From 3b2a6286e64157097ec8268bc943b4b3bcba2e9d Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Mon, 30 Jul 2012 17:01:04 -0400 Subject: [PATCH] Updated documentation again. --- ...397496c922665b55efc848d4359d67e574554e.png | Bin 1634 -> 0 bytes ...58b57a0a7ff49829bd762992dd29db49f83d6b.png | Bin 0 -> 805 bytes ...f8b0e83826e6a4328880ee19bd48c383b1c42a.png | Bin 923 -> 0 bytes ...237a127a96bfc4531a8f04aec0c9811718f4bd.png | Bin 0 -> 973 bytes ...30fbb8c3437fc5d744dccc33198f7da78a2451.png | Bin 1303 -> 0 bytes ...764bdb1f7b53d6b36b2444cf7803576073b024.png | Bin 1285 -> 0 bytes ...83a478ee4d87407045696a72b69baebc8164ae.png | Bin 443 -> 0 bytes ...e0abf0ecbd684740f6b693395eef8f3458c71d.png | Bin 205 -> 0 bytes ...86994c6d176a8a6c6efdd41dacea4e178b5486.png | Bin 872 -> 0 bytes ...8ec3447b53f7b165169a3eea8dcc8a70816ed3.png | Bin 1328 -> 0 bytes ...c6609efaa364753fa5a4e4107b712efc452d4f.png | Bin 1040 -> 0 bytes ...c733d766b73d1ed41269cdf741be04030b8ec7.png | Bin 358 -> 0 bytes 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A different -random number is used to sample a scattering cosine bin :math:`j` using the -cumulative distribution function: +:math:`\ell` be the chosen table where :math:`\ell = i` if :math:`\xi_1 > f` and +:math:`\ell = i + 1` otherwise where :math:`\xi_1` is a random number. Another +random number :math:`\xi_2` is used to sample a scattering cosine bin :math:`j` +using the cumulative distribution function: .. math:: :label: sample-cdf - c_{\ell,j} < \xi < c_{\ell,j+1} + c_{\ell,j} < \xi_2 < c_{\ell,j+1} The final scattering cosine will depend on whether histogram or linear-linear interpolation is used. In general, we can write the cumulative distribution @@ -123,7 +166,7 @@ after integration we have that .. math:: :label: cumulative-dist-histogram - c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi + c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi_2 Solving for the scattering cosine, we obtain the final form for histogram interpolation: @@ -131,7 +174,7 @@ interpolation: .. math:: :label: cosine-histogram - \mu = \mu_{\ell,j} + \frac{\xi - c_{\ell,j}}{p_{\ell,j}} + \mu = \mu_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}} For linear-linear interpolation, we represent the function :math:`p(\mu')` as a first-order polynomial in :math:`\mu'`. If we interpolate between successive @@ -159,7 +202,7 @@ Let us now make a change of variables using :label: introduce-eta \eta = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} - (\mu' - \mu_{\ell,j}) + (\mu' - \mu_{\ell,j}) + p_{\ell,j} Equation :eq:`cdf-linlin` then becomes @@ -182,7 +225,7 @@ Integrating equation :eq:`cdf-linlin-eta`, we have :label: cdf-linlin-integrated c(\mu) = c_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) + - p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi + p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi_2 Solving for :math:`\mu`, we have the final form for the scattering cosine using linear-linear interpolation: @@ -190,7 +233,7 @@ linear-linear interpolation: .. math:: :label: cosine-linlin - \mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi - + \mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi_2 - c_{\ell,j} )} - p_{\ell,j} \right ) .. _sample-energy: diff --git a/_sources/methods/statistics.txt b/_sources/methods/statistics.txt index 7563006f09..b4e6fa2295 100644 --- a/_sources/methods/statistics.txt +++ b/_sources/methods/statistics.txt @@ -57,7 +57,7 @@ where :math:`g`, :math:`c`, and :math:`M` are constants. The choice of these constants will have a profound effect on the quality and performance of the generator, so they should not be chosen arbitrarily. As Donald Knuth said in his seminal work *The Art of Computer Programming*, "random numbers should not be -generated with a method chosen at random". Some theory should be used." +generated with a method chosen at random. Some theory should be used." Typically, :math:`M` is chosen to be a power of two as this enables :math:`x \mod M` to be performed using the binary AND operator with a bit mask. The constants for the linear congruential generator used by default in OpenMC are diff --git a/_sources/usersguide/input.txt b/_sources/usersguide/input.txt index 89fb95222f..5dde07468e 100644 --- a/_sources/usersguide/input.txt +++ b/_sources/usersguide/input.txt @@ -151,6 +151,23 @@ problem. It has the following attributes/sub-elements: *Default*: None +```` Element +-------------------------- + +The ```` element indicates that a fixed source calculation should be +performed. It has the following attributes/sub-elements: + + :batches: + The total number of batches. For fixed source calculations, each batch + represents a realization of random variables for tallies. + + *Default*: None + + :particles: + The number of particles to simulate per batch. + + *Default*: None + ```` Element ----------------------- @@ -184,26 +201,91 @@ pseudo-random number generator. ```` Element -------------------- -The ``source`` element gives information on an initial source guess for -criticality calculations. It takes the following attributes: +The ``source`` element gives information on an external source distribution to +be used either as the source for a fixed source calculation or the initial +source guess for criticality calculations. It takes the following +attributes/sub-elements: - :type: - The type of source distribution. Setting this to "box" indicates that the - starting source should be sampled uniformly in a parallelepiped. Setting - this to "point" indicates that the starting source should be sampled from an - isotropic point source. Setting this to "file" indicates that the starting - source should be sampled from a ``source.binary`` file. + :file: + If this attribute is given, it indicates that the source is to be read from + a binary source file whose path is given by the value of this element - :coeffs: - For a "box" source distribution, ``coeffs`` should be given as six real - numbers, the first three of which specify the lower-left corner of a - parallelepiped and the last three of which specify the upper-right - corner. Source sites are sampled uniformly through that parallelepiped. + *Default*: None - For a "point" source distribution, ``coeffs`` should be given as three real - numbers which specify the (x,y,z) location of an isotropic point source + :space: + An element specifying the spatial distribution of source sites. This element + has the following attributes: - For a "file" source distribution, ``coeffs`` should not be specified. + :type: + The type of spatial distribution. Valid options are "box" and "point". A + "box" spatial distribution has coordinates sampled uniformly in a + parallelepiped. A "point" spatial distribution has coordinates specified + by a triplet. + + *Default*: None + + :parameters: + For a "box" spatial distribution, ``parameters`` should be given as six + real numbers, the first three of which specify the lower-left corner of a + parallelepiped and the last three of which specify the upper-right + corner. Source sites are sampled uniformly through that parallelepiped. + + For a "point" spatial distribution, ``parameters`` should be given as + three real numbers which specify the (x,y,z) location of an isotropic + point source + + *Default*: None + + :angle: + An element specifying the angular distribution of source sites. This element + has the following attributes: + + :type: + The type of angular distribution. Valid options are "isotropic" and + "monodirectional". The angle of the particle emitted from a source site is + isotropic if the "isotropic" option is given. The angle of the particle + emitted from a source site is the direction specified in the + attribute if "monodirectional" option is given. + + *Default*: isotropic + + :parameters: + For an "isotropic" angular distribution, ``parameters`` should not be + specified + + For a "monodirectional" angular distribution, ``parameters`` should be + given as three real numbers which specify the angular cosines with respect + to each axis. + + *Default*: None + + :energy: + An element specifying the energy distribution of source sites. This element + has the following attributes: + + :type: + + The type of energy distribution. Valid options are "monoenergetic", + "watt", and "maxwell". The "monoenergetic" option produces source sites at + a single energy. The "watt" option produces source sites whose energy is + sampled from a Watt fission spectrum. The "maxwell" option produce source + sites whose energy is sampled from a Maxwell fission spectrum + + *Default*: watt + + :parameters: + For a "monoenergetic" energy distribution, ``parameters`` should not be + given as the energy in MeV of the source sites. + + For a "watt" energy distribution, ``parameters`` should be given as two + real numbers :math:`a` and :math:`b` that parameterize the distribution + :math:`p(E) dE = c e^{-E/a} \sinh \sqrt{b \, E} dE`. + + For a "maxwell" energy distribution, ``parameters`` should be given as one + real number :math:`a` that parameterizes the distribution :math:`p(E) dE = + c E e^{-E/a} dE`. + + *Default*: 0.988 2.249 ```` Element ------------------------------ diff --git a/methods/criticality.html b/methods/criticality.html index 8d19a04e17..ffc2c5e75d 100644 --- a/methods/criticality.html +++ b/methods/criticality.html @@ -59,8 +59,8 @@ neutrons includes a fissionable material. Some common criticality calculations include the simulation of nuclear reactors, spent fuel pools, nuclear weapons, and other fissile systems. The term criticality calculation is also synonymous with the term eigenvalue calculation. The reason for this is that the transport -equation becomes an eigenvalue value equation if a fissionable source is present -since then the source of neutrons will depend on the flux of neutrons +equation becomes an eigenvalue equation if a fissionable source is present since +then the source of neutrons will depend on the flux of neutrons itself. Criticality simulations using Monte Carlo methods are becoming increasingly common with the advent of high-performance computing.

This section will explore the theory behind and implementation of criticality diff --git a/methods/geometry.html b/methods/geometry.html index 898fdfbf43..70987130b0 100644 --- a/methods/geometry.html +++ b/methods/geometry.html @@ -67,12 +67,12 @@ where f(x,y,z) > 0 can be called the positive half-space.

Let us take the example of a sphere centered at the point (x_0,y_0,z_0) with radius R. One would normally write the equation of the sphere as

-
-

(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2

-

By subtracting the right-hand term from both sides of the equation, we can then -write the surface equation:

-

If no solutions to equation (3) exist or the only solutions +are complex, then the particle’s direction of travel will not intersect the +surface. If the solution to equation (3) is negative, this +means that the surface is “behind” the particle, i.e. if the particle continues +traveling in its current direction, it will not hit the surface. The complete +derivation for different types of surfaces used in OpenMC will be presented in +the following sections.

+

Once a distance has been computed to a surface, we need to check if it is closer +than previously-computed distances to surfaces. Unfortunately, we cannot just +use the minimum function because some of the calculated distances, which should +be the same in theory (e.g. coincident surfaces), may be slightly different due +to the use of floating-point arithmetic. Consequently, we should first check for +floating-point equality of the current distance calculated and the minimum found +thus far. This is done by checking if

-

(2)\frac{| d - d_{min} |}{d_{min}} < \epsilon

+

(4)\frac{| d - d_{min} |}{d_{min}} < \epsilon

where d is the distance to a surface just calculated, d_{min} is the minimum distance found thus far, and \epsilon is a small number. In OpenMC, this parameter is set to \epsilon = 10^{-14} since all floating @@ -148,7 +149,7 @@ given level.

x - x_0 = 0. As such, we need to solve x + du - x_0 = 0. The solution for the distance is

-

(3)d = \frac{x_0 - x}{u}

+

(5)d = \frac{x_0 - x}{u}

Note that if the particle’s direction of flight is parallel to the x-axis, i.e. u = 0, the distance to the surface will be infinity. While the example here was for a plane perpendicular to the x-axis, the same formula can @@ -160,7 +161,7 @@ be applied for the surfaces A(x + du) + B(y + dv) + C(z + dw) = D. The solution to this equation for the distance is

-

(4)d = \frac{D - Ax - By - Cz}{Au + Bv + Cw}

+

(6)d = \frac{D - Ax - By - Cz}{Au + Bv + Cw}

Again, we need to check whether the denominator is zero. If so, this means that the particle’s direction of flight is parallel to the plane and it will therefore never hit the plane.

