JP2864097B2 - Method and apparatus for analyzing thermal diffusivity, biot number and specific heat data in laser flash method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for analyzing thermal diffusivity, biot number and specific heat data in a laser flash method capable of determining the thermal diffusivity and biot number of a sample with high accuracy.

[0002]

2. Description of the Related Art In recent years, the laser flash method has rapidly spread as a method for measuring the thermal diffusivity of a homogeneous material. In this method, a sample surface is irradiated with a heat pulse by a laser, the back surface temperature of the sample is measured using a radiation thermometer or the like, and the thermal diffusivity of the sample is obtained from the data of the temperature response characteristics. The thermal conductivity κ is related to a steady heat flow, whereas the thermal diffusivity is a heat constant related to an unsteady heat flow. When there is an unsteady heat flow inside a one-dimensional material, δT / δt = α ·
It is defined by α according to the equation (δ ² T / δx ² ). T is temperature and x is a one-dimensional coordinate variable. Further, the proportionality constant α in this equation is α = κ / (C _p · ρ) = κ / c. here,
C _p , ρ and c are constant pressure specific heat, density and volume specific heat. Therefore, if the specific heat and the density of the measurement sample are known, the thermal conductivity κ of the sample can be obtained using the thermal diffusivity α.

A general method of analyzing the thermal diffusivity measurement data in the laser flash method is obtained by making the theoretical solution coincide with the measured value data of the measured sample back surface temperature.
That is, assuming the theoretical solution of the sample back surface temperature by the one-dimensional heat conduction equation: T, the maximum rise temperature: T _m , the thermal diffusivity: α, and the Biot number, which is a parameter relating to the heat loss: h, time: t T = T (T _m , α, h, t) using
Exactly given by However, in order to determine the unknowns α, h, and T _m under the condition that the actual temperature response measurement data matches the pattern based on T, the equations include an implicit function including a trigonometric function and an exponential function, and Since the calculation becomes extremely difficult, a method of simplifying the calculation is generally used by using an approximate value for a specific variable and limiting data used for the calculation to only data in a specific area. I have. At present, the t _1/2 method and the area method as described below are known as typical analysis methods of the thermal diffusivity data by such a laser flash method.

[0004] t _1/2 method: α is established when it is assumed that there is no heat loss, using a time t _1/2 until the temperature rises to half the maximum rise temperature T _m on the temperature response curve. ₀ = 1.
The thermal diffusivity α ₀ is calculated according to the expression 36975L ² / (π ² · t _1/2 ). Here, L is the thickness of the measurement sample.
The maximum rise temperature _Tm is obtained by extrapolating the decay curve after the temperature rise to the pulse irradiation time. Then, the relaxation coefficient k of the decay curve (the reciprocal of the time constant τ of the decay curve) and the aforementioned t
_A heat loss correction coefficient is obtained from the product of _1/2 and the heat loss correction coefficient is multiplied by the heat diffusion coefficient α ₀ to obtain a heat diffusion coefficient α.

Area method: A method of calculating a deviation R between an average temperature in a section region set on a temperature response curve and an average temperature obtained from a theoretical formula to obtain a thermal diffusivity α. as the highest temperature increase T _m by extrapolating decay curve to the laser pulse irradiation time of the temperature response after the rising, expressed the Biot's number h as a function of the thermal diffusivity α using the time constant τ of the decay curve, said deviation R is obtained, and then the thermal diffusivity α is calculated by the conditional expression δR / δα = 0 that minimizes the deviation R.

[0006]

However, the above-described method of analyzing the thermal diffusivity in the laser flash method is as follows.
The effect of heat loss on the actual sample surface, or
Conditions such as the laser pulse waveform applied to the sample are not strictly reflected, and the results are obtained using data in a limited area within the measurement data. However, there is a problem that it is difficult to determine the thermal diffusivity and biot number of the sample. The present invention has been made in view of such circumstances, the thermal diffusivity of the measurement sample, the thermal diffusivity in the laser flash method that can determine the biot number and specific heat with high accuracy and efficiency, biot number and An object of the present invention is to provide a method and an apparatus for analyzing specific heat data.

[0007]

According to the present invention, there is provided a semiconductor device comprising:
The method of analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method described is to irradiate a laser flash from the front side of the plate-shaped measurement sample, measure the temperature on the back side, and measure the thermal diffusion of the measurement sample. A method for analyzing the ratio, biot number and specific heat data, wherein the data is set by the following equations (A) and (B) based on the time constants at which the initial values of the biot number and the thermal diffusivity of the measurement sample are measured. A first step of setting a laser pulse waveform together with the laser pulse waveform, biot number,
A second step of obtaining a theoretical temperature value on the back surface of the measurement sample from a heat conduction equation including a thermal diffusivity and a theoretical maximum temperature value as variables; and a theoretical temperature value in time space and a measurement temperature value on the back surface of the measurement sample. Is calculated under the conditions of the following equations (A) and (B) so as to reduce the deviation from
And a fourth step of determining the thermal diffusivity and biot number of the measurement sample by repeating the second to third steps until the deviation, the biot number and the thermal diffusivity are equal to or less than a specified value. And a fifth step of calculating the specific heat after obtaining the maximum theoretical temperature without radiation loss. h ₀ = β ₀ [1 / sin (β ₀ ) −1 / tan (β ₀ )] (A) β ₀ = L / sqrt (α ₀ · τ) (B) h ₀ : biot Number β ₀ : eigenvalue L: thickness of measurement sample α ₀ : thermal diffusivity τ: time constant Further, the method of analyzing thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 2 is described in claim 1.
In the method for analyzing thermal diffusivity, Biot number and specific heat data in the laser flash method according to the above, in place of the deviation between the theoretical temperature value in the time space and the measured temperature value on the back surface of the measurement sample, the theoretical value in the Laplace space is used. Forward square deviation between the temperature value and the measured temperature value on the back surface of the measurement sample,
It is either the squared deviation of the reciprocal or the squared deviation of the logarithm. The method for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 3, irradiates a laser flash from the surface side of the plate-shaped measurement sample,
A method for analyzing the thermal diffusivity, biot number and specific heat data of the measurement sample by measuring the temperature on the back side, wherein each initial value of the biot number and the thermal diffusivity of the measurement sample is set and a laser pulse waveform is set. A first step of setting a theoretical temperature value on the back surface of the measurement sample from a heat conduction equation including the laser pulse waveform, biot number, thermal diffusivity, and a theoretical maximum temperature value as variables. Assuming that one of the thermal diffusivity and the biot number is A and the other is B, a plurality of A's are set near the initial value of A, and for each of the plurality of A's, B that minimizes the deviation is individually obtained, Next, a third step of obtaining the deviation for the combination of the A and the B and updating a new A in a direction that gives a minimum value of the deviation, wherein the deviation, the Biot number and the thermal diffusivity are defined From the second step to the A fourth step of determining the thermal diffusivity and biot number of the measurement sample by repeating the steps up to and including a fifth step of calculating the specific heat after obtaining the maximum theoretical temperature when there is no radiation loss, and The deviation is any one of a forward squared deviation, a reciprocal squared deviation, or a logarithmic squared deviation between the theoretical temperature value in Laplace's space and the measured temperature value on the back surface of the measurement sample.

