Subsequently, Brandt and Sudbo extended these results for the case of a pair of twisted flux lines, they considered the tension and interaction between each flux-line or vortex. Given that cutting is energetically plausible, they found that flux-line cutting is an effective disentanglement mechanism of flux lines; and that the cutting energy barrier, in the case of twisted flux lines, is lower for the rigid two-flux-lines case [11, 12]. Several groups, for example, M.A.R. LeBlanc *et. al* [13–15], have done experiments in the last decades that have shown flux-cutting presence in materials with low or high *κ*, and low or high *Tc*. In the Clem *et. al* [16–18] and Brandt [19] papers, we can see flux-line cutting diagrams for the case of rigid-vortices arrangement. Furthermore, it includes diagrams for the first theoretical formulations for this often-studied and not completely understood phenomenon.

More recent theoretical studies have studied the scattering dynamics of vortices, and the resulting topology after a collision between two flux lines generated by an applied current. Using the time-dependent Ginzburg-Landau equations, numerical results yielded two generic collision types dependent on the initial angle: one local collision that induces changes in topology through recombination, and a double collision that can occur due to geometrical restrictions. The second case leads to a vortex-crossing type configuration, that is, it seems as if two vortices, while interacting, would cut themselves and join again. This can be seen in the simulations shown in paper [20]. Experiments have been proposed using a magnetic force microscope to monitor vortex-line dynamics and prove if these cut through each other when they are in a liquid-vortex phase [21].

In 2008, A. Palau *et al.* reported results with superconductive heterostructures subjected to an external magnetic field at a *θ* angle respect to the sample's normal. For this, they designed a device made out of a thin film of low-pinning amorphous material (*Mo*82*Si*18), sandwiched between two *Nb* films– a material characterized for strongly pinning the vortices.

They measured the critical current density *Jc* obtained as a function of *θ*, the applied field *µ*0*Ha*, and temperature *T*. Once obtained, they calculated the balance force between the Lorentz force, the pinning force, and a so-called breaking force. They found that the latter was necessary in order to consider vortex deformation and destabilization. Results showed that the breaking force is independent of *B*, and that the following cross joining neither limit vortex movement nor increase *Jc*. Even if a flux-line segment is strongly pinned to the area where *Nb* material is found, the cut induces other vortex segments to be liberated, thus reducing *Jc* [22].

Furthermore, Campbell's revision paper can be consulted to know the state of the art about experiments and critical state theories for flux-cutting in superconductors [23]. He included the last proposal of Clem to determine the electric field direction, for the flux transport regime in a type-II superconductor.

Here it is presented three critical state models created for the phenomenological study of type-II superconductors subject to magnetic fields that vary not only in magnitude, but also in direction. All models consider that flux pinning and flux-line cutting govern their answer. It is undeniable that both phenomena can occur in cases when a sample oscillates in presence of a static magnetic field, or when it is subjected to a DC magnetic field and a transversal sweeping magnetic field is superimposed.

#### **2. Other critical state models**

#### **2.1. Generalized Double Critical-State Model**

LeBlanc *et. al* proposed a model containing two critical-state equations based on their experimental observations on the magnetic response of a disc oscillating at low frequency, in presence of a magnetic field [24]:

$$
\frac{d\mathbf{B}}{d\mathbf{x}} = \pm \frac{F\_p(\mathbf{B})}{B}, \qquad \frac{d\mathbf{a}}{d\mathbf{x}} = \pm k(\mathbf{B}) \frac{d\mathbf{B}}{d\mathbf{x}}.\tag{4}
$$

This pair of equations is known as the Double Critical-State Model (DCSM). In their construction, the fact that the magnetic induction *B* and the orientation *α* of flux-lines varied spatially was considered. They assumed as well that gradients existed in critical states. In his model, *Fp*(*B*) is a parameter that characterizes pinning intensity; *k* is associated to the shearing coefficient of the flux lattice in the superconductive sample.

Clem and Perez-Gonzalez extended the DCSM based on the assumption that intersection and cross-joining of adjacent non-parallel vortices generate a electric field different to the electric field **E** = **B** × **v**. The latter field is associated to the flux flow with velocity **v**, for the case in which **J** is perpendicular to **B**. For this, they proposed a pair of constitutive laws of the form:

$$J\_{\perp} = J\_{\mathfrak{c},\perp} \operatorname{sign} E\_{\perp}$$

$$J\_{\parallel} = J\_{\mathfrak{c},\parallel} \operatorname{sign} E\_{\parallel \prime} \tag{5}$$

where parameters *Jc*,⊥ and *Jc*,� correspond to the depinning and the flux-line cutting thresholds, respectively. They considered that the electric field **E** components obey independently the vertical laws:

$$E\_{\perp} = \begin{cases} \rho\_{\perp} [|f\_{\perp}| - f\_{\text{c,\perp}}] \text{sign} \,(f\_{\perp})\_{\prime} & |f\_{\perp}| > f\_{\text{c,\perp}} \\ 0, & 0 \le |f\_{\perp}| \le f\_{\text{c,\perp}} \end{cases} \tag{6}$$

$$E\_{\parallel} = \begin{cases} \rho\_{\parallel} [||f\_{\parallel}| - f\_{c,\parallel}] \text{sign} \,(f\_{\parallel}), & |f\_{\parallel}| > f\_{c,\parallel} \\ 0, & 0 \le |f\_{\parallel}| \le f\_{c,\parallel} \text{'} \end{cases} \tag{7}$$

here, *ρ*⊥ and *ρ*� are the resistivities caused by flux transport and superconductor flux-line cutting, respectively. The group of equations (5)-(7) constitutes the Generalized Double Critical-State Model (GDCSM.) Clem and Perez-Gonzalez did numerical calculations considering as possible values for magnitude *Jc* all those defined within a rectangular region of *Jc*,� and *Jc*,⊥ sides. The model reproduced successfully the experimental distributions of magnetic induction and magnetization, when the magnetic field oscillates at great amplitudes [25].

The GDCSM was also used to try to reproduce *Magnetization Collapse* and *Paramagnetism*, phenomena encountered when a type-II superconductor is subjected first to a DC magnetic field on which an oscillating low-frequency magnetic field is superposed, perpendicular to the former.

#### **2.2. Two-velocities Hydrodynamic Model (TVHM)**

This macroscopic model considers that electrodynamics of a type-II superconductor depends on the translation of vortex planes and the interaction between them. It establishes two vortex subsystems, assuming they posses no elastic properties and that the flux cutting consists of the disappearance of interacting vortices, creating new vortices on a plane with an orientation different to the previous one. Gibbs energy varies through small disturbances on the vortices' coordinates considering the following: 1) magnetic energy; 2) work done by pinning forces given the translation of the vortex network; and 3) the work done by the pinning forces to straighten a vortex after its crossing [26–28]. Thus, the model is conformed by a continuity equation for total vortex density *n*(*x*, *t*), and the average angular distribution *α*(*x*, *t*) of the vortex planes:

$$\frac{\partial n}{\partial t} = -\frac{\partial}{\partial \mathbf{x}} \left[ n \frac{V\_A + V\_B}{2} \right],\tag{8}$$

$$\frac{\partial(na)}{\partial t} = -\frac{1}{2} \frac{\partial[na(V\_A + V\_B)]}{\partial \mathbf{x}} - \frac{1}{4} \frac{\partial[na(V\_A - V\_B)]}{\partial \mathbf{x}},\tag{9}$$

where

6 Superconductors

the form:

**2. Other critical state models**

presence of a magnetic field [24]:

independently the vertical laws:

*E*⊥ =

*E*� =

>

**2.1. Generalized Double Critical-State Model**

*dB dx* = ±

shearing coefficient of the flux lattice in the superconductive sample.

LeBlanc *et. al* proposed a model containing two critical-state equations based on their experimental observations on the magnetic response of a disc oscillating at low frequency, in

*B* , *dα*

This pair of equations is known as the Double Critical-State Model (DCSM). In their construction, the fact that the magnetic induction *B* and the orientation *α* of flux-lines varied spatially was considered. They assumed as well that gradients existed in critical states. In his model, *Fp*(*B*) is a parameter that characterizes pinning intensity; *k* is associated to the

Clem and Perez-Gonzalez extended the DCSM based on the assumption that intersection and cross-joining of adjacent non-parallel vortices generate a electric field different to the electric field **E** = **B** × **v**. The latter field is associated to the flux flow with velocity **v**, for the case in which **J** is perpendicular to **B**. For this, they proposed a pair of constitutive laws of

*J*⊥ = *Jc*,⊥ sign *E*⊥

where parameters *Jc*,⊥ and *Jc*,� correspond to the depinning and the flux-line cutting thresholds, respectively. They considered that the electric field **E** components obey

*ρ*⊥[|*J*⊥| − *Jc*,⊥] sign (*J*⊥), |*J*⊥| > *Jc*,⊥

0, 0 |*J*⊥| *Jc*,⊥

*ρ*�[|*J*�| − *Jc*,�] sign (*J*�), |*J*�| > *Jc*,�

0, 0 |*J*�| *Jc*,�;

here, *ρ*⊥ and *ρ*� are the resistivities caused by flux transport and superconductor flux-line cutting, respectively. The group of equations (5)-(7) constitutes the Generalized Double Critical-State Model (GDCSM.) Clem and Perez-Gonzalez did numerical calculations considering as possible values for magnitude *Jc* all those defined within a rectangular region of *Jc*,� and *Jc*,⊥ sides. The model reproduced successfully the experimental distributions

*dx* = ±*k*(*B*)

*dB*

*J*� = *Jc*,� sign *E*�, (5)

*dx* . (4)

(6)

(7)

*Fp*(*B*)

$$V\_A = V + \frac{U}{2}' \qquad V\_B = V - \frac{U}{2}' \tag{10}$$

correspond to the velocities of subsystems *A* and *B*, *V*(*x*, *t*) is the mean hydrodynamic velocity, and *U*(*x*, *t*) is the relative velocity. The TVHM requires additionally two equations obtained from force balance conditions in a superconductor, defined for the magnetic induction gradient and for angular distribution:

$$\begin{split} \frac{\partial B}{\partial \mathbf{x}} &= -\frac{\mu\_0 I\_{\mathcal{E}, \perp}}{2} [F(V\_A) + F(V\_B)]\_{\prime} \\ \Delta a B \frac{\partial \mathbf{a}}{\partial \mathbf{x}} &+ p \sqrt{\frac{n}{8}} \left[ B - \mu\_0 H\_{\mathcal{E}} \cos(\mathfrak{a} - \mathfrak{a}\_0) \right] \Delta a^2 \text{sign}(V\_A - V\_B)\_{\prime} \\ &= -\mu\_0 I\_{\mathcal{E}, \perp} [F(V\_A) - F(V\_B) + p \text{sign}(V\_A - V\_B)]\_{\prime} \end{split} \tag{12}$$

here *p* corresponds to the probability that flux-line cutting occurs. Finally, to resolve the equation system (8)-(12) for variables *VA*, *VB*, *B* = *n*Φ0, *α* and ∆*α*, it is required to introduce a phenomenological equation that relates ∆*α* to the mean orientation's spatial derivative of the form:

$$
\Delta \mathfrak{a} = -l \operatorname{sign}(V\_A - V\_B) \frac{\partial \mathfrak{a}}{\partial \mathfrak{x}'} \tag{13}
$$

where *l* is the vortex mean free path between two successive cuttings or crossings.

