[20] To make the calculations easier, I assume that the crystals are identical cuboids (rectangular prisms), evenly spaced with long axes parallel to the axis of the chain and to the 〈111〉 easy axis in each crystal. Some of these assumptions are more accurate than others. In most strains of bacteria the long axis of each crystal is always the 〈111〉 axis. Although in many images of magnetosome chains the chains have kinks, this is probably the result of shrinkage during preparation [*Shcherbakov et al.*, 1997], so it is reasonable to assume the chains are straight. The cuboidal shape with square cross section is convenient for calculating magnetostatic coupling. Although no natural magnetosome crystal has this shape, the approximation does not introduce much error (see section 3.1.2).

[21] The most critical assumption is that all the crystals are the same size. Through most of a chain, the crystals tend to have very similar sizes and shapes. The exception is at the ends of the chain, where the crystals are often smaller, probably because they are still growing [*Mann and Frankel*, 1989]. By forcing all the particles to have the same size, I am modeling a process in which all the crystals are growing at the same rate until they are large enough for the magnetization to be stable. This is not what actually happens. Usually a bacterium inherits part of a chain from its parent and starts growing new crystals at the ends. However, while the bacterium is waiting for new crystals to finish growing, the starting chain must have a stable moment, and most of the crystals in the chain have similar sizes. Thus, the critical size for identical crystals establishes a minimum size for most of the crystals.

#### 3.1. Energy and Equilibrium Equations

[22] Stable magnetic states in a ferromagnet are minima of the Gibbs free energy *G* under the constraint that the magnetization at any point in the body has fixed magnitude and can only rotate [*Brown*, 1978]. This implies that the moment of a uniformly magnetized crystal with volume *V* is *VM*_{s}_{i}, where *M*_{s} is the saturation magnetization and = (*α*, *β*, *γ*) is a unit vector with *α*^{2} + *β*^{2} + *γ*^{2} = 1. In this section, I develop an expression for the free energy, taking into account symmetries to simplify the expression.

##### 3.1.1. Energy Terms

[23] The energy of a chain of uniformly magnetized magnetosome crystals has two parts: the magnetostatic energy and the magnetocrystalline anisotropy energy. Suppose there are *N* crystals with identical dimensions Δ**r** = (*W*, *W*, *L*), positions **r**_{i} and unit vectors _{i} = (*α*_{i}, *β*_{i}, *γ*_{i}). The demagnetizing energy is [*Newell et al.*, 1993]

where *V* = *W*^{2}*L* is the volume and *μ*_{0} is the permeability of free space. The generalized demagnetizing tensor (Δ**r**, **r**_{i} − **r**_{j}) is a dimensionless coupling term that depends on the geometry of the chain: the shapes of the magnetosome crystals (represented by Δ**r**), their orientation and the positions **r**_{i} of their centers relative to some arbitrary reference point. In the rest of this article, the crystal dimensions are implied in reference to .

[24] Magnetite has a cubic anisotropy that adds fourth-order terms to the expression for the energy. Most magnetosome crystals are oriented so the 〈111〉 axis is parallel to the chain and is the axis of elongation. If the cubic anisotropy were large enough, it would greatly increase the complexity of the energy surface. Fortunately, if the cubic anisotropy is small enough it can be replaced by an effective uniaxial anisotropy.

[25] For a crystal elongated in a 〈111〉 direction, *Newell* [2006b] showed that for sufficient elongation the energy barrier is effectively uniaxial with energy barrier (*K*_{u}^{(d)} − *K*′_{1}/12), where *K*_{u}^{(d)} represents the shape anisotropy and *K*′_{1} is the cubic anisotropy parameter. The criterion for sufficient elongation is *K*_{u}^{(d)} > 0.2229∣*K*′_{1}∣, which only requires a length-to-width ratio of 1.3. This energy barrier is also correct for *K*′_{1} = 0. Since the critical size changes smoothly in between, this expression is also quite accurate for intermediate ratios.

##### 3.1.2. Simplifications

[26] Let the chain axis be in the *z* direction and the crystals be indexed by their position in the chain, with *i* = 1 at one end and *i* = *N* at the other end. If the distance between adjacent magnetosome centers is Δ*z*, all the demagnetizing tensors have the form ((*i* − *j*)Δ*z*) (where the dependence on Δ**r** is implicit).