According to a fourth aspect of the present invention, there is provided a method of analyzing thermal diffusivity, biot number and specific heat data in the laser flash method, wherein the method of analyzing thermal diffusivity, biot number and specific heat data in the laser flash method is provided. In the fifth step, the maximum theoretical temperature in the case where there is no radiation loss is determined by the following equation (C). T _m = ΔT / (A ₀ ∫exp (a ₀ t ′) · f (t ′) dt ′) (C) where And T _m : maximum theoretical temperature when there is no radiation loss ΔT: measurement temperature specification value β ₀ : eigenvalue α: thermal diffusivity L: thickness of measurement sample Thermal diffusivity in the laser flash method according to claim 5,
The method for analyzing biot number and specific heat data is described in claim 2 or 3
In the method for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method described above, the fifth step includes:
Corresponding to the forward squared deviation, the reciprocal squared deviation, or the logarithmic squared deviation, formula (D), formula (E), or formula (F), respectively.
The maximum theoretical temperature when there is no radiation loss is calculated by the formula. T _m : the maximum theoretical temperature in the absence of radiation loss. Here, the deviation is the difference between the measured data of the back surface temperature of the sample actually measured by the laser flash method and the theoretically calculated solution of the temperature response to the laser flash. Define by difference.

[0009]

[Function] Theoretical solution of the back surface temperature of the sample by the one-dimensional heat conduction equation: T is the maximum rise temperature: T _m , the thermal diffusivity: α, the biot number which is a parameter related to heat loss: h, the heat input pulse waveform: f When each (t) is set, it is strictly represented by time: t. The laser according to any one of claims 1 to 5, wherein the thermal diffusivity α, the biot number h, and the maximum temperature rise _Tm are determined under conditions that match the pattern calculated by the theoretical solution with actual temperature response measurement data. thermal diffusivity in the flash method, these three unknowns α is Biot number and analysis methods specific heat data, h, obtained in advance by comparing the deviation between the theoretical solution and the actual temperature response measurement data obtained by setting T _m The value of the unknown is sequentially and efficiently updated by reflecting the relationship data between certain variables, and the value of the unknown that satisfies the measured data of the actual temperature response is asymptotically determined.

In the method for analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method according to claims 1 to 5, each of the initial values of the biot number and the thermal diffusivity of the sample to be measured in the laser flash method as the first step. And a laser pulse waveform for irradiating the measurement sample. The initial value of the thermal diffusivity of the measurement sample may be a value measured by the conventional t _1/2 method, but in the subsequent calculations, the thermal diffusivity asymptotically approaches the true value each time the thermal diffusivity is updated. There is no particular need to set this value. The initial value of the Biot number by the time constant method is set by the time constant τ of the attenuation curve obtained from the temperature response measurement data and the thermal diffusivity α, and the Biot number is sequentially updated by this functional relationship in the subsequent repetitive operations. You. However, when the time constant method is not used, for example, an appropriate biot number is set, and a plurality of biot numbers are set in the vicinity of the initial biot number. Is determined, a deviation is determined for the combination of the thermal diffusivity and the Biot number, a new Biot number initial value is set in a direction that gives the minimum value of the deviation, and by this repetitive calculation, The thermal diffusivity and Biot number that give the minimum value of the deviation can also be obtained. As the pulse waveform, the waveform of the laser input actually irradiated on the sample is measured by a laser pulse detection device, and a representation of the waveform as a function of time t is used in the following calculation.

Next, as a second step, a theoretical temperature value on the back surface of the measurement sample is obtained from a heat conduction equation including the laser pulse waveform, the biot number, the thermal diffusivity, and the maximum temperature rise as variables. The heat conduction equation is
By setting each of the above variables, a basic equation of heat conduction that can strictly determine the temperature response of the back surface of the sample due to the laser pulse is adopted. When applying the Newton method and the time constant method to the third step of updating the biot number and the thermal diffusivity so that the deviation R between the theoretical temperature value and the measured temperature value on the back side of the measurement sample is reduced, , Where one of the thermal conductivity and the biot number is A and the other is B, a conditional expression δR / δB = 0 for minimizing the deviation R by utilizing that A is a function of B via a time constant The solution that satisfies is calculated by Newton's method. As a result, the updated values A _new , B of the B giving the minimum value of the deviation R
The values of _new and R (B _new ) are obtained. When the Newton method and the time constant method were applied to the third step, the minimum value of the deviation R was determined by utilizing the fact that the thermal diffusivity α and the Biot number h had a unique relationship via the time constant. However, in this method, the time constant needs to be obtained with high accuracy, and a data amount that is nearly 10 times the characteristic time is required. On the other hand, as a method of directly obtaining the minimum value of the deviation R (hereinafter, referred to as a direct method), the function structure of the deviation R and its variables h and α is grasped in advance, and based on the grasped function structure. There is a method of setting an appropriate update value and reaching the minimum value of the deviation R, whereby a highly accurate analysis can be performed even when the data amount is small. For example, for a plurality of biot numbers, δR
/ Δα = 0 is determined, a deviation R is calculated for each combination of α and h, and the next Biot number is set in a direction that gives a minimum value of the deviation R, and the above calculation is performed. Is repeated, it is possible to reach the minimum value of the deviation R, and the required thermal diffusivity α and the Biot number h are given as a set of the thermal diffusivity α and the Biot number h that give this minimum value.

In a fourth step, the steps from the second step to the third step are repeated until the value of the deviation R, the thermal diffusivity α or the Biot number h becomes equal to or less than a specified value of a level that is not substantially updated. Thus, the thermal diffusivity α and the biot number h of the measurement sample are finally determined. From the thermal diffusivity α, Biot number h, and the temperature response measurement data of the laser flash, the maximum rise temperature without radiation loss is obtained, and the data of the energy of the laser pulse absorbed by the sample and the sample weight are combined with C = The specific heat of the sample can be obtained from the formula Q / (VρT _m ). Here, Q is the laser pulse energy absorbed by the sample, V is the volume of the sample, and C is the specific heat of the sample. Although the heat transfer equation can use the theoretical solution of the temperature response in the one-dimensional heat conduction basic equation in time space, the calculation is simplified by using the temperature response theoretical solution of the one-dimensional heat conduction basic equation in Laplace space. As well as being able to use measurement data in a wide range, it is possible to analyze thermal diffusivity data with higher accuracy. Here, the deviation R is calculated by taking the average temperature or the sum of the measured temperatures in the section on the actual measurement data to simplify the calculation,
It can be the difference from the average temperature or the sum of the theoretical temperatures obtained from the theoretical solution in the corresponding section, and the forward squared deviation, the reciprocal squared deviation, or the logarithmic squared deviation can be used. it can. The Biot number h when the Newton method and the time constant method are applied is given by an equation set from the time constant τ and the thermal diffusivity α of the temperature decay curve on the back of the measurement sample,
By relating the biot number h as a function of the thermal diffusivity α and the time constant τ, it is possible to continuously update the biot number h based on the newly updated α _new. it can.

The laser pulse generator generates a necessary heat pulse on the surface of the sample to be measured, and measures the response temperature on the back surface of the sample by a temperature measuring device such as a radiation thermometer having a short response time. When measuring the back surface temperature of the sample with the radiation thermometer, it is desirable to attach the sample so that the condition of black body radiation is satisfied. The laser pulse detection device detects the waveform of a laser pulse actually applied to the sample surface. The computer has arithmetic means for determining the thermal diffusivity and the biot number based on the signals from the temperature measuring device and the laser pulse detecting device. The output device connected to the computer has the thermal diffusivity and the like. The calculation result is displayed. The calculation of the thermal diffusivity is performed in either the case where the time is represented in the time space represented by the time t and the coordinate variable x in one dimension, or the case where the one-dimensional heat conduction equation is subjected to the Laplace transform, that is, the case where the thermal diffusivity is treated in the Laplace space. The present invention can also be applied to this.