#### **3. Theoretical Description of the ECSM**

Now we introduce the characteristics of the Elliptic Critical-State Model (ECSM) used in this chapter.

#### **3.1. Geometrical aspects of a superconductor plate**

**Figure 2.** Diagram of a portion of infinite superconducting plate. It is shown, in an instant of time *t*, the orientation of the external magnetic field **H***a* , which is always parallel to plane *yz*.

The study system is a superconducting plate possessing an infinite surface parallel to a plane *yz* and a finite thickness 0 ≤ *x* ≤ *D*, as it is shown in Figure (2). The plate is subjected to a magnetic field **H***a* parallel to plane *yz* given by the expression:

$$\mathbf{H}\_{a} = H\_{ay}\mathbf{\hat{y}} + H\_{a2}\mathbf{\hat{z}} = H\_{a}(\sin\alpha\_{a}\mathbf{\hat{y}} + \cos\alpha\_{a}\mathbf{\hat{z}}),\tag{14}$$

where *αa* is an angle measured relative to *z* axis . This problem pertains to the *parallel geometry*. Demagnetization effects are not present, the current density **J**, electric field **E**, and magnetic induction **B** vectors are all coplanar with their components *y* and *z* depending only on variable *x* and time *t*. Given the applied magnetic field **H***a*, local magnetic induction in a superconducting sample is:

$$\mathbf{B} = B(\mathbf{x}, t)\mathbf{\hat{e}}\_{\parallel} = B(\sin a\mathbf{\hat{y}} + \cos a\mathbf{\hat{z}}).\tag{15}$$

The superconducting current circulates around planes *x* = constant. In region *x* ∈ [0, *D*/2) it circulates in the opposite direction from the one in region *x* ∈ (*D*/2, *D*].

Since we are studying flux-line cutting, ECSM postulates that the current density **J** and electric field **E** can possess components both parallel and perpendicular to **B**. The perpendicular component is associated to the flux-pinning effect, while parallel components are associated to flux-line cutting. Therefore, we wish to calculate how large is the force-free component parallel to B of the current density, and if the latter reaches its critical value *Jc*�.

It is convenient to work with a reference system that rotates (follows) with the (the) magnetic induction **B**. The current density and electric field components –parallel and perpendicular to **B**– are monitored:

$$\mathbf{J} = J\_{\parallel}\mathbf{\dot{e}}\_{\parallel} + J\_{\perp}\mathbf{\dot{e}}\_{\perp \prime} \qquad \mathbf{E} = E\_{\parallel}\mathbf{\dot{e}}\_{\parallel} + E\_{\perp}\mathbf{\dot{e}}\_{\perp \prime} \tag{16}$$

where the unit vector **ˆe**⊥ is built as **ˆe**⊥ = **ˆe**� × **ˆx** = cos *α* **ˆy** − sin *α***ˆz**.

8 Superconductors

the form:

chapter.

here *p* corresponds to the probability that flux-line cutting occurs. Finally, to resolve the equation system (8)-(12) for variables *VA*, *VB*, *B* = *n*Φ0, *α* and ∆*α*, it is required to introduce a phenomenological equation that relates ∆*α* to the mean orientation's spatial derivative of

∆*α* = −*l* sign(*VA* − *VB*) *∂α*

Now we introduce the characteristics of the Elliptic Critical-State Model (ECSM) used in this

**j**

**Figure 2.** Diagram of a portion of infinite superconducting plate. It is shown, in an instant of time *t*, the orientation of the

The study system is a superconducting plate possessing an infinite surface parallel to a plane *yz* and a finite thickness 0 ≤ *x* ≤ *D*, as it is shown in Figure (2). The plate is subjected to a

where *αa* is an angle measured relative to *z* axis . This problem pertains to the *parallel geometry*. Demagnetization effects are not present, the current density **J**, electric field **E**, and magnetic induction **B** vectors are all coplanar with their components *y* and *z* depending only on variable *x* and time *t*. Given the applied magnetic field **H***a*, local magnetic induction in a

where *l* is the vortex mean free path between two successive cuttings or crossings.

z

*D*

**3. Theoretical Description of the ECSM**

**3.1. Geometrical aspects of a superconductor plate**

x

magnetic field **H***a* parallel to plane *yz* given by the expression:

external magnetic field **H***a* , which is always parallel to plane *yz*.

superconducting sample is:

**H**

*∂x*

y

*x=D*

**H***a* = *Hay* **ˆy** + *Haz* **ˆz** = *Ha*(sin *αa* **ˆy** + cos *αa* **ˆz**), (14)

**j**

*x*=0

, (13)

#### **3.2. Physical considerations for a Type-II superconductor**

Type-II superconductor materials are found in vortex state and contain a dense distribution and random pinning centers. When applying a magnetic field **H***a*, Eq. (14), it is considered that the material is in a critic state (metastable state) when there is a balance between Lorentz force density **F***L* = **J***c*⊥ × **B** and the average pinning force. If the perpendicular component of the current overcomes its critical value *Jc*⊥, flux transport is begun until another material metastable state is reached. Also, vortex avalanches can occur; however, we will not cover here this phenomenon [29]. In addition, given that a **J** component is parallel to **B**, magnetic flux distribution will depend on flux line cutting, only if *J*� exceeds its critical value *Jc*�. Finally, the magnitude of an applied magnetic field is considered to be much larger than the first critical field *Hc*1, that is, the Meissner currents on the material's surface are neglected.

We are interested in the macroscopic electrodynamics of a type-II superconductor; thus, we work with Maxwell equations, given that low-frequency magnetic fields are considered, displacement current is neglected. In order to establish the critical state, this model considers quasi-stationary electromagnetic fields. This is why first the Faraday law is used, and subsequently stationary solutions are sought. Having chosen a reference system that rotates with magnetic induction **B**, Ampère and Faraday laws are written as follows:

$$-\frac{\partial B}{\partial \mathbf{x}} = \mu\_0 \mathbf{J}\_{\perp} \qquad \mathcal{B} \frac{\partial \mathbf{a}}{\partial \mathbf{x}} = \mu\_0 \mathbf{J}\_{\parallel \prime} \tag{17}$$

$$\frac{\partial E\_{\perp}}{\partial \mathbf{x}} + E\_{\parallel} \frac{\partial \alpha}{\partial \mathbf{x}} = -\frac{\partial B}{\partial t}, \qquad E\_{\perp} \frac{\partial \alpha}{\partial \mathbf{x}} - \frac{\partial E\_{\parallel}}{\partial \mathbf{x}} = -B \frac{\partial \alpha}{\partial t}. \tag{18}$$

In absence of a demagnetization factor, and considering that the applied magnetic field **H***a* is much larger than *Hc*1, the constitutive relation between **B** and **H** is modeled with the linear approximation **B** = *µ*0**H**, where the Bean-Livingston superficial barrier is neglected. Therefore, the boundary condition is written as follows:

$$
\mu\_0 H\_{ay}(t) = B\_y(0; t) = B\_y(D; t), \tag{19}
$$

$$
\mu\_0 H\_{a\mathbb{Z}}(t) = B\_{\mathbb{Z}}(0; t) = B\_{\mathbb{Z}}(D; t). \tag{20}
$$

#### **3.3. Elliptic Flux-Cutting Critical-State Model for a type-II superconductor in critical state**

The Elliptic Flux-Cutting Critical-State Model (ECSM) [1] considered the presence of an electric field in non-critical states. It models the material as a highly non-linear conductor, using a constitutive relation **E** = **E**(**J**, **B**), which completes the system of equations (17)-(18). Furthermore, it monitors the electric field's decrease through a vertical law. The model supposes that the material reaches a critical state when the magnetic induction distribution enters a stationary state where the electric field has decreased to zero.

