Mass defect and nuclear binding energy · 质量亏损与核结合能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| energy/ˈenədʒi/ | 能量 | néngliàng |
| nucleus/ˈnjuːklɪəs/ | 原子核 | yuán zǐ hé |
| mass-energy equivalence/mæs ˈenədʒi ɪˈkwɪvələns/ | 质能等价 | zhì néng děng jià |
| mass defect/mæs ˈdiːfekt/ | 质量亏损 | zhì liàng kuī sǔn |
| binding energy/ˈbaɪndɪŋ ˈenədʒi/ | 结合能 | jié hé néng |
| luminosity/ˌluːmɪˈnɒsɪti/ | 光度 | guāng dù |
| nuclear reaction/ˈnjuːklɪə rɪˈækʃn/ | 核反应 | hé fǎn yìng |
| nucleon number/ˈnjuːklɪən ˈnʌmbə/ | 核子数 | hé zǐ shù |
| conservation of charge/ˌkɒnsəˈveɪʃn ɒv tʃɑːdʒ/ | 电荷守恒 | diàn hè shǒu héng |
| positron/ˈpɒzɪtrɒn/ | 正电子 | zhèng diàn zi |
| neutrino/njuːˈtriːnəʊ/ | 中微子 | zhōng wēi zi |
| protons/ˈprəʊtɒnz/ | 质子 | zhì zi |
| neutrons/ˈnjuːtrɒnz/ | 中子 | zhōng zi |
| binding energy per nucleon/ˈbaɪndɪŋ ˈenədʒi pɜː ˈnjuːklɪən/ | 比结合能 | bǐ jié hé néng |
The Sun is four million tonnes lighter than a second ago
- Every second the Sun radiates $3.8\times10^{26}\ \text{J}$, and every second it is about four million tonnes lighter.
- Nothing is leaking out of it. The mass has become the energy, at the exchange rate $E = mc^2$.
- On the nuclear scale the same trade explains why a nucleus weighs less than the parts it is made of, and where the energy of a reactor comes from.
- This lesson is mass-energy equivalence 质能等价, mass defect 质量亏损 and binding energy 结合能.
太阳比一秒钟前轻了四百万吨
- 太阳每秒辐射 $3.8\times10^{26}\ \text{J}$,而每一秒它都轻了大约四百万吨。
- 并没有什么东西漏出去。是质量变成了能量,兑换率是 $E = mc^2$。
- 在核的尺度上,同一笔交易解释了原子核为什么比组成它的部件更轻,以及反应堆的能量从哪里来。
- 这一课讲质能等价(mass-energy equivalence)、质量亏损(mass defect)和结合能(binding energy)。
Mass and energy are the same currency
- $E = mc^2$, with $c = 3.00\times10^{8}\ \text{m/s}$. For a change of mass:
- In nuclear physics the mass changes are tiny and $c^2$ is enormous, so a minute mass change is a large energy 能量.
- Learn the conversion, because almost every question uses it:
One quantity, two units
质量和能量是同一种货币
- $E = mc^2$,其中 $c = 3.00\times10^{8}\ \text{m/s}$。对质量的变化:
- 在核物理中质量变化极小而 $c^2$ 极大,所以微小的质量变化对应很大的能量(energy)。
- 要记住这个换算,因为几乎每道题都用得上:
一个量,两种单位
The mass-energy relation is: · 质能关系是：
Mass and energy are equivalent: $E = mc^{2}$, so a mass change $\Delta m$ releases $c^{2}\Delta m$. · 质量与能量等价：$E = mc^{2}$，因此质量变化$\Delta m$释放$c^{2}\Delta m$。
A mass change of $1\ \text{u}$ corresponds to about how many MeV? · 质量变化$1\ \text{u}$大约对应多少MeV？
$1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$ gives $\Delta E = c^{2}\Delta m \approx 931\ \text{MeV}$. · $1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$给出$\Delta E = c^{2}\Delta m \approx 931\ \text{MeV}$。
Worked example: weighing the output of a star
- Sirius loses mass by fusion at $1.09\times10^{11}\ \text{kg/s}$. Find the power it radiates.
- $P = c^2 \times (\text{mass lost per second}) = (3.00\times10^{8})^2(1.09\times10^{11}) = 9.8\times10^{27}\ \text{W}$.
