1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
W1.0 환영(Welcome)
W1.1 물질(Matter)
W1.2 힘(Forces)
W1.2a 자연단위(Natural units)
W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
W1.2c 가상입자(Virtual Particles)
W1.3 확률과 단면(Probability and cross section)
W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
W1.4 러더포드 실험(Rutherford experiment)
W1.4a 러더포드 단면(Rutherford cross section)
W1.4b 산란율 계산(Counting rate Rutherford)
W1.5 양자 산란(Quantum scattering)
W1.6 러더포드 실험 실습(Rutherford experiment in practice)
W1.7 1주 평가문제(Graded quiz for module 1)
2주: 핵 물리학(Nuclear Physics)
W2.1 핵 질량 및 결합 에너지(Nuclear mass and binding energy)
W2.2 핵의 크기와 스핀(Nuclear size and Spin)/동영상/영문자막/슬라이드
[2.2-1]---------------------------------------------------------------------
During this second module, we deal with nuclear physics and its applications. In this second video, we will summarize what is known about the size and the spin of nuclei. The goals for you are
- To know how nuclear size is measured and what the results are.
- To know general facts about the spin of nuclei.
- To be able to describe the valley of stability of nuclei as a function of the number of protons and neutrons.
이번 강의의 목표,
- 핵의 크기 측정 방법과 측정 결과
- 핵의 스핀에 대한 이해
- 핵의 안정곡선(valley of stability)을 양성자와 중성자의 수에 따른 함수로 알아보기
[2.2-2]---------------------------------------------------------------------
- The size of a subatomic object must be carefully defined. In a quantum system, it is given by the root mean square of the eigenvalue of the coordinate operator in its ground state.
원자 이하의 크기를 재려면 먼저 조심스러운 정의가 필요하다. 양자 체제(quantum system)에서는 입자가 기저상태(ground state)일때 좌표 연산자(coordinate operator)의 고유치(eigenvalue)를 평균자승제곱근(root mean square)하여 정한다.
* ground state, 입자가 외부로부터 에너지를 흡수하여 흥분되면 고유 크기보다 커짐
* coordinate operator, 크기를 재려는 것이므로 당연히 좌표가 등장. 하지만 통계 '연산자'
* eigenvalue, 좌표 연산자는 시공간의 행렬로 주어질 것인데 크기는 스칼라(고유치)
* the root mean square, 통계 처리. 크기 스칼라는 음수가 없으므로 제곱하여 구함.
- For an atom, this is the root mean square of the radial position of the electron, which is farthest away from the nucleus. This distance can be calculated because we know perfectly the binding electromagnetic force and because it is defined with respect to a static reference point, the position of the nucleus.
원자(atom)의 경우 핵에서 멀리 떨어진 전자(구름)의 구형위치(radial position)을 자승평균으로 구함. 이 거리는 전자기력(electromagnetic force)에 의해 묶여있는 크기로서 정확히 계산해 낼 수 있다.
- For the nucleus, we do not have a simple description of these actions. So we must interpret the results of experiments, which probe the distribution of nucleons inside the nucleus.
핵(nucleus)의 경우 핵력(nuclear force)이 지배하고 있어 쉽게 계산 할 수 없음. 핵내의 핵자들의 배열(distribution of nucleons) 구조를 밝힐 수 있는 실험이 필요함.
- It will be unwise to use hadronic probes to do so because they are sensitive to the nuclear force. High energy electrons, on the contrary, can penetrate inside the nucleus and their scattering maps out the charge distribution of the target.
이 핵자들의 구조를 밝히는 실험에 핵력에 민감한 강입자류(hadronic probe)를 사용하는 것은 바람직하지 않음. 고 에너지 전자(electron) 역시 핵을 그냥 통과해 버리므로 역시 바람직 하지 않음.
- The cross section for a point-like target without spin is given by the Mott formula displayed here.
스핀(spin)을 감안하지 않은 점으로 간주한 목표입자(point-like target)에 대한 단면(cross section, dσ/dΩ)은 위의 그림에 보인 것처럼 모트 공식(Mott formula). 산란각의 사인 네제곱분의 일, 1/(sin θ/2)^4.
- If the charge instead is distributed according to a volume density ρ(x), leaving the total charge intact, the cross section will be reduced by form factor F(q), which is a function of the momentum transfer q.