The material law of the elliptic model considers, in contrast to multicomponent Bean and double critical-state models, that the critical current density is a rank-2 tensor determined by the equation:

$$\mathbf{J}\_{\mathbf{i}} = (\mathcal{J}\_{\mathbf{c}})\_{\mathbf{i}\mathbf{k}} \frac{E\_{\mathbf{k}}}{E}, \qquad (\mathcal{J}\_{\mathbf{c}})\_{\mathbf{i}\mathbf{k}} = \mathbf{J}\_{\mathbf{c},\mathbf{i}}(\mathcal{B}) \delta\_{\mathbf{i}\mathbf{k}\prime} \qquad \mathbf{i}, \mathbf{k} = \boldsymbol{\perp}, \parallel. \tag{21}$$

or, by the vector relation:

$$\mathbf{J} = J\_{c\parallel}(\mathcal{B})\frac{\mathbf{E}\_{\parallel}}{E} + J\_{c\perp}(\mathcal{B})\frac{\mathbf{E}\_{\perp}}{E}. \tag{22}$$

In critical state, the magnitude of **J**, that is, the critical current density *Jc*(*B*, *ϕ*) traces an ellipse on the plane *J*� − *J*⊥, defined by the equation:

$$J\_c(B, \varphi) = \left[ \left( \frac{\cos \varphi}{J\_{c\parallel}(B)} \right)^2 + \left( \frac{\sin \varphi}{J\_{c\perp}(B)} \right)^2 \right]^{-1/2} \text{.} \tag{23}$$

where the angle *ϕ* is measured respect to **B**. Notice that the behaviour of the vortex distribution is a function of the magnitude of **J** and its relative orientation respect to **B**. The model postulates that *Jc*(*B*, *ϕ*) is defined in terms of critical values *Jc*⊥ and *Jc*�, that is, *Jc*(*B*, *ϕ*) is restricted to the critical ellipse (23). The fact that *J*� and *J*⊥ are different produces an anisotropy on the plane *J*� − *J*⊥ because of the external magnetic field **H***a* and since such components are associated with different physical phenomena which are still not fully understood. Lastly, a vertical law is used to relate the current density and the electric field magnitudes as:

10 Superconductors

**critical state**

the equation:

or, by the vector relation:

*∂E*⊥ *∂x*

+ *E*� *∂α ∂x* = −*∂B ∂t*

Therefore, the boundary condition is written as follows:

, *E*⊥

In absence of a demagnetization factor, and considering that the applied magnetic field **H***a* is much larger than *Hc*1, the constitutive relation between **B** and **H** is modeled with the linear approximation **B** = *µ*0**H**, where the Bean-Livingston superficial barrier is neglected.

**3.3. Elliptic Flux-Cutting Critical-State Model for a type-II superconductor in**

enters a stationary state where the electric field has decreased to zero.

*Ek*

**J** = *Jc*�(*B*)

 � cos *ϕ Jc*�(*B*)

**E**�

In critical state, the magnitude of **J**, that is, the critical current density *Jc*(*B*, *ϕ*) traces an

�2 +

*E* + *Jc*⊥(*B*)

*Ji* = (J*c*)*ik*

ellipse on the plane *J*� − *J*⊥, defined by the equation:

*Jc*(*B*, *ϕ*) =

The Elliptic Flux-Cutting Critical-State Model (ECSM) [1] considered the presence of an electric field in non-critical states. It models the material as a highly non-linear conductor, using a constitutive relation **E** = **E**(**J**, **B**), which completes the system of equations (17)-(18). Furthermore, it monitors the electric field's decrease through a vertical law. The model supposes that the material reaches a critical state when the magnetic induction distribution

The material law of the elliptic model considers, in contrast to multicomponent Bean and double critical-state models, that the critical current density is a rank-2 tensor determined by

*∂α ∂x* − *∂E*�

*∂x* = −*B ∂α*

*µ*0*Hay*(*t*) = *By*(0; *t*) = *By*(*D*; *t*), (19) *µ*0*Haz*(*t*) = *Bz*(0; *t*) = *Bz*(*D*; *t*). (20)

*E* , (J*c*)*ik* = *Jc*,*i*(*B*)*δik*, *i*, *k* =⊥, �. (21)

*E* . (22)

, (23)

**E**⊥

� sin *ϕ Jc*⊥(*B*) �2

−1/2

*∂t*

. (18)

$$E(f) = \begin{cases} 0, & \text{for} \\ \rho(f - f\_c(B, \varphi)), & \text{for } f > f\_c(B, \varphi). \end{cases} \tag{24}$$

It can be considered that an electric field can exist in the superconductor only if the current density exceeds its critical value *Jc*(*B*, *ϕ*). Conventionally the parameter *ρ* acts as a resistivity, however, here it serves as an auxiliary parameter that counteracts the difference *J* − *Jc*(*B*, *ϕ*). The empirical relation between the critical current density **J***c* and magnetic induction **B** used is known as generalized relation, it is an adaptation to such deduced by Y.B. Kim *et al.* [30], and was proposed by M. Xu *et al.* [31], its explicit form is as follows:

*Jc*(*B*) = *Jc*(0)/ 1 + *B B*∗ *n* , (25)

where *n* and *B*∗ are fitting parameters, and the maximum critical current density value *Jc*(*B* = 0) is defined by the following relation:

$$J\_c(0) = \left[ \left( 1 + \frac{B\_p}{B^\*} \right)^{n+1} \right] \frac{2B^\*}{(n+1)\mu\_0 d}. \tag{26}$$

In our case, temperature dependency remains implicit in the parameters election.

All solutions that satisfy the system of equations formed by the Ampère's law (17), the Faraday's law (18), the critical current density *Jc*(*B*, *ϕ*) (23), and the material law (24) establish the superconductor's critical state. Such system is designated the *Elliptic Flux-Cutting Critical-State Model* (ECSM) [1].

#### **4. Results using ECSM for a type-II superconductor in vortex state**

Here we present theoretical curves obtained with the ECSM, corresponding to experimental results which depend on vortex dynamics. Specifically, numerical results are shown for magnetic-moment curves and magnetic-induction evolution, using type-II superconductor samples in mixed state, subject to transverse fields, or to a low-frequency rotating field.

#### **4.1. Type-II superconductor plates subjected to transverse DC and AC fields**

In this section we present the effect of an AC magnetic field on the static magnetization of a type-II superconductor plate. The purpose was to compare and explain experimental results - obtained by Fisher *et al.*- from YBCO samples subjected to two fields perpendicular to each other, in parallel geometry [28, 33].

**Figure 3.** Theoretical average magnetization curves �*µ*0*Mz* � vs *Hy* of an YBCO plate. The red (blue) curves were calculated for an initial paramagnetic (diamagnetic) state. In each figure, the magnitude of magnetic fields to which the sample was subjected to is shown: a static one *Hz* , and one with oscillation amplitude *Hy*,*max*. These fields are perpendicular to each other and coplanar to the specimen. We observe the dependency of �*µ*0*Mz* � to the first cycles of *Hy*. Results correspond to a texturized YBCO plate, cooled down on field, a *d* = 3mm thickness and a *µ*0*Hp* = 0.0856T penetration field. The specimen's larger sides are parallel to the crystallographic plane **ab**.

In the experiments of Fisher *et al.*, plates were cut out from the homogenous part of melt-textured YCBO ingots. This was done in such a way that the larger-sized sides were parallel to the crystallographic **ab** plane, which possesses isotropic properties respect to current conduction capacity. First they applied on the sample an *Hz* field generated by direct current, with a direction such that it was parallel to the plate's main surface. Afterward they applied a second field *Hy*, oscillating at low frequencies -of the order 1kOe- coplanar to the sample's surface and perpendicular to the first field. Under this configuration, and depending on the order of magnitude for both fields, they observed phenomena denominated *Magnetization Collapse* and *Paramagnetism*.

Results presented in Figure (3) correspond to a plate with a *d* = 3mm thickness that possesses a penetration field *µ*0*Hp* = 0.0856T. This last value is very close to the experimental *µ*0*Hp* ∼ 0.1T value for the YBCO samples [33]. The parameters used to model the perpendicular component of current critical density *Jc*⊥(*B*) (see Eq. 25) are: *n*⊥ = 0.5 y *B*∗ ⊥ = 0.2T, and a maximum value *Jc*⊥(0) = 5 × 108A/m2 (see Eq. 26). The parallel component *Jc*�(*B*) was modeled just as *Jc*⊥(*B*). Therefore, we use *B*∗ � = *B*∗ ⊥, *n*� = *n*⊥. The best fitting to the experimental curves was achieved assuming *Jc*�(0) = 2*Jc*⊥(0).

12 Superconductors

other, in parallel geometry [28, 33].

0

My/Hp

0.5

My/Hp

0

Hz = 5Hp |Hy,max| = 0.06(T)

Hz = 10Hp |Hy,max| = 0.07(T)

larger sides are parallel to the crystallographic plane **ab**.

0.5

Hy/Hy,max

Hy/Hy,max

phenomena denominated *Magnetization Collapse* and *Paramagnetism*.

component of current critical density *Jc*⊥(*B*) (see Eq. 25) are: *n*⊥ = 0.5 y *B*∗

**4.1. Type-II superconductor plates subjected to transverse DC and AC fields**

(A) (B)

(C) (D)

In this section we present the effect of an AC magnetic field on the static magnetization of a type-II superconductor plate. The purpose was to compare and explain experimental results - obtained by Fisher *et al.*- from YBCO samples subjected to two fields perpendicular to each

Hz = Hp |Hy,max| = Hp

0

Hz = 10Hp |Hy,max| = 0.06(T)

My/Hp

My/Hp

**Figure 3.** Theoretical average magnetization curves �*µ*0*Mz* � vs *Hy* of an YBCO plate. The red (blue) curves were calculated for an initial paramagnetic (diamagnetic) state. In each figure, the magnitude of magnetic fields to which the sample was subjected to is shown: a static one *Hz* , and one with oscillation amplitude *Hy*,*max*. These fields are perpendicular to each other and coplanar to the specimen. We observe the dependency of �*µ*0*Mz* � to the first cycles of *Hy*. Results correspond to a texturized YBCO plate, cooled down on field, a *d* = 3mm thickness and a *µ*0*Hp* = 0.0856T penetration field. The specimen's

In the experiments of Fisher *et al.*, plates were cut out from the homogenous part of melt-textured YCBO ingots. This was done in such a way that the larger-sized sides were parallel to the crystallographic **ab** plane, which possesses isotropic properties respect to current conduction capacity. First they applied on the sample an *Hz* field generated by direct current, with a direction such that it was parallel to the plate's main surface. Afterward they applied a second field *Hy*, oscillating at low frequencies -of the order 1kOe- coplanar to the sample's surface and perpendicular to the first field. Under this configuration, and depending on the order of magnitude for both fields, they observed

Results presented in Figure (3) correspond to a plate with a *d* = 3mm thickness that possesses a penetration field *µ*0*Hp* = 0.0856T. This last value is very close to the experimental *µ*0*Hp* ∼ 0.1T value for the YBCO samples [33]. The parameters used to model the perpendicular

a maximum value *Jc*⊥(0) = 5 × 108A/m2 (see Eq. 26). The parallel component *Jc*�(*B*) was

0.5

Hy/Hy,max

Hy/Hy,max

⊥ = 0.2T, and

Results for �*Mz*(*Hy*)� during the first cycles of the oscillating magnetic field *Hy* are shown in Figure (3). One can observe in (A)-(C) graphics the symmetric reduction of the average magnetization for both the diamagnetic initial state (blue curves) and paramagnetic initial state (red curves). These three cases are characterized by the fact that static magnetic field *Hz* is greater than *Hp* (*Hz* = 5, 10*Hp*), whereas the AC magnetic field oscillates at small amplitudes (*Hy*,*max* ≪ *Hz*). In graphic (D) we appreciate a different behavior of the diamagnetic branch of �*Mz*(*Hy*)� (blue curve). This is due to the order of magnitude of fields *Hz* and *Hy*, which have the same order of magnitude of the sample penetration field *Hp*. One can observe in graphics (A)-(C) that for the same number of cycles of the oscillating field *Hy*, the magnetization reduction prevails, however, it is asymmetric. Specifically, the diamagnetic branch of the average magnetization changes sign, this behavior is the so-called paramagnetism. All theoretical results successfully agree with experiments for large (|*Hy*| ∼ *Hz*) and small (|*Hy*| ≪ *Hz*) values of the transverse field amplitude.

