple virtual collisions.

== Advantages and limitations of delta-tracking ==

The advantage of delta-tracking over the surface-tracking algorithm is that there is no need to
calculate the surface distances or stop the particle at the material boundaries. This becomes
significant for computational performance in geometries where the particle mean-free-path is long
compared to dimensions. The difference between surface- and delta-tracking is emphasized in HTGR [[particle fuel geometries]] and advanced
[[CAD-based geometry type|CAD]] and an [[Unstructured mesh-based geometry type|mesh]] based geometry types. 

Since the majorant cross section used for sampling the path lengths does not depend on the spatial coordinates
and the material total is needed only at discrete locations, variations of delta-tracking can be used for modelling
inhomogeneous material compositions. This capability is utilized in the [[TMS on-the-fly temperature treatment routine]] and [[multi-physics interface]] in Serpent 2.

Delta-tracking also has its drawbacks. Since the majorant cross section reflects the largest interaction probability within the system, the efficiency of the rejection sampling loop may become
poor in the presence of localized heavy absorbers (control rods, burnable absorber pins, etc.) that
dominate the majorant cross section, but occupy a relatively small volume in the geometry.
Another drawback is that delta-tracking rules out the use of the [[Result estimators#Implicit estimators|track-length estimate (TLE)]] of
flux, and reaction rate estimates need to be calculated using the
potentially less efficient [[Result estimators#Implicit estimators|collision estimator (CFE)]].

== Hybrid method used in Serpent ==

== References ==

<references/>