전하가 총 전하량에는 변화가 없이 체적밀도(volume density) ρ(x)를 따라 분포한다면 단면(cross-section)은 전하량에 따른 크기를 함수 F(q)로 나타낼 수 있을 것임. (단면을 단순히 전하량의 함수로 나타냄)
[2.2-3]---------------------------------------------------------------------
(?)
- For a static target, F(q) is the Fourier transform of the spatial charge distribution.
(?) 정 전하분포(static charge distribution) F(q)는 공간상의 전하 분포로 퓨리어 변환(근사식)으로 계산
- For small momentum transfers, we can develop the form factor in a Taylor series.
(?) 운동량 전달은 테일러 급수(Taylor series)를 활용하여 전개
- If the distribution is spherically symmetric, the terms with an odd power drop out. The dominant second term is proportional to the mean square radius, <r^2> of the charge distribution. It can thus serve as a size estimator for the nucleus.
(?) (공간상에 분포하는 전하의 영향범위를 감안)구형 대칭성(spherically symmetric)을 고려하면 퓨리어 변환 중 거리 r의 홀수제곱 항은 사라지고, 거리의 제곱 평균 <r^2>이 핵의 크기를 평가 할 수 있는 인자로 남는다.
- For an exponential charge distribution, the form factor takes what is called a dipolar form.
(?) 지수형 전하분포(exponential charge distribution)를 크기측정 요인(form factor)으로 삼는데 이를 쌍극형(dipolar form)이라 함.
- The dependence of the electron-nucleus cross section on the momentum transfer for small angle scattering at high electron energies is thus used to measure the size of the nucleus.
(!) 고 에너지 전자를 입사 입자로 한 산란실험을 통해 얻은 작은 산란각을 가지고 운동량 전달을 계산하여 전자와 핵이 결합된(원자) 단면을 계산 할 수 있다. 이 (원자의)단면은 결국 핵의 크기를 측정 하는데 이용된다.
[2.2-4]---------------------------------------------------------------------
(?)
- Scattering experiments establish a simple relation between the radius of a nucleus R and the number of nucleons A. R ~ A^(1/3)
산란실험으로 핵의 반경과 핵자의 숫자의 관계를 얻었다. 핵의 반경 R은 핵자의 수 A의 세제곱근에 근사. R ~ A^(1/3)
- The radius R is proportional to the cube root of the number of nucleons A, with a universal proportionality constant of 1.2 femtometers. Nuclei are thus indeed small
compared to the atomic size.
핵의 반경 R은 핵자의 수의 세제곱근에 비례하는데 비레상수는 1.2 펨토미터, 1.2x10^(-15)m. 핵은 실제로 원자의 크기보다 아주 작다. 원자 반지름은 원자핵에서 가장 바깥 궤도의 전자까지의 거리로 수소원자의 크기는 25 피코메터, 25x10^(-9)m
- The nuclear volume is proportional to A. This corresponds to densely packed and
incompressible nucleons, which do not fuse.
핵의 부피는 핵자의 개수에 비례(당연히!). 이는 융합되지 않고 밀집된 핵자들에 해당함.
- The mass density of nuclear matter is of the order of 10^14 grams per cm^3.
핵물질의 질량밀도의 규모는 세제곱 센티미터당 10의 14승 그램. 10^14 grams per cm^3.
[2.2-5]---------------------------------------------------------------------
(?)
Nuclear spin is the sum of the spin S of all individual nucleons and the relative angular momentum, L.
- Protons and neutrons are fermions with spin one-half.
- Like in atoms, the nuclear angular momentum L follows an integer quantum number.
- The total should therefore be a half integer number if A is odd, an integer if A is even.
- Indeed, all nuclei with both N and Z even have nuclear spin 0. Heavy nuclei have rather small nuclear spin in their ground state. We conclude that neutrons and protons tend to arrange in pairs of opposite spin direction.
[2.2-6]---------------------------------------------------------------------
(...........)
Every charged particle has a magnetic dipole moment associated with its spin, including the nuclei.
The order of magnitude for an electron is given by the Bohr magneton, µ_B.
The nuclear magneton, e/2m_p, is three orders of magnitude smaller due to the larger proton mass.
- The gyromagnetic factor g measures the ratio between the angular momentum and the magnetic moment.
- For a point charge, g is about 2 with small deviation of order 10^-3 for electrons and
muons as we will see in module 4.