@@ -172,17 +173,17 @@ y_0)^2 + (z - z_0)^2 = R^2"/>. Thus, we need to solve . Let us define \bar{y} = y - y_0 and \bar{z} = z - z_0. We then have

-

(5)(\bar{y} + dv)^2 + (\bar{z} + dw)^2 = R^2

-

Expanding equation (5) and rearranging terms, we obtain

+

(7)(\bar{y} + dv)^2 + (\bar{z} + dw)^2 = R^2

+

Expanding equation (7) and rearranging terms, we obtain

-

(6)(v^2 + w^2) d^2 + 2 (\bar{y}v + \bar{z}w) d + (\bar{y}^2 + \bar{z}^2 - R^2) +<p><span class=(8)(v^2 + w^2) d^2 + 2 (\bar{y}v + \bar{z}w) d + (\bar{y}^2 + \bar{z}^2 - R^2) = 0

This is a quadratic equation for d. To simplify notation, let us define a = v^2 + w^2, k = \bar{y}v + \bar{z}w, and c = \bar{y}^2 + \bar{z}^2 - R^2. Thus, the distance is just the solution to ad^2 + 2kd + c = 0:

-

(7)d = \frac{-k \pm \sqrt{k^2 - ac}}{a}

+

(9)d = \frac{-k \pm \sqrt{k^2 - ac}}{a}

A few conditions must be checked for. If a = 0, this means the particle is parallel to the cylinder and will thus never intersect it. Also, if k^2 - ac < 0, this means that both solutions to the quadratic are @@ -205,21 +206,21 @@ the y- or z-axis with appropriate substitution of constants.

The equation for a sphere is (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2. Thus, we need to solve the equation

-

(8)(x + du - x_0)^2 + (y + dv - y_0)^2 + (z + dw - z_0)^2 = R^2

+

(10)(x + du - x_0)^2 + (y + dv - y_0)^2 + (z + dw - z_0)^2 = R^2

Let us define \bar{x} = x - x_0, \bar{y} = y - y_0, and \bar{z} = z - z_0. We then have

-

(9)(\bar{x} + du)^2 + (\bar{y} + dv)^2 + (\bar{z} - dw)^2 = R^2

-

Expanding equation (9) and rearranging terms, we obtain

+

(11)(\bar{x} + du)^2 + (\bar{y} + dv)^2 + (\bar{z} - dw)^2 = R^2

+

Expanding equation (11) and rearranging terms, we obtain

-

(10)d^2 + 2 (\bar{x}u + \bar{y}v + \bar{z}w) d + (\bar{x}^2 + \bar{y}^2 + +<p><span class=(12)d^2 + 2 (\bar{x}u + \bar{y}v + \bar{z}w) d + (\bar{x}^2 + \bar{y}^2 + \bar{z}^2 - R^2) = 0

This is a quadratic equation for d. To simplify notation, let us define k = \bar{x}u + \bar{y}v + \bar{z}w and c = \bar{x}^2 + \bar{y}^2 + \bar{z}^2 - R^2. Thus, the distance is just the solution to d^2 + 2kd + c = 0:

-

(11)d = -k \pm \sqrt{k^2 - c}

+

(13)d = -k \pm \sqrt{k^2 - c}

If the discriminant k^2 - c < 0, this means that both solutions to the quadratic are complex. In physical terms, this means that the ray along which the particle is traveling does not make any intersections with the sphere.

@@ -279,11 +280,11 @@ space of a sphere, the positive half-space of an x-plane, and the negative half-space of a y-plane. Said another way, any point inside this cell must satisfy the following equations

-

(12)x^2 + y^2 + z^2 - 10^2 < 0 \\ +<p><span class=(14)x^2 + y^2 + z^2 - 10^2 < 0 \\ x - (-3) > 0 \\ x - 2 < 0

So in order to determine if a point is inside the cell, we would plug its -coordinates into equation (12) and if the inequalities +coordinates into equation (14) and if the inequalities are satisfied, than the point is indeed inside the cell.

@@ -324,7 +325,7 @@ performed over all cells and the neighbor lists are populated for each surface.< the form f(x,y,z) = 0 with a reflective boundary condition, it can be shown based on geometric arguments that the velocity vector will then become

-

(13)\mathbf{v'} = \mathbf{v} - 2 (\mathbf{v} \cdot \hat{\mathbf{n}}) +<p><span class=(15)\mathbf{v'} = \mathbf{v} - 2 (\mathbf{v} \cdot \hat{\mathbf{n}}) \hat{\mathbf{n}}

where \hat{\mathbf{n}} is a unit vector normal to the surface at the point of the surface crossing. The rationale for this can be understood by @@ -333,22 +334,22 @@ projection of the velocity vector onto the normal vector. By subtracting two times this projection, the velocity is reflected with respect to the surface normal. Since the velocity of the particle will not change as it undergoes reflection, we can work with the direction of the particle instead, simplifying -equation (13) to

+equation (15) to

-

(14)\mathbf{\Omega'} = \mathbf{\Omega} - 2 (\mathbf{\Omega} \cdot +<p><span class=(16)\mathbf{\Omega'} = \mathbf{\Omega} - 2 (\mathbf{\Omega} \cdot \hat{\mathbf{n}}) \hat{\mathbf{n}}

The direction of the surface normal will be the gradient to the surface at the point of crossing, i.e. \mathbf{n} = \nabla f(x,y,z). Substituting this -into equation (14), we get

+into equation (16), we get

-

(15)\mathbf{\Omega'} = \mathbf{\Omega} - \frac{2 ( \mathbf{\Omega} \cdot \nabla +<p><span class=(17)\mathbf{\Omega'} = \mathbf{\Omega} - \frac{2 ( \mathbf{\Omega} \cdot \nabla f )}{|| \nabla f ||^2} \nabla f

If we write the initial and final directions in terms of their vector components, \mathbf{\Omega} = (u,v,w) and \mathbf{\Omega'} = (u', -v', w'), this allows us to represent equation (14) as a +v', w')"/>, this allows us to represent equation (16) as a series of equations:

-

(16)u' = u - \frac{2 ( \mathbf{\Omega} \cdot \nabla f )}{|| \nabla f ||^2} +<p><span class=(18)u' = u - \frac{2 ( \mathbf{\Omega} \cdot \nabla f )}{|| \nabla f ||^2} \frac{\partial f}{\partial x} \\ v' = v - \frac{2 ( \mathbf{\Omega} \cdot \nabla f )}{|| \nabla f ||^2} @@ -366,10 +367,10 @@ with the derivation to confirm that the rules of geometry agree with our intuition. The gradient of the surface <img class= is simply \nabla f = (1, 0, 0). Note that this vector is already normalized, i.e. || \nabla f || = 1. The second two equations in -(16) tell us that v and w do not change and +(18) tell us that v and w do not change and the first tell us that

-

(17)u' = u - 2u = -u

+

(19)u' = u - 2u = -u

We see that reflection for a plane perpendicular to an axis only entails negating the directional cosine for that axis.

@@ -379,36 +380,36 @@ negating the directional cosine for that axis.

gradient to the surface is simply \nabla f = (A,B,C) whose norm squared is A^2 + B^2 + C^2. This implies that

-

(18)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = \frac{2(Au + +<p><span class=(20)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = \frac{2(Au + Bv + Cw)}{A^2 + B^2 + C^2}

-

Substituting equation (18) into equation -(16) gives us the form of the solution. For example, the +

Substituting equation (20) into equation +(18) gives us the form of the solution. For example, the x-component of the reflected direction will be

-

(19)u' = u - \frac{2A(Au + Bv + Cw)}{A^2 + B^2 + C^2}

+

(21)u' = u - \frac{2A(Au + Bv + Cw)}{A^2 + B^2 + C^2}

4.7.3. Cylinder Parallel to an Axis

A cylinder parallel to, for example, the x-axis has the form f(x,y,z) = (y - y_0)^2 + (z - z_0)^2 - R^2 = 0. Thus, the gradient to the surface is

-

(20)\nabla f = 2 \left ( \begin{array}{c} 0 \\ y - y_0 \\ z - z_0 \end{array} +<p><span class=(22)\nabla f = 2 \left ( \begin{array}{c} 0 \\ y - y_0 \\ z - z_0 \end{array} \right ) = 2 \left ( \begin{array}{c} 0 \\ \bar{y} \\ \bar{z} \end{array} \right )

where we have introduced the constants \bar{y} and \bar{z}. Taking the square of the norm of the gradient, we find that

-

(21)|| \nabla f ||^2 = 4 \bar{y}^2 + 4 \bar{z}^2 = 4 R^2

+

(23)|| \nabla f ||^2 = 4 \bar{y}^2 + 4 \bar{z}^2 = 4 R^2

This implies that

-

(22)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = +<p><span class=(24)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = \frac{\bar{y}v + \bar{z}w}{R^2}

-

Substituting equations (22) and -(20) into equation (16) gives us +

Substituting equations (24) and +(22) into equation (18) gives us the form of the solution. In this case, the x-component will not change. The y- and z-components of the reflected direction will be

-

(23)v' = v - \frac{2 ( \bar{y}v + \bar{z}w ) \bar{y}}{R^2} \\ +<p><span class=(25)v' = v - \frac{2 ( \bar{y}v + \bar{z}w ) \bar{y}}{R^2} \\ w' = w - \frac{2 ( \bar{y}v + \bar{z}w ) \bar{z}}{R^2}

@@ -417,22 +418,22 @@ w' = w - \frac{2 ( \bar{y}v + \bar{z}w ) \bar{z}}{R^2}"/>

The surface equation for a sphere has the form f(x,y,z) = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 - R^2 = 0. Thus, the gradient to the surface is

-

(24)\nabla f = 2 \left ( \begin{array}{c} x - x_0 \\ y - y_0 \\ z - z_0 +<p><span class=(26)\nabla f = 2 \left ( \begin{array}{c} x - x_0 \\ y - y_0 \\ z - z_0 \end{array} \right ) = 2 \left ( \begin{array}{c} \bar{x} \\ \bar{y} \\ \bar{z} \end{array} \right )

where we have introduced the constants \bar{x}, \bar{y}, \bar{z}. Taking the square of the norm of the gradient, we find that

-

(25)|| \nabla f ||^2 = 4 \bar{x}^2 + 4 \bar{y}^2 + 4 \bar{z}^2 = 4 R^2

+

(27)|| \nabla f ||^2 = 4 \bar{x}^2 + 4 \bar{y}^2 + 4 \bar{z}^2 = 4 R^2

This implies that

-

(26)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = +<p><span class=(28)\frac{2 (\mathbf{\Omega} \cdot \nabla f)}{|| \nabla f ||^2} = \frac{\bar{x}u + \bar{y}v + \bar{z}w}{R^2}

-

Substituting equations (26) and -(24) into equation (16) gives us the +

Substituting equations (28) and +(26) into equation (18) gives us the form of the solution:

-

(27)u' = u - \frac{2 ( \bar{x}u + \bar{y}v + \bar{z}w ) \bar{x} }{R^2} \\ +<p><span class=(29)u' = u - \frac{2 ( \bar{x}u + \bar{y}v + \bar{z}w ) \bar{x} }{R^2} \\ v' = v - \frac{2 ( \bar{x}u + \bar{y}v + \bar{z}w ) \bar{y} }{R^2} \\ diff --git a/methods/index.html b/methods/index.html index c627904ba4..270619f631 100644 --- a/methods/index.html +++ b/methods/index.html @@ -100,28 +100,29 @@ </ul> </li> <li class=5. Physics

where \xi is a pseudorandom number sampled from a uniform +distribution on [0,1).

