The X-ray diffraction data (2θ) of a certain substance measured by Xiao Ming are as follows:

$32.2\degree$, $37.3\degree$, $53.9\degree$, $64.2\degree$, $67.3\degree$

It is known that the test was conducted using $\mathrm{Cu}$ - K $\alpha$ radiation. Now, there are several statements:

1. Assuming the relative molecular mass of the substance is 100, its crystal density is approximately $\mathrm{5.97}$ $\mathrm{g·cm^{-3}}$

2. Among the above diffraction peaks, there are second-order or higher-order diffraction peaks

3. To capture more diffraction peaks within $\mathrm{5-80\degree}$, the $\mathrm{Cu}$ target of the instrument needs to be replaced with $\mathrm{Mo}$

4. To capture more diffraction peaks within $\mathrm{5-80\degree}$, the $\mathrm{Cu}$ target of the instrument needs to be replaced with $\mathrm{Cr}$

5. Without changing the test conditions, the next diffraction peak theoretically appearing at a higher angle would be from the $\mathrm{(331)}$ plane

Among the above statements, the correct ones are
小明测定的某物质的X-射线衍射数据(2θ)如下：

$32.2\degree$,$37.3\degree$,$53.9\degree$,$64.2\degree$,$67.3\degree$

已知此次测试使用的是$\mathrm{Cu}$ - K $\alpha$射线，现在有以下几种说法：

1. 假设该物质的相对分子质量为100，则其晶体密度约为$\mathrm{5.97}$ $\mathrm{g·cm^{-3}}$

2. 上述衍射峰中有二级或二级以上衍射峰

3. 若要在$\mathrm{5-80\degree}$拍到更多的衍射峰，需要将仪器的$\mathrm{Cu}$ 靶换成$\mathrm{Mo}$

4. 若要在$\mathrm{5-80\degree}$拍到更多的衍射峰，需要将仪器的$\mathrm{Cu}$ 靶换成$\mathrm{Cr}$

5. 不改变测试条件，理论上继续增大测试角度下一个出现的衍射峰为$\mathrm{(331)}$ 面的衍射峰

上述说法中正确的有
All statements are incorrect
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1,3
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1,5
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1, 2, 3
1, 2, 4
1, 2, 5
1,3,5
1,4,5
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2,4,5
1, 2, 3, 5
1,2,4,5
所有说法均不正确
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1,2,4,5
By default, it is known that the wavelength of $\mathrm{Cu}$-K $\alpha$ radiation is $\mathrm{154.1pm}$.

First, according to the Bragg diffraction equation:

$\mathrm{2d\sin(\theta)=n\lambda}$

the interplanar spacings corresponding to the given angles can be calculated as:

$\mathrm{278}$ $\mathrm{pm}$, $\mathrm{241}$ $\mathrm{pm}$, $\mathrm{170}$ $\mathrm{pm}$, $\mathrm{145}$ $\mathrm{pm}$, $\mathrm{139}$ $\mathrm{pm}$

Since

$\mathrm{d_{(hkl)}=\frac{a}{\sqrt{h^2+k^2+l^2}}}$

by squaring the above interplanar spacings and taking their ratios, we obtain a set of approximate integer ratios, which correspond to the ratios of $\mathrm{h^2+k^2+l^2}$:

$\mathrm{3:4:8:11:12}$
<Checkpoint>[The ratios of $\mathrm{h^2+k^2+l^2}$ are approximately $\mathrm{3:4:8:11:12}$](1)</Checkpoint>

This matches the extinction rules for a face-centered cubic (FCC) structure, indicating that the material has an FCC lattice.

<Checkpoint>[The material has a face-centered cubic structure](2)</Checkpoint>

The five diffraction peaks correspond to the $\mathrm{(111)}$, $\mathrm{(200)}$, $\mathrm{(220)}$, $\mathrm{(311)}$, and $\mathrm{(222)}$ crystal planes, respectively.

<Checkpoint>[The five diffraction peaks correspond to the $\mathrm{(111)}$, $\mathrm{(200)}$, $\mathrm{(220)}$, $\mathrm{(311)}$, and $\mathrm{(222)}$ crystal planes](1)</Checkpoint>

For Statement 1:

Using the formula: $\mathrm{d_{(hkl)}=\frac{a}{\sqrt{h^2+k^2+l^2}}}$

we obtain the unit cell edge length $a=481$ $\mathrm{pm}$.

From the formula $\mathrm{ρ=\frac{ZM}{VN_A}}$, the density is calculated as $5.97$ $g·cm^{-3}$, which is correct.

<Checkpoint>[The density is $5.97$ $g·cm^{-3}$](1)</Checkpoint>

<Checkpoint>[Statement 1 is correct](0.5)</Checkpoint>

For Statement 2:

The diffraction peak of the $\mathrm{(222)}$ plane can be considered as the second-order diffraction of the $\mathrm{(111)}$ plane, which is correct.