Hereinafter, the basic concept of the theoretical solution of the one-dimensional heat conduction equation in the time space and the Laplace space will be described. The basic equation of heat conduction in one dimension is (1)
It is represented by an expression. Assuming that the heat input is a δ function type, that is, Qδ (x) giving a finite value with a time width of 0, the initial condition is expressed by equation (2) using the δ function. Further, the solution in the case where the boundary conditions are expressed by equations (3) and (4) is given by equation (5). That is, the solution in the time space of the one-dimensional heat conduction equation is represented by a series solution using β _n as an eigenvalue. Here, the eigenvalue β _n satisfies the expression (6), and is an infinite number sequence determined by the values of the biot numbers h ₀ and h ₁ on the front and back surfaces of the measurement sample.

Furthermore, the heat input is an arbitrary time function Qf (t)
Is given by the time function f
When (t) is normalized by time t, that is, assuming that the equation (7) is satisfied, the above-mentioned temperature response (5) when the heat input is a δ function form is expressed as Tδ (x, t). , Based on this, is given by equation (8). Expressions (9) to (13) are further expanded from these expressions, which are general expressions of the heat loss condition expressed by the Biot number and the temperature response of the laser pulse when the arbitrary laser pulse waveform condition is included. It is a basic formula that gives a solution.

[0016]

(Equation 1)

Next, the solution of the one-dimensional heat conduction equation in Laplace space to the theoretical temperature response will be described. The boundary conditions including the one-dimensional heat conduction basic expression, the initial condition, and the δ function δ (t) are given by Expressions (14) to (17), respectively.

[0018]

(Equation 2)

These are Laplace-transformed (18), (1)
9) and (20) are obtained. The general solution (21) including the integration constants A and B is obtained by solving the formula (18). This is differentiated, and the integral constants A and B are obtained by substituting into equations (19) and (20) to be substituted into equation (21). ) Get the equation. Next, the temperature response in the Laplace space when the heat input is given by the general time function Qf (t) is obtained. Here, the time function f (t) uses a function standardized by time, and satisfies the expression (23). The temperature response in the case where the heat input is given by an arbitrary time function is obtained by re-calculating the temperature response solution in the case where the heat input shown in the equation (22) is a δ function and f (p) is Tδ (x, p). By the expression described as (2
It is given by equation 5). The general solution of the temperature response in the Laplace space in which this is expanded is shown by the equations (26) to (29).

[0020]

(Equation 3)

Here, the eigenvalue of the one-dimensional heat conduction equation will be considered. Eigenvalue β determined by equation (6)
_n is given by equation (30) when the number of bios on the front surface and the back surface of the measurement sample surface is equal. Then, when the biot number is h <1, Cape & Lehman expresses (30) as (3
1) and (32) indicate that the approximation can be made. In the following, it was confirmed how the eigenvalue β _n has a quantitative relationship with the Biot number, and the applicable range of the approximate expressions (31) and (32) was examined. 5 and 6 show the relationship between the biot number and the eigenvalue β _n for n = 0 to 7, and FIGS.
FIG. 13 is a diagram illustrating calculation accuracy of approximate expressions (31) and (32) with respect to No. 5; In the approximation expressions given by the expressions (31) and (32), the calculation error increases as n decreases and the biot number increases. The precision of the eigenvalue in the Biot number h to 1 is n =
1 is the worst, about 4%. When it is desired to obtain the eigenvalue with a calculation accuracy of 1% or less using this approximate expression, it is understood that it is necessary to limit the range to h <0.1. In the present invention, this approximation formula is not used, and the eigenvalue obtained by the Newton method using the approximation eigenvalue given by the formulas (31) and (32) as an initial value and using the formula (33) as a deviation is adopted.

[0022]

(Equation 4)

In the method for analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method according to the first to fifth aspects, the pulse waveform and the heat loss conditions in the laser flash method are simultaneously and accurately reflected in the calculation. A highly accurate thermal diffusivity can be efficiently obtained. Then, in the theoretical formula of the temperature response on the back surface of the sample, starting from the initial values of the unknown Biot number, thermal diffusivity and maximum temperature rise, the temperature response data asymptotically measured by repeating the calculation is satisfied. The values of the thermal diffusivity and the Biot number are obtained, the highest temperature is obtained from the obtained thermal diffusivity and the Biot number, and the specific heat of the sample is obtained by combining with the data of the laser pulse energy absorbed by the sample. In particular, the thermal diffusivity in the laser flash method according to claim 1, the method of analysis Biot number and specific heat data for the use of temperature-responsive theoretical solution of one-dimensional heat conduction basic equation in the time-space, conventional t _1/2 method The use of a simple numerical value is possible, for example, the value of the thermal diffusivity can be used as an initial value.
In the method for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method according to claims 1 and 2, the biot number is set from the time constant of the temperature decay curve on the back surface of the measurement sample and the value of the thermal diffusivity. Since both values asymptotically approach the desired value while being updated, the analysis accuracy can be further improved. In the method for analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 2 or 3, the calculation is simplified because the heat conduction equation and the measured data are handled in the Laplace-transformed space. In addition, the accuracy of the thermal diffusivity analysis can be improved by reflecting a wide range of data with less restrictions on the setting of the measurement data area in the spatiotemporal method. The method of analyzing the thermal diffusivity, biot number and specific heat data in the same laser flash method,
Since the squared deviation is used as the deviation, the result of the calculation is highly reliable and always valid. The method of analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method uses the inverse square deviation, so that the calculation results are highly reliable and always valid. Thermal diffusivity in the laser flash method,
Since the analysis method of the Biot number and the specific heat data uses the square deviation of the logarithm, data with little error can be handled in a wide range and an accurate result can be obtained. In the method for analyzing the thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 3, the third step of updating the biot number and the thermal diffusivity so as to reduce the deviation includes, for example, the initial value of the biot number. A plurality of Biot numbers are set in the vicinity, a thermal diffusivity that minimizes the deviation with respect to the plurality of Biot numbers is determined, a deviation is determined with respect to a combination of the thermal diffusivity and the Biot number, and the deviation is calculated. A new biot number initial value is set in the direction to give the local minimum value, and the thermal diffusivity and biot number that give the local minimum value of the deviation are obtained by this repetitive calculation. , The biot number can be analyzed. Alternatively, a plurality of thermal diffusivities are set in the vicinity of the initial value of the thermal diffusivity, and a Biot number that minimizes a deviation with respect to the thermal diffusivities is determined,
It is also possible to obtain a deviation with respect to the combination of the thermal diffusivity and the Biot number, and update the numerical value in a direction that gives the minimum value of the deviation.