- -

  1. If the distance to the nearest boundary is less than the distance to the next collision, the particle is moved forward to this boundary. Then, the process is repeated from step 2. If the distance to collision is closer than the @@ -132,7 +130,7 @@ the nuclide with which the collision will happen is sampled based on the total cross-sections. If the total cross section of material i is \Sigma_{t,i}, then the probability that any nuclide is sampled is

    -

    P(i) = \frac{\Sigma_{t,i}}{\Sigma_t}

    +

    P(i) = \frac{\Sigma_{t,i}}{\Sigma_t}.

  2. Once the specific nuclide is sampled, the random samples a reaction for that nuclide based on the microscopic cross sections. If the microscopic @@ -140,7 +138,7 @@ cross-section for some reaction \sigma_t, then the probability that reaction x will occur is

    -

    P(x) = \frac{\sigma_x}{\sigma_t}

    +

    P(x) = \frac{\sigma_x}{\sigma_t}.

  3. If the sampled reaction is elastic or inelastic scattering, the outgoing energy and angle is sampled from the appropriate distribution. If the diff --git a/methods/physics.html b/methods/physics.html index 3b937cf0dd..d09f92fd9a 100644 --- a/methods/physics.html +++ b/methods/physics.html @@ -54,8 +54,34 @@

    5. Physics

    +
    +

    5.1. Sampling Distance to Next Collision

    +

    As a particle travels through a homogeneous material, the probability +distribution function for the distance to its next collision \ell is

    +
    +

    (1)p(\ell) d\ell = \Sigma_t e^{-\Sigma_t \ell} d\ell

    +

    where \Sigma_t is the total macroscopic cross section of the +material. Equation (1) tells us that the further the distance is +to the next collision, the less likely the particle will travel that distance, +which should agree with your intuition. In order to sample the probability +distribution function, we first need to convert it to a cumulative distribution +function

    +
    +

    (2)\int_0^{\ell} d\ell' p(\ell') = \int_0^{\ell} d\ell' \Sigma_t e^{-\Sigma_t +\ell'} = 1 - e^{-\Sigma_t \ell}

    +

    By setting the cumulative distribution function equal to \xi, a random +number on the unit interval, and solving for the distance \ell, we +obtain a formula for sampling the distance to next collision:

    +
    +

    (3)\ell = -\frac{\ln (1 - \xi)}{\Sigma_t}

    +

    Since \xi is uniformly distributed on [0,1), this implies that +1 - \xi is also uniformly distributed on [0,1) as well. Thus, +the formula usually used to calculate the distance to next collision is

    +
    +

    (4)\ell = -\frac{\ln \xi}{\Sigma_t}

    +
    -

    5.1. Secondary Angles and Energy Distributions

    +

    5.2. Secondary Angles and Energy Distributions

    For any reactions with secondary neutrons, it is necessary to sample secondary angle and energy distributions. This includes elastic and inelastic scattering, fission, and (n,xn) reactions. In some cases, the distributions may be specified @@ -64,7 +90,7 @@ angle-energy distribution. In this section, we will outline the methods used to sample secondary distributions as well as how they are used to modify the state of a particle.

    -

    5.1.1. Sampling Secondary Angle Distributions

    +

    5.2.1. Sampling Secondary Angle Distributions

    For elastic scattering, it is only necessary to specific a secondary angle distribution since the outgoing energy can be determined analytically. Other reactions may also have separate secondary angle and secondary energy @@ -76,27 +102,27 @@ distribution is represented as either

  4. A tabular distribution.
    5. -

    5.1.1.1. Isotropic Angular Distribution

    +

    5.2.1.1. Isotropic Angular Distribution

    In the first case, no data needs to be stored on the ACE table, and the cosine of the scattering angle is simply calculated as

    -

    (1)\mu = 2\xi - 1

    +

    (5)\mu = 2\xi - 1

    where \xi is a random number sampled uniformly on [0,1).

    -

    5.1.1.2. Equiprobable Angle Bin Distribution

    +

    5.2.1.2. Equiprobable Angle Bin Distribution

    For a 32 equiprobable bin distribution, the procedure to determine the scattering cosine is as follows. First, we select a random number \xi to sample a cosine bin i such that

    -

    (2)i = 1 + \lfloor 32\xi \rfloor

    +

    (6)i = 1 + \lfloor 32\xi \rfloor

    The same random number can then also be used to interpolate between neighboring \mu values to get the final scattering cosine:

    -

    (3)\mu = \mu_i + (32\xi - i) (\mu_{i+1} - \mu_i)

    +

    (7)\mu = \mu_i + (32\xi - i) (\mu_{i+1} - \mu_i)

    -

    5.1.1.3. Tabular Angular Distribution

    +

    5.2.1.3. Tabular Angular Distribution

    As the MCNP Manual points out, using an equiprobable bin distribution works well for high-probability regions of the scattering cosine probability, but for low-probability regions it is not very accurate. Thus, a more typical treatment @@ -108,71 +134,71 @@ probability distribution function and -

    (4)f = \frac{E - E_i}{E_{i+1} - E_i}

    +

    (8)f = \frac{E - E_i}{E_{i+1} - E_i}

    where E is the incoming energy of the particle. Then, statistical interpolation is performed to choose between using the cosines and distribution functions corresponding to energy E_i and E_{i+1}. Let -\ell be the chosen table where \ell = i if \xi > f and -\ell = i + 1 otherwise where \xi is a random number. A different -random number is used to sample a scattering cosine bin j using the -cumulative distribution function:

    +\ell be the chosen table where \ell = i if \xi_1 > f and +\ell = i + 1 otherwise where \xi_1 is a random number. Another +random number \xi_2 is used to sample a scattering cosine bin j +using the cumulative distribution function:

    -

    (5)c_{\ell,j} < \xi < c_{\ell,j+1}

    +

    (9)c_{\ell,j} < \xi_2 < c_{\ell,j+1}

    The final scattering cosine will depend on whether histogram or linear-linear interpolation is used. In general, we can write the cumulative distribution function as

    -

    (6)c(\mu) = \int_{-1}^\mu p(\mu') d\mu'

    +

    (10)c(\mu) = \int_{-1}^\mu p(\mu') d\mu'

    where c(\mu) is the cumulative distribution function and p(\mu) is the probability distribution function. Since we know that c(\mu_{\ell,j}) = c_{\ell,j}, this implies that for \mu > \mu_{\ell,j},

    -

    (7)c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} p(\mu') d\mu'

    +

    (11)c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} p(\mu') d\mu'

    For histogram interpolation, we have that p(\mu') = p_{\ell,j}. Thus, after integration we have that

    -

    (8)c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi

    +

    (12)c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi_2

    Solving for the scattering cosine, we obtain the final form for histogram interpolation:

    -

    (9)\mu = \mu_{\ell,j} + \frac{\xi - c_{\ell,j}}{p_{\ell,j}}

    +

    (13)\mu = \mu_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}}

    For linear-linear interpolation, we represent the function p(\mu') as a first-order polynomial in \mu'. If we interpolate between successive values on the probability distribution function, we know that

    -

    (10)p(\mu') - p_{\ell,j} = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - +<p><span class=(14)p(\mu') - p_{\ell,j} = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} (\mu' - \mu_{\ell,j})

    -

    Solving for p(\mu') in equation (10) and inserting it -into equation (7), we obtain

    +

    Solving for p(\mu') in equation (14) and inserting it +into equation (11), we obtain

    -

    (11)c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} \left [ \frac{p_{\ell,j+1} - +<p><span class=(15)c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} \left [ \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} (\mu' - \mu_{\ell,j}) + p_{\ell,j} \right ] d\mu'

    Let us now make a change of variables using

    -

    (12)\eta = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} -(\mu' - \mu_{\ell,j})

    -

    Equation (11) then becomes

    +

    (16)\eta = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} +(\mu' - \mu_{\ell,j}) + p_{\ell,j}

    +

    Equation (15) then becomes

    -

    (13)c(\mu) = c_{\ell,j} + \frac{1}{m} \int_{p_{\ell,j}}^{m(\mu - \mu_{\ell,j}) + +<p><span class=(17)c(\mu) = c_{\ell,j} + \frac{1}{m} \int_{p_{\ell,j}}^{m(\mu - \mu_{\ell,j}) + p_{\ell,j}} \eta \, d\eta

    where we have used

    -

    (14)m = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}

    -

    Integrating equation (13), we have

    +

    (18)m = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}

    +

    Integrating equation (17), we have

    -

    (15)c(\mu) = c_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) + -p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi

    +

    (19)c(\mu) = c_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) + +p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi_2

    Solving for \mu, we have the final form for the scattering cosine using linear-linear interpolation:

    -

    (16)\mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi - +<p><span class=(20)\mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi_2 - c_{\ell,j} )} - p_{\ell,j} \right )

    -

    5.1.2. Sampling Secondary Energy and Correlated Angle/Energy Distributions

    +

    5.2.2. Sampling Secondary Energy and Correlated Angle/Energy Distributions

    For a reaction with secondary neutrons, it is necessary to determine the outgoing energy of the neutrons. For anything other than elastic scattering, the outgoing energy must be determined based on tabulated or parameterized data. The @@ -194,7 +220,7 @@ if more than one is present.

    determining the outgoing energy will depend on which ACE law has been specified for the data.

    -

    5.1.2.1. ACE Law 1 - Tabular Equiprobable Energy Bins

    +

    5.2.2.1. ACE Law 1 - Tabular Equiprobable Energy Bins

    In the tabular equiprobable bin representation, an array of equiprobable outgoing energy bins is given for a number of incident energies. While the representation itself is simple, the complexity lies in how one interpolates @@ -207,11 +233,11 @@ reaction.

    scaled interpolation [Doyas]. First, we find the tabulated incident energies which bound the actual incoming energy of the particle, i.e. find i such that E_i < E < E_{i+1} and calculate the interpolation factor f -via (4). Then, we interpolate between the minimum and +via (8). Then, we interpolate between the minimum and maximum energies of the outgoing energy distributions corresponding to E_i and E_{i+1}:

    -

    (17)E_{min} = E_{i,1} + f ( E_{i+1,1} - E_i ) \\ +<p><span class=(21)E_{min} = E_{i,1} + f ( E_{i+1,1} - E_i ) \\ E_{max} = E_{i,M} + f ( E_{i+1,M} - E_M )

    where E_{min} and E_{max} are the minimum and maximum outgoing energies of a scaled distribution, E_{i,j} is the j-th outgoing energy @@ -224,26 +250,26 @@ between using the outgoing energy distributions corresponding to energy outgoing energy bin j and interpolate between successive values on the outgoing energy distribution:

    -

    (18)\hat{E} = E_{\ell,j} + \xi_2 (E_{\ell,j+1} - E_{\ell,j})

    +

    (22)\hat{E} = E_{\ell,j} + \xi_2 (E_{\ell,j+1} - E_{\ell,j})

    where \xi_2 is a random number sampled uniformly on [0,1). Since this outgoing energy may violate reaction kinematics, we then scale it to the minimum and maximum energies we calculated earlier to get the final outgoing energy:

    -

    (19)E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}} +<p><span class=(23)E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}} (E_{max} - E_{min})

    -

    5.1.2.2. ACE Law 3 - Inelastic Level Scattering

    +

    5.2.2.2. ACE Law 3 - Inelastic Level Scattering

    It can be shown [Foderaro] that in inelastic level scattering, the outgoing energy of the neutron E' can be related to the Q-value of the reaction and the incoming energy:

    -

    (20)E' = \left ( \frac{A}{A+1} \right )^2 \left ( E - \frac{A + 1}{A} Q \right )

    +

    (24)E' = \left ( \frac{A}{A+1} \right )^2 \left ( E - \frac{A + 1}{A} Q \right )

    where A is the mass of the target nucleus measured in neutron masses.

