Correlated Sampling Technique

Correlated sampling techniques force all the histories corresponding to the perturbed system to follow the same transition points in phase space as the unperturbed histories. Appropriate weight factors are then used to adjust the particle weights at the transition points. This can be explained mathematically by looking at the integral form of the neutron transport equation (equation 2.1), expressed in terms of collision density and its solution by the Neumann series [Spa69, Lux91]. The collision density equation is given by,

$$\varphi(x) = \int_V\varphi(y)\kappa(y,x)dy + Q(x) ,$$ (40)

where x and y are the coordinates of a particle in the six-dimensional phase space, $\kappa(x,y)$ is the transport kernel from y to x, $\varphi(x)$ is the collision density of particles entering a collision in x, and Q(x) is the external particle source in x. The Neumann series solution of equation (3.8) is given by,

$$\varphi(x) =\sum_{n=0}^{\infty}\varphi_{n}(x)$$

$$= Q(x) + \sum_{n=1}^{\infty}\int_V\cdot\cdot\cdot\cdot\cdot ... ...)\kappa(u_{n-1},u_n)..... \kappa(u_1,u_2)du_n.....du_1 Q(u_1) ,$$ (41)

where $\kappa(u_{i-1},u_i)$ is the probability that a particle entering into a collision at $\vec{r}_{i-1}$ with energy $\vec{E}_{i-1}$ will appear at $\vec{r}_i$ with energy $\vec{E}_i$,

$$\kappa(u_{i-1},u_{i}) = \kappa(\vec{r}_{i-1},\vec{E}_{i-1}\rightarrow\vec{r}_i,\vec{E}_i).$$ (42)

Here

$$\varphi_0(x) = Q(x) ,$$ (43)

is the direct source contribution term. For n = 1 we get the once-collided term,

$$\varphi_1(x) = \int_V\kappa(u_1,x)Q(u_1)du_1.$$ (44)

Similarly for n = 2,3... etc. the twice, thrice ... etc. collided terms can be found. The transport kernel $\kappa(u_{i-1},u_i)$ is expressed in terms of the product of a collision kernel, C($\vec{E}_{i-1}\rightarrow\vec{E}_i\vert \vec{r}_{i-1}$), and a translation kernel, T($\vec{r}_{i-1}\rightarrow\vec{r}_i\vert E_i$), as shown below,

$$\kappa(u_{i-1},u_{i}) = \kappa(\vec{r}_{i-1},\vec{E}_{i-1}\r... ...rt\vec{r}_{i-1})T(\vec{r}_{i-1} \rightarrow\vec{r}_i\vert E_i),$$ (45)

or,

$$\kappa_{i-1} = C_{i-1} T_{i-1}.$$ (46)

The kernel C_i-1 denotes the probabilities of particles that are coming out of a collision in $\vec{r}_{i-1}$with direction $\vec \Omega$ and energy E_i, i.e., $\vec{E}_i$ = $\vec\Omega_iE_i$. The collision kernel can be represented explicitly as,

$$C(\vec{E}_{i-1}\rightarrow\vec{E}_i\vert\vec{r}_{i-1}) = \su... ..._j} {C_j(\vec{E}_{i-1}\rightarrow\vec{E}_i\vert\vec{r}_{i-1})},$$ (47)

where p_j denotes the probability of scattering collision of type j, and C_j is the corresponding collision kernel. Each C_j can be normalized to the mean number of secondaries, $\nu$, per event,

$$\int{C_j(\vec{E}_{i-1}\rightarrow\vec{E}_i\vert\vec{r}_{i-1})}d\vec{E}_i = \nu_j.$$ (48)

For elastic scattering events, $\nu$ = 1; for fission $\nu$ > 1. The probabilities p_j can be written as,

$$p_j = \frac{\Sigma_{sj}(\vec{r}_{i-1},\vec{E}_{i-1})} {\Sigma_{t}(\vec{r}_{i-1},\vec{E}_{i-1})},$$ (49)

where $\Sigma_{sj}$ is the macroscopic scattering cross section for scattering type j.
The kernel T_i-1 represents the probability for the transport of particles from $\vec{r}_{i-1}$ to the next collision in $\vec{r}_i$. For example, if $\Sigma_t(\vec{r}_{i-1}, \vec{E}_{i})$,the total macroscopic cross section at ($\vec{r}_{i-1},\vec{E}_{i}$), is spatially constant along the direction $\vec\Omega_i$ then,

$$T(\vec{r}_{i-1}\rightarrow\vec{r}_i\vert E_i) = \Sigma_t(E_i)exp(-\Sigma_t(E_i)d),$$ (50)

where d is the distance from $\vec{r}_{i-1}$ to $\vec{r}_i$.

To develop expressions for correlated sampling tracking, we will denote the transport kernel of the unperturbed system by,

$$\kappa_i^{up} = \kappa(u_i,u_{i+1};\Sigma_x^{up}),$$ (51)

and that for the perturbed system by,

$$\kappa_i^p = \kappa(u_i,u_{i+1};\Sigma_x^p).$$ (52)

$\Sigma_x^{up}$ and $\Sigma_x^p$ denote a generic cross section for the unperturbed and the perturbed systems, respectively, and the perturbation in cross section can be expressed as,

$$\Delta\Sigma_x = \Sigma_x^p - \Sigma_x^{up}.$$ (53)

Now the collision densities for the unperturbed and the perturbed systems are respectively,

$$\varphi^{up}(x) = Q(x) + \sum_{n=1}^{\infty}\int_V\cdot\cdot\cdot\cdot\cdot\int_V(\prod_{i=1}^n \kappa_i^{up}du_i)Q(u_1),$$ (54)

and

$$\varphi^p(x) = Q(x) + \sum_{n=1}^{\infty}\int_V\cdot\cdot\cdot\cdot\cdot\int_V(\prod_{i=1}^n \kappa_i^{p}du_i)Q(u_1).$$ (55)

The difference between the two collision densities is a function of the cross section change $\Delta\Sigma_x$. As shown before, independent simulation of the unperturbed and perturbed systems and straightforward subtraction of the results is not sufficient for calculating perturbation effects, especially for small perturbations. In the correlated sampling method, the perturbed histories are forced to follow the same trajectories as the unperturbed histories including the same transition points in phase space [Rie84]. A weight factor is used for the perturbed histories to account for the resulting biasing due to the forced transition. The weight factor for the perturbed system is given by,

$$F^p(u_{i+1},u_{i};\Delta\Sigma_x)= F_i^p = \frac{\kappa_i^p}{\kappa_i^{up}}.$$ (56)

Now the perturbation effect is given by,

$$\Delta\varphi(x;\Delta\Sigma_x) = \varphi^p(x) - \varphi^{up}(x) =$$

$$\sum_{n=1}^{\infty}\int_V\cdot\cdot\cdot\cdot\cdot\int_V(\pr... ...=0}^n{F^p_i} -1){[\prod_{i=1}^n{\kappa_i^{up}{du_i}}{Q(u_1)}]}.$$ (57)

We notice in the expression for $\Delta\varphi(x)$ that the summation expression for the unperturbed collision density, $\varphi^{up}(x)$, has been multiplied by the weight factor

$$(\prod_{i=0}^n{{F^p_i}-1}).$$

$\Delta\varphi$ is tallied at each collision point and contributes to the calculation of the perturbation.