- That is its luminosity 光度, and it is about $26$ times the Sun's $3.8\times10^{26}\ \text{W}$.
- The same sum backwards: a $1\ \text{GW}$ power station running for a year converts $E/c^2 = 3.2\times10^{16}/9.0\times10^{16} = 0.35\ \text{kg}$ of mass. That is why the spent fuel weighs almost exactly what it did going in.
例题:称一颗恒星的输出
- 天狼星因聚变以 $1.09\times10^{11}\ \text{kg/s}$ 损失质量。求它辐射的功率。
- 每秒损失的质量乘以 $c^2$:$P = c^2 \times (\text{mass lost per second}) = (3.00\times10^{8})^2(1.09\times10^{11}) = 9.8\times10^{27}\ \text{W}$。
- 这就是它的光度(luminosity),约为太阳 $3.8\times10^{26}\ \text{W}$ 的 $26$ 倍。
- 反过来算同一笔账:一座 $1\ \text{GW}$ 的电站运行一年转化 $E/c^2 = 3.2\times10^{16}/9.0\times10^{16} = 0.35\ \text{kg}$ 的质量。这就是乏燃料的重量几乎与入堆时分毫不差的原因。
Nuclear equations balance twice
- A nuclear reaction 核反应 conserves nucleon number 核子数 (the top numbers) and charge (the bottom numbers), which is conservation of charge 电荷守恒:
- Alpha decay: $A$ falls by $4$, $Z$ by $2$. Beta-minus: a neutron becomes a proton, so $A$ is unchanged and $Z$ rises by $1$, with an electron and an antineutrino.
- Beta-plus: a proton becomes a neutron, $Z$ falls by $1$, emitting a positron 正电子 and a neutrino 中微子. Gamma changes neither number.
- The neutrino is in the mark scheme. Leaving it out of a beta equation costs a mark every time.
核方程要平衡两次
- 核反应(nuclear reaction)守恒核子数(nucleon number,上标)和电荷(下标),即电荷守恒(conservation of charge):
- α 衰变:$A$ 减 $4$,$Z$ 减 $2$。β⁻:中子变成质子,所以 $A$ 不变、$Z$ 加 $1$,并放出一个电子和一个反中微子。
- β⁺:质子变成中子,$Z$ 减 $1$,放出一个正电子(positron)和一个中微子(neutrino)。γ 两个数都不改变。
- 中微子在评分标准里。β 衰变方程漏掉它,每次都要扣一分。
In alpha decay the nucleon number falls by 4 and the proton number falls by . · 在α衰变中，核子数减少4，质子数减少。
An alpha particle is a helium-4 nucleus, 2 protons and 2 neutrons. Beta-minus leaves A unchanged and raises Z by 1, and the antineutrino must be written too. · α粒子是氦-4原子核，含2个质子和2个中子。β⁻衰变保持A不变并使Z增加1，同时必须写出反中微子。
Mass defect and binding energy
- A nucleus 原子核 has less mass than its separate protons 质子 and neutrons 中子. The difference is the mass defect:
- The mass defect is the difference between the total mass of the separate nucleons and the mass of the nucleus.
- The binding energy is the minimum energy required to separate the nucleus into its individual nucleons, and $B = \Delta m\, c^2$.
- Say "separate nucleons" or "individual protons and neutrons". "The energy holding the nucleus together" scores nothing.
- Binding energy is released when the nucleus forms, not stored in it. The nucleus has less energy than its parts, which is exactly why it stays together.
Assemble it and mass goes missing
质量亏损与结合能
- 原子核(nucleus)的质量比分开的质子(protons)和中子(neutrons)之和更小。这个差就是质量亏损:
- __质量亏损__是分开的核子的总质量与原子核质量之差。
- __结合能__是把原子核分离成一个个核子所需的最小能量,且 $B = \Delta m\, c^2$。
- 要说"分开的核子"或"单个的质子和中子"。"把原子核束缚在一起的能量"得零分。
- 结合能是原子核形成时放出的,不是储存在里面的。原子核的能量比它的部件更低,这正是它能待在一起的原因。
装到一起,质量就少了
A nucleus has ____ mass than its separate protons and neutrons added together. · 原子核的质量____其分离的质子和中子质量之和。
That missing mass (the mass defect) was released as energy when the nucleus formed. · 这部分缺失的质量（质量亏损）在原子核形成时以能量形式释放。
The binding energy is the energy needed to pull a nucleus completely apart. · 结合能是将原子核完全拆解所需的能量。
$B = \Delta m\,c^{2}$ — the energy released on forming the nucleus, which must be returned to separate it. · $B = \Delta m\,c^{2}$——原子核形成时释放的能量，拆解时必须重新提供。
Match each term to the definition the examiner marks. · 将每个术语与考官标记的定义匹配。
"The energy holding the nucleus together" scores nothing. The marked phrase is separating it into individual nucleons. · “将原子核束缚在一起的能”不得分。标准表述是将其分离成单个核子。
Worked example: helium-4
- A helium-4 nucleus has a mass defect of $0.0304\ \text{u}$. Find its binding energy.