    -

    5.1.2.3. ACE Law 4 - Continuous Tabular Distribution

    +

    5.2.2.3. ACE Law 4 - Continuous Tabular Distribution

    This representation is very similar to ACE Law 1 - Tabular Equiprobable Energy Bins except that instead of equiprobable outgoing energy bins, the outgoing energy distribution for each incoming energy is represented with a probability distribution function. For @@ -254,14 +280,14 @@ energy.

    We proceed first as we did for ACE Law 1, determining the bounding energies of the particle’s incoming energy such that E_i < E < E_{i+1} and calculating an interpolation factor f with equation -(4). Next, statistical interpolation is performed to +(8). Next, statistical interpolation is performed to choose between using the outgoing energy distributions corresponding to energy E_i and E_{i+1}. Let \ell be the chosen table where \ell = i if \xi_1 > f and \ell = i + 1 otherwise where \xi_1 is a random number. Then, we sample an outgoing energy bin j using the cumulative distribution function:

    -

    (21)c_{\ell,j} < \xi_2 < c_{\ell,j+1}

    +

    (25)c_{\ell,j} < \xi_2 < c_{\ell,j+1}

    where \xi_2 is a random number sampled uniformly on [0,1). At this point, we need to interpolate between the successive values on the outgoing energy distribution using either histogram or linear-linear interpolation. The @@ -269,11 +295,11 @@ formulas for these can be derived along the same lines as those found in Tabular Angular Distribution. For histogram interpolation, the interpolated outgoing energy on the \ell-th distribution is

    -

    (22)\hat{E} = E_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}}

    +

    (26)\hat{E} = E_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}}

    If linear-linear interpolation is to be used, the outgoing energy on the \ell-th distribution is

    -

    (23)\hat{E} = E_{\ell,j} + \frac{E_{\ell,j+1} - E_{\ell,j}}{p_{\ell,j+1} - +<p><span class=(27)\hat{E} = E_{\ell,j} + \frac{E_{\ell,j+1} - E_{\ell,j}}{p_{\ell,j+1} - p_{\ell,j}} \left ( \sqrt{p_{\ell,j}^2 + 2 \frac{p_{\ell,j+1} - p_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( \xi_2 - c_{\ell,j} )} - p_{\ell,j} \right )

    @@ -281,18 +307,18 @@ p_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( \xi_2 - c_{\ell,j} )} - p_{\ell,j} minimum and maximum energies interpolated between the neighboring outgoing energy distributions to get the final outgoing energy:

    -

    (24)E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}} +<p><span class=(28)E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}} (E_{max} - E_{min})

    where E_{min} and E_{max} are defined the same as in equation -(17).

    +(21).

    -

    5.1.2.4. ACE Law 7 - Maxwell Fission Spectrum

    +

    5.2.2.4. ACE Law 7 - Maxwell Fission Spectrum

    One representation of the secondary energies for neutrons from fission is the so-called Maxwell spectrum. A probability distribution for the Maxwell spectrum can be written in the form

    -

    (25)p(E') dE' = c E'^{1/2} e^{-E'/T(E)} dE'

    +

    (29)p(E') dE' = c E'^{1/2} e^{-E'/T(E)} dE'

    where E is the incoming energy of the neutron and T is the so-called nuclear temperature, which is a function of the incoming energy of the neutron. The ACE format contains a list of nuclear temperatures versus incoming @@ -301,24 +327,24 @@ energies using a specified interpolation law. Once the temperature Monte Carlo Sampler:

    -

    (26)E' = -T \left [ \log (\xi_1) + \log (\xi_2) \cos^2 \left ( \frac{\pi +<p><span class=(30)E' = -T \left [ \log (\xi_1) + \log (\xi_2) \cos^2 \left ( \frac{\pi \xi_3}{2} \right ) \right ]

    where \xi_1, \xi_2, \xi_3 are random numbers sampled on the unit interval. The outgoing energy is only accepted if

    -

    (27)0 \le E' \le E - U

    +

    (31)0 \le E' \le E - U

    where U is called the restriction energy and is specified on the ACE table. If the outgoing energy is rejected, it is resampled using equation -(26).

    +(30).

    -

    5.1.2.5. ACE Law 9 - Evaporation Spectrum

    +

    5.2.2.5. ACE Law 9 - Evaporation Spectrum

    Evaporation spectra are primarily used in compound nucleus processes where a secondary particle can “evaporate” from the compound nucleus if it has sufficient energy. The probability distribution for an evaporation spectrum can be written in the form

    -

    (28)p(E') dE' = c E' e^{-E'/T(E)} dE'

    +

    (32)p(E') dE' = c E' e^{-E'/T(E)} dE'

    where E is the incoming energy of the neutron and T is the nuclear temperature, which is a function of the incoming energy of the neutron. The ACE format contains a list of nuclear temperatures versus incoming @@ -327,17 +353,17 @@ energies using a specified interpolation law. Once the temperature Monte Carlo Sampler:

    -

    (29)E' = -T \log (\xi_1 \xi_2)

    +

    (33)E' = -T \log (\xi_1 \xi_2)

    where \xi_1, \xi_2 are random numbers sampled on the unit interval. The outgoing energy is only accepted according to a specified -restriction energy as in equation (27).

    +restriction energy as in equation (31).

    -

    5.1.2.6. ACE Law 11 - Energy-Dependent Watt Spectrum

    +

    5.2.2.6. ACE Law 11 - Energy-Dependent Watt Spectrum

    The probability distribution for a Watt fission spectrum can be written in the form

    -

    (30)p(E') dE' = c e^{-E'/a(E)} \sinh \sqrt{b(E) \, E'} dE'

    +

    (34)p(E') dE' = c e^{-E'/a(E)} \sinh \sqrt{b(E) \, E'} dE'

    where a and b are parameters for the distribution and are given as tabulated functions of the incoming energy of the neutron in the ACE format. These two parameters are interpolated on the incoming energy grid using @@ -346,16 +372,16 @@ sample a Maxwellian spectrum with nuclear temperature ACE Law 7 - Maxwell Fission Spectrum to get an energy W. Then, the outgoing energy is calculated as

    -

    (31)E' = W + \frac{a^2 b}{4} + (2\xi - 1) \sqrt{a^2 b W}

    +

    (35)E' = W + \frac{a^2 b}{4} + (2\xi - 1) \sqrt{a^2 b W}

    where \xi is a random number sampled on the interval [0,1). The outgoing energy is only accepted according to a specified restriction energy -U as defined in equation (27).

    +U as defined in equation (31).

    This algorithm can be found in Forrest Brown’s lectures on Monte Carlo methods and is an unpublished sampling scheme based on the original Watt spectrum derivation [Watt].

    -

    5.1.2.7. ACE Law 44 - Kalbach-Mann Correlated Scattering

    +

    5.2.2.7. ACE Law 44 - Kalbach-Mann Correlated Scattering

    This law is very similar to ACE Law 4 except now the outgoing angle of the neutron is correlated to the outgoing energy and is not sampled from a separate distribution. For each incident neutron energy E_i tabulated, there is @@ -367,38 +393,38 @@ as in the algorithm described in , statistically sampled an incoming energy bin \ell, and sampled an outgoing energy bin j based on the tabulated cumulative distribution function. Once the outgoing energy has been -determined with equation (24), we then need to calculate the +determined with equation (28), we then need to calculate the outgoing angle based on the tabulated Kalbach-Mann parameters. These parameters themselves are subject to either histogram or linear-linear interpolation on the outgoing energy grid. For histogram interpolation, the parameters are

    -

    (32)R = R_{\ell,j} \\ +<p><span class=(36)R = R_{\ell,j} \\ A = A_{\ell,j}

    If linear-linear interpolation is specified, the parameters are

    -

    (33)R = R_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( +<p><span class=(37)R = R_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( R_{\ell,j+1} - R_{\ell,j} ) \\ A = A_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( A_{\ell,j+1} - A_{\ell,j} )

    -

    where \hat{E} is defined in equation (23). With the +

    where \hat{E} is defined in equation (27). With the parameters determined, the probability distribution function for the cosine of the scattering angle is

    -

    (34)p(\mu) d\mu = \frac{A}{2 \sinh (A)} \left [ \cosh (A\mu) + R \sinh (A\mu) +<p><span class=(38)p(\mu) d\mu = \frac{A}{2 \sinh (A)} \left [ \cosh (A\mu) + R \sinh (A\mu) \right ] d\mu

    The rules for sampling this probability distribution function can be derived based on rules C39 and C40 in the Monte Carlo Sampler. First, we sample two random numbers \xi_3, \xi_4 on the unit interval. If \xi_3 > R then the outgoing angle is

    -

    (35)\mu = \frac{1}{A} \ln \left ( T + \sqrt{T^2 + 1} \right )

    +

    (39)\mu = \frac{1}{A} \ln \left ( T + \sqrt{T^2 + 1} \right )

    where T = (2 \xi_4 - 1) \sinh (A). If \xi_3 \le R, then the outgoing angle is

    -

    (36)\mu = \frac{1}{A} \ln \left ( \xi_4 e^A + (1 - \xi_4) e^{-A} \right )

    +

    (40)\mu = \frac{1}{A} \ln \left ( \xi_4 e^A + (1 - \xi_4) e^{-A} \right )

    -

    5.1.2.8. ACE Law 61 - Correlated Energy and Angle Distribution

    +

    5.2.2.8. ACE Law 61 - Correlated Energy and Angle Distribution

    This law is very similar to ACE Law 44 in the sense that the outgoing angle of the neutron is correlated to the outgoing energy and is not sampled from a separate distribution. In this case though, rather than being determined from an @@ -410,7 +436,7 @@ as in the algorithm described in , statistically sampled an incoming energy bin \ell, and sampled an outgoing energy bin j based on the tabulated cumulative distribution function. Once the outgoing energy has been -determined with equation (24), we then need to decide which +determined with equation (28), we then need to decide which angular distribution to use. If histogram interpolation was used on the outgoing energy bins, then we use the angular distribution corresponding to incoming energy bin \ell and outgoing energy bin j. If linear-linear @@ -421,13 +447,13 @@ sample the chosen tabular angular distribution has been previously described in Tabular Angular Distribution.