[0024]

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram showing an embodiment of the present invention; FIG. 1 is a block diagram of an apparatus for analyzing thermal diffusivity, biot number and specific heat data in a laser flash method according to one embodiment of the present invention, FIG. 2 is a flow diagram of a time-space analysis method of the data, and FIG. FIG. 4 is a flow chart of a Laplace space time constant analysis method of the same data, FIG. 4 is a flow chart of a Laplace space direct analysis method of the same data, FIG. 5 is a diagram showing the relationship between the biot number and the eigenvalue β _n, and FIG. FIG. 7 is a diagram showing the relationship between the number and the eigenvalue β _n , FIG. 7 is a diagram showing the relationship between the biot number and the calculation accuracy of the eigenvalue β _n , FIG. 8 is a diagram showing the relationship between the biot number and the calculation accuracy of the eigen value β _n , FIG. 9 is a diagram showing the structure of the squared deviation according to the equation (70) in the case of the forward squared deviation, FIG. 10 is a diagram showing the structure of the squared deviation according to the equation (71) in the case of the forward squared deviation, and FIG. FIG. 12 is a diagram showing the structure of the squared deviation according to the equation (74) in the case of the inverse squared deviation. FIG. FIG. 13 is a bird's-eye view showing the structure of the square deviation R according to equation (75), FIG. 13 is a bird's-eye view showing the structure of the square deviation R, and FIG. 14 is a diagram showing the projection of the square deviation R groove on the α-h plane. FIG. 15 is a diagram showing the mapping of the square deviation R groove on the R-α surface, FIG. 16 is a diagram showing the mapping of the square deviation R groove on the Rh surface, and FIG. 17 is a Laplace space. Fig. 18 is a diagram showing the calculation error in the time constant analysis method using the square deviation shown in the equation (56) with respect to the ratio between the characteristic time and the pulse width. Fig. 18 shows the equation (60) based on the logarithm in the Laplace space. FIG. 19 shows the calculation error in the time constant analysis method using the squared deviation shown in FIG. 19 with respect to the ratio between the characteristic time and the pulse width. FIG. 19 shows the squared deviation and the inverse squared deviation in the same Laplace space. FIG. 20 shows the calculation error in the direct analysis method with respect to the ratio between the characteristic time and the pulse width. FIG. 20 shows the calculation by the Laplace space direct analysis method. FIG. 21 shows an error with respect to a data amount in units of characteristic time, and FIG. 21 shows a calculation error when an average temperature difference in a specific time region is used as a deviation in time space with respect to a ratio between characteristic time and pulse width. FIG. 22 is a diagram showing pt _{min at} which the thermal diffusivity in Laplace space direct analysis can be calculated with better accuracy than 1% with respect to a data amount in units of characteristic time, and FIG. 23 is a diagram showing Laplace space. In the direct analysis method, when the data amount per unit of characteristic time is 10, the thermal diffusivity becomes 1
FIG. 9 is a diagram showing pt _min that can be calculated with good accuracy of not more than% with respect to the ratio between the characteristic time and the pulse width.

An analysis method in the case where the pulse heat input applied to the sample has a temporal width and there is a heat radiation loss proportional to the temperature from the sample surface will be developed in the time space and the Laplace space. In this analysis method, the analysis calculation was performed assuming that the following conditions were satisfied. There is no intensity distribution of the heat input pulse on the sample surface, and one-dimensional heat conduction is established. The temperature of the sample to be measured is the temperature of the back surface of the sample. The number of bios on the front and back surfaces of the sample is equal. The sample is uniform and the thermal diffusivity is constant during the measurement. An apparatus for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method shown in FIG. 1 will be described. The thermal diffusivity and the like are measured using a thin plate as a measurement sample using a laser pulse generator. The laser pulse generator can generate a pulse with an arbitrary waveform if necessary.Selecting a special waveform according to the range of the physical property value of the measurement sample and the method of analyzing the thermal diffusivity simplifies the calculation or makes it possible to specify a specific area. Measurement accuracy can be improved. The temperature response on the back of the sample is measured by a temperature measuring device such as a radiation thermometer. When measuring a sample with a radiation thermometer, the sample is mounted so that black body radiation conditions are established as much as possible. For example, a tubular body having the same shape as the outer shape of the sample is attached to the back side of the sample receiving the laser pulse, and the temperature response of the back side of the sample is measured by the radiation thermometer through the tubular body. However, when the specific heat is not measured, the black body furnace conditions are unnecessary. The waveform signal of the laser pulse applied to the measurement sample is captured by a laser pulse detection device via a half mirror 11 provided between the measurement sample and the laser pulse generator. Further, it is also possible to measure a pulse waveform by measuring leaked light in the middle of the optical path. Then, the signal data from the temperature measuring device and the laser pulse detecting device are taken into a computer, a calculation process for determining a thermal diffusivity and a biot number is performed based on the signal data, and the data and the calculation result are connected to a computer. The whole was configured so that it could be displayed on the device. Hereinafter, the spatio-temporal method according to the first embodiment will be described in detail.

FIG. 2 is a flow chart of a time-space analysis method of the thermal diffusivity and biot number data in the laser flash method according to one embodiment of the present invention. As shown in FIG. 2, the following analysis is performed based on the temperature data and pulse data on the back surface of the measurement sample. The theoretical solution of the back surface temperature of the sample in consideration of the waveform condition of the laser pulse and the Biot number which is a dimensionless number representing the heat loss of the front surface of the sample is given by Expression (9). This is modified and used as in equation (34) for convenience in the following development. Here, A _n and a _n (35), (3
6) The series given by the equations.

[0027]

(Equation 5)

Β _n is an eigenvalue, and tan (30)
The root of (β _n ) = 2hβ _n / (β _n ² −h ² ). First, the initial value of the thermal diffusivity α ₀ is determined using the conventional t _1/2 method, assuming that there is no heat radiation loss. Using the temperature data on the back side of the measurement sample, a time t _1/2 from the center of gravity of the pulse heat input to the half of the maximum temperature rise was determined, and from this value, α ₀ = 1.336975 L ² / (π ² · t _{1 / 2} ) Calculate the initial value of the thermal diffusivity α ₀ using the equation. Alternatively, an appropriate initial value of the thermal diffusivity may be set. Here, the time origin was set at the position of the center of gravity of the laser pulse waveform.

Next, the characteristic time t ₀ representing the time scale of the heat conduction phenomenon is represented by t ₀ = L ² / (π ² · α
₀ ), t = mt ₀ , where m is a positive constant.
In a time domain satisfying t ₀ (m is usually set to a value of 8 or more), the second term or less (n ≧
Since 1) is so small that it can be ignored compared to the first term, it can be approximated by Expression (37). Accordingly, in the time domain of t >> t ₀ where Expression (37) holds, the logarithm of the temperature data is plotted against time, and the time constant τ can be obtained from the slope. The time constant τ is τ = 1 / a
₀ = L ² / (β ₀ ² · α ₀ ), and the initial value of the biot number h
₀ is obtained by a formula (38) obtained by modifying the formula (30) by using the time constant τ and β ₀ which is a function of the initial value of the thermal diffusivity α ₀ .

[0030]

(Equation 6)

Using the time constant τ obtained above, the measured temperature data is extrapolated to the time origin (t = 0), and the temperature data is normalized with the value to obtain the equation (40). After normalizing the measured temperature data according to equation (40), a specific time region (t
_The average temperature is determined by ₃ ≦ t ₄ ), and this value is defined as sum.
This value is expressed by Expression (41) using the left side of the above expression.

[0032]

(Equation 7)

Using the theoretical temperature equation (40), a theoretical average temperature is obtained in the same time domain as the measured temperature data, and this value is defined as thtav. Here, as the heat input pulse function f (t) used for thav, a normalized measurement pulse waveform is used. thtav is represented by equation (42). Here, B _n is given by Expression (43). Here, assuming that the thermal diffusivity α is α = α ₀ · x, thtav is rewritten as a function of x as in equation (44).

[0034]

(Equation 8)

The difference between the theoretical average temperature thtav in a specific time region and the average sum of the measured temperatures is defined as R (x), and x that makes this value 0 is obtained by Newton's method. The thermal diffusivity α is given by α ₀ x. The equations for updating the thermal diffusivity by the Newton method are shown in the following equations (45) to (49).

