### A Pluto.jl notebook ###
# v0.17.5

using Markdown
using InteractiveUtils

# ╔═╡ 0c66d310-bf10-11ec-1be7-5d0878ec4a0b
begin
	using Markdown
	using Unitful
	using UnitfulRecipes
	using Plots
	using LsqFit
	using PlutoUI
	plotly()
end

# ╔═╡ ec84079f-cb49-4b60-a5b7-6ea1cf6a0240
md"""
### Fotoefekta un Komptona nobīdes šķērsgriezumi
Šodien apskatīsimies kādā veidā fotona enerģija un mērķa atomu skaitlis $Z$ ietekmē fotoefektu un komptona nobīdes varbūtību. 

#### Šķērsgriezums
Domājot tēlaini, kāda objekta šķērsgriezums ir lauka vienība, ko prezentē kādam virzienam. Šķērsgriezums mēra $\mathrm{cm}^2$, kā lauka vienība, bet nav obligāti saistīts ar reālo virsmu. Piemēram, elektrona šķērsgriezumu var attēlot kā aplis ar virsmas lauku $\sigma$ ap elektronu. Bet šis šķērsgriezums var mainīties ar enerģiju, tāpēc nav saistīts ar kaut ko fizikālo.    

Pieņemsim, ka kāda fotona plūsma $I$ ar enerģiju $E$ nonāk uz kāda materiāla slāni ar biezumu $t$  un $n$ atomiem uz lauka vienību. Atomu šķērsgriezums ir $\sigma$. Reakciju skaits sekundē $N_{reakcijas}$ var noteikt šādi:

$N_{reakcijas} = \sigma I n $


Parametru $n$ var iegūt šādi:

$n=\frac{\rho t N_A}{M_A}$,

kur $N_A$ ir Avogrado skaitlis un $M_A$ ir molāra masa. 

#### Dati
Mēs varam atrast dati par šķērsgriezumu atkārība no enerģijas [XCOM datubāzē](https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html) no NIST mājas lapas. 
Izvelēsim _Element_ un _Enter additional energies by hand_ un kliķšinām `Submit information`.

Nākamājā logā, rakstīsim vai nu _Atomic number_ $Z$ (piemēram "56") vai _Symbol_ (piemēram "Fe"). Zem _Graph options_, kliķšinām 
* Total Attenuation with Coherent Scattering
* Incoherent Scattering (Komptona izkliede)
* Photoelectric Absorption
Izvēlēsim vēlamo enerģijas diapazonu (pietiek līdz 10 MeV)
un kļiķšinām `Submit Information`. (Var arī citus, bet mēs pagaidām gribam zināt tikai par fotoefektu un Komptona izkliedi.) 

Parādās grafiks ar šķersgriezumu $\tilde{\sigma}$ [$\mathrm{g}\, \mathrm{cm}^{-2}$] y-asī pret fotona enerģiju x-asī.

#### Citi noderīgi dati
[Mendelejeva tabula](https://pubchem.ncbi.nlm.nih.gov/periodic-table/)
Spiežot uz katru elementu var iegūt papildus informāciju ($\rho$, $M_A$, utt. )


"""

# ╔═╡ e735388c-6631-4a83-84a3-458931dd63cf
md"""
### Uzdevums 1. 
#### Dažādi veidi, kā skatīties uz šķērsgriezumu.
a) Kā mēs varētu dabūt atoma šķērsgriezumu $\sigma$ no $\tilde{\sigma}$?
"""


# ╔═╡ 701363c4-3989-4f79-84cd-0972de60506c
md"""
b) Kāds ir fotoefekta šķērsgriezums pie $E=1$ MeV vienam dzelzs atomam?
"""

# ╔═╡ af8d9681-e622-4b2c-b7cb-7069835dc88e
md"""
c) Cik biezs dzelzs slānis ir vajadzīgs, lai samazinātu fotonu plūsmu uz $1 \%$ pie $E=10$ keV? Pie $E=1$ keV? Kas notiek, ja slānis ir no svīna?
"""

# ╔═╡ d69d9cf6-278f-4cdf-8afe-215d9028cf65
md"""
### Uzdevums 2. 
#### Komptona izkliedes šķērsgriezums atkarībā no $Z$.
a) Izvirziet hipotēzi par to, kādā veidā Komptona izkliedes šķērsgriezums ir atkarīgs no $Z$. 
"""