    -

    5.1.2.9. ACE Law 66 - N-Body Phase Space Distribution

    +

    5.2.2.9. ACE Law 66 - N-Body Phase Space Distribution

    Reactions in which there are more than two products of similar masses are sometimes best treated by using what’s known as an N-body phase distribution. This distribution has the following probability density function for outgoing energy of the i-th particle in the center-of-mass system:

    -

    (37)p_i(E') dE' = C_n \sqrt{E'} (E_i^{max} - E')^{(3n/2) - 4} dE'

    +

    (41)p_i(E') dE' = C_n \sqrt{E'} (E_i^{max} - E')^{(3n/2) - 4} dE'

    where n is the number of outgoing particles, C_n is a normalization constant, E_i^{max} is the maximum center-of-mass energy for particle i, and E' is the outgoing energy. The algorithm for @@ -435,7 +461,7 @@ sampling the outgoing energy is based on algorithms R28, C45, and C64 in the Monte Carlo Sampler. First we calculate the maximum energy in the center-of-mass using the following equation:

    -

    (38)E_i^{max} = \frac{A_p - 1}{A_p} \left ( \frac{A}{A+1} E + Q \right )

    +

    (42)E_i^{max} = \frac{A_p - 1}{A_p} \left ( \frac{A}{A+1} E + Q \right )

    where A_p is the total mass of the outgoing particles in neutron masses, A is the mass of the original target nucleus in neutron masses, and Q is the Q-value of the reaction. Next we sample a value x from @@ -445,16 +471,16 @@ will depend on how many outgoing particles there are. For n = 4, we use the equation

    -

    (39)y = -\ln ( \xi_1 \xi_2 \xi_3 )

    +

    (43)y = -\ln ( \xi_1 \xi_2 \xi_3 )

    where \xi_i are random numbers sampled on the interval [0,1). For n = 5, we use the equation

    -

    (40)y = -\ln ( \xi_1 \xi_2 \xi_3 \xi_4 ) - \ln ( \xi_5 ) \cos^2 \left ( +<p><span class=(44)y = -\ln ( \xi_1 \xi_2 \xi_3 \xi_4 ) - \ln ( \xi_5 ) \cos^2 \left ( \frac{\pi}{2} \xi_6 \right )

    After x and y have been determined, the outgoing energy is then calculated as

    -

    (41)E' = \frac{x}{x + y} E_i^{max}

    +

    (45)E' = \frac{x}{x + y} E_i^{max}

    There are two important notes to make regarding the N-body phase space distribution. First, the documentation (and code) for MCNP5 has a mistake in the algorithm for n = 4. That being said, there are no existing nuclear data @@ -465,7 +491,7 @@ the (n,2n) reaction with H-2.

    -

    5.1.3. Transforming a Particle’s Coordinates

    +

    5.2.3. Transforming a Particle’s Coordinates

    Once the cosine of the scattering angle \mu has been sampled either from a angle distribution or a correlated angle-energy distribution, we are still left with the task of transforming the particle’s coordinates. If the outgoing @@ -473,14 +499,14 @@ energy and scattering cosine were given in the center-of-mass system, then we first need to transform these into the laboratory system. The relationship between the outgoing energy in center-of-mass and laboratory is

    -

    (42)E' = E'_{cm} + \frac{E + 2\mu_{cm} (A + 1) \sqrt{EE'_{cm}}}{(A+1)^2}.

    +

    (46)E' = E'_{cm} + \frac{E + 2\mu_{cm} (A + 1) \sqrt{EE'_{cm}}}{(A+1)^2}.

    where E'_{cm} is the outgoing energy in the center-of-mass system, \mu_{cm} is the scattering cosine in the center-of-mass system, E' is the outgoing energy in the laboratory system, and E is the incident neutron energy. The relationship between the scattering cosine in center-of-mass and laboratory is

    -

    (43)\mu = \mu_{cm} \sqrt{\frac{E'_{cm}}{E'}} + \frac{1}{A + 1} +<p><span class=(47)\mu = \mu_{cm} \sqrt{\frac{E'_{cm}}{E'}} + \frac{1}{A + 1} \sqrt{\frac{E}{E'}}.

    where \mu is the scattering cosine in the laboratory system. The scattering cosine still only tells us the cosine of the angle between the @@ -492,7 +518,7 @@ post-collision components. We first need to uniformly sample an azimuthal angle \phi in [0, 2\pi). After the azimuthal angle has been sampled, the post-collision direction is calculated as

    -

    (44)u' = \mu u + \frac{\sqrt{1 - \mu^2} ( uw \cos\phi - v \sin\phi )}{\sqrt{1 - +<p><span class=(48)u' = \mu u + \frac{\sqrt{1 - \mu^2} ( uw \cos\phi - v \sin\phi )}{\sqrt{1 - w^2}} \\ v' = \mu v + \frac{\sqrt{1 - \mu^2} ( vw \cos\phi + u \sin\phi )}{\sqrt{1 - @@ -502,7 +528,7 @@ w' = \mu w - \sqrt{1 - \mu^2} \sqrt{1 - w^2} \cos\phi

    -

    5.2. Elastic Scattering

    +

    5.3. Elastic Scattering

    Elastic scattering refers to the process by which a neutron scatters off a nucleus and does not leave it in an excited. It is referred to as “elastic” because in the center-of-mass system, the neutron does not actually lose @@ -519,18 +545,18 @@ velocity of the target nucleus are described later in section target velocity v_t. The velocity of the center-of-mass system is calculated as

    -

    (45)\mathbf{v}_{cm} = \frac{\mathbf{v}_n + A \mathbf{v}_t}{A + 1}

    +

    (49)\mathbf{v}_{cm} = \frac{\mathbf{v}_n + A \mathbf{v}_t}{A + 1}

    where \mathbf{v}_n is the velocity of the neutron and A is the atomic mass of the target nucleus measured in neutron masses (commonly referred to as the atomic weight ratio). With the velocity of the center-of-mass calculated, we can then determine the neutron’s velocity in the center-of-mass system:

    -

    (46)\mathbf{V}_n = \mathbf{v}_n - \mathbf{v}_{cm}

    +

    (50)\mathbf{V}_n = \mathbf{v}_n - \mathbf{v}_{cm}

    where we have used uppercase \mathbf{V} to denote the center-of-mass system. The direction of the neutron in the center-of-mass system is

    -

    (47)\mathbf{\Omega}_n = \frac{\mathbf{V}_n}{|| \mathbf{V}_n ||}

    +

    (51)\mathbf{\Omega}_n = \frac{\mathbf{V}_n}{|| \mathbf{V}_n ||}

    At low energies, elastic scattering will be isotropic in the center-of-mass system, but for higher energies, there may be p-wave and higher order scattering that leads to anisotropic scattering. Thus, in general, we need to sample a @@ -544,11 +570,11 @@ procedure in Tra the speed of the neutron in the center-of-mass system to obtain the new velocity vector in the center-of-mass:

    -

    (48)\mathbf{V}'_n = || \mathbf{V}_n || \mathbf{\Omega}'_n.

    +

    (52)\mathbf{V}'_n = || \mathbf{V}_n || \mathbf{\Omega}'_n.

    Finally, we transform the velocity in the center-of-mass system back to lab coordinates:

    -

    (49)\mathbf{v}'_n = \mathbf{V}'_n + \mathbf{v}_{cm}

    +

    (53)\mathbf{v}'_n = \mathbf{V}'_n + \mathbf{v}_{cm}

    In OpenMC, the angle and energy of the neutron are stored rather than the velocity vector itself, so the post-collision angle and energy can be inferred from the post-collision velocity of the neutron in the lab system.

    @@ -556,14 +582,14 @@ from the post-collision velocity of the neutron in the lab system.

    in the lab system. If we know the scattering cosine in the center-of-mass, the scattering cosine in the lab system can be calculated as

    -

    (50)\mu_{lab} = \frac{1 + A\mu}{\sqrt{A^2 + 2A\mu + 1}}.

    +

    (54)\mu_{lab} = \frac{1 + A\mu}{\sqrt{A^2 + 2A\mu + 1}}.

    However, this formula is only valid if the target was at rest. When the target nucleus does have thermal motion, the cosine of the scattering angle can be determined by simply taking the dot product of the neutron’s initial and final direction in the lab system.

    -

    5.3. Inelastic Scattering

    +

    5.4. Inelastic Scattering

    The major algorithms for inelastic scattering were described in previous sections. First, a scattering cosine is sampled using the algorithms in Sampling Secondary Angle Distributions. Then an outgoing energy is sampled using the algorithms in @@ -576,7 +602,7 @@ of the particle is changed also using the procedure in secondary photons from nuclear de-excitation are tracked in OpenMC.

    -

    5.4. (n,xn) Reactions

    +

    5.5. (n,xn) Reactions

    These types of reactions are just treated as inelastic scattering and as such are subject to the same procedure as described in Inelastic Scattering. Rather than tracking multiple secondary neutrons, the @@ -585,7 +611,7 @@ neutrons, e.g. for (n,2n), only one outgoing neutron is tracked but its weight is doubled.

    -

    5.5. Fission

    +

    5.6. Fission

    While fission is normally considered an absorption reaction, as far as it concerns a Monte Carlo simulation it actually bears more similarities to inelastic scattering since fission results in secondary neutrons in the exit @@ -610,7 +636,7 @@ representations exist for c_0,c_1,\dots. If \nu_t has this format, we can evaluate it at incoming energy E by using the equation

    -

    (51)\nu_t (E) = \sum_{i = 0}^N c_i E^i

    +

    (55)\nu_t (E) = \sum_{i = 0}^N c_i E^i

    where N is the order of the polynomial. The other representation is just a tabulated function with a specified interpolation law. The number of prompt neutrons released per fission event \nu_p is also given as a function of @@ -620,12 +646,12 @@ specified in a tabular format. In practice, we only need to determine nu_t and nu_d. Once these have been determined, we can calculated the delayed neutron fraction

    -

    (52)\beta = \frac{\nu_d}{\nu_t}

    +

    (56)\beta = \frac{\nu_d}{\nu_t}

    We then need to determine how many total neutrons should be emitted from fission. If no survival biasing is being used, then the number of neutrons emitted is

    -

    (53)\nu = \frac{w \nu_t}{k_{eff}}

    +

    (57)\nu = \frac{w \nu_t}{k_{eff}}

    where w is the statistical weight and k_{eff} is the effective multiplication factor from the previous generation. The number of neutrons produced is biased in this manner so that the expected number of fission @@ -645,7 +671,7 @@ bank. In a subsequent generation, these fission bank sites are used as starting source sites.

    -

    5.6. (n,\gamma) and Other Disappearance Reactions

    +

    5.7. (n,\gamma) and Other Disappearance Reactions

    All absorption reactions other than fission do not produce any secondary neutrons. As a result, these are the easiest type of reactions to handle. When a collision occurs, the first step is to sample a nuclide within a material. Once @@ -656,7 +682,7 @@ whether a “disappearance” reaction occurs where no secondary neutron produced. This is done by sampling a random number \xi on the interval [0,1) and checking whether

    -

    (54)\xi \sigma_t (E) < \sigma_a (E) - \sigma_f (E)

    +

    (58)\xi \sigma_t (E) < \sigma_a (E) - \sigma_f (E)

    where \sigma_t is the total cross section, \sigma_a is the absorption cross section (this includes fission), and \sigma_f is the total fission cross section. If this condition is met, then the neutron is @@ -667,7 +693,7 @@ heating in a problem, it would be necessary to explicitly track photons originating from (n,\gamma) and other reactions.

    -

    5.7. Survival Biasing

    +

    5.8. Survival Biasing

    In problems with highly absorbing materials, a large fraction of neutrons may be killed through absorption reactions thus leading to tallies with very few events scoring in them. To remedy this situation, an algorithm known as survival @@ -677,16 +703,16 @@ is a misnomer) is commonly used.

    collision, the weight of neutron is reduced by probability of absorption occurring, i.e.