[0036]

(Equation 9)

The next time a constant expression that relates the thermal diffusivity and eigenvalues (39) to by substituting the updated thermal diffusivity alpha _{new new} updated eigenvalues beta _0, the equation (38) using the eigenvalues beta ₀ To update the number of bios. Hereinafter, this procedure is repeated until the difference between the theoretical average temperature thtav in the specific time region and the average sum of the measured temperatures becomes equal to or less than a specified value, and the finally obtained thermal diffusivity α and the value of the Biot number are obtained. The thermal diffusivity and biot number of the measurement data are used. The specific heat is determined as follows. The maximum temperature rise is determined from the finally obtained thermal diffusivity, biot number and standard value of the measured temperature.
The maximum rise temperature T _m is given by ΔT as the measured temperature specification value.
It is given by the following equation from the denominator of the equation (40). T _m = ΔT / (A ₀ ∫exp (a ₀ t ′) · f (t ′)
dt ′) Further, the specific heat C of the sample is given by the formula C = Q / (wT _m ) using the maximum temperature rise T _m , the sample weight w, and the laser pulse energy Q absorbed by the sample. It absorbed laser pulse energy to the sample specific heat measurement sample and the absorption rate was set such that the same is determined by the maximum temperature increase value T _m for a known standard sample. In the above, the thermal diffusivity was treated as an independent variable.
A data analysis method in which the thermal diffusivity is treated as a function of the Biot number via a time constant also holds. Although the deviation R is used as a criterion in the flow of FIG. 2, a criterion using the thermal diffusivity shown in FIG. 3 or a similar criterion using the biot number may be used. In the time constant calculation area, the characteristic time is determined as a unit of time, but if the characteristic time using the initial value of the thermal diffusivity α based on the t _1/2 method is used, the characteristic time is usually affected by radiation loss, This is a small value, which deviates from the target time constant calculation area. For this reason, by calculating the characteristic time again using the thermal diffusivity once calculated, the time constant can be calculated in the target time region, and the time constant with a predetermined accuracy can be obtained. However, if the accuracy of the time constant does not matter, if m is set to have a sufficient margin to satisfy mt ₀ >> t ₀ using the characteristic time based on the thermal diffusivity α initial value based on the t _1/2 method, This repetition is unnecessary. For example, to determine the thermal diffusivity α with an accuracy better than 1%, m> 8
Preferably, m is set to about 15 to 20.
FIG. 2 shows a flow in which the characteristic time is calculated again based on the thermal diffusivity once calculated and can be repeatedly calculated by the time constant method. FIG. 21 shows the calculation errors of the thermal diffusivity α and the Biot number h when the Biot number is 0.10, with and without the correction of the pulse waveform. Even in a region where the characteristic time / pulse width is small, the thermal diffusivity α and the Biot number h can be analyzed with high accuracy. On the other hand, in the case where the pulse waveform is not taken into account, it can be seen that the accuracy gradually deteriorates when the characteristic time / pulse width is smaller than 3.

Next, a method of analyzing a time constant in a Laplace space according to a second embodiment shown in FIG. 3 will be described. In this method, in order to analyze the thermal diffusivity and the Biot number in Laplace space, both the measured data and the theoretical solution are subjected to the Laplace transform, and the analysis is performed using the least squares method, the Newton method, and the time constant method. As an example of the squared deviation, a squared deviation of the Laplace transform of the measured data and the theoretical solution and a squared deviation of the logarithm of each of them (hereinafter referred to as a logarithmic method) will be described. The biot number is determined by the time constant method (determined from the thermal diffusivity using the eigenvalue equation and the time constant equation in the time space). This analysis is performed in a real space where the Laplace variable is positive.

An outline of a method of analyzing the thermal diffusivity, biot number and specific heat data in the Laplace space in the laser flash method shown in FIG. 3 will be described below. Calculation of the initial value of thermal diffusivity α ₀ : The time required to rise from the center of gravity of pulse heat input to half the maximum temperature is calculated according to the usual t _1/2 method, and the initial value α of thermal diffusivity α is obtained from this value. Calculate ₀ . Alternatively, an appropriate initial value of the thermal diffusivity may be set. Calculation of time constant τ: In the time domain in which sufficient time has elapsed after pulse heating and only the first term of the time-space theoretical solution of the sample backside temperature is approximated, the time constant τ is obtained from the decay curve of the measured sample temperature. . Calculation of Biot number initial value h ₀ : An initial value of the Biot number is obtained from the thermal diffusivity initial value α ₀ , the time constant τ, and the eigenvalue formula. Calculation of thermal diffusivity by Newton's method: The thermal diffusivity deviation Δα is determined so that the square deviation between the Laplace space theoretical temperature using the measurement pulse and the Laplace-transformed measured temperature becomes zero. The calculation is repeated until Δα / α becomes sufficiently small. Iterative calculation is performed by setting α _new = α + Δα, and this α
_The biot number h is updated by _new , the time constant expression, and the eigenvalue expression. (Time constant method) In addition, the maximum temperature rise is determined from the finally obtained thermal diffusivity and Biot number, and the specific heat is determined from the maximum temperature rise.

In the flowchart shown in FIG. 3, the calculation of the initial value of the thermal diffusivity α ₀ , the calculation of the time constant τ, and the calculation of the Biot number based on the time constant are the same as those in the above-described time-space analysis method. In the following, a thermal diffusivity analysis method using the least squares method and the Newton method will be described. The Laplace transform of the measured data is performed according to the equation (50) using the measured data as E (t). Here, t _data is a measurement data capture time. For the temperature data and the pulse data, the t _data is commonly used, and the Laplace transform variable p of both data is used.
Is set according to equation (51). Where pt _min , p
t _max is the minimum value and the maximum value of pt _data , and n Laplace variables are set within this range. The degree of this approximation generally depends on the product of p and t _data , and pt _data ≧ 8
And if the value is too large, the result will be evaluated with emphasis on only the initial data of the temperature change.
It is said that a Laplace variable should be selected in the range of 8 ≦ pt _data ≦ 12, but it is important to grasp the function structure between the variables and select an appropriate value. The Laplace transform of the measurement data may be performed once in a series of data processing. In the flow shown in FIG. 3, the process is in a repetitive loop, but if Laplace transform has been completed, this step is omitted. Alternatively, the Laplace transform of the measurement data may be performed before entering this loop. In the flow of FIG. 3, the calculation region is set again using the characteristic time t ₀ calculated based on the thermal diffusivity α obtained in the same manner as the flow of FIG. 2, and the time constant is calculated again. It is possible flow. Also in this case, if the data satisfies m> 8 (for example, m = 15 to 20), the thermal diffusivity α and the biot number h can be obtained with high accuracy without repeating the above.

[0041]

(Equation 10)

The theoretical solution in the Laplace space uses the functions (52) and (53) which give the temperature response of the back surface of the sample using the above-mentioned equation (26). Here, the Laplace variable sets a discrete value, and p _j (j = 1, 2,...)
k). Further, f (p) in this theoretical solution is obtained by Laplace transform of the measured pulse data. (5
Uj and gj in the expression 3) are given by the expressions (54) and (55).

