CN113051794A - Method and device for calculating concentration diffusion of impurity elements of quartz and storage medium - Google Patents

Method and device for calculating concentration diffusion of impurity elements of quartz and storage medium Download PDF

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CN113051794A
CN113051794A CN202110270105.9A CN202110270105A CN113051794A CN 113051794 A CN113051794 A CN 113051794A CN 202110270105 A CN202110270105 A CN 202110270105A CN 113051794 A CN113051794 A CN 113051794A
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李育彪
马强
裴振宇
柯春云
雷绍民
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Wuhan University of Technology WUT
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Abstract

The invention relates to a method and a device for calculating the concentration diffusion of impurity elements of quartz and a computer readable storage medium, wherein the method comprises the following steps: acquiring vacancy concentration distribution of impurity elements in quartz, and equating the vacancy concentration distribution to concentration distribution of the impurity elements; determining the boundary concentration of the impurity element, wherein the boundary concentration of the impurity element is used as a boundary condition of a finite difference method; and acquiring the concentration profile distribution of the impurity elements at different diffusion times through the boundary conditions, the finite difference method and the concentration distribution of the impurity elements. The quartz impurity element concentration diffusion calculation method provided by the invention realizes the calculation of the concentration diffusion of the quartz impurity element.

Description

Method and device for calculating concentration diffusion of impurity elements of quartz and storage medium
Technical Field
The invention relates to the technical field of quartz impurity removal, in particular to a method and a device for calculating concentration diffusion of quartz impurity elements and a computer readable storage medium.
Background
The natural quartz resources generally have lattice substitution and gap filling impurities and interfacial ultra-fine mineral inclusion, and the quartz lattice impurities, especially the lattice substitution impurity elements, are difficult to realize effective separation by a conventional mineral separation method, so that the method is a leading-edge technical problem which always troubles the high-purity quartz manufacturing industry in China. Today, no breakthrough progress is made in the key technology for manufacturing high-purity ultrahigh-purity quartz by using natural quartz resources. The reason for this is that the effective separation technology of quartz lattice impurities and the basic theory research thereof are delayed, which results in slow development and unsatisfactory development of new technologies.
The leaching can effectively dissolve and separate impurities on the surface and interface of the quartz, but is difficult to act on lattice substitution and interstitial impurities. Although hydrofluoric acid can break silicon-oxygen bonds and dissolve quartz, the hydrofluoric acid cannot directly act on impurities in quartz crystal grains, and after the impurities which are intensively distributed on the surface and the interface are completely removed, lattice impurities are uniformly distributed in the quartz, so that even if the mixed acid solution containing hydrofluoric acid is continuously leached, the content of residual impurities can hardly be reduced. The content of the impurity element in the quartz can be further reduced only by changing the distribution state of the impurity element so that the concentration of the impurity element on the surface is greater than the concentration of the impurity element in the interior. Diffusion and segregation play a very important role in the process of mineral formation, and if the phenomena of diffusion and segregation can be utilized to make impurity elements in quartz gather on the surface, the concentration of the impurity elements in the quartz can be reduced by leaching and denudating the surface of the quartz. Therefore, obtaining the diffusion state of the quartz impurity element is important for removing the impurity element, and a scheme for calculating the concentration diffusion of the quartz impurity element is lacked in the prior art.
Disclosure of Invention
In view of the above, it is desirable to provide a method and an apparatus for calculating the concentration diffusion of a quartz impurity element, and a computer-readable storage medium, which are used to solve the problem in the prior art that the concentration diffusion of the quartz impurity element cannot be calculated.
The invention provides a method for calculating the concentration diffusion of impurity elements of quartz, which comprises the following steps:
acquiring vacancy concentration distribution of impurity elements in quartz, and equating the vacancy concentration distribution to concentration distribution of the impurity elements;
determining the boundary concentration of the impurity element, wherein the boundary concentration of the impurity element is used as a boundary condition of a finite difference method;
and acquiring the concentration profile distribution of the impurity elements at different diffusion times through the boundary conditions, the finite difference method and the concentration distribution of the impurity elements.
Further, acquiring the vacancy concentration distribution of impurity elements in the quartz specifically comprises the following steps: the vacancy concentration distribution of Al, Na or K elements in the quartz is obtained through a vacancy concentration distribution equation
Figure BDA0002973966960000021
Wherein N is the distance from a far boundary point of the quartz surface, X is the dimensionless distance from the surface, T is the dimensionless time, and T is the diffusion time; c is a dimensionless concentration.