# ╔═╡ 04ce5b6e-f60b-4c73-b98a-548d5e9ca36a
md"""
b) Jūsu modelī, vai labāk izmantot $\sigma$ [$\mathrm{cm}^2$] vai $\tilde{\sigma}$ $\mathrm{cm}^2 \, \mathrm{g}^{-1}$]
"""

# ╔═╡ 71933852-57e2-4e24-8ca2-7dc91577d061
md"""
c) Izmantojiet [XCOM datubāzi](https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html), lai iegūtu šķērsgriezumu vērtības pie vairākām $Z$ vērtībām pie kādas konkrētas enerģijas. 
"""

# ╔═╡ af0029ec-251c-4451-9ec3-4b217e739a90
md"""
d) Pārbaudiet Jūsu modeli ar iegūtajiem datiem un komentējiet iespējamas atšķirības. _Julia_ valodā modeli var rakstīt šādi

```@. model(x,p) =p[1]*f1(x)+p[2]*f2(x)+p[3]*f3(x)+...```

Tad, definējiet minējumu Jūsu parametriem

``` p0=[a,b,c]```

Beidzot, ja X=[$X_1,X_2,...,X_n$] ir X-dati un Y=[$Y_1,Y_2,...,Y_n$] ir Y-dati, izmantojiet pavēles

```	
fit=curve_fit(model,xdata,ydata,p0)
println("p = ",fit.param)
println("stderr = ", stderror(fit))
plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
plot!(X,model(X,fit.param))
```
"""

# ╔═╡ adadb085-e762-41b9-a2a9-9982035fb8af
md"""
### Uzdevums 3. 
#### Fotoefekta šķērsgriezums atkarībā no $Z$.

"""

# ╔═╡ e7540bb4-2a31-452e-94b3-127715d0f69b
md"""
a) Izmantojiet XCOM datubāzē lai iegūtu fotoefekta šķērsgriezumu vērtības pie vairākām  vērtībām pie kādas konkrētas enerģijas. Pie kuras enerģijas vērtības ņemsiet Jūsu mērījumus? Kāpēc?
"""

# ╔═╡ a52de69b-3fda-4616-88b1-236aa601704c
md"""
b) Zīmējiet Jūsu "datus" ar $Z$ vērtību x-asī un šķērsgriezumu uz $y$-asi. Ja X=[$X_1,X_2,...,X_n$] un Y=[$Y_1,Y_2,...,Y_n$], izmantojiet pavēli ```plot(X,Y)```
"""

# ╔═╡ 95f0e1eb-9461-4db6-a98e-6803ad9c67d8
md"""
c) Izvirziet modeli, kas varētu aprakstīt datus. (Skatieties logaritmiskajā skalā ar pavēli ```plot(X,Y,yaxis=:log, xaxis=:log)```. Ko tas nozīmē?)
"""


# ╔═╡ 419e92ea-47b6-4365-8136-fd775813f042
md"""
d) Pārbaudiet Jūsu modeli un nosakiet parametrus.
"""

# ╔═╡ 65c443af-c62c-469f-9c63-559542868652
md"""
### Uzdevums 4. 
#### Fotoefekta šķērsgriezums atkarībā no $E$.
a) Izvirziet hipotēzi par pēkšņiem šķērsgriezuma palielinājumu pie dažādām enerģijām. (Atcerēties fotoefekta būtību un Einšteina aprakstu.)
"""

# ╔═╡ 905896c6-aa45-4957-9ed9-e18288016b46
md"""
b) Izvelēties kādu elementu. No tabulas, ko veido XCOM programma, rakstiet šķērsgriezumu pie dažādām enerģijām. Kādu energijas diapazonu Jūs izvēlējāt? Kāpēc?
"""

# ╔═╡ b4fe3ae0-7f91-4a49-9b39-8035200ecfdd
md"""
c) Līdzīgi iepriekšējam uzdevumam, veidojiet un pārbaudiet modeli, kas apraksta fotoefekta šķērsgriezumu atkarībā no enerģijas.
"""

# ╔═╡ 7921514e-6fde-45a9-b275-0efeb588e286
md"""
### Uzdevums 5. 
#### Radiācijas ekranēšana.
a) Kāpēc svīns tiek izmantots, lai ekranētu rentgena starus?
"""

# ╔═╡ 3ab2cf7b-f2d1-41c2-8530-3e419980a5dd
md"""
b) Pēc kuras funkcijas samazināsies fotonu plūsma, kas šķērso vielas slāni, atkarībā no slāņa biezumu. Pamatojiet Jūsu atbildi ar teorētiskiem vai skaitliskiem aprēķiniem vai argumentiem.) 
"""

# ╔═╡ 5ec2c1f9-983e-4236-aa8a-b7f76f059b2d
md"""
## Piezīmes par fitēšanu
Fitēšanas rūtīnas ir tikai programmas. Tādēļ viņas var nonākt strupceļā, un ir vajadīga cilvēka iesaistīšanas, lai nonāktu pie risinājuma. 