[0043]

[Equation 11]

Calculations of the thermal diffusivity α and the Biot number h by the least squares method and the Newton method are shown below. Using the Laplace transform measurement data equation (50) and the theoretical data equation (52), square deviation R using discrete Laplace variables as parameters
Ask for. (50), (52)
The Laplace space method using the equation as it is and the logarithmic method using the logarithm are formulated. Where E (p _j )
= E _j . In the Laplace space method, the squared deviation R is obtained by the equation (56) using the measured data and the theoretical data, and the partial differentiation is performed with the independent variables T _m and α.
Is the value to be obtained. From δR / δT _m = 0,
Equation (57) is obtained, and Equation (57) is substituted for δR / δα = 0 to obtain Equation (58). The thermal diffusivity α satisfying the equation (58) is determined by the Newton method using the thermal diffusivity determined by the t _1/2 method or the initial value of the biot number determined by the time constant method. First, Δα is obtained using Expression (59). Therefore, the thermal diffusivity α is α _new = α ₀
+ Δα. Update the number of Biot's this α _new
By using the time constant method in the time space. The above procedure is repeated, and the values obtained when Δα / α becomes sufficiently small are the thermal diffusivity and the biot number to be obtained. Further, the specific heat C of the sample is obtained by substituting the finally obtained thermal diffusivity, Biot number and Laplace-transformed measurement data into the equation (57), the maximum temperature rise _Tm , the sample weight w, and the sample weight. Using the absorbed laser pulse energy Q, C = Q / (wT _m ). It absorbed laser pulse energy to the sample specific heat measurement sample and the absorption rate was set such that the same is determined by the maximum temperature increase value T _m for a known standard sample. Further, in FIG. 3, the determination criterion is applied to the difference between before and after the update of the thermal diffusivity value. However, R <ε may be set using the same equation using the biot number or the deviation R.
In this case, ε corresponds to the specified value.

[0045]

(Equation 12)

Next, the logarithmic method will be described. Using the logarithm of the measured data and the theoretical data, the square deviation R of the equation (60)
Ask for. Here, partial differentiation is performed with the independent variables T _m and α, and the respective equations are set to 0 to obtain the values. That is, Expression (61) is obtained from δR / δT _m = 0.
Therefore, T _m is given by equation (62). Equation 2 of Equation (60)
Equation (63) is obtained from δR / δα = 0 using the power deviation R. Here, T _m is expressed by equation (62). The thermal diffusivity satisfying the equation (63) is obtained by the Newton method. The initial values of the thermal diffusivity and biot number are the methods used in the analysis in time and space in the same manner as the method described above, that is, the initial value of the thermal diffusivity is a value obtained by using the t _1/2 method, or set appropriately. The initial value of the biot number is determined by the time constant method. The specific heat is determined as follows. The highest temperature rise is determined from the finally obtained thermal diffusivity, Biot number and Laplace-transformed measurement data. That is, the maximum rise temperature T
_m is given by equation (62). Further, the specific heat C of the sample is
C = Q / (wT _m ) using the maximum temperature T _m , the sample weight w, and the laser pulse energy Q absorbed by the sample. It absorbed laser pulse energy to the sample specific heat measurement sample and the absorption rate was set such that the same is determined by the maximum temperature increase value T _m for a known standard sample. In the Laplace spatial time constant analysis method described above, the thermal diffusivity is treated as an independent variable. However, the data analysis method in which the Biot number is treated as an independent variable and the thermal diffusivity is treated as a function of the Biot number via the time constant is also used. To establish. Further, as another example of the squared deviation, one using the reciprocal of the temperature shown in Expression (72) is also applicable.

[0047]

(Equation 13)

In the time space analysis method and the time constant analysis method in the Laplace space described above, the time constant is obtained, and the fact that the thermal diffusivity α and the Biot number h have a unique relationship via the time constant is used. The thermal diffusivity and the biot number h were simultaneously updated. However, this method has a problem that the thermal diffusivity α and the biot number h cannot be determined with high accuracy unless the data amount is at least 10 times the characteristic time. Therefore, if the minimum value of the squared deviation can be directly obtained, the calculation of the time constant becomes unnecessary, and the analysis with a smaller data amount becomes possible. Hereinafter, the structure of the square deviation R, which is particularly important when applying the least square method in the Laplace space, will be described in detail. The square deviation R has three independent variables (maximum rise temperature T _m , thermal diffusivity α, Biot number h), and the value to be taken for each independent variable is
If the squared deviation is simply a downwardly convex structure, a partial solution of the squared deviation to obtain a value that sets each equation to 0 will be the respective solution. However, in the case of this square deviation, an analysis was attempted by this method, but a convergent solution could not be obtained.
From this, it is considered that this square deviation is not simply a function of convex downward. Therefore, it is understood that the structure of the squared deviation needs to be clarified in advance in order to obtain the solution of each independent variable using the squared deviation.

Here, the theoretical equation of the sample back surface temperature in the Laplace space and the Laplace transform equation of the measured data are expressed by (6)
4), (65), (66), and (67). Here, the measurement data as E (t), displayed as a material obtained by this Laplace transform ^# E (p), Biot number of the sample surface and the back surface were equal. (64) as the squared deviation,
The case where the equation (67) is used as it is (forward square deviation) and the case where the reciprocals thereof are taken and the square deviation (reverse square deviation) is taken will be described below.

[0050]

[Equation 14]

The forward squared deviation is given to equation (68). This takes a partial differential in the independent variable T _m, the value as 0 (6
9) Obtain the equation. Next, the equation (69) is substituted into the equation (68), and the squared deviation equation (7
0) is obtained. Further, taking only the molecule of the formula (70), R ′
And R ′ is represented by equation (71).

[0052]

(Equation 15)

The inverse square deviation is shown in equation (72). This is partially differentiated with the independent variable _Tm , and its value is set to 0 to obtain the equation (73). By substituting the equation (73) into the equation (72), an inverse square deviation equation (74) expressed only by the thermal diffusivity and the biot number is obtained. Alternatively, the formula (75) is obtained as R 'by taking only the molecule.

[0054]

(Equation 16)

If the squared deviation is a forward squared deviation (70)
In equation (71) or equation (71), (7)
The functional relationship between the respective deviations is expressed by equation 4) or equation (75). The structure of the square deviation was determined to be 0.497 in sample thickness.
mm, thermal diffusivity 1.0 × 10 ⁻⁴ m ² / s, biot number 0.
2. A saw-tooth pulse waveform was confirmed based on theoretical data created under the condition of a pulse width of 1 ms.

FIG. 9 shows the structure of the square deviation according to the equation (70) in the case of the forward square deviation, and FIG. 10 shows the structure of the square deviation according to the equation (71). FIG. 11 shows the structure of the squared deviation according to the equation (74) in the case of the inverse squared deviation, and FIG. 12 shows the structure of the squared deviation according to the equation (75). In each case, the structure of the square deviation is basically the same. Further, the detailed structure of the squared deviation was examined using a squared deviation formula (74) using a reciprocal. FIG. 13 shows a bird's-eye view in which the thermal diffusivity α and Biot number h of the square deviation are plotted on the XY axes, and the logarithm of the square deviation is plotted on the Z axis. The square deviation R has a downwardly convex groove structure. FIG. 14 shows the groove projected on the plane of the thermal diffusivity and the Biot number, and FIG. 14 shows the groove projected on the plane of the square deviation and the thermal diffusivity. FIG. 15 and FIG. 16 show the square deviation and the projection onto the Biot number plane. From the above results, it has been clarified that the function structure of the square deviation has the following characteristics.