Further, equating the vacancy concentration profile to an impurity element self concentration profile specifically includes: for Al element in quartz, the vacancy concentration distribution is equivalent to the concentration distribution of impurity element by a concentration conversion formula
Figure BDA0002973966960000022
Wherein D isAl,VaIs the diffusion coefficient at vacancy concentration, DAl,AlDiffusion coefficient at self concentration, CvaIs the vacancy concentration.
Further, determining the boundary concentration of the impurity element specifically includes: and carrying out mirror symmetry on the concentration of the impurity elements in the bulk along the boundary of the quartz surface to obtain the boundary concentration of the impurity elements.
Further, the method for obtaining the impurity element concentration profile distribution at different diffusion times through the boundary conditions, the finite difference method and the impurity element concentration distribution per se specifically comprises the following steps:
and determining initial conditions of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity elements according to the initial conditions and the boundary conditions to obtain the concentration profile distribution of the impurity elements at different diffusion times.
Further, the finite difference method corresponds to the equation set of
-rCX-1,T+1+(2+2r)CX,T+1-rCX+1,T+1＝rCX-1,T+(2-2r)CX,T+rCX+1,T
Wherein, r is delta T/(delta X)2T is time and X is distance.
Further, for Al element, determining an initial condition of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity element itself according to the initial condition and the boundary condition to obtain the concentration profile distribution of the impurity element at different diffusion times, specifically including:
initial condition for determining the finite difference method is the surface concentration CVa,X＝0Is 1, bulk internal concentration CVa,X>0Is 0, and the boundary condition is CVa,X＝0Constant is 1, CVa,X＝NIs always 0; the vacancy concentration at each point is C when T is increased in each iteration finite difference calculationVa,T＝t+0.01Calculating the equivalent Al element concentration C from the vacancy concentrationAl′,T＝t+0.01Concentration of Al element CAl,T＝tPlus CAl′,T＝t+0.01Then carrying out finite difference calculation and then subtracting CAl′,T＝t+0.01To obtain the Al element concentration C when T is increasedAl,T＝t+0.01Thereby obtaining the concentration profile distribution of impurity elements at different diffusion times.
Further, for Na and K elements, determining initial conditions of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity element itself according to the initial conditions and the boundary conditions to obtain the concentration profile distribution of the impurity element at different diffusion times, specifically comprising:
initial condition for determining the finite difference method is the surface concentration CVa,X＝0Is 1, bulk internal concentration CVa,X>0Is 0, and the boundary condition is CVa,X＝0Constant is 1, CVa,X＝NIs always 0; the initial condition for the diffusion of Na and K elements is CX≥00.01, the boundary condition is that the surface boundary C is at X-0X＝-0.01＝CX＝0.01The quartz inner boundary being C at X-NX＝N＝CX＝N-0.02And carrying out iterative calculation by using a finite difference method and the concentration distribution of the impurity elements to obtain the concentration profile distribution of the impurity elements at different diffusion times.
The invention also provides a quartz impurity element concentration diffusion calculation device, which comprises a processor and a memory, wherein the memory is stored with a computer program, and when the computer program is executed by the processor, the quartz impurity element concentration diffusion calculation device of any technical scheme is realized.
The present invention also provides a computer-readable storage medium having stored thereon a computer program which, when executed by a processor, implements the method for calculating the diffusion of a concentration of an impurity element in quartz according to any one of the above-mentioned embodiments.
Compared with the prior art, the invention has the beneficial effects that: acquiring vacancy concentration distribution of impurity elements in quartz, and equating the vacancy concentration distribution to concentration distribution of the impurity elements; determining the boundary concentration of the impurity element, wherein the boundary concentration of the impurity element is used as a boundary condition of a finite difference method; acquiring impurity element concentration profile distributions at different diffusion times through the boundary conditions, the finite difference method and the concentration distribution of the impurity element; the calculation of the concentration diffusion of the quartz impurity elements is realized.
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FIG. 1 is a schematic flow chart of a method for calculating the diffusion of the concentration of impurity elements in quartz according to the present invention;
FIG. 2 is a cross-sectional diagram of Al element concentration at different diffusion times according to the present invention;
FIG. 3 shows different scaling factors R provided by the present inventionAUnder the condition, a Na and K element concentration profile map of the surface gasification factor exists.
Detailed Description
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate preferred embodiments of the invention and together with the description, serve to explain the principles of the invention and not to limit the scope of the invention.