The asymmetrical suppression of average magnetization �*Mz*(*Hy*)� — caused by an AC magnetic field for diamagnetic and paramagnetic initial states as well as the change of sign in its diamagnetic branch — is only possible to reproduce if the effects of flux-line cutting and anisotropy between the perpendicular and parallel components of **J***c*(*B*) are incorporated. Also, it is necessary to consider that the effect of flux-line cutting has greater influence on the sample's magnetic induction behavior than the pinning effect.

**Figure 4.** Theoretical magnetic induction curves *Bz*/*µ*0*Hp* vs *x*/*d* for a YBCO plate (in Fig. (3) specimen characteristics are specified). The red curves (blue) were calculated for an initial paramagnetic (diamagnetic) state. The magnitude of the static magnetic field *Hz* , and the amplitude of the oscillating magnetic field |*Hy* | are shown. In all panels (A)-(D) it is initiated with a magnetic field *Hy* = *Hy*,*max*, each profile is generated every half cycle of *Hy*.

We can observe in Figure (4) the evolution of the magnetic induction profiles *Bz*(*x*) as *Hy* ocillates. They correspond to the paramagnetic and diamagnetic branches of the average magnetization �*Mz*� (see Fig.(3). In graphics (A)-(C) we can see the reduction of magnetic induction component *Bz*(*x*) as the transverse field *Hy* describes small-amplitude cycles (|*Hy*| < *Hp* ≪ *Hz*). The ECSM predicts that *Bz*(*x*) has a tendency towards a fixed distribution (see (A), (B)) or quasi-uniform (see (C)), regardless of the number of cycles. Close to the sample's borders, the distribution *Bz*(*x*) collapses to an almost homogeneous value (*Bz* ∼ *B*) as *Hy* oscillates. However, the graphic (D) shows that when the magnitude of *Hz* and *Hy* are comparable to *Hp* (|*Hy*| ∼ *Hz* = *Hp*), the slope of *Bz*(*x*) changes (conserves) its sign close to the plate borders if the initial state is diamagnetic (paramagnetic). Moreover, *Bz* collapse areas are not present in the region close to the plate border. Therefore, the average magnetization value �*Mz*�, after several *Hy* cycles becomes positive for the initial diamagnetic state.

In more recent experiments using the magneto-optic technique and Hall sensors, the effect of crossed magnetic fields in the perpendicular geometry was studied [34]. In this case, the sample is pre-magnetized in z-direction. Once the magnetic field along the z-axis is removed, an AC magnetic field, parallel to the sample's *ab* plane, is applied. Experimental results show that in this perpendicular geometry, given the magnitude of external magnetic fields *Hy* and *Hz*, only the symmetrical collapse effect of the average magnetization �*Mz*� is present. This occurs regardless of the transverse field *Hy* magnitude respect to the sample's penetration field *µ*0*Hp*. It was employed a finite-element numerical model and a power law function *E*(*J*). They reproduced the experimental curves of �*Mz*� considering that the flux-line cutting effect is not present in the superconductor dynamics. Indeed, their results do not contradict those obtained with the ESCM, because we have reproduced symmetrical collapse of �*Mz*� in a type-II superconductor with the Bean multicomponent model, which does not consider flux-line cutting.

It has also studied results of magnetization collapse for dissipative states using both ECSM [1] and the **extended ECSM**, the latter proposed by Clem [32]. In the experiments, it was measured the remanent magnetization of a PbBi superconductor disk. First, the disk is magnetized with a single pulse of a *Hz* field, and subsequently it is subjected to an oscillating magnetic field *Hy*. Even though the theoretical curves �*Mz*� vs *Hy* obtained with each model are very similar, the profile evolution of *Bz* obtained by using the extended ECSM contradicts the cases when the magnitude of fields *Hy* and *Hz* are comparable to *Hp* [5].

#### **4.2. Rotating type-II superconductors in presence of a magnetic field**

Another instance in which flux-line cutting occurs is when a superconductor sample rotates or oscillates in presence of an external magnetic field. We will focus on experiments conducted by Cave and LeBlanc [24] where a low-*Tc* type-II superconductor disk oscillates slowly in presence of a static magnetic field in parallel geometry. Cave and LeBlanc realized extensive research on the magnetic behavior of such sample, for a series of rotation angles, different initial states i.e. magnetic, non-magnetic, diamagnetic, paramagnetic and hybrid, and investigated the energy dissipation during oscillations.

Here we present theoretical calculations both for magnitude and direction of the magnetic induction **B**, as the sample rotates under different angles, also present the average component �*Bz*� and the hysteresis cycles. Results were obtained considering that the parallel and perpendicular components of current density possess a dependency on the magnetic induction of the form:

$$J\_{c\perp(\parallel)}(B) = J\_{c\perp(\parallel)}(0)\left(1 - \frac{B}{B\_{c2}}\right). \tag{27}$$

We used the second critical field *Bc*2 = 0.35T reported in Ref.[24], and the maximum value of critical current density *Jc*⊥(0) = 1.6 × 109A/m2. We found that, with the anisotropy parameter *Jc*�(0)/*Jc*⊥(0) = 4, we obtain the best agreement with the experimental curves.

14 Superconductors

state.

does not consider flux-line cutting.

induction of the form:

induction component *Bz*(*x*) as the transverse field *Hy* describes small-amplitude cycles (|*Hy*| < *Hp* ≪ *Hz*). The ECSM predicts that *Bz*(*x*) has a tendency towards a fixed distribution (see (A), (B)) or quasi-uniform (see (C)), regardless of the number of cycles. Close to the sample's borders, the distribution *Bz*(*x*) collapses to an almost homogeneous value (*Bz* ∼ *B*) as *Hy* oscillates. However, the graphic (D) shows that when the magnitude of *Hz* and *Hy* are comparable to *Hp* (|*Hy*| ∼ *Hz* = *Hp*), the slope of *Bz*(*x*) changes (conserves) its sign close to the plate borders if the initial state is diamagnetic (paramagnetic). Moreover, *Bz* collapse areas are not present in the region close to the plate border. Therefore, the average magnetization value �*Mz*�, after several *Hy* cycles becomes positive for the initial diamagnetic

In more recent experiments using the magneto-optic technique and Hall sensors, the effect of crossed magnetic fields in the perpendicular geometry was studied [34]. In this case, the sample is pre-magnetized in z-direction. Once the magnetic field along the z-axis is removed, an AC magnetic field, parallel to the sample's *ab* plane, is applied. Experimental results show that in this perpendicular geometry, given the magnitude of external magnetic fields *Hy* and *Hz*, only the symmetrical collapse effect of the average magnetization �*Mz*� is present. This occurs regardless of the transverse field *Hy* magnitude respect to the sample's penetration field *µ*0*Hp*. It was employed a finite-element numerical model and a power law function *E*(*J*). They reproduced the experimental curves of �*Mz*� considering that the flux-line cutting effect is not present in the superconductor dynamics. Indeed, their results do not contradict those obtained with the ESCM, because we have reproduced symmetrical collapse of �*Mz*� in a type-II superconductor with the Bean multicomponent model, which

It has also studied results of magnetization collapse for dissipative states using both ECSM [1] and the **extended ECSM**, the latter proposed by Clem [32]. In the experiments, it was measured the remanent magnetization of a PbBi superconductor disk. First, the disk is magnetized with a single pulse of a *Hz* field, and subsequently it is subjected to an oscillating magnetic field *Hy*. Even though the theoretical curves �*Mz*� vs *Hy* obtained with each model are very similar, the profile evolution of *Bz* obtained by using the extended ECSM contradicts

Another instance in which flux-line cutting occurs is when a superconductor sample rotates or oscillates in presence of an external magnetic field. We will focus on experiments conducted by Cave and LeBlanc [24] where a low-*Tc* type-II superconductor disk oscillates slowly in presence of a static magnetic field in parallel geometry. Cave and LeBlanc realized extensive research on the magnetic behavior of such sample, for a series of rotation angles, different initial states i.e. magnetic, non-magnetic, diamagnetic, paramagnetic and hybrid,

Here we present theoretical calculations both for magnitude and direction of the magnetic induction **B**, as the sample rotates under different angles, also present the average component �*Bz*� and the hysteresis cycles. Results were obtained considering that the parallel and perpendicular components of current density possess a dependency on the magnetic

the cases when the magnitude of fields *Hy* and *Hz* are comparable to *Hp* [5].

**4.2. Rotating type-II superconductors in presence of a magnetic field**

and investigated the energy dissipation during oscillations.

Figures (5) and (6) show the theoretical curves of �*By*(*θ*)� and �*Mz*� = *µ*0*Ha* − �*Bz*� for a non magnetic initial state and four rotation angles *θmax* = 45*o*, 120*o*, 270*o*, 360*o*. All theoretical curves show the main characteristics of the corresponding experimental measurements.