The minimum value of the square deviation exists at the position of the required thermal diffusivity and biot number. When the Biot number is fixed, the thermal diffusivity showing the minimum value is
The ratio of the change is small with respect to the change in the biot number, and a stable extreme value is obtained. When the thermal diffusivity is fixed, the Biot number showing the minimum value is
The rate of change is large relative to the change in thermal diffusivity. The position of the curve where the partial differential based on the thermal diffusivity of the square deviation is 0 and the position of the curve where the partial differential based on the Biot number are 0 are both located very near the groove. Thermal diffusivity α of square deviation
The thermal diffusivity α is updated by setting the partial differential by 0 to 0, and based on the updated thermal diffusivity α, the procedure of further setting the partial differential by the biot number to 0 is repeated to obtain the thermal diffusivity and the biot number. In some cases, the thermal diffusivity and the Biot number cannot be obtained without reaching the minimum value of the square deviation. This is because, when the curve with the partial differential according to the thermal diffusivity of the square deviation being 0 and the curve with the partial differential due to the Biot number being 0 are shifted from each other near the groove,
When the initial value of the Biot number is set to a value smaller than the value to be obtained, and the thermal diffusivity with the partial differential by the thermal diffusivity set to 0 is calculated, a thermal diffusivity larger than the value to be obtained is obtained. It is considered that when a biot number with the partial differential based on the biot number being 0 is obtained using the thermal diffusivity, a value smaller than the biot number used first is obtained. For this reason, this series of calculations does not converge but diverges, and the thermal diffusivity and Biot number cannot be obtained. In order to overcome the above problem, for example, a plurality of biot numbers are set as follows, and δR /
The thermal diffusivity that satisfies δα = 0 is determined individually, and then the deviation R is determined for each combination of the thermal diffusivity α and the Biot number h. By setting the number and repeating the above calculation, the minimum value of the deviation R and the thermal diffusivity α and the Biot number h that give this minimum value can be obtained.

Here, the direct analysis method in the Laplace space shown in FIG. 4 which is the third embodiment will be described.
For the deviation R expressed by three independent variables of the maximum rise temperature T _m , the thermal diffusivity α and the Biot number h, δR / δα = 0
FIG. 4 shows a procedure for obtaining the thermal diffusivity α and the Biot number h satisfying the minimum value of the deviation R. The initial value of the thermal diffusivity α is determined by the t _1/2 method. Alternatively, an appropriate initial value may be set. The Biot number with the partial differential equation based on the Biot number of the squared deviation being 0 is defined as the Biot number initial value. Alternatively, an appropriate initial value may be set. A plurality of biot numbers are set near the initial value of the biot number, and partial differentiation of each biot number by the thermal diffusivity of the square deviation is set to 0.
Is determined. The square deviation is calculated for the combination of the Biot number and the thermal diffusivity, and the combination of the thermal diffusivity and the Biot number that gives the minimum value is a value closer to the one to be obtained. A biot number for the next iteration calculation is newly set near the biot number giving the smallest square deviation based on knowledge of the function structure. In this procedure, for example, the initial biot number is set to 5
In the case where points are obtained, the procedure corresponds to the determination of five deviations R for each combination of the thermal diffusivity α and the biot number h. By examining the magnitude relation of the deviation, if the minimum value of the deviation R is given by the maximum or minimum value of the set biot number, a new biot number including the maximum biot number or the minimum biot number is calculated as the biot number for the next calculation. Set the number of bios at 5 points in the vicinity. Also, when the biot number giving the minimum value of the deviation is other than the maximum biot number and the minimum biot number,
A biot number is newly set near the biot number giving the minimum value, at an interval smaller than the interval of the previous biot number. What gives the minimum value of the square deviation by repeating the above procedure is the thermal diffusivity and the biot number to be obtained. The maximum temperature rise is determined from the finally obtained thermal diffusivity, Biot number and Laplace-transformed measurement data, and the specific heat is determined from the maximum temperature rise.
The maximum rise temperature is given by equation (69) for forward squared deviation, equation (73) for inverse squared deviation, and equation (62) for logarithmic squared deviation. The method of obtaining the specific heat from the maximum temperature rise is performed according to the procedure shown in the above-mentioned time-space analysis method.

FIGS. 17 and 18 show the results of analysis by the time constant method among the calculations in the Laplace space. The Laplace space method using equations (50) and (52) as the squared deviation and the logarithm thereof are used. FIG. 5 is a diagram showing each calculation error by a logarithmic method with respect to a ratio between a characteristic time and a pulse width, wherein a thermal diffusivity α is 1.0 × 10 ⁻⁴ m ² / s, and a biot number h is 0.1. For the ten data, the values of the thermal diffusivity α and Biot number h corrected for the pulse waveform and those not corrected are compared. In any of the figures, the effect of the correction by the pulse waveform, that is, the calculation error is smaller in the region where the characteristic time / pulse width is small, and in the region where the value of the characteristic time / pulse width is 2 or less, in the region where the characteristic time / pulse width is small. The calculation error can be reduced to 0.5% or less by incorporating the correction of the pulse waveform. Also,
It can be seen that in the region where the characteristic time / pulse width is greater than 4, the value of the calculation error as an absolute value decreases, but the effect of correcting the pulse waveform is reduced. FIGS. 19 to 2
0 is the analysis result by the direct analysis method in the Laplace space, and FIG. 19 shows that the thermal diffusivity α is 1.0 × 10 ⁻⁴ m ² /
FIG. 9 is a diagram showing calculation errors of respective data of s and the biot number of 0.10 with respect to the characteristic time / pulse width, and the thermal diffusivity α and the biot number h can be analyzed with good accuracy in all regions. . FIG. 20 shows that α = 1.0 × 10 ⁻⁴ m ^{2 / s and} h = 0.1
The relationship between the data amount and the calculation error in units of the characteristic time is shown using data of 0, characteristic time / pulse width = 6.0. As can be seen from the figure, unlike the time constant method described above, the thermal diffusivity α and the biot number h can be accurately analyzed with a small amount of data. In particular, the thermal diffusivity α can be analyzed with better accuracy than 1% even if the data amount is 2 to 3 times the characteristic time. In the flow chart shown in FIG. 4, Δh <ε is set using the biot interval Δh as a criterion.
The criterion may be | Δh | / h <ε, or a similar criterion using the thermal diffusivity, or as shown in FIG.
Using the deviation R, R <ε can also be set. FIG. 22,
As shown in FIG. 23, among the Laplace variables, especially pt _min is
Since it is affected by the characteristic time / pulse width, the data amount, and the number of bios, the flow is calculated by resetting based on one calculation result. FIGS. 22 and 23 focus on the thermal diffusivity, but the same relationship exists for the biot number, and it is better to set different Laplace variables for the thermal diffusivity α and the biot number h. Is required. Therefore, whether to calculate the thermal diffusivity α or the Biot number h,
Also, by setting the optimal Laplace variable by looking at the characteristic time / pulse width, data amount, biot number, etc., the thermal diffusivity α and the biot number h can be obtained with good accuracy. Where pt _min lower limit is characteristic time / pulse width, biot number,
It does not depend much on the amount of data and which one of the thermal diffusivity α and the Biot number h is calculated. Therefore, if it is close to this, the thermal diffusivity α and the Biot number can be obtained with good accuracy without recalculating the Laplace variable. h can be calculated. The above-described analysis method can be applied to a logarithmic square deviation as shown in Expression (60).