Example 1
The embodiment of the invention provides a method for calculating concentration diffusion of impurity elements of quartz, which has a flow schematic diagram, and as shown in fig. 1, the method comprises the following steps:
s1, acquiring vacancy concentration distribution of impurity elements in quartz, and equating the vacancy concentration distribution to concentration distribution of the impurity elements;
s2, determining the boundary concentration of the impurity elements, and taking the boundary concentration of the impurity elements as the boundary condition of the finite difference method;
and S3, acquiring the impurity element concentration profile distribution at different diffusion times through the boundary conditions, the finite difference method and the impurity element concentration distribution.
Preferably, obtaining the vacancy concentration profile of the impurity element in the quartz specifically comprises: obtaining the vacancy concentration distribution of Al, Na or K elements in the quartz by a vacancy concentration distribution equation, wherein the vacancy concentration distribution equation is (1)
Figure BDA0002973966960000051
Wherein N is the distance from a far boundary point of the quartz surface, X is the dimensionless distance from the surface, T is the dimensionless time, and T is the diffusion time; c is a dimensionless concentration.
In one embodiment, the diffusion coefficient of Al element can be expressed as the diffusion coefficient of Al element neglecting impurity metal element with very small concentration according to the effective binary method of the multi-element diffusion system
Figure BDA0002973966960000061
Here, it is assumed that Ca is uniformly distributed, and Si and O elements as the matrix should be uniformly distributed, so that a concentration gradient exists only in the vacant sites, and the formula (2) can be simplified to (3)
Figure BDA0002973966960000062
The diffusion flux of Al in quartz can be expressed as (4)
Figure BDA0002973966960000063
It can be seen that the diffusion flux of Al is only influenced by the diffusion coefficient D resulting from the vacancy concentration differenceAl,VaAnd vacancy concentration CvaThe effect is that the Al diffusion in quartz is mainly determined by the vacancy concentration gradient.
From the formula (2), it is understood that the diffusion of the Al element in the quartz crystal body during the firing is mainly determined by the concentration gradient of the vacancies, and therefore the diffusion of the Al element can be estimated and calculated from the diffusion of the vacancies. The finite difference method can only calculate the diffusion of a single element, so that it is necessary here to convert the vacancy concentration gradient that gives rise to the diffusion coefficient of Al element into an equivalent Al element concentration gradient. According to formula (4) having (5), (6)
JAI＝-DAI,Va(CVa,X＝n-CVa,X＝n-1)＝-DAI,AI(CAI、,X＝n-CAI、,X＝n-1) (5)
Figure BDA0002973966960000064
Wherein, CAI、,X＝n-CAI、,X＝n-1) The concentration gradient of the equivalent Al element of the vacancy concentration gradient is shown, and Al is the equivalent Al element.
Preferably, the equalizing the vacancy concentration profile to the concentration profile of the impurity element itself includes: for Al element in the quartz, the vacancy concentration profile is equivalent to the concentration profile of the impurity element itself by the concentration conversion formula (7)
Figure BDA0002973966960000065
Wherein D isAl,VaIs the diffusion coefficient under the influence of vacancies, DAl,AlDiffusion coefficient at self concentration, CvaIs the vacancy concentration.
In one embodiment, the boundary away from the surface is defined as X ═ N, and
Figure BDA0002973966960000071
under such conditions, the following equation (6) can be simplified to equation (7), and the vacancy concentration at each point is converted to an equivalent Al element concentration.
Converting the vacancy concentration distribution into equivalent Al element concentration distribution according to the formula (7), and adding the equivalent Al element concentration distribution and the Al element concentration to obtain the total Al element concentration as the initial concentration of each calculation. Furthermore, the boundary condition needs to be determined before the calculation of the Al element diffusion.
Preferably, determining the boundary concentration of the impurity element specifically includes: and carrying out mirror symmetry on the concentration of the impurity elements in the bulk along the boundary of the quartz surface to obtain the boundary concentration of the impurity elements.
Preferably, the obtaining of the impurity element concentration profile distributions at different diffusion times by the boundary condition, the finite difference method, and the impurity element concentration distribution itself includes:
and determining initial conditions of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity elements according to the initial conditions and the boundary conditions to obtain the concentration profile distribution of the impurity elements at different diffusion times.
Preferably, the finite difference method corresponds to the equation set (8)
-rCX-1,T+1+(2+2r)CX,T+1-rCX+1,T+1＝rCX-1,T+(2-2r)CX,T+rCX+1,T (8)
Wherein, r is delta T/(delta X)2R is a proportionality coefficient, T is time, and X is distance.
In one embodiment, as the firing time increases, some metal elements near the surface of the quartz may be vaporized or sublimated to leave the surface of the quartz, so that the concentration of some metal elements near the surface of the quartz decreases, thereby forming a concentration gradient between the surface and the inside, or some metal elements may diffuse due to the influence of other components and form a concentration gradient between different positions of the quartz.