    -

    (55)w' = w \left ( 1 - \frac{\sigma_a (E)}{\sigma_t (E)} \right )

    +

    (59)w' = w \left ( 1 - \frac{\sigma_a (E)}{\sigma_t (E)} \right )

    where w' is the weight of the neutron after adjustment and w is the weight of the neutron before adjustment. A few other things need to be handled differently if survival biasing is turned on. Although fission reactions never actually occur with survival biasing, we still need to create fission sites to preserve the basic criticality algorithm. The algorithm for sampling fission sites is the same as that described in Fission. The only -difference is in equation (53). We now need to produce

    +difference is in equation (57). We now need to produce

    -

    (56)\nu = \frac{w}{k} \frac{\nu_t \sigma_f(E)}{\sigma_t (E)}

    +

    (60)\nu = \frac{w}{k} \frac{\nu_t \sigma_f(E)}{\sigma_t (E)}

    fission sites, where w is the weight of the neutron before being adjusted. One should note this is just the expected number of neutrons produced per collision rather than the expected number of neutrons produced given that @@ -705,7 +731,7 @@ weight is -

    5.8. Effect of Thermal Motion on Cross-Sections

    +

    5.9. Effect of Thermal Motion on Cross-Sections

    When a neutron scatters off of a nucleus, many times it is assumed that the target nucleus is at rest. However, if the material is at a temperature greater than 0 K, it will have motion associated with the thermal vibration. Thus, the @@ -714,7 +740,7 @@ same as the velocity of the neutron entering the collision.

    The effect of the thermal motion on the interaction probability can be written as

    -

    (57)v_n \bar{\sigma} (v_n, T) = \int d\mathbf{v}_T v_r \sigma(v_r) +<p><span class=(61)v_n \bar{\sigma} (v_n, T) = \int d\mathbf{v}_T v_r \sigma(v_r) M (\mathbf{v}_T)

    where v_n is the magnitude of the velocity of the neutron, \bar{\sigma} is an effective cross section, T is the temperature @@ -744,7 +770,7 @@ treatment for secondary distributions.

    nuclide and then use it directly in the kinematic calculations. However, this calculation is a bit more nuanced than it might seem at first glance. One might be tempted to simply sample a Maxwellian distribution for the velocity of the -target nuclide. Careful inspection of equation (57) however +target nuclide. Careful inspection of equation (61) however tells us that target velocities that produce relative velocities which correspond to high cross sections will have a greater contribution to the effective reaction rate. This is most important when the velocity of the @@ -754,26 +780,26 @@ small target velocity can cause the relative velocity to correspond to the peak of the resonance, thus making a disproportionate contribution to the reaction rate. The conclusion is that if we are to sample a target velocity in the Monte Carlo code, it must be done in such a way that preserves the thermally-averaged -reaction rate as per equation (57).

    +reaction rate as per equation (61).

    The method by which most Monte Carlo codes sample the target velocity for use in elastic scattering kinematics is outlined in detail by [Gelbard]. The derivation here largely follows that of Gelbard. Let us first write the reaction rate as a function of the velocity of the target nucleus:

    -

    (58)R(\mathbf{v}_T) = || \mathbf{v}_n - \mathbf{v}_T || \sigma ( || +<p><span class=(62)R(\mathbf{v}_T) = || \mathbf{v}_n - \mathbf{v}_T || \sigma ( || \mathbf{v}_n - \mathbf{v}_T || ) M ( \mathbf{v}_T )

    where R is the reaction rate. Note that this is just the right-hand side -of equation (57). Based on the discussion above, we want to +of equation (61). Based on the discussion above, we want to construct a probability distribution function for sampling the target velocity to preserve the reaction rate – this is different from the overall probability distribution function for the target velocity, M ( \mathbf{v}_T ). This probability distribution function can be found by integrating equation -(58) to obtain a normalization factor:

    +(62) to obtain a normalization factor:

    -

    (59)p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{R(\mathbf{v}_T) d\mathbf{v}_T}{\int +<p><span class=(63)p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{R(\mathbf{v}_T) d\mathbf{v}_T}{\int d\mathbf{v}_T \, R(\mathbf{v}_T)}

    Let us call the normalization factor in the denominator of equation -(59) C.

    +(63) C.

    It is normally assumed that \sigma (v_r) is constant over the range of relative velocities of interest. This is a good assumption for almost all cases since the elastic scattering cross section varies slowly with velocity for light @@ -782,13 +808,13 @@ scattering, the moderating effect is rather small. Nonetheless, this assumption may cause incorrect answers in systems with U-238 where the low-lying resonances can cause a significant amount of up-scatter that would be ignored by this assumption. Nevertheless, with this assumption, we write \sigma (v_r) = -\sigma_s which simplifies (59) to

    +\sigma_s"/> which simplifies (63) to

    -

    (60)p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{\sigma_s}{C} || \mathbf{v}_n - +<p><span class=(64)p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{\sigma_s}{C} || \mathbf{v}_n - \mathbf{v}_T || M ( \mathbf{v}_T ) d\mathbf{v}_T

    The Maxwellian distribution in velocity is

    -

    (61)M (\mathbf{v}_T) = \left ( \frac{m}{2\pi kT} \right )^{3/2} \exp \left ( +<p><span class=(65)M (\mathbf{v}_T) = \left ( \frac{m}{2\pi kT} \right )^{3/2} \exp \left ( \frac{-m || \mathbf{v}_T^2 ||}{2kT} \right )

    where m is the mass of the target nucleus and k is Boltzmann’s constant. Notice here that the term in the exponential is dependent only on the @@ -796,23 +822,23 @@ speed of the target, not on the actual direction. Thus, we can change the Maxwellian into a distribution for speed rather than velocity. The differential element of velocity is

    -

    (62)d\mathbf{v}_T = v_T^2 dv_T d\mu d\phi

    +

    (66)d\mathbf{v}_T = v_T^2 dv_T d\mu d\phi

    Let us define the Maxwellian distribution in speed as

    -

    (63)M (v_T) dv_T = \int_{-1}^1 d\mu \int_{0}^{2\pi} d\phi \, dv_T \, v_T^2 +<p><span class=(67)M (v_T) dv_T = \int_{-1}^1 d\mu \int_{0}^{2\pi} d\phi \, dv_T \, v_T^2 M(\mathbf{v}_T) = \sqrt{ \frac{2}{\pi} \left ( \frac{m}{kT} \right )^3} v_T^2 \exp \left ( \frac{-m v_T}{2kT} \right ) dv_T

    To simplify things a bit, we’ll define a parameter

    -

    (64)\beta = \sqrt{\frac{m}{2kT}}

    -

    Substituting this into equation (63), we get

    +

    (68)\beta = \sqrt{\frac{m}{2kT}}

    +

    Substituting this into equation (67), we get

    -

    (65)M (v_T) dv_T = \frac{4}{\sqrt{\pi}} \beta^3 v_T^2 \exp \left ( -\beta^2 +<p><span class=(69)M (v_T) dv_T = \frac{4}{\sqrt{\pi}} \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) dv_T

    -

    Now, changing variables in equation (60) by using the result from -equation (63), our new probability distribution function is

    +

    Now, changing variables in equation (64) by using the result from +equation (67), our new probability distribution function is

    -

    (66)p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} || \mathbf{v}_n - +<p><span class=(70)p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} || \mathbf{v}_n - \mathbf{v}_T || \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) dv_T d\mu

    Again, the Maxwellian distribution for the speed of the target nucleus has no dependence on the angle between the neutron and target velocity vectors. Thus, @@ -822,21 +848,21 @@ of magnitudes of the velocity vectors and the angle rather than the vectors themselves. We can establish this relation based on the law of cosines which tells us that

    -

    (67)2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2

    +

    (71)2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2

    Thus, we can infer that

    -

    (68)|| \mathbf{v}_n - \mathbf{v}_T || = || \mathbf{v}_r || = v_r = \sqrt{v_n^2 + +<p><span class=(72)|| \mathbf{v}_n - \mathbf{v}_T || = || \mathbf{v}_r || = v_r = \sqrt{v_n^2 + v_T^2 - 2v_n v_T \mu}

    -

    Inserting equation (68) into (66), we obtain

    +

    Inserting equation (72) into (70), we obtain

    -

    (69)p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} \sqrt{v_n^2 + +<p><span class=(73)p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} \sqrt{v_n^2 + v_T^2 - 2v_n v_T \mu} \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) dv_T d\mu

    This expression is still quite formidable and does not lend itself to any natural sampling scheme. We can divide this probability distribution into two parts as such:

    -

    (70)p(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\ +<p><span class=(74)p(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\ f_1(v_T, \mu) &= \frac{4\sigma_s}{\sqrt{\pi} C'} \frac{ \sqrt{v_n^2 + v_T^2 - 2v_n v_T \mu}}{v_n + v_T} \\ @@ -846,33 +872,33 @@ f_2(v_T) &= (v_n + v_T) \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) with f_1(x) bounded can be sampled by sampling x' from the distribution

    -

    (71)q(x) dx = \frac{f_2(x) dx}{\int f_2(x) dx}

    +

    (75)q(x) dx = \frac{f_2(x) dx}{\int f_2(x) dx}

    and accepting it with probability

    -

    (72)p_{accept} = \frac{f_1(x')}{\max f_1(x)}

    +

    (76)p_{accept} = \frac{f_1(x')}{\max f_1(x)}

    The reason for dividing and multiplying the terms by v_n + v_T is to ensure that the first term is bounded. In general, || \mathbf{v}_n - \mathbf{v}_T || can take on arbitrarily large values, but if we divide it by its maximum value v_n + v_T, then it ensures that the function will be bounded. We now must come up with a sampling scheme for equation -(71). To determine q(v_T), we need to integrate f_2 -in equation (70). Doing so we find that

    +(75). To determine q(v_T), we need to integrate f_2 +in equation (74). Doing so we find that

    -

    (73)\int_0^{\infty} dv_T (v_n + v_T) \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 +<p><span class=(77)\int_0^{\infty} dv_T (v_n + v_T) \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) = \frac{1}{4\beta} \left ( \sqrt{\pi} \beta v_n + 2 \right )

    Thus, we need to sample the probability distribution function

    -

    (74)q(v_T) dv_T = \left ( \frac{4\beta^2 v_n v_T^2}{\sqrt{\pi} \beta v_n + 2} + +<p><span class=(78)q(v_T) dv_T = \left ( \frac{4\beta^2 v_n v_T^2}{\sqrt{\pi} \beta v_n + 2} + \frac{4\beta^4 v_T^3}{\sqrt{\pi} \beta v_n + 2} \right ) exp \left ( -\beta^2 v_T^2 \right )

    Now, let us do a change of variables with the following definitions

    -

    (75)x = \beta v_T \\ +<p><span class=(79)x = \beta v_T \\ y = \beta v_n.

    -

    Substituting equation (75) into equation (74) along +

    Substituting equation (79) into equation (78) along with dx = \beta dv_T and doing some crafty rearranging of terms yields

    -

    (76)q(x) dx = \left [ \left ( \frac{\sqrt{\pi} y}{\sqrt{\pi} y + 2} \right ) +<p><span class=(80)q(x) dx = \left [ \left ( \frac{\sqrt{\pi} y}{\sqrt{\pi} y + 2} \right ) \frac{4}{\sqrt{\pi}} x^2 e^{-x^2} + \left ( \frac{2}{\sqrt{\pi} y + 2} \right ) 2x^3 e^{-x^2} \right ] dx

    It’s important to make note of the following two facts. First, the terms outside @@ -881,27 +907,27 @@ can be sampled directly. Secondly, the terms inside the parentheses are always less than unity. Thus, the sampling scheme for q(x) is as follows. We sample a random number \xi_1 on the interval [0,1) and if

    -

    (77)\xi_1 < \frac{2}{\sqrt{\pi} y + 2}

    +

    (81)\xi_1 < \frac{2}{\sqrt{\pi} y + 2}

    then we sample the probability distribution 2x^3 e^{-x^2} for x using rule C49 in the Monte Carlo Sampler which we can then use to determine the speed of the target nucleus v_T from equation -(75). Otherwise, we sample the probability distribution +(79). Otherwise, we sample the probability distribution \frac{4}{\sqrt{\pi}} x^2 e^{-x^2} for x using rule C61 in the Monte Carlo Sampler.