[27] Given this geometry, the principal axes of the tensors coincide with the coordinate system. The off-diagonal components are zero because contributions from opposing faces cancel out [*Newell et al.*, 1993]. The self-demagnetizing tensors (*i* = *j*) have trace 1 while the coupling tensors (*i* ≠ *j*) have trace 0. Finally, I will further simplify the problem by assuming that the magnetosome crystals have square cross sections, so *N*_{xx}(**r**) = *N*_{yy}(**r**). Then

and for *i* ≠ *j*,

[28] The difference *N*_{d} ≡ (*N*_{xx}(0) − *N*_{zz}(0)) is the usual demagnetizing factor for an isolated SD crystal and is nonnegative for lengths *L* and widths *W* such that *L* ≥ *W*. If we define

the total normalized energy is (aside from a constant that can be ignored)

Since *N*_{d}(−**r**) = *N*_{d}(**r**) [*Newell et al.*, 1993], *k*_{i}^{(d)} = *k*_{−i}^{(d)}. Note that, under the constraints of this problem, the *k*_{i} are functions of the aspect ratio *q* = *L*/*W*. In this article *k*_{1} varies between about 0.02 and 10, and *k*_{i} is monotonically decreasing with increasing *i*. The *k*_{i} are calculated for cuboids, but in the single-domain approximation details of the shape aside from aspect ratio do not matter much. For example, the self-demagnetizing factor (*i* = 0) for a spheroid [*Chikazumi*, 1997] is at most 20% larger than the same factor for a cuboid [*Rhodes and Rowlands*, 1954; *Newell et al.*, 1993].

##### 3.1.3. Planarity Constraint

[29] Because of the rotational symmetry of the energy, a given solution can be rotated about the chain axis by an arbitrary angle to get another solution. To make the solution unique, the longitudinal angle of at least one moment must be specified. However, we can do better than that. Each coupling term has the form *k*_{i−j}(*α*_{i}*α*_{j} + *β*_{i}*β*_{j} − 2 *γ*_{i}*γ*_{j}). If spherical polar coordinates are defined such that θ_{i} ∈ [0, *π*) is the angle of moment *i* with respect to the chain axis and ϕ_{i} ∈ [0, 2*π*) is the longitudinal angle, the coupling term is *g*_{ij} = *k*_{i−j}(cos(ϕ_{i} − ϕ_{j}) sinθ_{i}sinθ_{j} − 2cosθ_{i}cosθ_{j}). The derivatives of this term with respect to ϕ_{i} and ϕ_{j} are zero if ϕ_{i} = ϕ_{j} or ϕ_{i} = ϕ_{j} + *π*. If all the moments are in the same plane, then any state satisfying ∂*g*/∂θ_{i} = 0 for all *i* is an equilibrium state because all the derivatives with respect to ϕ_{i} are also zero.

[30] Now suppose that all the ϕ_{i} are in the same plane and have values 0 or *π*. Each nearest-neighbor term has a Hessian {∂^{2}*g*_{i,i+1}/∂ϕ_{i}∂ϕ_{i+1}} with determinant 1, consistent with this term being a minimum with respect to ϕ_{i} and ϕ_{i+1}. Since that is true for all indices *i*, the planar solutions are minima with respect to out-of-plane rotations. This argument only shows that the solutions are local minima. It is possible that there are nonplanar energy minima, but it is reasonable to assume that the moments are in the *XZ* plane. The reduced energy is now

#### 3.3. Master Equation

[42] The moments in a chain of ferromagnets are constantly fluctuating. If there are *N* moments, the collective state can be represented by a vector in a 2*N*-dimensional space. However, near the SP critical size this vector is rarely far away from a direction of minimum energy. For such a system the high-energy barrier approximation [*Brown*, 1979] is appropriate. In this approximation, a local equilibrium is established near each minimum. The local distribution of moments is determined by the Boltzmann distribution, and this distribution is a very steep function of distance from the minimum. Transitions between states are rare. The energy surface is also steep near transition states, so during a transition the moment is very likely to pass near the transition state. It follows that transitions only occur between states that are directly connected by a transition state.