The element concentration at the quartz surface is set approximately constant over a certain period of time, while the element concentration at a point in the bulk sufficiently far from the surface, at which point the diffusion proceeds gradually and slowly from the surface to the interior, remains low for a certain period of time after the start, is approximately constant. The two boundary points are used as boundary conditions of diffusion, and a Crank-Nicolson implicit finite difference method can be used for approximately calculating a theoretical diffusion curve of the diffusion curve, as shown in a formula (9).
Figure BDA0002973966960000081
The error function is formula (10)
O{(δT)2+(δX)2} (10)
Wherein X is a dimensionless distance from the surface, i.e. X/X0X is the distance to the surface, x0Is the length of the diffusion region; t is dimensionless time, i.e. Dt/(x)0 2) D is a diffusion coefficient, and t is diffusion time; c is a dimensionless concentration, i.e. C/C0C is concentration, c0Surface concentration is the standard concentration.
Let r be δ T/(δ X)2The formula (9) can be converted into a multivariate nonhomogeneous linear equation system
-rCX-1,T+1+(2+2r)CX,T+1-rCX+1,T+1＝rCX-1,T+(2-2r)CX,T+rCX+1,T
The finite difference method requires knowledge of initial conditions and boundary conditions to make the unknown parameters the same as the equation numbers and to obtain each unknown number by eliminating the variables, whereas only the surface concentration c is known during the sheet source diffusion process of the voids0The number of unknowns is greater thanThe number of the equation set cannot be obtained; however, the diffusion of vacancies proceeds gradually and slowly from the surface to the interior, and as long as the other boundary is far enough away from the surface, the vacancy concentration at this boundary will be low all the time, which can be approximated as 0, under which conditions the initial conditions and the boundary conditions are known and the solution of the equation set can be obtained; take δ T as 0.01, δ X as 0.1, r as δ T/(δ X)21 is ═ 1; when T is 0, the initial concentration of vacancies at each point is shown in table 1.
TABLE 1 initial concentration of vacancies at various points
Figure BDA0002973966960000082
According to the formula (8), the time T is T, and the equation system of the concentration of each point when the distance from the surface to the boundary point is X is N is calculated as
Figure BDA0002973966960000091
After the solution of the T-T time equation set is obtained through calculation, the solution of T-T +0.01 is calculated by using the solution of T-T, and the solution is repeatedly substituted until the solution of the needed T time equation set is obtained; the whole calculation process involves more unknown elements, needs more times of repeated iterative calculation, and has very large calculation amount, so that the calculation is carried out by utilizing a Python programming language and a NumPy library. When the boundary is approximately 0, the farther from the boundary, the smaller the influence, and the more accurate the calculation result. Therefore, when the Python linear inequality is used, in order to make the influence of the boundary approximate to 0 on the result small enough, 51 points with the value range of X of 0-5 are calculated, and only the first 11 calculation results with X of 0-1 are output.
Is provided with CX＝0Constant is 1, CX＝5Constant 0, converting the system of equations into matrix Ax ═ b form has (11), (12)
Figure BDA0002973966960000092
Figure BDA0002973966960000093
Solve () function is used to calculate the solution x of the equation and then assign the concentration in the result to CX＝0.1～CX＝4.9This process is repeated again using while cycle until the desired number of times, whereby the calculation of the diffusion at the concentration of the quartz impurity itself can be realized.
Al element is diffused only in bulk phase in the roasting process and hardly crosses the surface, so that the concentration of Al element in bulk phase can be mirror-symmetrical along the surface boundary, the surface is set as X-0, and when X-0.01 is taken as the boundary, the concentration at the boundary can pass through CX＝-0.01＝CX＝0.01The other boundary, which is far from the surface, is determined as in the case of calculating the voids, and a point X N, which is sufficiently far from the surface, is selected as the boundary point, and the concentration at the boundary can be determined by the boundary condition C required for the finite difference methodX＝N＝CX＝N-0.02Determining that the two boundaries are boundary conditions required by a finite difference method; formula (1) is changed to (13)
Figure BDA0002973966960000101
The matrix Ax ═ b corresponding to the equation set t becomes (14), (15)
Figure BDA0002973966960000102
Figure BDA0002973966960000103
Preferably, for Al element, determining an initial condition of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity element itself according to the initial condition and the boundary condition to obtain the concentration profile distribution of the impurity element at different diffusion times, specifically including:
the initial condition for determining the finite difference method is the surface concentration CVa,X＝0Constant 1, bulk concentration CVa,X>0Constant 0, boundary condition CVa,X＝0Constant is 1, CVa,X＝NIs always 0; the vacancy concentration at each point is C when T is increased in each iteration finite difference calculationVa,T＝t+0.01Calculating the equivalent Al element concentration C from the vacancy concentrationAl′,T＝t+0.01Concentration of Al element CAl,T＝tPlus CAl′,T＝t+0.01Then carrying out finite difference calculation and then subtracting CAl′,T＝t+0.01To obtain the Al element concentration C when T is increasedAl,T＝t+0.01Thereby obtaining the concentration profile distribution of impurity elements at different diffusion times.