### Tolerance
Piemēram, kad mums ir ļoti mazi skaitli, ļoti vienkāršs fits ir neveiksmīgs. Izskatās, ka funkcija `curve_fit` no pakates LsqFit.jl nestrādā labi ar mazām vērtībām. 

"""

# ╔═╡ 70a934e4-b76c-40a6-84ec-dbeb43869722
begin
	xdata=[1,2,3,4,5]
	ydata=2.0e-24 .* xdata
	@. myModel1(x,p)=p[1]+p[2]*x
	p0=[0,3e-24]
	fit=curve_fit(myModel1,xdata,ydata,p0)
	println("p = ",fit.param)
	println("stderr = ", stderror(fit))
	plot(xdata,ydata,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red,ylims=(0,2e-23))
	plot!(xdata,myModel1(xdata,fit.param))
end

# ╔═╡ a4fd6ed0-87da-4e84-a9ae-4aaad315b377
	with_terminal() do
		println("p = ",fit.param)
		println("stderr = ", stderror(fit))
	end

# ╔═╡ 44459c45-38f9-453c-b2a5-726ca1610330
md"""
Vienkāršs risinājums būtu reizināt mūsu $y$ vērtības ar $10^{24}$, un pēc tam pievienot apgriezto reizinātāju ($10^{-24}$) fita rezultātam. Tomēr šī metode nav ļoti apmierinoša. Vai tā strādās, ja mums vienkārši ir liels diapazons?

Cits risinājums ir izmantot "keyword arguments" (`kwargs`) `x_tol` un `g_tol`. Funkcija `curve_fit` meklē fita parametrus iteratīvi. Šie parametri nosaka, kad funkcia `curve_fit` būs "apmierināta" ar savu rezultātu, t.i., kad būs pēdēja iterācija. Parametrs `x_tol` ir saistīts ar relatīvu parametru vērtību izmaiņu starp iterācijām. Kad relatīva parametru izmaiņa ir mazākā par `x_tol`, tad `curve_fit` beidzās. Savukārt parametrs `g_tol` ir saistīts ar relatīvā $\Xi^2$ funkcijas gradientu. Atkaļ, kad tas gradients būs mazāk par `g_tol`, iterācijas beidzās. 

Parametriem `x_tol` un `g_tol` ir _default_ vērtības, bet izskatās, ka tie nav adekvāti. Varam mēģināt citus.

`fit=curve_fit(myModel1,xdata,ydata,p0,x_tol=1e-24,g_tol=1e-24)`
Precīzas vērtības ir jāmeklē. Ja izvēlēsimies parāk lielas vērtības, fits varētu būt nepareizs. Izvēlojoties pārāk mazāk vērtības, fits varētu prasīt ļoti daudz laika, vai vispār nekonverģēt. 
"""

# ╔═╡ 844b8483-bcc9-4a91-9be5-4784f3002d7a
begin
	fit2=curve_fit(myModel1,xdata,ydata,p0,x_tol=1e-24,g_tol=1e-24)
	plot(xdata,ydata,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red,ylims=(0,2e-23))
	plot!(xdata,myModel1(xdata,fit2.param))
end

# ╔═╡ c4fd2179-f382-4d35-8799-2a3efef9f321
	with_terminal() do
		println("p = ",fit2.param)
		println("stderr = ", stderror(fit2))
	end

# ╔═╡ dbbab02a-8a4f-4b44-937c-d1ef38ab966a
md"""
### Eksponentiālas funkcijas
Pieņemsim, ka vēlāmies pielīdzināt modeli $y=ax^n$ kādiem datiem. Mēs zinām, ka tādu vienādojumu varam pārvērst pa lineāru vienādojumu šādi: $\log y =\log a+n \log x$.
Mēģināsim abus variantus:
"""