**Figure 5.** Theoretical curves of �*Bz* � vs *θ* for a non magnetic initial state for an *Nb* plate with thickness *d* = 0.25mm, *Bc*2 = 0.25T , and *Jc*⊥(0) = 1.6 × 109A/m2. The relation *Jc*�(0)/*Jc*⊥(0) = 4 was used. Curves are shown for different static external magnetic fields and rotation angles: *µ*0*Ha* = 261mT and *θmax* = 45*o* ; *µ*0*Ha* = 149mT and *θmax* = 120*o* ; *µ*0*Ha* = 172mT and *θmax* = 270*o* ; *µ*0*Ha* = 149mT and *θmax* = 360*o* .

The most outstanding phenomenon in this experiment is the *magnetic flux consumption*. This phenomenon is analyzed with the help of the spatial evolution distribution of *B*(*x*) and *α*(*x*) predicted with the ESCM. For instance, Fig. 7 presents the results for an amplitude oscillation *θmax* = 45*o* of the sample in presence of a static field *µ*0*Ha* = 261mT. We can observe in the left panel that, as the sample rotates, the two U-shaped minima of the profile of *B*(*x*) move away from the surface of the sample, as well as the local magnetic flux reduction. The right panel shows how *α*(*x*)(B) decreases from a maximum value at the sample's surface to a zero value at the region 0.4 *x*/*d* 0.6. Therefore, the ECSM can predict the existence of three areas within the material: (1) an exterior area where both flux cutting and flux transport occur, (*α* �= 0 y *dB*/*dx* �= 0); (2) an internal area where only flux transport occurs (*α* = 0 y *dB*/*dx* �= 0) and (3) a central area were neither transport flux nor flux cutting occur (*α* = *dB*/*dx* = 0).

**Figure 6.** Theoretical curves of −*µ*0�*Mz* � vs *θ* corresponding to the graphs presented in Fig.(5).

**Figure 7.** Theoretical curves for the evolution of *B*(*x*) vs *x*/*d* and *θ*(*x*) vs *x*/*d* corresponding to *µ*0*Ha* = 261mT y *θmax* = 45*o* .

## **5. Conclusions and expectations**

16 Superconductors

(*α* = *dB*/*dx* = 0).

0.01 0.02 0.03 0.04

0

0.05

0.252 0.253 0.254 0.255 0.256 0.257 0.258 0.259 0.26 0.261 0.262

B(T)

!

70Mz(T)

0.1

0.15

!

70Mz(T)

(*α* = 0 y *dB*/*dx* �= 0) and (3) a central area were neither transport flux nor flux cutting occur

0.05

0.05

**Figure 7.** Theoretical curves for the evolution of *B*(*x*) vs *x*/*d* and *θ*(*x*) vs *x*/*d* corresponding to *µ*0*Ha* = 261mT y *θmax* = 45*o* .

R1

R0

R3

0

3(degrees)

0.2

0.4

0.6

!

70Mz(T)

0.1

0.15

!

70Mz(T)

0.1

0.15

3max = 120

3max = 360

3(degrees)

3(degrees)

0 0.5 1

R2

R4

x/d

3max = 45

3(degrees)

3max = 270

3(degrees)

0 0.5 1

R5

R3

x/d

**Figure 6.** Theoretical curves of −*µ*0�*Mz* � vs *θ* corresponding to the graphs presented in Fig.(5).

R1

As J.R. Clem [32] emphasized, in contrast with other critical-state models, the Elliptic Flux-line Cutting Critical-state Model (ECSM) contains important new physical characteristics: it considers that the depinning threshold decreases as magnitude *J*� increases and, in the same way, the flux cutting threshold decreases as magnitude *J*⊥ increases. The model also reproduces the experimentally observed soft angular dependency of *Jc*(*B*). In this chapter, we have shown that the ECSM satisfactorily reproduces the response of type-II superconductors subjected to external magnetic field varying in magnitude and direction when the flux-line cutting effect is considered. Following this line of research, and based on recently results reported [32, 35, 36], the effects of the flux transport will be studied, as well as energy dissipation. This can be achieved incorporating an electric field with a well-defined direction. Our next study will consider the anisotropy in the current carrying capacity, since this characteristic is present in high-*Tc* superconductors, and flux line cutting as well. It is also desirable to study more complex phenomena in high-*Tc* superconductors, such as Meissner holes or the turbulent structures that occur in the perpendicular geometry when the external magnetic field is rotated. For this, it is essential to implement the ECSM for the case of finite geometries.

#### **Acknowledgments**

This work was supported by SEP-CONACYT under Grant No.CB-2008-01-106433

## **Author details**

Carolina Romero-Salazar\* and Omar Augusto Hernández-Flores

School of Sciences, Autonumous University "Benito Juárez" of Oaxaca, Oaxaca, Mexico

∗Address all correspondence to<EMAIL_ADDRESS>
#### **References**

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18 Superconductors

250-253.

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3309-3312.

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Low Temperature Physics 1980; 39(1/2) 173-189.

Temperature Physics 1980; 38(3/4) 353-370.

Model. Physica B + C 1981; 107(1-3) 453-545.

[6] Bean C. P. Magnetization of Hard Superconductors. Physical Review Letters 1962; 8(6)

[7] Bean C. P., Schmitt R.W. The Physics of High-Field Superconductors. Science 1963;

[8] Bean C. P. Magnetization of High-Field Superconductors. Review of Modern Physics

[9] Bean C.P. Rotational Hysteresis Loss in High-Field Superconductors. Journal of Applied

[10] Walmsley D. G. Force Free Magnetic Fields in a Type-II Superconducting Cylinder.

[11] Brandt E.H., Clem J.R., Walmsley D.G. Flux-Line Cutting in Type-II Superconductors.

[12] Sudbo A., Brandt E.H. Flux-Line Cutting in Superconductors. Physical Review Letters

[13] Fillion G., Gauthier R., LeBlanc M.A.R. Direct Evidence of Flux Cutting in Type-II

[14] LeBlanc M.A.R., LeBlanc D., Golebiowski A., Fillion G. New Observations on the Dynamics of Flus Lines in Type-II Superconductors. Physical Review Letters 1991; 66(25)

[15] LeBlanc M.A.R., Celebi S., Wang S. X., Plechác*e*ˆk. Cross-Flow of Flux-Lines in the Weak Link Regime of High-T*c* Superconductors. Physical Review Letters 1993; 71(20)

[16] Clem J. R., Yeh S. Flux-Line-Cutting Threshold in Type II Superconductors. Journal of

[17] Clem J. R. Steady-State Flux-Line Cutting in Type II Superconductors. Journal of Low

[18] Clem J.R. Steady-State Flux-Line Cutting in Type II Superconductors: A Double-Cutting

[19] Brandt E. H. Continuous Vortex Cutting in Type-II Superconductors with Longitudinal

[20] Bou-Diab M., Dodgson M. J. W., Blatter G. Vortex Collisions: Crossing or

[22] Palau A., Dinner R., Durrell J.H., Blamire M. G. Vortex Breaking and Cutting in Type II

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[21] Nelson D.R. Vortices Weave Tangled Web. Nature 2004; 430() 839-840.

[36] Cortés-Maldonado R., Espinosa-Reyes J.E., Carballo-Sánchez A.F., and Pérez-Rodríguez F. Flux-cutting and Flux-transport Effects in Type-II Superconductor Slabs in a Parallel Rotating Magnetic Field. Low Temperature Physics 2011; 37(11), 947.

20 Superconductors

270 Superconductors – New Developments

[36] Cortés-Maldonado R., Espinosa-Reyes J.E., Carballo-Sánchez A.F., and Pérez-Rodríguez F. Flux-cutting and Flux-transport Effects in Type-II Superconductor Slabs in a Parallel

Rotating Magnetic Field. Low Temperature Physics 2011; 37(11), 947.

## *Edited by Alexander Gabovich*

The chapters included in the book describe recent developments in the field of superconductivity. The book deals with both the experiment and the theory. Superconducting and normal-state properties are studied by various methods. The authors presented investigations of traditional and new materials. In particular, studies of oxides, pnictides, chalcogenides and intermetallic compounds are included. The superconducting order parameter symmetry is discussed and consequences of its actual non-conventional symmetry are studied. Impurity and tunneling effects (both quasiparticle and Josephson ones) are among topics covered in the chapters. Special attention is paid to the competition between superconductivity and other instabilities, which lead to the Fermi surface gapping.

Superconductors - New Developments

Superconductors

New Developments

*Edited by Alexander Gabovich*

Photo by cybrain / iStock


**1. Introduction**

Bare soil albedo has been widely used in climate models and remote sensing estimates of surface energy balance as one key component of the surface albedo by determining the amount of solar radiation reflected and absorbed at the Earth's surface [1,2]. Bare soil albedo has also been widely used in ecological research as a controlling factor in algorithms for deriving leaf area index (LAI) and fraction of Photosynthesis Active Radiation (fPAR) from satellite observations, which requires fine resolution data for satellite applications, e.g., [3,4]. A limited number of prescribed bare soil albedo values are assigned based on global soil color maps [5–7]. However, soil reflectivity varies both temporally and spatially. Besides soil color, bare soil albedo is also a function of moisture content [8,9], organic

matter [10], texture/roughness [11], and other surface characteristics. Oversimplified soil albedo parameterization has been reported to introduce uncertainties both in climate modeling [12,13] and satellite LAI/fPAR products [14–17]. Significant differences in soil albedo were found in comparisons of land surface models and satellite products [12], which lead to substantially different estimations of surface energy balance and hydrologic budget partitioning [18].

Efforts have been made towards generating global or regional soil albedo datasets that are independent of coarse resolution soil color maps. Zhou et al. [19] analyzed the Moderate Resolution Imaging Spectroradiometer (MODIS) 1 km albedo products over desert areas using principle component analysis and found the extracted spatial pattern could improve the soil albedo parameterization in climate models. In a recent study, a method has been proposed to estimate soil albedo by removing impacts of solar zenith angle and soil moisture empirically from MODIS albedo products [20]. In their method, coarse resolution (~15–100 km) soil moisture data from the North American and Global Land Data Assimilation System (LDAS) were used.