In a specific embodiment, after determining the initial condition and the boundary condition of Al element diffusion, the influence of additional calculation parameters generated due to different vacancy and Al element diffusion properties on the diffusion result needs to be considered: the concentration of the vacancy and Al elements may be different, so that the unit concentration C of the vacancy and Al elements may be different, and the ratio is set as Rc＝CAl/Cva(ii) a The diffusion rates of vacancies and Al at unit concentration may differ, i.e., δ T, δ X, where a vacancy δ X is providedvaAnd Al element δ XAlSame, but a vacancy δ TvaAnd Al element delta TAlThe diffusion speed in unit concentration is differentiated by the ratio RT＝δTAl/δTvaIterating 1 vacancy diffusion in the calculation, iterating RTDiffusing secondary Al element; the ratio of the concentration gradient of the vacancy and the concentration gradient of the equivalent Al element generated by the concentration gradient of the vacancy is RD＝DAl,va/DAl,Al。
When the finite difference method is used for calculation, the vacancy concentration distribution is calculated in each iterative calculation process, and the calculation is carried out by taking the value of delta T as 0.01, the value of delta X as 0.1 and the value of r as delta T/(delta X)2Initial condition is surface concentration C ═ 1Va,X＝0Is 1, bulk internal concentration CVa,X>0Is 0; boundary condition is CVa,X＝0Constant is 1, CVa,X＝NIs always 0; when calculating the diffusion of Al element, take delta T as 0.01, delta X as 0.1, r as delta T/(delta X)21 is ═ 1; each iteration calculates each point when T is increased by finite differenceConcentration of vacancies CVa,T＝t+0.01Calculating the equivalent Al element concentration C from the vacancy concentrationAl`,T＝t+0.01Concentration of Al element CAl,T＝tPlus CAl`,T＝t+0.01Then carrying out finite difference calculation and then subtracting CAl`,T＝t+0.01To obtain the Al element concentration C when T is increasedAl,T＝t+0.01The profile of Al element concentration at different diffusion times is shown in FIG. 2.
Preferably, for Na and K elements, determining initial conditions of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity element itself according to the initial conditions and the boundary conditions to obtain the concentration profile distribution of the impurity element at different diffusion times, specifically comprising:
initial condition for determining the finite difference method is the surface concentration CVa,X＝0Is 1, bulk internal concentration CVa,X>0Is 0, and the boundary condition is CVa,X＝0Constant 1, the initial condition for Na and K element diffusion is CX≥00.01, the boundary condition is that the surface boundary C is at X-0X＝-0.01＝CX＝0.01The quartz inner boundary being C at X-NX＝N＝CX＝N-0.02And carrying out iterative calculation by using a finite difference method and the concentration distribution of the impurity elements to obtain the concentration profile distribution of the impurity elements at different diffusion times.
In one embodiment, according to an Effective binding approach treatment method of a multi-element diffusion system, impurity metal elements with very small concentrations are ignored, and the diffusion coefficients of the Na element and the K element can be expressed as (16) and (17)
Figure BDA0002973966960000121
Figure BDA0002973966960000122
After firing, the concentration of Na and K elements near the surface is greater than the internal concentration, so there is a climbing diffusion, while under firing conditions, Na and K elements cannot generate a diffusion tendency from low concentration to high concentration region only by their own concentration difference, so Na and K elements are also affected by other factors in the diffusion process, and Na and K elements may diffuse directly through a ring mechanism, or diffuse in pairs with Al element, or move to the vicinity of Al element in order to balance the local nucleus imbalance in the crystal due to Al element diffusion.