[43] If a pair of states *i* and *j* is directly connected, there is a transition rate *ν*_{ij} from state *i* to state *j* and another *ν*_{ji} from state *j* to state *i*. If the states are not directly connected, *ν*_{ij} = *ν*_{ji} = 0. If the normalized energies of the connected states are *g*_{i} and *g*_{j}, and that of the transition state is *g*_{s}^{ij}, then

where *k*_{B} is the Boltzmann constant. The prefactor *ν*_{0} is determined by the detailed shape of the energy surface and the level of damping [*Brown*, 1979; *Newell*, 2006a], but a value of *ν*_{0} = 10^{9}Hz is accurate enough for blocking calculations.

[44] Let *n*_{i}(*V*, *t*) be the probability of state *i* at time *t* for particles of volume *V*. The probability depends on a lot of other variables such as the temperature *T*, but the goal is to solve for a critical volume *V* with all the other variables fixed. Let *m*_{i} be its moment in the *z* direction. The time evolution of the system is given by

This can be converted [*Brown*, 1979] to a master equation for **n**:

where *W*_{ij} = *ν*_{ij} for *i* ≠ *j*, *W*_{ii} = −∑_{j}*ν*_{ji}. Formally, the solution of this equation is

where the exponential is interpreted as a matrix exponential [e.g., *Horn and Johnson*, 1994, p. 204]. Such solutions have multiple decay times, not all of which affect the moment of the chain [*Newell*, 2006a, 2006b]. However, the point of Néel's theory is that, for sizes slightly larger than the critical size, the moment does not change significantly over the time scale of interest, *t*_{b}, while for slightly smaller sizes the moment rapidly goes to zero. It is therefore better to look at the time dependence of the moment:

[45] The Néel relaxation time is the time at which the moment has decayed to 1/*e* of the original moment:

To solve for the SP critical volume, substitute *t* = *t*_{b} in the above equation to get an implicit equation for *V*. The solution of this equation is the critical volume. In practice, I express *m* as a function of *L* = *V*^{1/3} because *m*(*V*, *t*) is an extremely sensitive function of volume. For the same reason, the initial guess must be quite accurate to avoid overflow or underflow. One way to obtain a good initial guess is to use the solution for a slightly larger or smaller aspect ratio. For an elongated crystal, the noninteracting limit can be used. To check that the critical volume is not sensitive to the fraction of the original moment, I also calculate it for *m*(*V*, *t*) = 0.01 *m*(*V*, 0).

[46] Most studies of magnetosome crystal size include SP critical size curves for time scales of 100 s and 4.5 Ga. These were used by *Butler and Banerjee* [1975] to represent a laboratory time scale and the upper limit of paleomagnetic time scales. The latter is certainly not relevant to magnetotactic bacteria, while the former is too short. A magnetotactic bacterium that loses its magnetic moment in 100 s will not thrive. The moment should last at least long enough for cell division to occur. The chain is divided between the offspring and must provide a moment to orient new magnetosome crystals when they are grown. The cell division time depends on nutrient availability, but a reasonable estimate is 10^{5} s, or about 1 day. This estimate can be off by orders of magnitude without much effect on the SP critical size.

#### 3.4. Lowest-Energy Paths

[47] I use two approaches to calculating critical sizes. The first is find all the stable states and determine the network of paths connecting them across transition states. This is the most accurate approach, but the network can be very complicated. Usually the decay of the moment is dominated by the lowest-energy path between the states of positive and negative saturation. My alternate approach is to find a lowest-energy path and use it to calculate the critical size. This requires some adjustment. For each stable state with moments **m**(*z*_{i}) another can be obtained by reflection about a plane through the center of the chain: **m**′(*z*_{i}) = **m**(−*z*_{i}), where the origin is equidistant from the ends of the chain. This transformation maps the two saturated states onto each other, but maps all other states on minimum-energy paths onto other states with the same energy and total moment. Furthermore, each transition has an image under a 180° rotation about the chain axis. Using these symmetries I use one path to build up the complete minimum-energy network. For example, in Figure 2b, the route 1 2 4 can be used to construct 1 3 4. The transition matrix for this system is

If this system begins in the state of positive saturation, the initial (normalized) moment is 2 and the time evolution of the moment is

This is the same time evolution that one would get in a two-state system with barrier rate 2*ν*_{1}. In this section, I discuss the states on the lowest-energy path, but calculate the critical sizes using the full network.