- $B = 0.0304 \times 931 = 28\ \text{MeV}$.
- Per nucleon that is $28/4 = 7.1\ \text{MeV}$, already high for such a light nucleus, which is why helium-4 shows as a spike on the curve.
- Binding energy per nucleon 比结合能 is $B/A$, and it is the fair way to compare two nuclides of very different size.
例题:氦-4
- 氦-4 核的质量亏损是 $0.0304\ \text{u}$。求它的结合能。
- $B = 0.0304 \times 931 = 28\ \text{MeV}$。
- 每核子是 $28/4 = 7.1\ \text{MeV}$,对这么轻的核来说已经很高了,这就是氦-4 在曲线上呈现为一个尖峰的原因。
- 比结合能(binding energy per nucleon)是 $B/A$,它是比较大小悬殊的两种核素的公平办法。
Worked example: polonium-212
- Proton $1.007276\ \text{u}$, neutron $1.008665\ \text{u}$, $^{212}_{84}\text{Po}$ nucleus $211.9454\ \text{u}$. Find the mass defect and the binding energy per nucleon.
- $Z = 84$ protons and $N = 212 - 84 = 128$ neutrons: $84(1.007276) + 128(1.008665) = 213.7203\ \text{u}$.
- $\Delta m = 213.7203 - 211.9454 = 1.7749\ \text{u}$.
- $B = 1.7749 \times 931.5 = 1653\ \text{MeV}$, so $B/A = 1653/212 = 7.80\ \text{MeV}$ per nucleon, on the falling part of the curve.
- Keep every decimal place until the subtraction. The defect is a small difference between two large numbers, and rounding early destroys it.
例题:钋-212
- 质子 $1.007276\ \text{u}$,中子 $1.008665\ \text{u}$,$^{212}_{84}\text{Po}$ 核 $211.9454\ \text{u}$。求质量亏损与比结合能。
- $Z = 84$ 个质子,$N = 212 - 84 = 128$ 个中子:$84(1.007276) + 128(1.008665) = 213.7203\ \text{u}$。
- $\Delta m = 213.7203 - 211.9454 = 1.7749\ \text{u}$。
- $B = 1.7749 \times 931.5 = 1653\ \text{MeV}$,所以 $B/A = 1653/212 = 7.80\ \text{MeV}$ 每核子,位于曲线的下降段。
- **相减之前每一位小数都要保留。**亏损是两个大数之间的小差值,过早取近似会把它毁掉。
Polonium-212 has a mass defect of 1.7749 u. What is its binding energy per nucleon, in MeV? (Use 931.5 MeV per u.) · 钋-212的质量亏损为1.7749 u。其比结合能为多少MeV？（使用每u 931.5 MeV。）
B = 1.7749 x 931.5 = 1653 MeV, then divide by A = 212 to get 7.80 MeV per nucleon, on the falling side of the curve. · B = 1.7749 x 931.5 = 1653 MeV，然后除以A = 212得到7.80 MeV/核子，位于曲线下降段。
The curve everything hangs on
- Plot $B/A$ against $A$ and you get a dome: a steep rise for light nuclei, a maximum near $A = 56$ (iron) at about $8.8\ \text{MeV}$, then a slow fall to about $7.5\ \text{MeV}$ at uranium.
- Iron-56 is the most stable nucleus, and everything else can release energy by moving towards it.
- Sketching rules the exam marks: do not start at the origin, do not make the fall as steep as the rise, and do not let the curve reach zero on the right.
- Mark a nucleus that alpha decays on the far right ($A > 200$), and one that fuses on the far left ($A < 10$). Both sit low and both move up the curve when they react.