Na and K elements generate concentration gradient due to gasification escape and generate diffusion tendency due to concentration difference, so the diffusion coefficients of the Na elements and the K elements can be simplified into (18) and (19)
Figure BDA0002973966960000131
Figure BDA0002973966960000132
From the formulas (18) and (19), it is understood that, in comparison with the Al element, the diffusion of Na and K elements in the quartz body during firing is influenced by the concentration gradient caused by self-vaporization escape in addition to the vacancy concentration gradient. The evaporation rate of the solid surface can be expressed as (20)
J＝KEvaporation ofAFixing device-KAgglomerationAQi (Qi) (20)
Wherein J is the net evaporation rate of the surface, KEvaporation ofAnd KAgglomerationIs the proportionality constant of the evaporation process and the condensation process, respectively, AFixing deviceAnd AQi (Qi)Is the density of the substance in the gas at and near the surface of the solid, if AQi (Qi)Very small, the agglomeration term in equation (20) can be ignored, and simplified to (21)
J＝KEvaporation ofAFixing device (21)
Under the condition that the density of the substance on the solid surface is proportional to the amount of evaporation, the concentration of an element on the solid surface and the remaining concentration on the solid surface after evaporation per unit time can be represented as (22)
A′Fixing device＝RAAFixing device (22)
Wherein A isFixing deviceAnd A'Fixing deviceThe concentration of an element on the surface of the solid and the residual concentration of the element on the surface of the solid after evaporation per unit time, RAIs a scaling factor.
In one embodiment, vacancy characterization, particularly high temperature in-situ characterization of vacancies during firing, is very difficult, so theoretical analytical calculations are used herein to derive the vacancy diffusion characteristics during firing. The Schottky type vacancies generated in the roasting process are mainly on the surface and interface of the quartz, the vacancies concentrated on the surface are diffused inwards due to the concentration difference between the surface and the internal part of the quartz body, the diffusion of the vacancies in the quartz body can be simplified into the sheet source diffusion in the radial direction from the surface to the internal part of the quartz body, if the generation rate of the vacancies under constant temperature is not changed, the diffusion of the vacancies in the quartz body can be further simplified into the sheet source diffusion with constant surface concentration, and the theoretical diffusion curve can be approximately calculated by using a Crank-Nicolson implicit finite difference method, as shown in the formula (9);
Figure BDA0002973966960000133
an error function of
O{(δT)2+(δX)2}
Wherein X is a dimensionless distance to the surface, i.e. X/X0X is the distance to the surface, x0Is the length of the diffusion region; t is dimensionless time, i.e. Dt/(x)0 2) D is a diffusion coefficient, and t is diffusion time; c is a dimensionless concentration, i.e. C/C0C is concentration, c0Is the surface concentration as the standard concentration. Let r be δ T/(δ X)2The formula (9) can be converted into a multivariate nonhomogeneous linear equation system
-rCX-1,T+1+(2+2r)CX,T+1-rCX+1,T+1＝rCX-1,T+(2-2r)CX,T+rCX+1,T
The finite difference method needs to know initial conditions and boundary conditions to enable unknown parameters to be identical to equation numbers, and obtains each unknown number by eliminating variables, and the sheet source diffusion process of the vacancyOnly the surface concentration c is known0The unknown number is greater than the number of the equation set, and the solution of the equation set cannot be obtained. However, the vacancy diffusion proceeds gradually from the surface to the interior, and as long as the other boundary is sufficiently far from the surface, the vacancy concentration at this boundary will always be low, which can be approximated by 0, for a certain time, under which the initial conditions and the boundary conditions are known, and the solution of the equation set can be obtained. Take δ T as 0.01, δ X as 0.1, r as δ T/(δ X)21. When T is 0, the initial concentration of vacancies at each point is shown in Table 2, and the initial concentration of vacancies at each point is shown in Table 2
Figure BDA0002973966960000141
According to the formula (8), the time T is T, and the equation system of the concentration of each point when the distance from the surface to the boundary point is X is N is calculated as
Figure BDA0002973966960000142
And after the solution of the T-T time equation set is obtained through calculation, the solution of the T-T +0.01 is calculated by using the solution of the T-T, and the solution is repeatedly substituted until the solution of the required T time equation set is obtained. The number of unknown elements involved in the whole calculation process is large, the number of times of repeated iterative calculation is large, and the calculation is carried out by utilizing a Python programming language and a NumPy library. When the boundary is approximately 0, the farther from the boundary, the smaller the influence, and the more accurate the calculation result. Therefore, when the Python solution linear inequality is used, in order to make the influence of the boundary approximate to 0 on the result small enough, 51 points with the value range of X of 0-5 are calculated, and only the first 11 calculation results with X of 0-1 are output. Is provided with CX＝0Constant is 1, CX＝5Constant 0, converting the equation set into a matrix Ax ═ b form
Figure BDA0002973966960000151
Figure BDA0002973966960000152
The solution x of the equation is calculated using the numpy, linear, solution () function, then the concentrations in the result are assigned CX 0.1 to CX 4.9, and the process is repeated using while loop until the desired number of times.