    With a target speed sampled, we must then decide whether to accept it based on -the probability in equation (72). The cosine can be sampled +the probability in equation (76). The cosine can be sampled isotropically as \mu = 2\xi_2 - 1 where \xi_2 is a random number on the unit interval. Since the maximum value of f_1(v_T, \mu) is 4\sigma_s / \sqrt{\pi} C', we then sample another random number \xi_3 and accept the sampled target speed and cosine if

    -

    (78)\xi_3 < \frac{\sqrt{v_n^2 + v_T^2 - 2 v_n v_T \mu}}{v_n + v_T}

    +

    (82)\xi_3 < \frac{\sqrt{v_n^2 + v_T^2 - 2 v_n v_T \mu}}{v_n + v_T}

    If is not accepted, then we repeat the process and resample a target speed and cosine until a combination is found that satisfies equation -(78).

    +(82).

    -

    5.9. S(\alpha,\beta) Tables

    +

    5.10. S(\alpha,\beta) Tables

    For neutrons with thermal energies, generally less than 4 eV, the kinematics of scattering can be affected by chemical binding and crystalline effects of the target molecule. If these effects are not accounted for in a simulation, the @@ -939,7 +965,7 @@ scattering in hydrogenous solids such as polyethylene. As it occurs in ACE data, thermal inelastic scattering includes both coherent and incoherent effects and is dominant for most other materials including hydrogen in water.

    -

    5.9.1. Calculating Integrated Cross Sections

    +

    5.10.1. Calculating Integrated Cross Sections

    The first aspect of using S(\alpha,\beta) tables is calculating cross-sections to replace the data that would normally appear on the incident neutron data, which do not account for thermal binding effects. For incoherent elastic and inelastic @@ -947,17 +973,17 @@ scattering, the cross-sections are stored as linearly interpolable functions on a specified energy grid. For coherent elastic data, the cross section can be expressed as

    -

    (79)\sigma(E) = \frac{\sigma_c}{E} \sum_{E_i < E} f_i e^{-4WE_i}.

    +

    (83)\sigma(E) = \frac{\sigma_c}{E} \sum_{E_i < E} f_i e^{-4WE_i}.

    where \sigma_c is the effective bound coherent scattering cross section, W is the effective Debye-Waller coefficient, E_i are the energies of the Bragg edges, and f_i are related to crystallographic structure factors. Since the functional form of the cross-section is just 1/E and the proportionality constant changes only at Bragg edges, the proportionality constants are stored and then the cross-section can be -calculated analytically based on equation (79).

    +calculated analytically based on equation (83).

    -

    5.9.2. Outgoing Angle for Coherent Elastic Scattering

    +

    5.10.2. Outgoing Angle for Coherent Elastic Scattering

    The other aspect of using S(\alpha,\beta) tables is determining the outgoing energy and angle of the neutron after scattering. For incoherent and coherent elastic scattering, the energy of the neutron does not actually change, but the angle @@ -965,15 +991,15 @@ does change. For coherent elastic scattering, the angle will depend on which Bragg edge scattered the neutron. The probability that edge i will scatter then neutron is given by

    -

    (80)\frac{f_i e^{-4WE_i}}{\sum_j f_j e^{-4WE_j}}.

    +

    (84)\frac{f_i e^{-4WE_i}}{\sum_j f_j e^{-4WE_j}}.

    After a Bragg edge has been sampled, the cosine of the angle of scattering is given analytically by

    -

    (81)\mu = 1 - \frac{E_i}{E}

    +

    (85)\mu = 1 - \frac{E_i}{E}

    where E_i is the energy of the Bragg edge that scattered the neutron.

    -

    5.9.3. Outgoing Angle for Incoherent Elastic Scattering

    +

    5.10.3. Outgoing Angle for Incoherent Elastic Scattering

    For incoherent elastic scattering, the probability distribution for the cosine of the angle of scattering is represent as a series of equally-likely discrete cosines \mu_{i,j} for each incoming energy E_i on the thermal @@ -981,13 +1007,13 @@ elastic energy grid. First the outgoing angle bin E_i < E < E_{i+1} the final cosine is

    -

    (82)\mu = \mu_{i,j} + f (\mu_{i+1,j} - \mu_{i,j})

    +

    (86)\mu = \mu_{i,j} + f (\mu_{i+1,j} - \mu_{i,j})

    where the interpolation factor is defined as

    -

    (83)f = \frac{E - E_i}{E_{i+1} - E_i}.

    +

    (87)f = \frac{E - E_i}{E_{i+1} - E_i}.

    -

    5.9.4. Outgoing Energy and Angle for Inelastic Scattering

    +

    5.10.4. Outgoing Energy and Angle for Inelastic Scattering

    On each S(\alpha,\beta) table, there is a correlated angle-energy secondary distribution for neutron thermal inelastic scattering. While the documentation for the ACE format implies that there are a series of equiprobable outgoing energies, the @@ -997,24 +1023,24 @@ and last outgoing energies have a relative probability of 1, the second and second to last energies have a relative probability of 4, and all other energies have a relative probability of 10. The procedure to determine the outgoing energy and angle is as such. First, the interpolation factor is determined from -equation (83). Then, an outgoing energy bin is sampled +equation (87). Then, an outgoing energy bin is sampled either from a uniform distribution or from a skewed distribution as discussed. The outgoing energy is then interpolated between values corresponding to neighboring incoming energies:

    -

    (84)E = E_{i,j} + f (E_{i+1,j} - E_{i,j})

    +

    (88)E = E_{i,j} + f (E_{i+1,j} - E_{i,j})

    where E_{i,j} is the j-th outgoing energy corresponding to the i-th incoming energy. For each combination of incoming and outgoing energies, there is a series equiprobable outgoing cosines. An outgoing cosine bin is sampled uniformly and then the final cosine is interpolated on the incoming energy grid:

    -

    (85)\mu = \mu_{i,j,k} + f (\mu_{i+1,j,k} - \mu_{i,j,k})

    +

    (89)\mu = \mu_{i,j,k} + f (\mu_{i+1,j,k} - \mu_{i,j,k})

    where \mu_{i,j,k} is the k-th outgoing cosine corresponding to the j-th outgoing energy and the i-th incoming energy.

    -

    5.10. Unresolved Resonance Region Probability Tables

    +

    5.11. Unresolved Resonance Region Probability Tables

    In the unresolved resonance energy range, resonances may be so closely spaced that it is not possible for experimental measurements to resolve all resonances. To properly account for self-shielding in this energy range, OpenMC @@ -1051,16 +1077,16 @@ capture cross-sections from the probability tables interpolating between neighboring incoming energies. If interpolation is specified, then the cross sections are calculated as

    -

    (86)\sigma = \sigma_{i,j} + f (\sigma_{i+1,j} - \sigma{i,j})

    +

    (90)\sigma = \sigma_{i,j} + f (\sigma_{i+1,j} - \sigma{i,j})

    where f is the interpolation factor defined in the same manner as -(83). If logarithmic interpolation is specified, the +(87). If logarithmic interpolation is specified, the cross sections are calculated as

    -

    (87)\sigma = \exp \left ( \log \sigma_{i,j} + f \log +<p><span class=(91)\sigma = \exp \left ( \log \sigma_{i,j} + f \log \frac{\sigma_{i+1,j}}{\sigma_{i,j}} \right )

    where the interpolation factor is now defined as

    -

    (88)f = \frac{\log \frac{E}{E_i}}{\log \frac{E_{i+1}}{E_i}}

    +

    (92)f = \frac{\log \frac{E}{E_i}}{\log \frac{E_{i+1}}{E_i}}

    A flag is also present in the probability table that specifies whether an inelastic cross section should be calculated. If so, this is done from a normal reaction cross section (either MT=51 or a special MT). Finally, if the @@ -1071,7 +1097,7 @@ section is calculated as the sum of the elastic, fission, capture, and inelastic cross sections.

    -

    5.11. References

    +

    5.12. References

    diff --git a/methods/statistics.html b/methods/statistics.html index 460833b7b5..c7cd98a7e7 100644 --- a/methods/statistics.html +++ b/methods/statistics.html @@ -95,7 +95,7 @@ of random numbers is generated using the following recurrence relation:

    constants will have a profound effect on the quality and performance of the generator, so they should not be chosen arbitrarily. As Donald Knuth said in his seminal work The Art of Computer Programming, “random numbers should not be -generated with a method chosen at random”. Some theory should be used.” +generated with a method chosen at random. Some theory should be used.” Typically, M is chosen to be a power of two as this enables x \mod M to be performed using the binary AND operator with a bit mask. The constants for the linear congruential generator used by default in OpenMC are diff --git a/objects.inv b/objects.inv index 80a5a258abb0daf8ea3bc8a4e981579b815d2e03..4e366bd3fea07e13e17b6e11404de85c2f738db9 100644 GIT binary patch delta 1058 zcmV+-1l{|b362Sndw=6N5dQ98A-KYQa+|)u!m%v0Nm>fcZf=uh_ghiyi6c&ye3HCv z?$__gl5E*|!Fy1MJ+GQ)W*)1Vl3KrN$PAN06K(RpINbmz={2b}C|%Snhbtzk+Fb5T z$^Xc#N7{OP%z6<%kvWGFgh@&;JDH=#P{Xug*&w>tUv+vH-+#-~$Zyp?hN#CwF_FW& zi*iBMJCJHmsL08p(Mv9ieNxK2G3!UD_GwT!f3lEy&4D~yQBu1U`L9j6}Q+fI!4Si0Y+&VMuYMJRnk`Jb9RSxxe9&9Z-v z!V_zP>)%!u(C`s*_~}5#IjA#Jt6{%_!BuDA;sr$-dCsVzNEG3{g^3BHaFrW&I>j2D zMIW%9?{oBG&kkHxh||H5y*yMta}rC5WDT_J%6>0-x|PZ(Do)fLSqut~edA}hs>lc$ zS1zF$q<@^gVaVShu4^Q&HQm&lDl*4rR!VN@1~+ytKoxuPfW1u$5JzY8zR+V>HB`zb9TrF$e796vZJ$VLW z=<|YnY;3!)p8!_uHd3V=1D>e10Ev%kA7VVzjDP30)X(0TyjbVJx!~acpQP)X=Qv0q z6(V-Iqv9*C>oYPFXk$ORsm`b|phjhW4qjyCyqY<*S&*n5#G^XzZSj1EZ3$a&VrzPw zH46*b5?93T8z`6%2TL6FxV1Q7_|Es9Jox93gBVG^R3H}9mWD#F$4o+?pVsHeKO`Ec z%71qK7*6IlZsiP?w16bTn+L9(FqDar6F39x7;c*#9k#`l>-LN=O_;DE42vJxEr&Cl zIt={P+y4aeAZs^~xP3Wuv&LBlLJu11g(1X?x4ADAxEJk~_?hyoc9#K--+qy&*~wew5Wz<@ zcQj}lK%Njv+l{4UmTm04<#lp-cX@s0$`e_+sEr_n><-AxE&uj%cU_FK@}W3Q+zxsrUz z_SfIYl5E-gg0B#WcV?ck=IzlqQ&Q_U4Vht5XrfL2E2lf)B%PB+gVJTga=2llZp`)G zRQ#vRBGL}=IU9KRjjT9SAWTw$*~=U?h8m^~%SPV4{-)PE|9@VdMz~G)7`&c##Y7Gt z&dU|q?m?;pp&}>iRxi0M4@o8S)@7cJetE4a6TU)j% zMeU`j!~CU~VSl0|f3IIK2UhaPIOs&vs^*YVQDSnYq+??N*3$+uDRPP>xo^Qstf*qU zrfr)pAqGQXaHhUqIm|;qY~BAv0SBrrSkHlbJ@1>l6%bdIRS#DAEYIq=q6v zgx_77m@o=QxMihtY|&c`6E?s;M?dV@fy)~6baZ4d50NjO#8M&I0na}ga8NAXG5+;CCdq<`^qsD-ml$GazA{&Ql+sE=V@~X?sq=I={WcK51 zhUPuf>;tM@P2JN5`Ano_{P8Xe}z zKYxVF65Ra!&)d>ZpWR~?QsEVZ&7ZqF^v3IkS80YD2acRDl!=KQxCAUcPMdu*D{$oc zIU`IHCaehE;^kU!xWK7n$6vkuHy{tPaRK4<<;)$GvkZhDb%qxP7cbuCzEt35ENTg) z<_0NB@1;Dy=fp-wjWO`Bjegi%jJhJTOMg*DIWuRf85s!bP`91TC+#rv@eVV;yu-|+H_RrzVK#ol?D89C!$tPQ z!ZG0FXzds*vkvDskT%hNltjVGM!A)_KjrPDSZ>F$++L35HmR|%u;?egvSLjr?I~*{ qOA74!l1{rhS4Dy2Mr2_>e^f;|e#f*i - +
    @@ -40,7 +40,7 @@