Everything climbs towards iron
一切都挂在这条曲线上
- 把 $B/A$ 对 $A$ 作图会得到一个穹顶:轻核处陡升,在 $A = 56$(铁)附近达到约 $8.8\ \text{MeV}$ 的最大值,然后缓慢下降到铀处的约 $7.5\ \text{MeV}$。
- 铁-56 是最稳定的核,而其余一切都能靠向它靠拢来释放能量。
- 考试评分的作图要点:不要从原点起笔,不要把下降画得和上升一样陡,也不要让曲线在右端落到零。
- 标出发生 α 衰变的核要在最右端($A > 200$),发生聚变的核在最左端($A < 10$)。两者都处在低位,反应时都沿曲线向上走。
一切都往铁那里爬
Mass defect energy lab · 质量亏损能量实验
E = delta m c^2
Change mass defect and see binding energy rise with E = mc^2. · 改变质量亏损并观察结合能随 E = mc^2 上升。
The binding energy per nucleon is greatest for: · 比结合能最大的是：
Iron ($A \approx 56$) sits at the peak — the most tightly bound, most stable nucleus. · 铁（$A \approx 56$）位于峰值——结合最紧密、最稳定的原子核。
Which are true of the binding-energy-per-nucleon curve? Select all · 所有 that apply. · 关于比结合能曲线的描述哪些是正确的？选择所有适用项。
It falls only to about 7.5 MeV at uranium. Drawing it back down to zero is a marked error, as is starting the curve exactly at the origin. · 在铀处仅下降至约 7.5 MeV。将其拉回零点是明显的错误，从原点开始曲线也是如此。
Sketch the curve: put the features in order from left to right. · 绘制曲线：将特征按从左到右的顺序排列。
A fusing nucleus is marked at the far left and an alpha emitter at the far right. Both are low on the curve and both move up it when they react. · 最左侧标记了一个正在聚变的原子核，最右侧标记了一个α衰变发射体。两者在曲线上都较低，反应时都会沿曲线向上移动。
Marks that slip away
- Binding energy is the energy to pull the nucleus apart into separate nucleons, not "the energy holding it together".
- Keep full precision until you subtract. A mass defect is a small difference of large numbers.
- $1\ \text{u} \leftrightarrow 931\ \text{MeV}$ works only when $\Delta m$ is in u. In kilograms you must use $c^2$ and you get joules.
- Balance both lines of a nuclear equation, and include the neutrino in beta decay.
- The nucleus is lighter than its parts. Writing the mass defect the other way round makes the binding energy negative.
容易丢掉的分
- 结合能是把原子核拆成一个个分开的核子所需的能量,不是"把它束缚在一起的能量"。
- 相减之前保持全部精度。质量亏损是大数之间的小差值。
- $1\ \text{u} \leftrightarrow 931\ \text{MeV}$ 只在 $\Delta m$ 以 u 为单位时成立。用千克就必须乘 $c^2$,得到的是焦耳。
- 核方程的两行都要平衡,β 衰变里要写上中微子。
- 原子核比它的部件更轻。质量亏损写反会让结合能变成负的。
You've got it
- $\Delta E = c^2 \Delta m$, and $1\ \text{u} \leftrightarrow 931\ \text{MeV}$ is the conversion nearly every question needs
- a nuclear equation conserves nucleon number and charge, and beta decay emits a neutrino or antineutrino
- the mass defect is the mass of the separate nucleons minus the mass of the nucleus, and the binding energy $B = \Delta m c^2$ is the minimum energy to separate it into individual nucleons
- binding energy per nucleon peaks near iron-56 at about $8.8\ \text{MeV}$, so light nuclei fusing and heavy nuclei splitting both climb the curve
你掌握了
- $\Delta E = c^2 \Delta m$,而 $1\ \text{u} \leftrightarrow 931\ \text{MeV}$ 是几乎每道题都要用的换算
- 核方程守恒核子数与电荷,而 β 衰变要放出中微子或反中微子
- 质量亏损是分开核子的质量减去原子核的质量,结合能 $B = \Delta m c^2$ 是把它分离成单个核子所需的最小能量
- 比结合能在铁-56 附近达到约 $8.8\ \text{MeV}$ 的峰值,所以轻核聚变与重核裂变都是沿曲线向上爬