In another embodiment, after the concentration distributions of the Na and K elements under the action of the vacancy concentration gradient are obtained according to the formulas (1) and (7), a surface gasification escape factor can be added into the boundary condition of the Na and K elements diffusion according to the formula (22), and finally the concentration distributions of the Na and K elements in the presence of surface evaporation are obtained. Calculating vacancy concentration distribution in a single iteration process, converting the vacancy concentration distribution into equivalent Al concentration distribution, adding the equivalent Al concentration distribution with actual Na and K element concentration distribution to obtain total Na and K element concentration distribution, calculating the total Na and K element concentration distribution after unit time T according to the total Na and K element concentration distribution, subtracting the equivalent Na and K element concentration distribution to obtain actual Na and K element concentration distribution after unit time, and multiplying the surface concentration of the actual Na and K element concentration distribution by a proportionality coefficient RAAnd obtaining the concentration distribution of Na and K elements added with surface gasification factors after unit time.
When the finite difference method is used for calculation, δ T is 0.01, δ X is 0.1, and r is δ T/(δ X)2The initial condition for vacancy diffusion is surface concentration C ═ 1Va,X＝0Is 1, bulk internal concentration CVa,X>0Is 0, and the boundary condition is CVa,X＝0Constant is 1, CVa,X＝N(ii) a The initial condition for the diffusion of Na and K elements is CX≥00.01, the boundary condition is that the surface boundary C is at X-0X＝-0.01＝CX＝0.01The internal boundary is C at X-NX＝N＝CX＝N-0.02Proportionality constant RDIs 0.5, RTIs 1, RC0.1, diffusion time T0.2, different proportionality coefficients RAThe profile of the concentration profile of Na and K elements with surface gasification factors under the conditions is shown in FIG. 3.
In specific implementation, vacancy concentration gradient of diffusion coefficients of Al, Na and K elements is converted into equivalent concentration gradient of Al, Na and K elements, and the concentration gradient can be represented by (23) and (24) according to formula (4)
JMe＝-DMe,Va(CVa,X＝n-CVa,X＝n-1)＝-DMe,Me(CMe、,X＝n-CMe、,X＝n-1) (23)
Figure BDA0002973966960000161
Wherein C isMe′,X＝n-CMe′,X＝n-1The concentration gradient of Al, Na and K elements equivalent to the concentration gradient of vacancy is set as that the boundary far away from the surface is X-N, and
Figure BDA0002973966960000162
under such conditions, the following equation (24) can be simplified to equation (7), and the vacancy concentration of each dot is converted to equivalent Al, Na, and K element concentrations.
Example 2
An embodiment of the present invention provides a quartz impurity element concentration diffusion calculation apparatus, including a processor and a memory, where the memory stores a computer program, and when the computer program is executed by the processor, the quartz impurity element concentration diffusion calculation apparatus according to embodiment 1 is implemented.
Example 3
The present invention provides a computer-readable storage medium having stored thereon a computer program characterized in that the computer program, when executed by a processor, implements the high purity quartz impurity element concentration diffusion calculation method as described in embodiment 1.
The invention discloses a method, a device and a computer readable storage medium for calculating the concentration diffusion of impurity elements in quartz, wherein the vacancy concentration distribution of the impurity elements in the quartz is obtained and is equivalent to the concentration distribution of the impurity elements; determining the boundary concentration of the impurity element, wherein the boundary concentration of the impurity element is used as a boundary condition of a finite difference method; acquiring impurity element concentration profile distributions at different diffusion times through the boundary conditions, the finite difference method and the concentration distribution of the impurity element; the calculation of the concentration diffusion of the quartz impurity elements is realized. The finite difference method of the technical scheme of the invention constructs a calculation method and a model of diffusion segregation concentration gradient, which shows that the diffusion of quartz crystal lattice substituted Al element follows the diffusion of vacancy concentration gradient leading, and the diffusion of lattice interstitial Na and K elements is mainly influenced by gasification volatilization speed and efficiency.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention.

Claims (10)

1. A method for calculating the concentration diffusion of impurity elements in quartz is characterized by comprising the following steps:
acquiring vacancy concentration distribution of impurity elements in quartz, and equating the vacancy concentration distribution to concentration distribution of the impurity elements;
determining the boundary concentration of the impurity element, wherein the boundary concentration of the impurity element is used as a boundary condition of a finite difference method;
and acquiring the concentration profile distribution of the impurity elements at different diffusion times through the boundary conditions, the finite difference method and the concentration distribution of the impurity elements.