    - «  6. Parallelization + «  7. Parallelization   ::   Contents   ::   @@ -81,15 +81,16 @@ essential aspects of using OpenMC to perform neutronic simulations.

  5. 3.2.3. <cutoff> Element
  6. 3.2.4. <energy_grid> Element
  7. 3.2.5. <entropy> Element
    8. -
  8. 3.2.6. <no_reduce> Element
    9. -
  9. 3.2.7. <ptables> Element
    10. -
  10. 3.2.8. <seed> Element
    11. -
  11. 3.2.9. <source> Element
    12. -
  12. 3.2.10. <survival_biasing> Element
    13. -
  13. 3.2.11. <trace> Element
    14. -
  14. 3.2.12. <uniform_fs> Element
    15. -
  15. 3.2.13. <verbosity> Element
    16. -
  16. 3.2.14. <write_source> Element
    17. +
  17. 3.2.6. <fixed_source> Element
    18. +
  18. 3.2.7. <no_reduce> Element
    19. +
  19. 3.2.8. <ptables> Element
    20. +
  20. 3.2.9. <seed> Element
    21. +
  21. 3.2.10. <source> Element
    22. +
  22. 3.2.11. <survival_biasing> Element
    23. +
  23. 3.2.12. <trace> Element
    24. +
  24. 3.2.13. <uniform_fs> Element
    25. +
  25. 3.2.14. <verbosity> Element
    26. +
  26. 3.2.15. <write_source> Element
  27. 3.3. Geometry Specification – geometry.xml
      @@ -141,7 +142,7 @@ essential aspects of using OpenMC to perform neutronic simulations.

      - «  6. Parallelization + «  7. Parallelization   ::   Contents   ::   diff --git a/usersguide/input.html b/usersguide/input.html index c750ac29f9..9750dff597 100644 --- a/usersguide/input.html +++ b/usersguide/input.html @@ -202,8 +202,30 @@ problem. It has the following attributes/sub-elements:

    28. +
    +

    3.2.6. <fixed_source> Element

    +

    The <fixed_source> element indicates that a fixed source calculation should be +performed. It has the following attributes/sub-elements:

    +
    +
    +++ + + + + + +
    batches:

    The total number of batches. For fixed source calculations, each batch +represents a realization of random variables for tallies.

    +

    Default: None

    +
    particles:

    The number of particles to simulate per batch.

    +

    Default: None

    +
    +
    +
    -

    3.2.6. <no_reduce> Element

    +

    3.2.7. <no_reduce> Element

    The <no_reduce> element has no attributes and has an accepted value of “on” or “off”. If set to “on”, all user-defined tallies and global tallies will not be reduced across processors in a parallel calculation. This means that the @@ -214,7 +236,7 @@ tally data, this option can significantly improve the parallel efficiency.

    Default: off
    -

    3.2.7. <ptables> Element

    +

    3.2.8. <ptables> Element

    The <ptables> element determines whether probability tables should be used in the unresolved resonance range if available. This element has no attributes or sub-elements and can be set to either “off” or “on”.

    @@ -222,35 +244,108 @@ or sub-elements and can be set to either “off” or “on”.<
    Default: on
    -

    3.2.8. <seed> Element

    +

    3.2.9. <seed> Element

    The seed element is used to set the seed used for the linear congruential pseudo-random number generator.

    Default: 1
    -

    3.2.9. <source> Element

    -

    The source element gives information on an initial source guess for -criticality calculations. It takes the following attributes:

    +

    3.2.10. <source> Element

    +

    The source element gives information on an external source distribution to +be used either as the source for a fixed source calculation or the initial +source guess for criticality calculations. It takes the following +attributes/sub-elements:

    - - + + + + @@ -258,7 +353,7 @@ numbers which specify the (x,y,z) location of an isotropic point source

    -

    3.2.10. <survival_biasing> Element

    +

    3.2.11. <survival_biasing> Element

    The <survival_biasing> element has no attributes and has an accepted value of “on” or “off”. If set to “on”, this option will enable the use of survival biasing, otherwise known as implicit capture or absorption.

    @@ -266,7 +361,7 @@ biasing, otherwise known as implicit capture or absorption.

    Default: off
    -

    3.2.11. <trace> Element

    +

    3.2.12. <trace> Element

    The <trace> element can be used to print out detailed information about a single particle during a simulation. This element should be followed by three integers: the batch number, generation number, and particle number.

    @@ -274,7 +369,7 @@ integers: the batch number, generation number, and particle number.

    Default: None
    -

    3.2.12. <uniform_fs> Element

    +

    3.2.13. <uniform_fs> Element

    The <uniform_fs> element describes a mesh that is used for re-weighting source sites at every generation based on the uniform fission site methodology described in Kelly et al., “MC21 Analysis of the Nuclear Energy Agency Monte @@ -303,7 +398,7 @@ problem. It has the following attributes/sub-elements:

    -

    3.2.13. <verbosity> Element

    +

    3.2.14. <verbosity> Element

    The <verbosity> element tells the code how much information to display to the standard output. A higher verbosity corresponds to more information being displayed. This element takes the following attributes:

    @@ -321,7 +416,7 @@ displayed. This element takes the following attributes:

    -

    3.2.14. <write_source> Element

    +

    3.2.15. <write_source> Element

    The <write_source> element has no attributes and has an accepted value of “on” or “off”. If set to “on”, a binary source file will be written to diskat the end of the run that can be used as a starting source for another run.

    type:

    The type of source distribution. Setting this to “box” indicates that the -starting source should be sampled uniformly in a parallelepiped. Setting -this to “point” indicates that the starting source should be sampled from an -isotropic point source. Setting this to “file” indicates that the starting -source should be sampled from a source.binary file.

    +
    file:

    If this attribute is given, it indicates that the source is to be read from +a binary source file whose path is given by the value of this element

    +

    Default: None
    coeffs:

    For a “box” source distribution, coeffs should be given as six real -numbers, the first three of which specify the lower-left corner of a +

    space:

    An element specifying the spatial distribution of source sites. This element +has the following attributes:

    + +++ + + + + + +
    type:

    The type of spatial distribution. Valid options are “box” and “point”. A +“box” spatial distribution has coordinates sampled uniformly in a +parallelepiped. A “point” spatial distribution has coordinates specified +by a triplet.

    +

    Default: None

    +
    parameters:

    For a “box” spatial distribution, parameters should be given as six +real numbers, the first three of which specify the lower-left corner of a parallelepiped and the last three of which specify the upper-right corner. Source sites are sampled uniformly through that parallelepiped.

    -

    For a “point” source distribution, coeffs should be given as three real -numbers which specify the (x,y,z) location of an isotropic point source

    -

    For a “file” source distribution, coeffs should not be specified.

    +

    For a “point” spatial distribution, parameters should be given as +three real numbers which specify the (x,y,z) location of an isotropic +point source

    +

    Default: None

    +
    +
    angle:

    An element specifying the angular distribution of source sites. This element +has the following attributes:

    + +++ + + + + + +
    type:

    The type of angular distribution. Valid options are “isotropic” and +“monodirectional”. The angle of the particle emitted from a source site is +isotropic if the “isotropic” option is given. The angle of the particle +emitted from a source site is the direction specified in the <parameters> +attribute if “monodirectional” option is given.

    +

    Default: isotropic

    +
    parameters:

    For an “isotropic” angular distribution, parameters should not be +specified

    +

    For a “monodirectional” angular distribution, parameters should be +given as three real numbers which specify the angular cosines with respect +to each axis.

    +

    Default: None

    +
    +
    energy:

    An element specifying the energy distribution of source sites. This element +has the following attributes:

    + +++ + + + + + +
    type:

    The type of energy distribution. Valid options are “monoenergetic”, +“watt”, and “maxwell”. The “monoenergetic” option produces source sites at +a single energy. The “watt” option produces source sites whose energy is +sampled from a Watt fission spectrum. The “maxwell” option produce source +sites whose energy is sampled from a Maxwell fission spectrum

    +

    Default: watt

    +
    parameters:

    For a “monoenergetic” energy distribution, parameters should not be +given as the energy in MeV of the source sites.

    +

    For a “watt” energy distribution, parameters should be given as two +real numbers a and b that parameterize the distribution +p(E) dE = c e^{-E/a} \sinh \sqrt{b \, E} dE.

    +

    For a “maxwell” energy distribution, parameters should be given as one +real number a that parameterizes the distribution p(E) dE = +c E e^{-E/a} dE.

    +

    Default: 0.988 2.249

    +
-

f(x,y,z) = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 - R^2 = 0

+
+

(1)(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2

+

By subtracting the right-hand term from both sides of equation +(1), we can then write the surface equation for the sphere:

+
+

(2)f(x,y,z) = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 - R^2 = 0

One can confirm that any point inside this sphere will correspond to f(x,y,z) < 0 and any point outside the sphere will correspond to f(x,y,z) > 0.

@@ -119,22 +119,23 @@ surface. Suppose we have a particle at u,v,w. To find the distance d to a surface f(x,y,z) = 0, we need to solve the equation:

-

(1)f(x + du, y + dv, z + dw) = 0

-

If no solutions to equation (1) exists or the only -solutions are complex, then the particle’s direction of travel will not -intersect the surface. If the solution to equation (1) is -negative, this means that the surface is “behind” the particle, i.e. if the -particle continues traveling in its current direction, it will not hit the -surface. The complete derivation for different types of surfaces used in OpenMC -will be presented in the following sections.

-

Once a distance has been computed to a boundary, we need to check if it is -closer than previously-computed distances to surfaces. Unfortunately, we cannot -just use the minimum function because some distances may be almost identical but -still different due to the use of floating-point arithmetic. Consequently, we -should first check for floating-point equality of the current distance -calculated and the minimum found thus far. This is done by checking if

+

(3)f(x + du, y + dv, z + dw) = 0

+