<Checkpoint>[The diffraction of the $\mathrm{(222)}$ plane can be regarded as the second-order diffraction of the $\mathrm{(111)}$ plane](0.5)</Checkpoint>

<Checkpoint>[Statement 2 is correct](0.5)</Checkpoint>

For Statements 3 and 4:

According to the formula: $\mathrm{2d\sin(\theta)=n\lambda}$

For the same material where $\mathrm{d}$ remains unchanged, a smaller $\mathrm{\lambda}$ corresponds to a smaller $\mathrm{\theta}$.

<Checkpoint>[For the same material where $\mathrm{d}$ remains unchanged, a smaller $\mathrm{\lambda}$ corresponds to a smaller $\mathrm{\theta}$](1)</Checkpoint>

The wavelength of $\mathrm{Cu}$ target is 1.54 Å; $\mathrm{Mo}$ target is 0.71 Å; $\mathrm{Cr}$ target is 2.29 Å.

Thus, Statement 3 is correct, and Statement 4 is incorrect.

<Checkpoint>[Statement 3 is correct, Statement 4 is incorrect](1)</Checkpoint>

For Statement 5:

Theoretically, the next crystal plane to exhibit diffraction should be the $\mathrm{(400)}$ plane with $\mathrm{h^2+k^2+l^2=16}$, not the $\mathrm{(331)}$ plane with $\mathrm{h^2+k^2+l^2=19}$. Therefore, Statement 5 is incorrect.

<Checkpoint>[The next diffraction plane should be the $\mathrm{(400)}$ plane](1)</Checkpoint>

<Checkpoint>[Statement 5 is incorrect](0.5)</Checkpoint>

Choose P.
默认可以知道$\mathrm{Cu}$-K $\alpha$射线的波长为$\mathrm{154.1pm}$

首先根据 Bragg 衍射方程：

$\mathrm{2d\sin(\theta)=n\lambda}$

可以计算出题目中各角度对应的晶面间距：

$\mathrm{278}$ $\mathrm{pm}$、$\mathrm{241}$ $\mathrm{pm}$、$\mathrm{170}$ $\mathrm{pm}$、$\mathrm{145}$ $\mathrm{pm}$、$\mathrm{139}$ $\mathrm{pm}$

由于

$\mathrm{d_{(hkl)}=\frac{a}{\sqrt{h^2+k^2+l^2}}}$

因此将上述晶面间距平方并作比可以得到如下的一组近似的整数比，即$\mathrm{h^2+k^2+l^2}$的比：

$\mathrm{3：4：8：11：12}$
<Checkpoint>[$\mathrm{h^2+k^2+l^2}$的比近似为$\mathrm{3：4：8：11：12}$](1)</Checkpoint>

符合面心立方消光规律，因此该物质是面心立方结构

<Checkpoint>[该物质是面心立方结构](2)</Checkpoint>

上述五个衍射峰也分别对应$\mathrm{(111)、(200)、(220)、(311)、(222)}$ 晶面

<Checkpoint>[上述五个衍射峰也分别对应$\mathrm{(111)、(200)、(220)、(311)、(222)}$ 晶面](1)</Checkpoint>

对于说法1：

根据公式：$\mathrm{d_{(hkl)}=\frac{a}{\sqrt{h^2+k^2+l^2}}}$

我们可以得到晶胞边长$a=481$ $\mathrm{pm}$

由公式 $\mathrm{ρ=\frac{ZM}{VN_A}}$得到密度为$5.97$ $g·cm^{-3}$，正确

<Checkpoint>[密度为$5.97$ $g·cm^{-3}$](1)</Checkpoint>

<Checkpoint>[说法1正确](0.5)</Checkpoint>

对于说法2：

上述衍射峰中$\mathrm{(222)}$ 面的衍射可看作$\mathrm{(111)}$ 面的二级衍射，正确

<Checkpoint>[$\mathrm{(222)}$ 面的衍射可看作$\mathrm{(111)}$ 面的二级衍射](0.5)</Checkpoint>

<Checkpoint>[说法2正确](0.5)</Checkpoint>

对于说法3和4：

根据公式：$\mathrm{2d\sin(\theta)=n\lambda}$

同一物质$\mathrm{d}$ 不变的情况下，$\mathrm{\lambda}$ 越小对应的 $\mathrm{\theta}$ 越小

<Checkpoint>[同一物质$\mathrm{d}$ 不变的情况下，$\mathrm{\lambda}$ 越小对应的 $\mathrm{\theta}$ 越小](1)</Checkpoint>

$\mathrm{Cu}$ 靶波长1.54 埃；$\mathrm{Mo}$ 靶波长0.71 埃；$\mathrm{Cr}$ 靶波长2.29 埃

因此说法3正确，说法4错误

<Checkpoint>[说法3正确，说法4错误](1)</Checkpoint>

对于说法5：

理论上下一个发生衍射的晶面应该是$\mathrm{h^2+k^2+l^2=16}$ 的$\mathrm{(400)}$ 面，而不是$\mathrm{h^2+k^2+l^2=19}$的$\mathrm{(331)}$ 面，因此说法5错误

<Checkpoint>[下一个发生衍射的晶面应该是$\mathrm{(400)}$ 面](1)</Checkpoint>

<Checkpoint>[说法5错误](0.5)</Checkpoint>

选P。