To develop a bare soil albedo dataset over densely vegetated area and without the support of reanalysis data, a method has been developed based on an empirical relationship between broadband albedos and the Normalized Difference Vegetation Index (NDVI) [21]. In their method, broadband albedo of bare soil over densely vegetated area was predicted through extrapolation of linear logarithmic relationship between albedo and NDVI. It was proposed that bare soil albedos were estimated at NDVI = 0.09 globally, based on the empirical relationship. However, we found that this empirical relationship could generate larger values for visible albedo than near infrared albedo due to lack of training data for extrapolation purposes. In addition, previous studies have proved that NDVI may not be a good quantitative indicator of surfaces with sparse vegetation coverage because of mutual shadow effects on surface anisotropy [22,23].

Pisek and Chen [16] proposed a method using multi-angular satellite observations to map background spectral reflectance in forested pixels and found significant variations between coniferous and deciduous forests, particularly in the near infrared wavelengths.

In some recent studies, MODIS LAI and fPAR products were introduced to calculate vegetation fraction used in a linear regression to estimate soil albedo and the difference of vegetation and soil albedos based on the assumption that these two variables were invariant within a certain period [24,25]. This was later improved by using a Kalman filter to generate dynamic albedo on a daily basis [26]. However, MODIS LAI/fPAR products were derived assuming a prescribed bare soil albedo based on a soil type map, and these products were believed to have lower reliability during the vegetation growing season [14,15].

Existing bare soil albedo datasets are subject to several limitations. First, most suffer from uncertainties in ancillary input products used in the albedo retrieval algorithm due to, for example residual cloud contamination, inaccurate soil maps, or scale differences between model inputs and the retrieval scale. Second, most of the existing global bare soil albedo datasets are only available at a spatial resolution coarser than 5 km, which cannot satisfy the increasing demand for high resolution soil parameterizations, especially in agricultural and ecological applications, e.g., [27]. Third, validation of these datasets has been generally limited to inter-comparison against model-simulated results.

In addition to mapping the broadband albedo for bare soil, there is also a need to generate spatially dynamic background spectral reflectance/albedo to improve the LAI/fPAR estimates from remote sensing data [16]. Thus, it is important to develop an approach that can generate both spectral and broadband albedos at a fine spatial resolution (e.g., 30–500 m) to satisfy the needs of climate modeling and ecosystem monitoring purposes. Significant impacts of soil moisture content on soil reflectivity have been reported based on laboratory measurements [28,29], ground measurements [8,9,30], and satellite products [31–33]. However, few studies have demonstrated the relationship over a large spatial domain, which is critical for land surface energy balance and hydrological modeling purposes.

The concept of a "soil line", a linear relationship between red and near infrared (NIR) reflectances typically observed over bare soils, has been widely used to discriminate vegetation from soil using spectral information [34–39]. The soil line has been commonly used to develop the vegetation indices, such as the simple ratio (SR), NDVI, Enhanced Vegetation Index (EVI), and Soil-Adjusted Vegetation Index (SAVI), to identify dense vegetation. However, this concept had mostly been applied to extracting information about the vegetation canopy rather than the understory and soil.

The purpose of this study is to build a 500 m bare soil albedo dataset over the contiguous United States (CONUS) on a pixel-basis from multi-year MODIS data based on the soil line feature and to explore the relationship between bare soil albedo and other surface properties such as soil moisture, soil type, and vegetation type. A description of the data and methodology is given in Section 2. Comparison results against Landsat data are presented in Section 3 and are followed by the evaluation of soil albedo with regards to soil moisture content, soil type, and land cover type.

### **2. Materials and Methods**

#### *2.1. MODIS Albedo Anisotropy Products*

The MODIS albedo and bidirectional reflectance distribution function (BRDF) products (MCD43A) [40] are available at a 500 m resolution globally using a 16-day temporal acquisition window. Of the available albedo anisotropy products, this study primarily used three datasets: Spectral/broadband albedos, nadir view corrected surface spectral reflectance, and quality control (QC) flags (overall accuracy and snow flag). Two types of albedos were included in the MODIS products: Black-sky albedo (BSA), also called directional-hemispherical albedo, and white-sky albedo (WSA), also called bi-hemispherical albedo. Datasets covering CONUS were chosen for the 13-year period from 2000 to 2012. Information of the datasets used in this study is summarized in Table 1.

To assist in the removal of possible vegetation, water, snow, and residual cloud shadow contaminated albedo values, the NDVI and normalized difference water index (NDWI) [41] were used, calculated from nadir BRDF-adjusted reflectances (NBAR) for the spectral bands (Equations (1) and (2)).

$$NDVI = \frac{\rho\_{b2} - \rho\_{b1}}{\rho\_{b2} + \rho\_{b1}},\tag{1}$$

$$NDWI = \frac{\rho\_{b2} - \rho\_{b5}}{\rho\_{b2} + \rho\_{b5}},\tag{2}$$

where *ρb*1, *ρb*2, and *ρb*5 are the spectral reflectances for MODIS band 1 (620–670 nm), band 2 (841–876 nm), and band 5 (1230–1250 nm), respectively.

**Table 1.** Information of the datasets used in this study.

## *2.2. Soil Type Map*

Current climate and ecological modeling applications generally use maps of soil type to assign albedo values to certain locations [42] because soil type is believed to be the dominant factor controlling soil reflectivity. The Natural Resources Conservation Service (NRCS) in the U.S. Department of Agriculture (USDA) provides a general soil taxonomy map for the CONUS (Figure 1). In the soil taxonomy map, a dominant soil type is given for each pixel at 2 arcmin resolution (~4 km) from the 12 soil types, including: Alfisols, Andisols, Aridisols, Entisols, Gelisols, Histosols, Inceptisols, Mollisols, Oxisols, Spodosols, Ultisols, and Vertisols [43]. This soil type map was used in the soil albedo derivation methodology described in the following sections.

**Figure 1.** Dominant soil type map over the contiguous United States (CONUS) extracted from U.S. Department of Agriculture (USDA) soil survey data.

#### *2.3. Using the Soil Line to Detect Bare Soil*

In the solar shortwave range, bare soil surfaces normally have small reflectivity in shorter wavelength, with reflectance gradually increasing with wavelength [44,45]. In contrast, a vegetation canopy has very strong absorption in the visible spectrum and back scattering in the near infrared spectrum. The soil line is one of the key and stable spectral signatures of soil and has been examined in many experiments [34–36]. The soil line describes the linear feature (the "envelope") of surface reflectivity in multi-band space over a landscape of a given soil type. For example, the soil line is the lower envelope in the red-NIR space and has been commonly used to identify areas with dense vegetation. However, in practice, the red-NIR soil line can exhibit significant variation because both residual vegetation (leaf litter) and soil moisture may affect the NIR reflectivity greatly. This can cause problems for deriving a universal threshold to separate soil from vegetation, especially when sparse vegetation cover is not available for certain surfaces. In contrast, the green-red based soil line is very stable [35,46]. To verify the green-red soil line feature, surface spectra from Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) and US Geological Survey (USGS) spectral libraries [44,45] were collected and used to identify soil line features with different combinations of spectral albedo and vegetation/soil related indices including NDVI, EVI, SR, NDWI, and Visible Atmospherically Resistant Index (VARI). Of all combinations tested, the green-red band combination could detect soil line with the highest R2 of 0.92, while the others ranged from 0.6 to 0.8.

In this study, thirteen years of MODIS albedo/BRDF data (2000–2012) were used to identify bare soil albedo characteristics over the CONUS using the following procedure. First, the soil line was generated for each of the 12 major soil types in the NRCS classification using MODIS data in 2005. To reduce uncertainties and minimize effects of vegetation and snow, only high quality snow-free albedo values (based on the QC flag in MODIS albedo product) collected between November and March were used in the soil line generation. MODIS NDVI and NDWI products were used to further exclude possible vegetation, snow/ice, water, and residual cloud/shadow pixels. Once the soil line was detected, the second step was to calculate statistics describing the distance of the bare soil albedo from the soil line for each pixel over the whole period 2000–2012. Considering the accuracy of the estimated soil line and the uncertainty in MODIS data products, the soil albedos were considered as candidates if their distance to the soil line was smaller than 20% of the magnitude of their albedo

values. Our sensitivity test showed that the 20% threshold did not introduce any significant difference in the soil albedo climatology results, and, at the same time, it significantly increased the number of samples that were used in the calculation. The bare soil albedos (both spectral and broadband) were generated at a 16-day interval. The third step was to calculate the mean and standard deviation (SDEV) of the 16-day bare soil albedos over the entire thirteen years for each location. Figure 2 illustrates the procedure used to generate the multiyear mean bare soil albedos from the MODIS datasets. In the final bare soil albedo map, the soil albedo represents the average state of the soil condition, regardless of the differences in moisture content and organic matter. The SDEV maps contain information regarding impacts of variation in soil moisture, organic matter, and residual vegetation on the apparent soil reflectivity.

**Figure 2.** Flowchart for mapping bare soil albedo from multiyear Moderate Resolution Imaging Spectroradiometer (MODIS) data based on soil line.

### **3. Results and Discussion**

#### *3.1. Charateristics of Soil Lines Derived from MODIS Albedo*

It has been demonstrated that the visible spectral space is useful in estimating vegetation fraction and indicating crop development stages tested at multiple crop sites with various soil backgrounds [35,46]. Based on field data, the green-red soil line feature has been proven to be a robust method for separating bare soil and vegetated surface conditions [35]. However, this feature has not yet been well documented using long-term satellite data. Figure 3 shows the soil line feature in green-red space from MODIS albedo samples over different land cover types based on MODIS land cover product during 2000–2012 using red and green bands.

In both crop and forest MODIS-derived samples, there was an obvious linear-line feature with relatively high green and red reflectance values in the spectral space (black line shown in Figure 3), which was in accordance with in-situ observations [46]. Based on the observations, this line was occupied mostly by samples in winter when vegetation had not sprouted. Such samples tend to be

bare soil samples with low NDVI and positive NDWI. The variations in surface reflectivity at this stage were likely due to changes of soil moisture [28].