- The magnetic moments of proton and neutron are considerably different, +2.79 and -1.91 nuclear magnetons, respectively. This is the first indication of a charged substructure of the nucleons. In fact, since the neutron has zero net charge, it must contain charged particles.
- All nuclei have measured magnetic moments between minus three and ten nuclear magnetons, thus, relatively small ones. This is a consequence of the spin pairings of nucleons, leading to a limited total nuclear angular momentum.
[2.2-7]---------------------------------------------------------------------
(............)
- Most nuclei and their isotopes are unstable. Stable nuclei are found in a narrow band, in the N-Z diagram which is called the valley of stability.
- The valley has the following shape.
> For lighter nuclei with atomic mass less than 40, the number of neutrons equals
the number of protons.
> For heavy nuclei with atomic mass bigger than 40, the number of neutrons is about 1.7 times the number of protons.
- This indicates that for heavy nuclei, the charge density, and thus the Coulomb repulsion must be diluted by additional neutrons.
- The decay of unstable nuclei is the source of nuclear radioactivity.
> Alpha radioactivity is the emission of He-4 nuclei or so-called alpha particles.
> Beta radioactivity is the emission of electrons or positrons together with neutrinos.
> Gamma radioactivity is the emission of photons.
- Nuclei with a surplus of neutrons can be stabilized by converting a neutron into a proton. And those with the surplus of protons can convert a proton into a neutron. These are isobar decays of type beta plus or minus.
- Heavy nuclei often decay into a pair of lighter nuclei. This corresponds to spontaneous fission, often by emitting a He-4 nucleus. This is the alpha decay.
- This decay is normally related to an excited state of the daughter nucleus. And they're often followed by a gamma decay towards the ground states.
- We will enter into more detail on these processes in the fourth and fifth video of this module.
[2.2-8]---------------------------------------------------------------------
(결론만이라도 이해? 아니면 외울까?)
So let us summarize what we have learned about the nuclear force.
핵력에 관해 배운 내용을 요약해보면 다음과 같다.
- It has a very short range limited to the nuclear size.
핵력의 영향 범위는 매우 좁은 핵의 크기에 국한 된다. 약 10^(-14) m
- The binding energy per nucleon it leads to is independent of the size of the nucleus. The nucleon thus interacts only with its nearest neighbors.
(핵을 구성하는) 핵자당 결합 에너지는 핵의 크기와는 상관 없이 핵자들 끼리 작동한다.
- The nuclear force is attractive and much stronger than the Coulomb repulsion between protons. But it must also have a repulsive component at distances compatible to the size of the nucleon, which is about 1 femtometer. This is due to the existence of quarks inside the nucleon. The repulsive component is necessary to prevent the fusion among nucleons.
핵력(nuclear force)은 양성자들 사이의 응집력으로 쿨롱 반발력을 능가한다. 하지만 핵력도 핵자들의 크기를 벗어나면 반발력을 띈다. 그 범위는 1 펨토미터(femto-meter), 10^(-15)m 다. 핵력이 반발력으로 발현되는 이유는 핵자(nucleon) 내부에 쿼크(quark)가 존재하기 때문이다. 핵 융합(nuclear fusion)을 어렵게 하는 요인이 바로 이 핵력의 반발력이다.
- QCD(quantum chromodynamics) is a quantum field theory which describes the interaction among colored particles, so quarks inside hadrons, via the exchange of equally colored gluons. But the strong force acts only inside hadrons, which thus have a zero net color.
QCD(quantum chromodynamics)는 색입자(colored particle)사이에 작용하는 상호작용을 설명하는 양자장 이론이다. 즉, 강입자(양성자)내의 쿼크는 글루온을 통해 힘을 전달한다. 이런 강력(strong force)는 오직 강입자(hadron)내에서만 작동한다. 강입자 전체의 색전하(net color) 값은 0이다.
- Nucleons cannot exchange gluons. It is by the exchange of colorless objects like mesons that they bind together.
- This makes the nucleus a complex multi-body object, which is difficult to understand. The appropriate theoretical methods are effective field theories like Chiral Perturbation Theory or a numerical calculation like Lattice QCD. All of this goes beyond the scope of this course.
In the next video, we will rather concentrate on much simpler models, which describe the gross features of nuclei.
[W2.2 연습문제]--------------------------------------------------------------
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