2. The quartz impurity element concentration diffusion calculation method according to claim 1, wherein obtaining a vacancy concentration distribution of an impurity element in quartz specifically includes: the vacancy concentration distribution of Al, Na or K elements in the quartz is obtained through a vacancy concentration distribution equation
Figure FDA0002973966950000011
Wherein N is the distance from a far boundary point of the quartz surface, X is the dimensionless distance from the surface, T is the dimensionless time, and T is the diffusion time; c is a dimensionless concentration.
3. The quartz impurity element concentration diffusion calculation method according to claim 2, wherein equating the vacancy concentration profile to an impurity element self concentration profile specifically includes: for Al element in quartz, the vacancy concentration distribution is equivalent to the concentration distribution of impurity element by a concentration conversion formula
Figure FDA0002973966950000012
Wherein D isAl,VaIs the diffusion coefficient under the influence of vacancies, DAl,AlDiffusion coefficient at self concentration, CvaIs the vacancy concentration.
4. The quartz impurity element concentration diffusion calculation method according to claim 3, wherein determining the impurity element boundary concentration specifically includes: and carrying out mirror symmetry on the concentration of the impurity elements in the bulk along the boundary of the quartz surface to obtain the boundary concentration of the impurity elements.
5. The method for calculating the diffusion of the impurity element concentration of quartz according to claim 4, wherein the step of obtaining the impurity element concentration profile distributions at different diffusion times by using the boundary conditions, the finite difference method and the impurity element concentration distribution itself includes:
and determining initial conditions of a finite difference method, and performing iterative computation by using the finite difference method and the concentration distribution of the impurity elements according to the initial conditions and the boundary conditions to obtain the concentration profile distribution of the impurity elements at different diffusion times.
6. The method according to claim 5, wherein the finite difference method corresponds to an equation set of-rCX-1,T+1+(2+2r)CX,T+1-rCX+1,T+1＝rCX-1,T+(2-2r)CX,T+rCX+1,T
Wherein, r is delta T/(delta X)2T is time, XIs a distance.
7. The method according to claim 6, wherein initial conditions of a finite difference method are determined for the Al element, iterative calculations are performed using the finite difference method and the self concentration distribution of the impurity element according to the initial conditions and the boundary conditions to obtain the impurity element concentration profile distributions at different diffusion times, and the method specifically comprises:
the initial condition for determining the finite difference method is the surface concentration CVa,X＝0Is 1, bulk internal concentration CVa,X>0Is 0; boundary condition is CVa,X＝0Constant is 1, CVa,X＝NConstant 0, and the vacancy concentration at each point is C when T is increased in each iterative finite difference calculationVa,T＝t+0.01Calculating the equivalent Al element concentration C from the vacancy concentrationAl′,T＝t+0.01Concentration of Al element CAl,T＝tPlus CAl′,T＝t+0.01Then carrying out finite difference calculation and then subtracting CAl′,T＝t+0.01Obtaining the Al element concentration C when T is increasedAl,T＝t+0.01Thereby obtaining the concentration profile distribution of impurity elements at different diffusion times.
8. The method according to claim 6, wherein initial conditions of a finite difference method are determined for Na and K elements, and iterative calculations are performed using the finite difference method and the concentration distribution of impurity elements themselves according to the initial conditions and boundary conditions to obtain the impurity element concentration profile distributions at different diffusion times, and the method specifically comprises:
initial condition for determining the finite difference method is the surface concentration CVa,X＝0Is 1, bulk internal concentration CVa,X>0Is 0, and the boundary condition is CVa,X＝0Constant is 1, CVa,X＝NConstant 0, the initial condition for Na and K element diffusion is CX≥00.01, the boundary condition is that the surface boundary C is at X-0X＝-0.01＝CX＝0.01The quartz inner boundary being C at X-NX＝N＝CX＝N-0.02By using finite difference method and self concentration of impurity elementsAnd performing iterative calculation to obtain the concentration profile distribution of the impurity elements in different diffusion times.
9. A quartz impurity element concentration diffusion calculation apparatus comprising a processor and a memory, wherein the memory stores a computer program, and the computer program is executed by the processor to realize the quartz impurity element concentration diffusion calculation apparatus according to any one of claims 1 to 8.
10. A computer-readable storage medium having stored thereon a computer program, wherein the computer program, when executed by a processor, implements the quartz impurity element concentration diffusion calculation method according to any one of claims 1 to 8.
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