There are several reasons why data obtained in winter are preferable for deriving the soil line, as demonstrated in Figure 3. First, the effect of vegetation is minimized during the fall and winter dormant months. As vegetation grows such as in spring, surface albedos for green and red bands rapidly decrease due to the strong absorption by chlorophyll pigment in leaves, while the red band has a larger decrease rate because chlorophyll absorbance coefficient is much higher in red than in green [47]. Samples were moving downwards in spring when vegetation began to grow and ended up by a short "vegetation line" with red reflectance almost invariant occupied mostly by samples in summer when vegetation density was quite high. During the fall at the onset of vegetation senescence, green and red reflectance increased due to less absorption by the degraded chlorophyll pigments. As such, the samples at this stage were moving upwards from "vegetation line" back to "soil line" (red points in Figure 3). Thus, we can see from Figure 3 that albedos in winter are on the soil line and those in summer are on the vegetation line. For spring and fall, the albedo values lie between these extremes due to partial (sub-pixel) vegetation cover.

**Figure 3.** MODIS white-sky albedo (or diffuse reflectance) for the red and green bands over crop and forest samples in MODIS tile h11v04 during 2000–2012. The left panel shows the albedo values for one pixel for the time period; the right panel shows albedo values for the pixels in the 5-by-5 surrounding spatial window with the same land cover type to that in the left panel.

Second, soil albedo varies with soil moisture. During drier seasons (fall and winter) soil albedo tends to have a higher value, while during wetter seasons (spring and summer), it decreases as soil moisture increases. As soil moisture is usually small in winter (excluding the snow albedo), a small increase/decrease in moisture can cause large decrease/increase in albedo. Thus, a large range of soil albedo would produce robust soil line estimation.

Therefore, the soil line features were estimated by fitting the albedo values obtained in winter after excluding the data with large NDVI (>0.3) and/or negative NDWI to reduce the impacts of dense vegetation, clouds, and snow/ice.

The scatter plots in the left panel in Figure 3 are based on data at a single 500 m pixel; the plots in the right panel include data from the 5-by-5 surrounding pixels with the same land cover type. Because soil albedo varies both seasonally and inter-annually as the averaging area increases, including more surrounding pixels of the same land cover type as shown in the right panel, the slope of the soil line does not change much, but the R2 increases. This procedure was tested over other locations with different land cover types, yielding results similar to those presented in Figure 3.

Soil line characteristics estimated from MODIS data from 2005 are listed in Table 2. The variability in these soil lines indicates that a global unified soil line may not be good enough to extract bare soil albedo. Most of the soil line estimations have an R2 greater than 0.9 except for Alfisols, which demonstrates that the soil type information can support soil line estimations and other soil properties (e.g., soil moisture, organic matter, and surface texture/roughness) may have a secondary impact to the soil line [34].

**Table 2.** Parameters of soil lines estimated for the major soil types over the CONUS

\*Statisticsarebased on 95%confidencelevel.

Following the procedure described in Figure 2, bare soil albedo over the CONUS was first generated on a 16-day interval including black-sky (direct) and white-sky (diffuse) albedos for the seven spectral bands and three broadbands, and then aggregated temporally to generate the climatological mean and SDEV for each of the bands. Statistics for broadband white-sky albedos are presented in Figure 4. Based on visual comparison, a general agreement can be reached in terms of magnitude and spatial pattern between the bare soil shortwave albedo map generated in this study and the data aggregated from MODIS data during 2001–2010 in a recent study [26]. The shortwave albedo for soil is lower in the East and Northwest CONUS (around 0.15~0.18) where dense vegetation is located with a wetter climate and higher in the central and Southwest CONUS from 0.20 to more than 0.40 where much of the land is covered with crops, grass, and bare soil with a drier climate. A more detailed spatial pattern in the soil albedo over the CONUS is provided in this study.

**Figure 4.** *Cont.*

**Figure 4.** Bare soil albedo over the CONUS: Total shortwave (0.3–5.0 μm) mean value (**a**) and standard deviation (SDEV) (**b**); visible (0.3–0.7 μm) mean value (**c**) and SDEV (**d**); and near infrared (NIR) (0.7–5.0 μm) mean value (**e**) and SDEV (**f**).

#### *3.2. Validation of MODIS Bare Soil Albedo*

As bare soil albedo ground measurements are not widely available, validation of the bare soil albedo extracted from MODIS data at 500 m resolution is difficult. In previous studies, bare soil albedo was either directly compared with model inputs or indirectly validated against MODIS albedo products using pure vegetation albedo and vegetation fraction as inputs to calculate soil/vegetation mosaic albedos. In this study, we proposed an approach using finer resolution satellite data to help verify the 500 m MODIS-based bare soil albedo estimations.

Landsat data are available from the USGS at 16-day intervals and at a spatial resolution of 30 m. Though it is usually difficult to find pure bare soil pixels at the MODIS resolution, they are considerably easier to identify at the Landsat resolution. The finer resolution offers a unique capability to observe the bare soil directly, which is otherwise mixed with vegetation at MODIS resolution. In this study, it was assumed that if the Landsat pixels that belong to a MODIS pixel with a 500m-by-500m nominal spatial coverage can be identified as bare soil, the averaged soil albedo value of the Landsat pixels will represent the bare soil albedo for the MODIS pixel. The Landsat shortwave broadband albedo was estimated following the procedure described by He et al. [48] for the atmospheric correction and narrow-to-broadband conversion. Clouds in Landsat data were screened using the Landsat Ecosystem Disturbance Adaptive Processing System (LEDAPS) tool [49].

**Figure 5.** Comparison of 16-day bare soil shortwave albedos derived from MODIS data with estimates from Landsat data.

Potential soil pixels were extracted from the Landsat datasets based on land cover class designated in the National Land Cover Database 2006 (NLCD2006). Only pixels classified as sparse vegetation–including developed, barren, and shrubland–were used [50]. Both NDVI and NDWI were calculated for each Landsat scene to exclude cloud/shadow, water, snow, and vegetation pixels, similar to the MODIS data processing discussed in previous sections. In this study, we used the Landsat scenes (path 042, row 034) centered at 119.0◦W, 37.5◦N for validation, which have a variety of land cover types, including forest, grassland, and cropland. There are four major soil types in this region, including Alfisols, Aridisols, Entisols, and Ultisols. All the available Landsat 5 TM scenes for the period 2005–2007 with nominal cloud coverage less than 30% were used. As the NLCD class and vegetation indices thresholds (NDVI<0.3 same as in Section 3.1) were applied to the Landsat data, it was assumed that the generated Landsat soil albedo can represent the bare soil surfaces with/without vegetation cover in a larger area, e.g., a MODIS pixel. As demonstrated in Figure 5, a general agreement can be found for bare soil shortwave black-sky albedos derived from MODIS and Landsat data with a bias of 0.003, a root-mean-square-error (RMSE) of 0.036, and an R2 of 0.316. There are several possible reasons for the differences between these two datasets found in the comparison: Landsat albedo estimations are instantaneous and thus more sensitive to soil moisture changes; Landsat and MODIS surface reflectance may not fit to 1-to-1 line exactly due to the differences in sensor characteristics and data processing; and Landsat pixels may not match MODIS pixels exactly due to the different registration accuracy and adjacency effects in the coarser resolution data.

#### *3.3. Impacts of Soil Moisture Content on Soil Albedo*

The water content in the top layer of soil is believed to have great impacts on the soil albedo. Previous studies have tried to establish the relationship between soil moisture content and soil reflectivity based on various datasets. Some studies reported that the relationship was linear, while others found an exponential equation provided a better fit (Equation (3)):

$$\mathfrak{a} = A \exp(-B\theta) + \mathbb{C},\tag{3}$$

where *α* is soil albedo and *θ* is soil moisture. *A*, *B*, and *C* are regression coefficients.

Gascoin et al. [9] found that the best fit coefficients (A= 0.31, B =12.7 and C =0.15) with an RMSE of 0.030 using various soil samples from field measurements, and Wang et al. [30] reported a lower value for the B coefficient (B = 3.52, clay soil type). Gascoin et al. [9] also pointed it out that the parameters must be carefully examined before application to other regions.

In this study, a relationship between bare soil albedo and soil moisture was derived using remote sensing data over a large spatial domain. White-sky albedo was used exclusively to minimize the solar zenith impacts on albedo data. In addition, the Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) Level 3 daily soil moisture products were used as estimates of soil moisture in the top ~1 cm of soil averaged over the retrieval footprint [51,52]. The 16-day averaged soil moisture values were generated from the daily AMSR-E products to match the temporal resolution of MODIS albedo during 2002–2011 (due to demise of AMSR-E in 2011) over the CONUS. The MODIS soil albedo data were rescaled to 25 km to match AMSR-E soil moisture products. Comparisons between the soil moisture and broadband white-sky albedos for bare soil in the Southwest U.S. (MODIS tile h08v05) indicate that soil albedo generally decreases with an increase in soil moisture (Figure 6). The rate of decrease for all three broadband albedos are very similar, which suggests that the change of soil moisture is likely the major factor of albedo change and that the soil line approach is resistant to soil moisture changes. Variation in NIR broadband albedo is larger than that in visible and total shortwave albedos, which is likely due to the fact that: (1) NIR albedo is larger than visible, and thus the absolute values are more variant; and (2) soil moisture may cause a larger variation in the longer wavelength part of the NIR spectral domain [34].

The relationship between bare soil albedo and soil moisture on a 16-day interval was found to be relatively stable across different seasons and locations over the CONUS (0.18 < A < 0.24; 9.86 < B < 21.33; 0.12 < C < 0.14; 0.031 < RMSE < 0.037). However, as the range of the 16-day averaged soil moisture variation is relatively small (most soil moisture values are between 0.08 and 0.12 g/cm3), the exponential relationship between these two variables cannot be verified without simultaneous instantaneous small soil moisture values (<0.05 g/cm3) and large bare soil albedos (>0.3). Nevertheless, the relationship found in this study is very close to the results presented in Gascoin et al. [9], considering that their soil moisture samples were taken at a ~5 cm depth from the surface.

Soil moisture is among the most important factors that can affect soil reflectivity, which should be considered in bare soil albedo estimations. However, soil moisture varies both spatially and temporally. Thus, it is difficult to remove the impacts of soil moisture in the final 500 m bare soil albedo map without the support of soil moisture products available at a similar resolution.

**Figure 6.** Relationship between bare soil broadband white-sky albedos and soil skin moisture over MODIS tile h08v05 for January 2002–2011. Red line is the best fit curve; its regression coefficients and RMSEs are shown above.