[0060]

According to the method for analyzing the thermal diffusivity measurement data in the laser flash method according to the first to fifth aspects, the pulse waveform, the thermal diffusivity α, the biot number h, and the maximum rise temperature T _m
By comparing the deviation between the theoretical solution obtained by setting and the actual temperature response measurement data and reflecting the relation data between each variable obtained in advance, the value of the unknown is updated sequentially and efficiently, and asymptotically The value of the unknown that satisfies the actual temperature response measurement data can be determined with good accuracy of 0.5% or less.

[0061]

[Brief description of the drawings]

FIG. 1 is a configuration diagram of an apparatus for analyzing thermal diffusivity, biot number, and specific heat data in a laser flash method according to an embodiment of the present invention.

FIG. 2 is a flowchart of a spatio-temporal analysis method of the data.

FIG. 3 is a flowchart of a Laplace space time constant analysis method of the data.

FIG. 4 is a flowchart of a Laplace space direct analysis method of the data.

FIG. 5 is a diagram illustrating a relationship between a biot number and an eigenvalue β _n .

FIG. 6 is a diagram illustrating a relationship between a biot number and an eigenvalue β _n .

FIG. 7 is a diagram showing a relationship between the biot number and the calculation accuracy of the eigenvalue β _n .

FIG. 8 is a diagram illustrating a relationship between the biot number and the calculation accuracy of the eigenvalue β _n .

FIG. 9 is a diagram showing a structure of a square deviation according to equation (70) in the case of a forward square deviation.

FIG. 10 is a diagram showing a structure of a square deviation according to equation (71) in the case of a forward square deviation.

FIG. 11 is a diagram showing a structure of a squared deviation according to equation (74) in the case of an inverse squared deviation.

FIG. 12 is a diagram showing a structure of a squared deviation according to equation (75) in the case of an inverse squared deviation.

FIG. 13 is a bird's-eye view showing the structure of the square deviation R;

FIG. 14 is a diagram showing a projection of a square deviation R groove on an α-h plane.

FIG. 15 is a diagram showing the mapping of the square deviation R groove on the R-α plane.

FIG. 16 is a diagram showing the mapping of the square deviation R groove on the Rh surface.

FIG. 17 is a diagram showing a calculation error in a time constant method using a square deviation in a Laplace space with respect to a ratio between a characteristic time and a pulse width.

FIG. 18 is a diagram showing a calculation error in a time constant method using a square deviation due to a logarithm in the Laplace space with respect to a ratio between a characteristic time and a pulse width.

FIG. 19 is a diagram showing a calculation error in a direct analysis method using a square deviation and an inverse square deviation in the Laplace space with respect to a ratio between a characteristic time and a pulse width.

FIG. 20 is a diagram illustrating a calculation error in the Laplace spatial direct analysis method with respect to a data amount in units of characteristic time.

FIG. 21 is a diagram showing a calculation error in a case where an average temperature difference in a specific time region is used as a deviation in a time space with respect to a ratio between a characteristic time and a pulse width.

FIG. 22 is a diagram illustrating the relationship between pt _min and the amount of data in a Laplace space.

FIG. 23 shows pt _min and characteristic time in Laplace space /
FIG. 4 is a diagram illustrating a relationship with a pulse width.

[Explanation of symbols]

 10 Measurement sample 11 Half mirror

Continuation of front page (56) References JP-A-57-157146 (JP, A) JP-A-62-263454 (JP, A) JP-A-63-159740 (JP, A) JP-A-64-86049 (JP) , A) JP-A-5-142169 (JP, A) (58) Fields investigated (Int. Cl. ⁶ , DB name) G01N 25/00-25/72 JICST file (JOIS)

Claims (5)

(57) [Claims] 1. A method for analyzing the thermal diffusivity, biot number and specific heat data of a plate-shaped measurement sample by irradiating a laser flash from the front surface side and measuring the temperature of the back surface side, A first step of setting a laser pulse waveform based on the following time constants (A) and (B) and setting a laser pulse waveform on the basis of a time constant obtained by measuring each initial value of the biot number or the thermal diffusivity of the measurement sample; A second step of obtaining a theoretical temperature value on the back surface of the measurement sample from a heat conduction equation including a waveform, a biot number, a thermal diffusivity, and a theoretical maximum temperature value as variables; The following (A) is used so that the deviation from the measured temperature value on the back surface is small.
And a third step of performing a numerical calculation under the conditions of the formula (B) and updating the biot number and the thermal diffusivity; and until the deviation, the biot number and the thermal diffusivity become equal to or less than a specified value. A fourth step of determining the thermal diffusivity and biot number of the measurement sample by repeating the second step to the third step; and a fifth step of calculating a specific heat after obtaining a maximum theoretical temperature without radiation loss. A method for analyzing thermal diffusivity, biot number and specific heat data in a laser flash method, comprising a step. h ₀ = β ₀ [1 / sin (β ₀ ) −1 / tan (β ₀ )] (A) β ₀ = L / sqrt (α ₀ · τ) (B) h ₀ : biot Number β ₀ : eigenvalue L: thickness of measurement sample α ₀ : thermal diffusivity τ: time constant 2. The order of the theoretical temperature value in the Laplace space and the measured temperature value on the back surface of the measurement sample instead of the deviation between the theoretical temperature value in the time space and the measurement temperature value on the back surface of the measurement sample. The method for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 1, wherein the deviation is any one of a square deviation, a reciprocal square deviation, and a logarithmic square deviation. 3. A method for analyzing the thermal diffusivity, biot number and specific heat data of a plate-shaped measurement sample by irradiating a laser flash from the front surface side and measuring the temperature of the rear surface side, A first step of setting each initial value of the biot number and thermal diffusivity of the measurement sample and setting a laser pulse waveform; and heat conduction including the laser pulse waveform, biot number, thermal diffusivity and theoretical maximum temperature value as variables. From the equation, a second step of obtaining a theoretical temperature value on the back surface of the measurement sample, setting one of the thermal diffusivity and the biot number to A and the other to B, and setting a plurality of A in the vicinity of the initial value of A For each of the plurality of A, B that minimizes the deviation is individually obtained, and then the deviation is obtained for the combination of A and B, and a new A is obtained in a direction that gives the minimum value of the deviation. A third step of updating, the deviation, A fourth step of determining the thermal diffusivity and biot number of the measurement sample by repeating the second to third steps until the biot number and the thermal diffusivity become equal to or less than a specified value; A fifth step of calculating the specific heat after obtaining the maximum theoretical temperature of the sample, and wherein the deviation is a forward square deviation between the theoretical temperature value in the Laplace space and the measured temperature value on the back surface of the measurement sample, A method for analyzing thermal diffusivity, biot number, and specific heat data in a laser flash method, wherein the deviation is any one of a reciprocal square deviation and a logarithmic square deviation. 4. The method for analyzing thermal diffusivity, biot number and specific heat data in the laser flash method according to claim 1, wherein the fifth step is to obtain a maximum theoretical temperature when there is no radiation loss by the following equation (C). . T _m = ΔT / (A ₀ ∫exp (a ₀ t ′) · f (t ′) dt ′) (C) where And T _m : maximum theoretical temperature without radiation loss ΔT: measured temperature specification value β ₀ : eigenvalue α: thermal diffusivity L: thickness of measurement sample 5. The fifth step corresponds to the forward squared deviation, the reciprocal squared deviation, or the logarithmic squared deviation according to the formula (D), the formula (E), or the formula (F), respectively. 4. The method for analyzing thermal diffusivity, biot number and specific heat data in a laser flash method according to claim 2 or 3, wherein a maximum theoretical temperature in the case where there is no radiation loss is obtained. T _m : maximum theoretical temperature without radiation loss