# ╔═╡ aa9d574c-b58d-42ea-a861-125807674b28
begin
	X=[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0]
	Y=2.5 .* X.^(3)
	Y=Y .+ sqrt.(Y).*randn(length(Y))
	plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red) #,ylims=(0,2e-23)))
	@. myExpModel1(x,p)=p[1]*x^p[2]
	P0=[1.0,1.0]
	fit3=curve_fit(myExpModel1,X,Y,P0)
	plot!(X,myExpModel1(X,fit3.param))
end

# ╔═╡ 84d1d9a0-054b-4e2a-a90a-d27380276521
with_terminal() do
	println("p = ",fit3.param)
	println("stderr = ", stderror(fit3))
end


# ╔═╡ 4565a5f8-7bb9-47ec-a656-1e73edd391e1
begin
	lX=log.(X)
	lY=log.(Y)
	plot(lX,lY,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
	@. myLogModel1(x,p)=p[1]+p[2]*x
	Pl0=[1.0,1.0]
	fit4=curve_fit(myLogModel1,lX,lY,Pl0)
	plot!(lX,myLogModel1(lX,fit4.param))
end


# ╔═╡ b8d74e23-653d-4d37-9707-854d64cbc3c1
with_terminal() do
	println("p = ",fit4.param)
	println("stderr = ", stderror(fit4))
end

# ╔═╡ 51dec9ac-ab2d-430c-b277-38bcfaed52f3
md"""
Izskatās, ka pirmas fits ($y=ax^n$) atbilst kļūdu robežas. Otrais fits ($\log y = \log a + n\log x$) ir nedaudz sliktāks. 
Kāpēc? Lai pielīdzinātu parametrus, tiek minimizēti attiecīgi lielumi $\sqrt ( \sum_i (f(x,p)-y_i)^2 )$ vai $\sqrt ( \sum_i (\log (f(x,p)) - \log y_i)^2 )$, kur f(x,p) ir modelis un $y_i=y_i^0+y_i^{err}$, bet kļūdas to ietekmē savādāk zem logaritma funkcijas. 
Varam mēģināt uzlabot izmantojot svarus (_weights_): 
```
weights=Y
fit5=curve_fit(myExpModel1,X,Y,weights,PP0)
```
Tāds svara faktors ņēm vērā, ka kļūdas tika definētas, ka $\propto \sqrt(y)$, līdz ar to, relatīva kļūda samazinās, jo lielāka $y$ vērtība. Tāpēc, vēlāmies, lai lielākas $y$ vērtības ietekmē vairāk.  
Skatīsimies, kas notiek:
"""

# ╔═╡ 94c7c9b7-2de8-4a93-8b42-56127012fedb
begin
	weights=Y
	plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red) #,ylims=(0,2e-23)))
	PP0=[1.0,1.0]
	fit5=curve_fit(myExpModel1,X,Y,weights,PP0)
	plot!(X,myExpModel1(X,fit5.param))
end

# ╔═╡ d04bbf04-ab8b-4905-b14c-05578426ed55
with_terminal() do
	println("p = ",fit5.param)
	println("stderr = ", stderror(fit5))
end

# ╔═╡ f48e07bf-0d5e-462e-9013-15a07100d5f3
md"""
Kā var redzēt, fits funkcijai $y=ax^n$ ir ļoti tuvu patiesībai un novērtēta kļūda ir maza.
"""

# ╔═╡ 456093b4-f9ef-412d-8950-34ec661f02fd
begin
	plot(lX,lY,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
	PPl0=[1.0,1.0]
	fit6=curve_fit(myLogModel1,lX,lY,weights,PPl0)
	plot!(lX,myLogModel1(lX,fit6.param))
end

# ╔═╡ 492fddf6-d464-47bc-bc9b-0d41be2564b5
with_terminal() do
	println("p = ",fit6.param)
	println("stderr = ", stderror(fit6))
end

# ╔═╡ e013b711-339b-44ef-87df-d476b2e231d1
md"""
Fits funkcijai $\log y = \log a + n\log x$ arī uzlabojās, bet tomēr nav tik labs. 
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╠═0c66d310-bf10-11ec-1be7-5d0878ec4a0b
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# ╟─84d1d9a0-054b-4e2a-a90a-d27380276521
# ╠═4565a5f8-7bb9-47ec-a656-1e73edd391e1
# ╠═b8d74e23-653d-4d37-9707-854d64cbc3c1
# ╟─51dec9ac-ab2d-430c-b277-38bcfaed52f3
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# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