#### *3.4. Relationship of Bare Soil Albedo with Major Soil Types and Land Cover Types*

Traditional methods directly relate soil albedo with soil types, which are currently used by many land surface models to simplify the parameterization of soil albedo besides correcting the illumination angle effects [20].

The climatological bare soil white-sky albedos for different soil types derived for the period 2000–2012 (Table 3) were compared between each other. The Aridisols were found to have the largest soil albedo values because they tend to be the driest among the 12 major soil types—too dry for mesophytic plants to grow. Entisols, many of which are found in steep and rocky settings, have the second largest albedo values. Since the soil reflectivity is very sensitive to moisture changes under dry conditions, albedos of both Aridisols and Entisols have the largest variations. Among three broadband albedos, the near infrared albedos have the highest mean and SDEV values, and the visible albedos have the lowest mean and SDEV values for each soil type.

**Table 3.** Statistics of broadband soil albedo distribution for each of the major soil types.

**Table 4.** *p*-value of two tailed *t*-test on the soil albedos among different soil types.

\*: *p* < 0.05; \*\*: *p* < 0.01; \*\*\*: *p* < 0.001.

Histosols have the lowest reflectivity, probably because of their high organic matter content. Oxisols and Spodosols also have low albedo values. Possible reasons are that the Oxisols are mainly distributed in tropical and subtropical areas where precipitation increases soil moisture, and the Spodosols are acid soils mainly under forest.

The two tailed *t*-test results shown in Table 4 suggest that differences between soil albedo values are statistically significant for most soil types. However, there are several exceptions. Oxisols are quite inseparable from all the other types. It is also difficult to separate the albedos of Gelisols from those of Alfisols and Entisols. Their small sample size (Oxisols have less than 0.004% of the total samples; Gelisols have 0.147% of the total samples) is likely to have resulted in the difficulty in the separation of their soil albedos because a 4 km soil map grid may contain multiple soil types.

Bare soil albedo can also be linked with the type of vegetation above the ground because that the soil moisture–precipitation feedback [53] and vegetation–soil moisture feedback [54] will determine the relationship between vegetation type and magnitude of soil albedo (because of soil moisture). Table 5 lists the climatological values of bare soil albedo for each of the International Geosphere-Biosphere Programme (IGBP) land cover types from MODIS land cover data. In this study, it was found that bare soil albedo decreases as the vegetation density increases from open shrubland to evergreen needle leaf forest.

**Table 5.** Statistics of broadband soil albedo distribution for each of the land cover classes.

ENF: Evergreen needleleaf forest; EBF: Evergreen broadleaf forest; DNF: Deciduous needleleaf forest; DBF: Deciduous broadleaf forest; MIX: Mixed forest; CSH: Closed shrubland; OSH: Open shrublands; WSV: Woody savannas; SAV: Savannas; GRA: Grasslands; WET: Permanent wetlands; CRO: Croplands; URB: Urban and built-up; CRC: Cropland/natural vegetation mosaic; GLA: Snow and ice; BRN: Barren or sparsely vegetated.

Compared with the statistics of albedo–soil types in Tables 2 and 3, bare soil albedos have smaller within-class SDEVs for the albedo–land cover types, except for open shrublands and barren surface. This suggests that land cover type might be a better indicator to quantify the magnitude of bare soil albedo than soil type for the vegetated areas. For sparsely vegetated areas, the soil type map provides spatial variation that is necessary to characterize the relationship between bare soil albedo and soil type.


**3. Focus on Mast Cells, SARS-CoV-2 and Microbiota**

In the context of the wide variety of cells involved in SARS-CoV-2, two types of cells that are at the forefront of the pathogenesis are mast cells (MCs) and platelets, and a control over them could represent biomarkers and targets at the same time for an interesting therapeutic strategy.

MCs are innate immunity cells present in mucous membranes and connective tissue, strategically located at the interface with the external environment such as the skin, lungs and intestines, where they act as gatekeepers for attack of pathogens [70,88], and they play, themselves, pathogenic roles in many inflammatory responses. MCs organize the inflammatory response and are crucial early participants in responses to viral infection. Once activated, in a very short time MCs release mediators classified as dependent or independent of degranulation. These molecules contribute to inflammation and changes at the site of infection. Mast cells also can be activated by a variety of both bacterial and viral products, and, consequently, to release a very wide spectrum of proinflammatory and immuno-regulatory molecules. In addition, many studies have analysed the ability of MCs to contract common viruses and release molecules such as histamine and leukotrienes [89,90]. MCs are resistant to productive infection with respiratory syncytial virus but have a protective response that includes the production of cytokines and chemokines that promote the recruitment of antiviral effector cells [89]. MCs can be activated directly by active viral infection or by contact with viral particles. Activation of MCs leads to the production of a variety of mediators, including large amounts of interferons (IFNs) by human virus-infected cells. In addition to initiating an antiviral state in neighbouring cells, a storm of chemokines and cytokines promote the local recruitment of effector cells. IFN also acts in an autocrine manner to further promote the production of MCs. The molecules released by MCs also act

by improving lymph nodes hypertrophy. Furthermore, the involvement of local dendritic cells promotes the development of a subsequent acquired immune response [91]. Mast cells can influence T cell proliferation and cytokine production [92]. In addition, MCs produce proteases that are increased in COVID-19 sera and lung districts [93]. Mast cells contain the serine protease ACE2 [94].

MCs play a leading role in many pathophysiological conditions, in which there is a condition of chronic silent inflammation. IL-1β, IL-6 and IL-8 are typical of silent chronic inflammation and MCs are both producers and effectors of these cytokines. Furthermore, MCs are involved in inflammatory responses and psychological stress [70]. MCs are about 2–3% of the immune cellular pool of the lamina propria, and in the muscular and serous layers (3000–25,000 MCs/mm3) [10]. Variations in the number of MCs are observed in the elderly, as with aging there are changes in connective and mucous tissues as well as other changes closely related to CNS disorders, including depression and anxiety [95]. An increase in the number of MCs during aging has also been observed in human organs and organs of other mammals and vertebrate animals.

MCs play a crucial role in host–microbiota communication, as they can help influence microbiota status and host conditions by modifying their activation [10,96]. MCs can contribute to the maintenance of intestinal homeostasis and their activation is linked to a variety of factors, motor abnormalities and dysfunctions of the intestinal epithelial barrier [10,26]. MCs establish functional signalling pathways with the nervous system and nerves in the gut. Their activation induces sensitization of the nerves, and these, in turn, can condition the release of mediators from MCs. This crosstalk is critical in the generation of symptoms or in the pathogenesis of inflammatory disorders [97,98].

MC responses to virus and other pathogens provide excellent tools for modifying local immune responses and could represent an attractive target for COVID-19 treatment, vaccination, and other immunotherapeutic uses. As is known, MCs can be activated by PAMPS through TLRs. Interestingly, MCs have been shown to express the renin–angiotensin system, the angiotensin 2 converting enzyme ectoprotease required for binding of SARS-CoV-2 and serine proteases [70,99]. This could lead to the secretion of proinflammatory mediators in a targeted and selective manner, without release of histamine or tryptase, as has already been described for the release of IL-6 in response to IL-1β from human MC cultures [70].

MCs could be a potential target to control SARS–CoV-2, for example employing known MC stabilizing agents [96]. Interestingly, some specific probiotic strains are able to stabilize MCs, especially *L. rhamnosus* GG [100]. Oral administration with *L. rhamnosus* JB-1 induces inhibition of peritoneal MC degranulation [101].

Relationships established between gut microbiota composition, cytokine storm, and MC activation in SARS-CoV-2 patients suggest that the gut microbiome is extremely involved in the severity of the pathology. Finally, an intestinal dysbiosis condition could then contribute to the condition of feeding those persistent symptoms that characterize the outcomes of SARS-CoV-2.


*AI in Healthcare: Implications for Family Medicine and Primary Care DOI: http://dx.doi.org/10.5772/intechopen.111498*

Family Medicine practitioners operate between extensive delivery systems practitioners such as mental health, home health, and public health. The familiarity with various healthcare stakeholders will enhance AI performance tools.

Because the amount of data we will manage will only increase, there needs to be a strategy both for the here-and-now and beyond. Without this, Primary Care risks becoming incapacitated, subjugated to metrics, and more prone to burnout. Ultimately, AI should be used to enhance the time we spend with patients and complement the Family Medicine experience. NLP aids computers to comprehend, deduce, and use an individual's vernacular. Moreover, AI may unearth material from prior visits, images, labs, and health data to composite them into the right documents so providers can focus on the human-connection [8]. AI chatbots can replicate human dialog, assist individuals in receiving the optimal care, utilizing advanced technology through patient surveillance between visits and provider consultations [3]. Patients suffering from congestive heart failure (CHF) may broadcast their weight through internet-enabled scales, have their diuretic doses titrated, or ensure that their worsening symptoms are being examined by their PCP. Patients may be reminded of health maintenance services like breast and colon cancer screening, provided education for shared decision making, create referrals, book appointment, and organize tests that need to be performed [9]. AI may data mine environmental, EMR, claims, and pharmaceutical data and integrate these to identify and treat high risk patients afflicted with asthma, MI's, and opioid overdoses to aid appropriate management. AI may explore massive amount of data, convey measures, close care gaps, and most importantly allow providers to spend more time with patients.

While these tools are fascinating, they are not yet equipped to being put into practice. They necessitate improvement, investigation, and substantiation. Privacy, malpractice, and overtreatment must all be carefully weighed and dealt with. Without consideration of fitting payment models AI will be imperfect to influence healthcare delivery. **Figure 1** provides a guide for Family Medicine providers on how to get better involved in leveraging this technology.

#### **Figure 1.**

*Steps for family physicians to get involved in artificial intelligence research.*

Family physicians pride themselves on the personal relationships they form with their patients. Computers will outdo physicians when it pertains to the performance of complicated undertakings. Nevertheless, creating and upholding strong relationships, recognizing and handling their intricacies, and eliciting and integrating preferences into medical decisions are difficult for technology to replicate. Humans and computers must complement each-other to enable physicians to spend more time with their patients.